,,the ,,logic ,,of ,,modern ,,physics ,,by ,p4 ,w4 ,,bridgman ,,hollis ,,professor ,,of ,,mathematics ,,and ,,natural ,,history ,,in ,,harvard ,,university ,,new ,,york ,,the ,,macmillan ,,company ,,copyright1 #aibg1 ,by ,,the ,,macmillan ,,company4 ,,preface ,this excursion into the field of fundamental criticism by one whose activities have hitherto been confined almost entirely to experiment is not evidence of senile decay1 as might be cynically assumed4 ,i have always1 throughout all my experimental work1 felt an imperative need of a better understanding of the foundations of our physical thought and have for a long time made more or less unsystematic attempts to reach such an understanding4 ,only now1 however1 has a half sabbatical year given me leisure to attempt a more or less orderly exposition4 ,in spite of previous writings on the broad fundamentals by ,clifford1 ,stallo1 ,mach1 and ,poincar1 to mention only a few1 ,i believe a new essay of this critical character needs no apology4 ,for entirely apart from the question of whether many of the points of view of these essays can be maintained1 the discovery of new facts in the domain of relativity and quantum theory has shifted the center of interest and emphasis4 ,all the quite recent activity with the new quantum mechanics seems to call for a new examination of fundamental matters which shall recognize1 at least by implication1 the existence of the special phenomena of the quantum domain4 ,however1 the necessity for re-examination does not mean at all that many of the results of previous criticism may not still be accepted2 some of these results have become so thoroughly incorporated into physical thinking that we can assume them without mention4 ,thus the fundamental attitude of this essay is empiricism1 which is now justified as the attitude of the physicist in large part by the inquiry into the physiological origin of our concepts of space1 time1 and mechanics with which the previous essays were largely concerned4 ,none of the previous essays have consciously or immediately affected the details of this2 in fact ,i have not read any of them within several years4 ,if passages here recall passages already written1 it is because the ideas have been assimilated and the precise origin forgotten2 it is probably worth while to let such passages stand without revision1 because such ideas gain in plausibility through having been found acceptable to independent thought4 ,i am much indebted to ,professor ,r4 ,f4 ,alfred ,hoernl of the ,department of ,philosophy of ,johannesburg ,university1 ,south ,africa1 for suggesting several modifications to make the text more acceptable to a philosopher1 and slight amplifications for the benefit of readers not familiar with all the details of recent technical developments in physics4 ,,introduction ,one of the most noteworthy movements in recent physics is a change of attitude toward what may be called the interpretative aspect of physics4 ,it is being increasingly recognized1 both in the writings and the conversation of physicists1 that the world of experiment is not understandable without some examination of the purpose of physics and of the nature of its fundamental concepts4 ,it is no new thing to attempt a more critical understanding of the nature of physics1 but until recently all such attempts have been regarded with a certain suspicion or even sometimes contempt4 ,the average physicist is likely to deprecate his own concern with such questions1 and is inclined to dismiss the speculations of fellow physicists with the epithet metaphysical4 ,this attitude has no doubt had a certain justification in the utter unintelligibility to the physicist of many metaphysical speculations and the sterility of such speculations in yielding physical results4 ,however1 the growing reaction favoring a better understanding of the interpretative fundamentals of physics is not a pendulum swing of the fashion of thought toward metaphysics1 originating in the upheaval of moral values produced by the great war1 or anything of the sort1 but is a reaction absolutely forced upon us by a rapidly increasing array of cold experimental facts4 ,this reaction1 or rather new movement1 was without doubt initiated by the restricted theory of relativity of ,einstein4 ,before ,einstein1 an ever increasing number of experimental facts concerning bodies in rapid motion required increasingly complicated modifications in our nave notions in order to preserve self-consistency1 until ,einstein showed that everything could be restored again to a wonderful simplicity by a slight change in some of our fundamental concepts4 ,the concepts which were most obviously touched by ,einstein were those of space and time1 and much of the writing consciously inspired by ,einstein has been concerned with these concepts4 ,but that experiment compels a critique of much more than the concepts of space and time is made increasingly evident by all the new facts being discovered in the quantum realm4 ,the situation presented to us by these new quantum facts is twofold4 ,in the first place1 all these experiments are concerned with things so small as to be forever beyond the possibility of direct experience1 so that we have the problem of translating the evidence of experiment into other language4 ,thus we observe an emission line in a spectroscope and may infer an electron jumping from one energy level to another in an atom4 ,in the second place1 we have the problem of understanding the translated experimental evidence4 ,now of course every one knows that this problem is making us the greatest difficulty4 ,the experimental facts are so utterly different from those of our ordinary experience that not only do we apparently have to give up generalizations from past experience as broad as the field equations of electrodynamics1 for instance1 but it is even being questioned whether our ordinary forms of thought are applicable in the new domain2 it is often suggested1 for example1 that the concepts of space and time break down4 ,the situation is rapidly becoming acute4 ,since ,i began writing this essay1 there has been a striking increase in critical activity inspired by the new quantum mechanics of #aibe-bf1 and it is common to hear expositions of the new ideas prefaced by analysis of what experiment really gives to us or what our fundamental concepts really mean4 ,the change in ideas is now so rapid that a number of the statements of this essay are already antiquated as expressions of the best current opinion2 however ,i have allowed these statements to stand1 since the fundamental arguments are in nowise affected and we have no reason to think that present best opinions are in any way final4 ,we have the impression of being in an important formative period2 if we are1 the complexion of physics for a long time in the future will be determined by our present attitude toward fundamental questions of interpretation4 ,to meet this situation it seems to me that something more is needed than the hand-to-mouth philosophy that is now growing up to meet special emergencies1 something approaching more closely to a systematic philosophy of all physics which shall cover the experimental domains already consolidated as well as those which are now making us so much trouble4 ,it is the attempt of this essay to give a more or less inclusive critique of all physics4 ,our problem is the double one of understanding what we are trying to do and what our ideals should be in physics1 and of understanding the nature of the structure of physics as it now exists4 ,these two ends are together furthered by an analysis of the fundamental concepts of physics2 an understanding of the concepts we how have discloses the present structure of physics and a realization of what the concepts should be involves the ideals of physics4 ,this essay will be largely concerned with the fundamental concepts2 it will appear that almost all the concepts can profit from re-examination4 ,the material of this essay is largely obtained by observation of the actual currents of opinion in physics2 much of what ,i have to say is more or less common property and doubtless every reader will find passages that he will feel have been taken out of his own mouth4 ,on certain broad tendencies in present day physics1 however1 ,i have put my own interpretation1 and it is more than likely that this interpretation will be unacceptable to many4 ,but even if not acceptable1 ,i hope that the stimulus of combatting the ideas offered here may be of value4 ,certain limitations will have to be set to our inquiry in order to keep it within manageable compass4 ,it is of course the merest truism that all our experimental knowledge and our understanding of nature is impossible and non-existent apart from our own mental processes1 so that strictly speaking no aspect of psychology or epistemology is without pertinence4 ,fortunately we shall be able to get along with a more or less naive attitude toward many of these matters4 ,we shall accept as significant our common sense judgment that there is a world external to us1 and shall limit as far as possible our inquiry to the behavior and interpretation of this external world4 ,we shall rule out inquiries into our states of consciousness as such4 ,in spite1 however1 of the best intentions1 we shall not be able to eliminate completely considerations savoring of the metaphysical1 because it is evident that the nature of our thinking mechanism essentially colors any picture that we can form of nature1 and we shall have to recognize that unavoidable characteristics of any outlook of ours are imposed in this way4 ,,contents ,preface ,introduction ,,chapter ,i4 ,,broad ,,points ,,of ,,view ,new ,kinds of ,experience ,always ,possible ,the ,operational ,character of ,concepts ,einstein's ,contribution in ,changing ,our ,attitude toward ,concepts ,detailed ,discussion of the ,concept of ,length ,the ,relative ,character of ,knowledge ,meaningless ,questions ,general ,comments on the ,operational ,point of ,view ,,ii4 ,,other ,,general ,,considerations ,the ,approximate ,character of ,empirical ,knowledge ,explanations and ,mechanisms ,models and ,constructs ,the ,rle of ,mathematics in ,physics ,,iii4 ,,detailed ,,consideration ,,of ,,various ,,concepts ,,of ,,physics ,the ,concept of ,space ,the ,concept of ,time ,the ,causality ,concept ,the ,concept of ,identity ,the ,concept of ,velocity ,the ,concepts of ,force and ,mass ,the ,concept of ,energy ,the ,concepts of ,thermodynamics ,electrical ,concepts ,the ,nature of ,light and the ,concepts of ,relativity ,other ,relativity ,concepts ,rotational ,motion and ,relativity ,quantum ,concepts ,,iv4 ,,special ,,views ,,of ,,nature ,the ,simplicity of ,nature ,determinism ,on the ,possibility of ,describing ,nature ,completely in ,terms of ,analysis ,a ,glimpse ,ahead ,,index ,,the ,,logic ,,of ,,modern ,,physics ,,chapter ,i ,,broad ,,points ,,of ,,view ,,whatever may be one's opinion as to our permanent acceptance of the analytical details of ,einstein's restricted and general theories of relativity1 there can be no doubt that through these theories physics is permanently changed4 ,it was a great shock to discover that classical concepts1 accepted unquestioningly1 were inadequate to meet the actual situation1 and the shock of this discovery has resulted in a critical attitude toward our whole conceptual structure which must at least in part be permanent4 ,reflection on the situation after the event shows that it should not have needed the new experimental facts which led to relativity to convince us of the inadequacy of our previous concepts1 but that a sufficiently shrewd analysis should have prepared us for at least the possibility of what ,einstein did4 ,looking now to the future1 our ideas of what external nature is will always be subject to change as we gain new experimental knowledge1 but there is a part of our attitude to nature which should not be subject to future change1 namely that part which rests on the permanent basis of the character of our minds4 ,it is precisely here1 in an improved understanding of our mental relations to nature1 that the permanent contribution of relativity is to be found4 ,we should now make it our business to understand so thoroughly the character of our permanent mental relations to nature that another change in our attitude1 such as that due to ,einstein1 shall be forever impossible4 ,it was perhaps excusable that a revolution in mental attitude should occur once1 because after all physics is a young science1 and physicists have been very busy1 but it would certainly be a reproach if such a revolution should ever prove necessary again4 ,,new ,,kinds ,,of ,,experience ,,always ,,possible ,the first lesson of our recent experience with relativity is merely an intensification and emphasis of the lesson which all past experience has also taught1 namely1 that when experiment is pushed into new domains1 we must be prepared for new facts1 of an entirely different character from those of our former experience4 ,this is taught not only by the discovery of those unsuspected properties of matter moving with high velocities1 which inspired the theory of relativity1 but also even more emphatically by the new facts in the quantum domain4 ,to a certain extent1 of course1 the recognition of all this does not involve a change of former attitude2 the fact has always been for the physicist the one ultimate thing from which there is no appeal1 and in the face of which the only possible attitude is a humility almost religious4 ,the new feature in the present situation is an intensified conviction that in reality new orders of experience do exist1 and that we may expect to meet them continually4 ,we have already encountered new phenomena in going to high velocities1 and in going to small scales of magnitude3 we may similarly expect to find them1 for example1 in dealing with relations of cosmic magnitudes1 or in dealing with the properties of matter of enormous densities1 such as is supposed to exist in the stars4 ,implied in this recognition of the possibility of new experience beyond our present range1 is the recognition that no element of a physical situation1 no matter how apparently irrelevant or trivial1 may be dismissed as without effect on the final result until proved to be without effect by actual experiment4 ,the attitude of the physicist must therefore be one of pure empiricism4 ,he recognizes no a priori principles which determine or limit the possibilities of new experience4 ,experience is determined only by experience4 ,this practically means that we must give up the demand that all nature be embraced in any formula1 either simple or complicated4 ,it may perhaps turn out eventually that as a matter of fact nature can be embraced in a formula1 but we must so organize our thinking as not to demand it as a necessity4 ,,the ,,operational ,,character ,,of ,,concepts ,einstein's ,contribution in ,changing ,our ,attitude ,toward ,concepts ,recognizing the essential unpredictability of experiment beyond our present range1 the physicist1 if he is to escape continually revising his attitude1 must use in describing and correlating nature concepts of such a character that our present experience does not exact hostages of the future4 ,now here it seems to me is the greatest contribution of ,einstein4 ,although he himself does not explicitly state or emphasize it1 ,i believe that a study of what he has done will show that he has essentially modified our view of what the concepts useful in physics are and should be4 ,hitherto many of the concepts of physics have been defined in terms of their properties4 ,an excellent example is afforded by ,newton's concept of absolute time4 ,the following quotation from the ,scholium in ,book ,i of the ,principia is illuminating3 ,i do not define ,time1 ,space1 ,place or ,motion1 as being well known to all4 ,only ,i must observe that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects4 ,and thence arise certain prejudices1 for the removing of which1 it will be convenient to distinguish them into ,absolute and ,relative1 ,true and ,apparent1 ,mathematical and ,common4 7,i7 ,absolute1 ,true1 and ,mathematical ,time1 of itself1 and from its own nature flows equably without regard to anything external1 and by another name is called ,duration4 ,now there is no assurance whatever that there exists in nature anything with properties like those assumed in the definition1 and physics1 when reduced to concepts of this character1 becomes as purely an abstract science and as far removed from reality as the abstract geometry of the mathematicians1 built on postulates4 ,it is a task for experiment to discover whether concepts so defined correspond to anything in nature1 and we must always be prepared to find that the concepts correspond to nothing or only partially correspond4 ,in particular1 if we examine the definition of absolute time in the light of experiment1 we find nothing in nature with such properties4 ,the new attitude toward a concept is entirely different4 ,we may illustrate by considering the concept of length3 what do we mean by the length of an object8 ,we evidently know what we mean by length if we can tell what the length of any and every object is1 and for the physicist nothing more is required4 ,to find the length of an object1 we have to perform certain physical operations4 ,the concept of length is therefore fixed when the operations by which length is measured are fixed3 that is1 the concept of length involves as much as and nothing more than the set of operations by which length is determined4 ,in general1 we mean by any concept nothing more than a set of operations2 the concept is synonymous with the corresponding set of operations4 ,if the concept is physical1 as of length1 the operations are actual physical operations1 namely1 those by which length is measured2 or if the concept is mental1 as of mathematical continuity1 the operations are mental operations1 namely those by which we determine whether a given aggregate of magnitudes is continuous4 ,it is not intended to imply that there is a hard and fast division between physical and mental concepts1 or that one kind of concept does not always contain an element of the other2 this classification of concept is not important for our future considerations4 ,we must demand that the set of operations equivalent to any concept be a unique set1 for otherwise there are possibilities of ambiguity in practical applications which we cannot admit4 ,applying this idea of concept to absolute time1 we do not understand the meaning of absolute time unless we can tell how to determine the absolute time of any concrete event1 i4e41 unless we can measure absolute time4 ,now we merely have to examine any of the possible operations by which we measure time to see that all such operations are relative operations4 ,therefore the previous statement that absolute time does not exist is replaced by the statement that absolute time is meaningless4 ,and in making this statement we are not saying something new about nature1 but are merely bringing to light implications already contained in the physical operations used in measuring time4 ,it is evident that if we adopt this point of view toward concepts1 namely that the proper definition of a concept is not in terms of its properties but in terms of actual operations1 we need run no danger of having to revise our attitude toward nature4 ,for if experience is always described in terms of experience1 there must always be correspondence between experience and our description of it1 and we need never be embarrassed1 as we were in attempting to find in nature the prototype of ,newton's absolute time4 ,furthermore1 if we remember that the operations to which a physical concept are equivalent are actual physical operations1 the concepts can be defined only in the range of actual experiment1 and are undefined and meaningless in regions as yet untouched by experiment4 ,it follows that strictly speaking we cannot make statements at all about regions as yet untouched1 and that when we do make such statements1 as we inevitably shall1 we are making a conventionalized extrapolation1 of the looseness of which we must be fully conscious1 and the justification of which is in the experiment of the future4 ,there probably is no statement either in ,einstein or other writers that the change described above in the use of concept has been self-consciously made1 but that such is the case is proved1 ,i believe1 by an examination of the way concepts are now handled by ,einstein and others4 ,for of course the true meaning of a term is to be found by observing what a man does with it1 not by what he says about it4 ,we may show that this is the actual sense in which concept is coming to be used by examining in particular ,einstein's treatment of simultaneity4 ,before ,einstein1 the concept of simultaneity was defined in terms of properties4 ,it was a property of two events1 when described with respect to their relation in time1 that one event was either before the other1 or after it1 or simultaneous with it4 ,simultaneity was a property of the two events alone and nothing else2 either two events were simultaneous or they were not4 ,the justification for using this term in this way was that it seemed to describe the behavior of actual things4 ,but of course experience then was restricted to a narrow range4 ,when the range of experience was broadened1 as by going to high velocities1 it was found that the concepts no longer applied1 because there was no counterpart in experience for this absolute relation between two events4 ,einstein now subjected the concept of simultaneity to a critique1 which consisted essentially in showing that the operations which enable two events to be described as simultaneous involve measurements on the two events made by an observer1 so that simultaneity is1 therefore1 not an absolute property of the two events and nothing else1 but must also involve the relation of the events to the observer4 ,until therefore we have experimental proof to the contrary1 we must be prepared to find that the simultaneity of two events depends on their relation to the observer1 and in particular on their velocity4 ,einstein1 in thus analyzing what is involved in making a judgment of simultaneity1 and in seizing on the act of the observer as the essence of the situation1 is actually adopting a new point of view as to what the concepts of physics should be1 namely1 the operational view4 ,of course ,einstein actually went much further than this1 and found precisely how the operations for judging simultaneity change when the observer moves1 and obtained quantitative expressions for the effect of the motion of the observer on the relative time of two events4 ,we may notice1 parenthetically1 that there is much freedom of choice in selecting the exact operations2 those which ,einstein chose were determined by convenience and simplicity with relation to light beams4 ,entirely apart from the precise quantitative relations of ,einstein's theory1 however1 the important point for us is that if we had adopted the operational point of view1 we would1 before the discovery of the actual physical facts1 have seen that simultaneity is essentially a relative concept1 and would have left room in our thinking for the discovery of such effects as were later found4 ,detailed ,discussion of the ,concept of ,length ,we may now gain further familiarity with the operational attitude toward a concept and some of its implications by examining from this point of view the concept of length4 ,our task is to find the operations by which we measure the length of any concrete physical object4 ,we begin with objects of our commonest experience1 such as a house or a house lot4 ,what we do is sufficiently indicated by the following rough description4 ,we start with a measuring rod1 lay it on the object so that one of its ends coincides with one end of the object1 mark on the object the position of the other end of the rod1 then move the rod along in a straight line extension of its previous position until the first end coincides with the previous position of the second end1 repeat this process as often as we can1 and call the length the total number of times the rod was applied4 ,this procedure1 apparently so simple1 is in practice exceedingly complicated1 and doubtless a full description of all the precautions that must be taken would fill a large treatise4 ,we must1 for example1 be sure that the temperature of the rod is the standard temperature at which its length is defined1 or else we must make a correction for it2 or we must correct for the gravitational distortion of the rod if we measure a vertical length2 or we must be sure that the rod is not a magnet or is not subject to electrical forces4 ,all these precautions would occur to every physicist4 ,but we must also go further and specify all the details by which the rod is moved from one position to the next on the object--its precise path through space and its velocity and acceleration in getting from one position to another4 ,practically of course1 precautions such as these are not mentioned1 but the justification is in our experience that variations of procedure of this kind are without effect on the final result4 ,but we always have to recognize that all our experience is subject to error1 and that at some time in the future we may have to specify more carefully the acceleration1 for example1 of the rod in moving from one position to another1 if experimental accuracy should be so increased as to show a measurable effect4 ,in principle the operations by which length is measured should be uniquely specified4 ,if we have more than one set of operations1 we have more than one concept1 and strictly there should be a separate name to correspond to each different set of operations4 ,so much for the length of a stationary object1 which is complicated enough4 ,now suppose we have to measure a moving street car4 ,the simplest1 and what we may call the nave procedure1 is to board the car with our meter stick and repeat the operations we would apply to a stationary body4 ,notice that this procedure reduces to that already adopted in the limiting case when the velocity of the street car vanishes4 ,but here there may be new questions of detail4 ,how shall we jump on to the car with our stick in hand8 ,shall we run and jump on from behind1 or shall we let it pick us up from in front8 ,or perhaps does now the material of which the stick is composed make a difference1 although previously it did not8 ,all these questions must be answered by experiment4 ,we believe from present evidence that it makes no difference how we jump on to the car1 or of what material the rod is made1 and that the length of the car found in this way will be the same as if it were at rest4 ,but the experiments are more difficult1 and we are not so sure of our conclusions as before4 ,now there are very obvious limitations to the procedure just given4 ,if the street car is going too fast1 we can not board it directly1 but must use devices1 such as getting on from a moving automobile2 and1 more important still1 there are limitations to the velocity that can be given to street cars or to meter sticks by any practical means in our control1 so that the moving bodies which can be measured in this way are restricted to a low range of velocity4 ,if we want to be able to measure the length of bodies moving with higher velocities such as we find existing in nature 7stars or cathode particles71 we must adopt another definition and other operations for measuring length1 which also reduce to the operations already adopted in the static case4 ,this is precisely what ,einstein did4 ,since ,einstein's operations were different from our operations above1 his length does not mean the same as our length4 ,we must accordingly be prepared to find that the length of a moving body measured by the procedure of ,einstein is not the same as that above2 this of course is the fact1 and the transformation formulas of relativity give the precise connection between the two lengths4 ,einstein's procedure for measuring the length of bodies in motion was dictated not only by the consideration that it must be applicable to bodies with high velocities1 but also by mathematical convenience1 in that ,einstein describes the world mathematically by a system of cordinate geometry1 and the length of an object is connected simply with quantities in the analytic equations4 ,it is of interest to describe briefly ,einstein's actual operations for measuring the length of a body in motion2 it will show how operations which may be simple from a mathematical point of view may appear complicated from a physical viewpoint4 ,the observer who is to measure the length of a moving object must first extend over his entire plane of reference 7for simplicity the problem is considered two-dimensional7 a system of time coordinates1 i4e41 at each point of his plane of reference there must be a clock1 and all these clocks must be synchronized4 ,at each clock an observer must be situated4 ,now to find the length of the moving object at a specified instant of time 7it is a subject for later investigation to find whether its length is a function of time71 the two observers who happen to coincide in position with the two ends of the object at the specified time on their clocks are required to find the distance between their two positions by the procedure for measuring the length of a stationary object1 and this distance is by definition the length of the moving object in the given reference system4 ,this procedure for measuring the length of a body in motion hence involves the idea of simultaneity1 through the simultaneous position of the two ends of the rod1 and we have seen that the operations by which simultaneity are determined are relative1 changing when the motion of the system changes4 ,we hence are prepared to find a change in the length of a body when the velocity of the measuring system changes1 and this in fact is what happens4 ,the precise numerical dependence is worked out by ,einstein1 and involves other considerations1 in which we are not interested at present4 ,the two sorts of length1 the naive one and that of ,einstein1 have certain features in common4 ,in either case in the limit1 as the velocity of the measuring system approaches zero1 the operations approach those for measuring the length of a stationary object4 ,this1 of course1 is a requirement in any good definition1 imposed by considerations of convenience1 and it is too obvious a matter to need elaboration4 ,another feature is that the operations equivalent to either concept both involve the motion of the system1 so that we must recognize the possibility that the length of a moving object may be a function of its velocity4 ,it is a matter of experiment1 unpredictable until tried1 that within the limits of present experimental error the naive length is not affected by motion1 and ,einstein's length is4 ,so far1 we have extended the concept of length in only one way beyond the range of ordinary experience1 namely to high velocities4 ,the extension may obviously be made in other directions4 ,let us inquire what are the operations by which we measure the length of a very large object4 ,in practice we probably first meet the desirability of a change of procedure in measuring large pieces of land4 ,here our procedure depends on measurements with a surveyor's theodolite4 ,this involves extending over the surface of the land a system of coordinates1 starting from a base line measured with a tape in the conventional way1 sighting on distant points from the extremities of the line1 and measuring the angles4 ,now in this extension we have made one very essential change3 the angles between the lines connecting distant points are now angles between beams of light4 ,we assume that a beam of light travels in a straight line4 ,furthermore1 we assume in extending our system of triangulation over the surface of the earth that the geometry of light beams is ,euclidean4 ,we do the best we can to check the assumptions1 but at most can never get more than a partial check4 ,thus ,gauss#a checked whether the angles of a large terrestrial triangle add to two right angles and found agreement within experimental error4 ,we now know from the experiments of ,michelson#b that if his measurements had been accurate enough he would not have got a check1 but would have had an excess or defect according to the direction in which the beam of light travelled around the triangle with respect to the rotation of the earth4 ,but if the geometry of light beams is ,euclidean1 then not only must the angles of a triangle add to two right angles1 but there are definite relations between the lengths of the sides and the angles1 and to check these relations the sides should be measured by the old procedure with a meter stick4 ,such a check on a large scale has never been attempted1 and is not feasible4 ,it seems1 then1 that our checks on the ,euclidean character of optical space are all of restricted character4 ,we have apparently proved that up to a certain scale of magnitude optical space is ,euclidean with respect to measures of angle1 but this may not necessarily involve that space is also ,euclidean with respect to measures of length1 so that space need not be completely ,euclidean4 ,there is a further most important restriction in that our studies of non-,euclidean geometry have shown that the percentage excess of the angles of a non-,euclidean triangle over #ahj may depend on the magnitude of the triangle1 so that it may well be that we have not detected the non-,euclidean character of space simply because our measurements have not been on a large enough scale4 ,footnote #a3 ,c4 ,f4 ,gauss1 ,gesammelte ,werke1 especially vol4 ,,iv4 ,footnote #b3 ,see a discussion of the theory of this experiment by ,l4 ,silberstein1 ,jour4 ,opt4 ,soc4 ,amer4 #e1 #bia-cjg4 #aiba4 ,we thus see that the concept of length has undergone a very essential change of character even within the range of terrestrial measurements1 in that we have substituted for what ,i may call the tactual concept an optical concept1 complicated by an assumption about the nature of our geometry4 ,from a very direct concept we have come to a very indirect concept with a most complicated set of operations4 ,strictly speaking1 length when measured in this way by light beams should be called by another name1 since the operations are different4 ,the practical justification for retaining the same name is that within our present experimental limits a numerical difference between the results of the two sorts of operations has not been detected4 ,we are still worse off when we make the extension to solar and stellar distances4 ,here space is entirely optical in character1 and we never have an opportunity of even partially comparing tactual with optical space4 ,no direct measures of length have ever been made1 nor can we even measure the three angles of a triangle and so check our assumption that the use of ,euclidean geometry in extending the concept of space is justified4 ,we never have under observation more than two angles of a triangle1 as when we measure the distance of the moon by observation from the two ends of the earth's diameter4 ,to extend to still greater distance our measures of length1 we have to make still further assumptions1 such as that inferences from the ,newtonian laws of mechanics are valid4 ,the accuracy of our inferences about lengths from such measurements is not high4 ,astronomy is usually regarded as a science of extraordinarily high accuracy1 but its accuracy is very restricted in character1 namely to the measurement of angles4 ,it is probably safe to say that no astronomical distance1 except perhaps that of the moon1 is known with an accuracy greater than #j4a4 ,when we push our estimates to distances beyond the confines of the solar system in which we are assisted by the laws of mechanics1 we are reduced in the first place to measurements of parallax1 which at best have a quite inferior accuracy1 and which furthermore fail entirely outside a rather restricted range4 ,for greater stellar distances we are driven to other and much rougher estimates1 resting for instance on the extension to great distances of connections found within the range of parallax between brightness and spectral type of a star1 or on such assumptions as that1 because a group of stars looks as if it were all together in space and had a common origin1 it actually is so4 ,thus at greater and greater distances not only does experimental accuracy become less1 but the very nature of the operations by which length is to be determined becomes indefinite1 so that the distances of the most remote stellar objects as estimated by different observers or by different methods may be very divergent4 ,a particular consequence of the inaccuracy of the astronomical measures of great distances is that the question of whether large scale space is ,euclidean or not is merely academic4 ,we thus see that in the extension from terrestrial to great stellar distances the concept of length has changed completely in character4 ,to say that a certain star is #aje light years distant is actually and conceptually an entire different kind of thing from saying that a certain goal post is #ajj meters distant4 ,because of our conviction that the character of our experience may change when the range of phenomena changes1 we feel the importance of such a question as whether the space of distances of #aje light years is ,euclidean or not1 and are correspondingly dissatisfied that at present there seems no way of giving meaning to it4 ,we encounter difficulties similar to those above1 and are also compelled to modify our procedures1 when we go to small distances4 ,down to the scale of microscopic dimensions a fairly straightforward extension of the ordinary measuring procedure is sufficient1 as when we measure a length in a micrometer eyepiece of a microscope4 ,this is of course a combination of tactual and optical measurements1 and certain assumptions1 justified as far as possible by experience1 have to be made about the behavior of light beams4 ,these assumptions are of a quite different character from those which give us concern on the astronomical scale1 because here we meet difficulty from interference effects due to the finite scale of the structure of light1 and are not concerned with a possible curvature of light beams in the long reaches of space4 ,apart from the matter of convenience1 we might also measure small distances by the tactual method4 ,as the dimensions become smaller1 certain difficulties become increasingly important that were negligible on a larger scale4 ,in carrying out physically the operations equivalent to our concepts1 there are a host of practical precautions to be taken which could be explicitly enumerated with difficulty1 but of which nevertheless any practical physicist is conscious4 ,suppose1 for example1 we measure length tactually by a combination of ,johanssen gauges4 ,in piling these together1 we must be sure that they are clean1 and are thus in actual contact4 ,particles of mechanical dirt first engage our attention4 ,then as we go to smaller dimensions we perhaps have to pay attention to adsorbed films of moisture1 then at still smaller dimensions to adsorbed films of gas1 until finally we have to work in a vacuum1 which must be the more nearly complete the smaller the dimensions4 ,about the time that we discover the necessity for a complete vacuum1 we discover that the gauges themselves are atomic in structure1 that they have no definite boundaries1 and therefore no definite length1 but that the length is a hazy thing1 varying rapidly in time between certain limits4 ,we treat this situation as best we can by taking a time average of the apparent positions of the boundaries1 assuming that along with the decrease of dimensions we have acquired a corresponding extravagant increase in nimbleness4 ,but as the dimensions get smaller continually1 the difficulties due to this haziness increase indefinitely in percentage effect1 and we are eventually driven to give up altogether4 ,we have made the discovery that there are essential physical limitations to the operations which defined the concept of length4 ,we perhaps do not regard the substitution of optical for tactual space on the astronomical scale as compelled by the same sort of physical necessity1 because ,i suppose the possible eventual landing of men in the moon will always be one of the dreams of humanity4 ,at the same time that we have come to the end of our rope with our ,johanssen gauge procedure1 our companion with the microscope has been encountering difficulties due to the finite wave length of light2 this difficulty he has been able to minimize by using light of progressively shorter wave lengths1 but he has eventually had to stop on reaching ,x-rays4 ,of course this optical procedure with the microscope is more convenient1 and is therefore adopted in practice4 ,let us now see what is implied in our concept of length extended to ultramicroscopic dimensions4 ,what1 for instance1 is the meaning of the statement that the distance between the planes of atoms in a certain crystal is #c x #aj-h cm48 ,what we would like to mean is that x #ajh of these planes piled on top of each other give a thickness of #a cm42 but of course such a meaning is not the actual one4 ,the actual meaning may be found by examining the operations by which we arrived at the number #c x #aj-h4 ,as a matter of fact1 #c x #aj-h was the number obtained by solving a general equation derived from the wave theory of light1 into which certain numerical data obtained by experiments with ,x-rays had been substituted4 ,thus not only has the character of the concept of length changed from tactual to optical1 but we have gone much further in committing ourselves to a definite optical theory4 ,if this were the whole story1 we would be most uncomfortable with respect to this branch of physics1 because we are so uncertain of the correctness of our optical theories1 but actually a number of checks can be applied which greatly restore our confidence4 ,for instance1 from the density of the crystal and the grating space1 the weight of the individual atoms may be computed1 and these weights may then be combined with measurements of the dimensions of other sorts of crystal into which the same atoms enter to give values of the densities of these crystals1 which may be checked against experiment4 ,all such checks have succeeded within limits of accuracy which are fairly high4 ,it is important to notice that1 in spite of the checks1 the character of the concept is changing1 and begins to involve such things as the equations of optics and the assumption of the conservation of mass4 ,we are not content1 however1 to stop with dimensions of atomic order1 but have to push on to the electron with a diameter of the order of #aj-ah cm4 ,what is the possible meaning of the statement that the diameter of an electron is #aj-ah cm48 ,again the only answer is found by examining the operations by which the number #aj-ah was obtained4 ,this number came by solving certain equations derived from the field equations of electrodynamics1 into which certain numerical data obtained by experiment had been substituted4 ,the concept of length has therefore now been so modified as to include that theory of electricity embodied in the field equations1 and1 most important1 assumes the correctness of extending these equations from the dimensions in which they may be verified experimentally into a region in which their correctness is one of the most important and problematical of present-day questions in physics4 ,to find whether the field equations are correct on a small scale1 we must verify the relations demanded by the equations between the electric and magnetic forces and the space coordinates1 to determine which involves measurement of lengths4 ,but if these space coordinates cannot be given an independent meaning apart from the equations1 not only is the attempted verification of the equations impossible1 but the question itself is meaningless4 ,if we stick to the concept of length by itself1 we are landed in a vicious circle4 ,as a matter of fact1 the concept of length disappears as an independent thing1 and fuses in a complicated way with other concepts1 all of which are themselves altered thereby1 with the result that the total number of concepts used in describing nature at this level is reduced in number4 ,a precise analysis of the situation is difficult1 and ,i suppose has never been attempted1 but the general character of the situation is evident4 ,until at least a partial analysis is attempted1 ,i do not see how any meaning can be attached to such questions as whether space is ,euclidean in the small scale4 ,it is interesting to observe that any increased accuracy in knowledge of large scale phenomena must1 as far as we now can see1 arise from an increase in the accuracy of measurement of small things1 that is1 in the measurement of small angles or the analysis of minute differences of wave lengths in the spectra4 ,to know the very large takes us into the same field of experiment as to know the very small1 so that operationally the large and the small have features in common4 ,this somewhat detailed analysis of the concept of length brings out features common to all our concepts4 ,if we deal with phenomena outside the domain in which we originally defined our concepts1 we may find physical hindrances to performing the operations of the original definition1 so that the original operations have to be replaced by others4 ,these new operations are1 of course1 to be so chosen that they give1 within experimental error1 the same numerical results in the domain in which the two sets of operations may be both applied2 but we must recognize in principle that in changing the operations we have really changed the concept1 and that to use the same name for these different concepts over the entire range is dictated only by considerations of convenience1 which may sometimes prove to have been purchased at too high a price in terms of unambiguity4 ,we must always be prepared some day to find that an increase in experimental accuracy may show that the two different sets of operations which give the same results in the more ordinary part of the domain of experience1 lead to measurably different results in the more unfamiliar parts of the domain4 ,we must remain aware of these joints in our conceptual structure if we hope to render unnecessary the services of the unborn ,einsteins4 ,the second feature common to all concepts brought out by the detailed discussion of length is that1 as we approach the experimentally attainable limit1 concepts lose their individuality1 fuse together1 and become fewer in number1 as we have seen that at dimensions of the order of the diameter of an electron the concepts of length and the electric field vectors fuse into an amorphous whole4 ,not only does nature as experienced by us become different in character on its horizons1 but it becomes simpler1 and therefore our concepts1 which are the building stones of our descriptions1 become fewer in number4 ,this seems to be an entirely natural state of affairs4 ,how the number of concepts is often kept formally the same as we approach the horizon will be discussed later in special cases4 ,a precise analysis of our conceptual structure has never been attempted1 except perhaps in very restricted domains1 and it seems to me that there is room here for much important future work4 ,such an analysis is not to be attempted in this essay1 but only some of the more important qualitative aspects are to be pointed out4 ,it will never be possible to give a clean-cut logical analysis of the conceptual situation1 for the nature of our concepts1 according to our operational point of view1 is the same as the nature of experimental knowledge1 which is often hazy4 ,thus in the transition regions where nature is getting simpler and the number of operationally independent concepts changes1 a certain haziness is inevitable1 for the actual change in our conceptual structure in these transition regions is continuous1 corresponding to the continuity of our experimental knowledge1 whereas formally the number of concepts should be an integer4 ,the ,relative ,character of ,knowledge ,two other consequences of the operational point of view must now be examined4 ,first is the consequence that all our knowledge is relative4 ,this may be understood in a general or a more particular sense4 ,the general sense is illustrated in ,haldane's book on the ,reign of ,relativity4 ,relativity in the general sense is the merest truism if the operational definition of concept is accepted1 for experience is described in terms of concepts1 and since our concepts are constructed of operations1 all our knowledge must unescapably be relative to the operations selected4 ,but knowledge is also relative in a narrower sense1 as when we say there is no such thing as absolute rest 7or motion7 or absolute size1 but rest and size are relative terms4 ,conclusions of this kind are involved in the specific character of the operations in terms of which rest or size are defined4 ,an examination of the operations by which we determine whether a body is at rest or in motion shows that the operations are relative operations3 rest or motion is determined with respect to some other body selected as the standard4 ,in saying that there is no such thing as absolute rest or motion we are not making a statement about nature in the sense that might be supposed1 but we are merely making a statement about the character of our descriptive processes4 ,similarly with regard to size3 examination of the operations of the measuring process shows that size is measured relative to the fundamental measuring rod4 ,the absolute therefore disappears in the original meaning of the word4 ,but the absolute may usefully return with an altered meaning1 and we may say that a thing has absolute properties if the numerical magnitude is the same when measured with the same formal procedure by all observers4 ,whether a given property is absolute or not can be determined only by experiment1 landing us in the paradoxical position that the absolute is absolute only relative to experiment4 ,in some cases1 the most superficial observation shows that a property is not absolute1 as1 for example1 it is at once obvious that measured velocity changes with the motion of the observer4 ,but in other cases the decision is more difficult4 ,thus ,michelson thought he had an absolute procedure for measuring length1 by referring to the wave length of the red cadmium line as standard2#c it required difficult and accurate experiment to show that this length varies with the motion of the observer4 ,even then1 by changing the definition of the length of a moving object1 we believe that length might be made to re-assume its desired absolute character4 ,footnote #c3 ,a4 ,a4 ,michelson1 ,light ,waves and ,their ,uses1 ,university of ,chicago ,press1 #aijc1 ,chap4 ,v4 ,to stop the discussion at this point might leave the impression that this observation of the relative character of knowledge is of only a very tenuous and academic interest1 since it appears to be concerned mostly with the character of our descriptive processes1 and to say little about external nature4 ,what this means we leave to the metaphysician to decide4 ,but ,i believe there is a deeper significance to all this4 ,it must be remembered that all our argument starts with the concepts as given4 ,now these concepts involve physical operations2 in the discovery of what operations may be usefully employed in describing nature is buried almost all physical experience4 ,in erecting our structure of physical science1 we are building on the work of all the ages4 ,there is then this purely physical significance in the statement that all motion is relative1 namely that no operations of measuring motion have been found to be useful in describing simply the behavior of nature which are not operations relative to a single observer2 in making this statement we are stating something about nature4 ,it takes an enormous amount of real physical experience to discover relations of this sort4 ,the discovery that the number obtained by counting the number of times a stick may be applied to an object can be simply used in describing natural phenomena was one of the most important and fundamental discoveries ever made by man4 ,meaningless ,questions ,another consequence of the operational character of our concepts1 almost a corollary of that considered above1 is that it is quite possible1 nay even disquietingly easy1 to invent expressions or to ask questions that are meaningless4 ,it constitutes a great advance in our critical attitude toward nature to realize that a great many of the questions that we uncritically ask are without meaning4 ,if a specific question has meaning1 it must be possible to find operations by which an answer may be given to it4 ,it will be found in many cases that the operations cannot exist1 and the question therefore has no meaning4 ,for instance1 it means nothing to ask whether a star is at rest or not4 ,another example is a question proposed by ,clifford1 namely1 whether it is not possible that as the solar system moves from one part of space to another the absolute scale of magnitude may be changing1 but in such a way as to affect all things equally1 so that the change of scale can never be detected4 ,an examination of the operations by which length is measured in terms of measuring rods shows that the operations do not exist 7because of the nature of our definition of length7 for answering the question4 ,the question can be given meaning only from the point of view of some imaginary superior being watching from an external point of vantage4 ,but the operations by which such a being measures length are different from the operations of our definition of length1 so that the question acquires meaning only by changing the significance of our terms--in the original sense the question means nothing4 ,to state that a certain question about nature is meaningless is to make a significant statement about nature itself1 because the fundamental operations are determined by nature1 and to state that nature cannot be described in terms of certain operations is a significant statement4 ,it must be recognized1 however1 that there is a sense in which no serious question is entirely without meaning1 because doubtless the questioner had in mind some intention in asking the question4 ,but to give meaning in this sense to a question1 one must inquire into the meaning of the concepts as used by the questioner1 and it will often be found that these concepts can be defined only in terms of fictitious properties1 as ,newton's absolute time was defined by its properties1 so that the meaning to be ascribed to the question in this way has no connection with reality4 ,i believe that it will enable us to make more significant and interesting statements1 and therefore will be more useful1 to adopt exclusively the operational view1 and so admit the possibility of questions entirely without meaning4 ,this matter of meaningless questions is a very subtle thing which may poison much more of our thought than that dealing with purely physical phenomena4 ,i believe that many of the questions asked about social and philosophical subjects will be found to be meaningless when examined from the point of view of operations4 ,it would doubtless conduce greatly to clarity of thought if the operational mode of thinking were adopted in all fields of inquiry as well as in the physical4 ,just as in the physical domain1 so in other domains1 one is making a significant statement about his subject in stating that a certain question is meaningless4 ,in order to emphasize this matter of meaningless questions1 ,i give here a list of questions1 with which the reader may amuse himself by finding whether they have meaning or not4 7#a7 ,was there ever a time when matter did not exist8 7#b7 ,may time have a beginning or an end8 7#c7 ,why does time flow8 7#d7 ,may space be bounded8 7#e7 ,may space or time be discontinuous8 7#f7 ,may space have a fourth dimension1 not directly detectible1 but given indirectly by inference8 7#g7 ,are there parts of nature forever beyond our detection8 7#h7 ,is the sensation which ,i call blue really the same as that which my neighbor calls blue8 ,is it possible that a blue object may arouse in him the same sensation that a red object does in me and vice versa8 7#i7 ,may there be missing integers in the series of natural numbers as we know them8 7#aj7 ,is a universe possible in which #b #b #d8 7#aa7 ,why does negative electricity attract positive8 7#ab7 ,why does nature obey laws8 7#ac7 ,is a universe possible in which the laws are different8 7#ad7 ,if one part of our universe could be completely isolated from the rest1 would it continue to obey the same laws8 7#ae7 ,can we be sure that our logical processes are valid8 ,,general ,,comments ,,on ,,the ,,operational ,,point ,,of ,,view ,to adopt the operational point of view involves much more than a mere restriction of the sense in which we understand concept1 but means a far-reaching change in all our habits of thought1 in that we shall no longer permit ourselves to use as tools in our thinking concepts of which we cannot give an adequate account in terms of operations4 ,in some respects thinking becomes simpler1 because certain old generalizations and idealizations become incapable of use2 for instance1 many of the speculations of the early natural philosophers become simply unreadable4 ,in other respects1 however1 thinking becomes much more difficult1 because the operational implications of a concept are often very involved4 ,for example1 it is most difficult to grasp adequately all that is contained in the apparently simple concept of time1 and requires the continual correction of mental tendencies which we have long unquestioningly accepted4 ,operational thinking will at first prove to be an unsocial virtue2 one will find oneself perpetually unable to understand the simplest conversation of one's friends1 and will make oneself universally unpopular by demanding the meaning of apparently the simplest terms of every argument4 ,possibly after every one has schooled himself to this better way1 there will remain a permanent unsocial tendency1 because doubtless much of our present conversation will then become unnecessary4 ,the socially optimistic may venture to hope1 however1 that the ultimate effect will be to release one's energies for more stimulating and interesting interchange of ideas4 ,not only will operational thinking reform the social art of conversation1 but all our social relations will be liable to reform4 ,let any one examine in operational terms any popular present-day discussion of religious or moral questions to realize the magnitude of the reformation awaiting us4 ,wherever we temporize or compromise in applying our theories of conduct to practical life we may suspect a failure of operational thinking4 ,,chapter ,,ii ,,other ,,general ,,considerations ,,the ,,approximate ,,character ,,of ,,empirical ,,knowledge ,,although many aspects of the processes by which we obtain knowledge of the external physical world are much beyond the scope of our present inquiry1 one matter must be mentioned in detail because it tacitly underlies all our discussion1 the fact1 namely1 that all results of measurement are only approximate4 ,that such is true is evident after the most superficial examination of any measuring process2 any statement about numerical relations between measured quantities must always be subject to the qualification that the relation is valid only within limits4 ,furthermore1 all experience seems to be of this character2 we never have perfectly clean-cut knowledge of anything1 but all our experience is surrounded by a twilight zone1 a penumbra of uncertainty1 into which we have not yet penetrated4 ,this penumbra is as truly an unexplored region as any other region beyond experiment1 such as the region of high velocities1 for example1 and we must hold no preconceived notions as to what will be found within the region4 ,the penumbra is to be penetrated by improving the accuracy of measurement4 ,within what was at one time penumbra has been found the displacement of angular position of the stars near the edge of the solar disc1 and within the penumbra as yet unpenetrated we look for such effects as the equivalence of mass and energy4 ,many of the great discoveries of the future will probably be made within the penumbra3 we have already mentioned that increased knowledge of phenomena of a cosmic scale is to be obtained by increasing the accuracy of measurement of the very small4 ,it is a general consequence of the approximate character of all measurement that no empirical science can ever make exact statements4 ,this was fairly obvious in the case of mechanics1 but it required a ,gauss#d to convince us that the geometry in which we are interested as physicists is an empirical subject1 and that one cannot say that actual space is ,euclidean1 but only that actual space approaches to ideal ,euclidean space within a certain degree of approximation4 ,i believe that we are compelled to go still further1 and recognize that arithmetic1 so far as it purports to deal with actual physical objects1 is also affected with the same penumbra of uncertainty as all other empirical science4 ,a typical statement of empirical arithmetic is that #b objects plus #b objects makes #d objects4 ,this statement acquires physical meaning only in terms of certain physical operations1 and these operations must be performed in time4 ,footnote #d3 ,c4 ,f4 ,gauss1 ,gesammelte ,werke1 especially vols4 ,,iv and ,,viii4 ,now the penumbra gets into this situation through the concept of object4 ,if the statement of arithmetic is to be an exact statement in the mathematical sense the object must be a definite clean-cut thing1 which preserves its identity in time with no penumbra4 ,but this sort of thing is never experienced1 and as far as we know does not correspond exactly to anything in experience4 ,it is of course true that in most experience the penumbra is so very thin and snug-fitting that it requires special effort to recognize its presence at all2 but scrutiny1 ,i believe1 shows that it is always there4 ,if our experience had been restricted to phenomena in a vacuum1 and the objects we were trying to count had been spheres of a gas which expand and interpenetrate1 it is obvious that the concept of object as a thing with identity would have been much more difficult to form4 ,or1 if our objects are tumblers of water1 we discover when our observation reaches a certain stage of refinement that the amount of water is continually changing by evaporation and condensation1 and we are bothered by the question whether the object is still the same after it has waxed and waned4 ,coming to solids1 we eventually discover that even solids evaporate1 or condense gases on them1 and we see that an object with identity is an abstraction corresponding exactly to nothing in nature4 ,of course the penumbra of uncertainty which surrounds our arithmetical statements because of this property of physical objects is so exceedingly tenuous that practically we are not aware of its existence1 and expect never to find undiscovered phenomena within the penumbra4 ,but in principle we must recognize its presence1 and must further recognize that all empirical science must be of this character4 ,in most empirical sciences1 the penumbra is at first prominent1 and becomes less important and thinner as the accuracy of physical measurement is increased4 ,in mechanics1 for example1 the penumbra is at first like a thick obscuring veil at the stage where we measure forces only by our muscular sensations1 and gradually is attenuated1 as the precision of measurements increases4 ,but with the arithmetical concept of an individual identifiable object it is the exact reverse2 a crude point of view does not suspect the existence of the penumbra at all1 and we discover it only by highly refining our methods4 ,doubtless arithmetic owes its early development to this property4 ,we may now go still further4 ,operations themselves are1 of course1 derived from experience1 and would be expected also to have a nebulous edge of uncertainty4 ,we have to ask such questions as whether the operations of arithmetic are clean-cut things4 ,is the operation of multiplying #b objects by #b a definite operation1 with no enveloping haze8 ,all our physical experience convinces us that if there is a penumbra about the concept of operations of this sort it is so tenuous as to be negligible1 at least for the present2 but the question affords an interesting topic for speculation4 ,we also have to ask whether mental operations may similarly be enveloped in a haze4 ,,explanations ,,and ,,mechanisms ,perhaps the climax of our task of interpreting and correlating nature is reached when we are able to find an explanation of phenomena2 with the finding of the explanation we are inclined to feel that our understanding of the situation is complete4 ,we now have to ask what is the nature of the explanation which we set as the goal of our efforts4 ,the answer is not easy to give1 and there may be difference of opinion about it4 ,we shall get the best answer to this1 as to so many other questions1 by adopting the operational point of view1 and examining what we do in giving an explanation4 ,i believe that examination will show that the essence of an explanation consists in reducing a situation to elements with which we are so familiar that we accept them as a matter of course1 so that our curiosity rests4#e ,reducing a situation to elements means1 from the operational point of view1 discovering familiar correlations between the phenomena of which the situation is composed4 ,footnote #e3 ,the ultimate elements of explanation are analogous to the axioms of formal mathematics4 ,there is involved here the thesis that it is possible to analyze nature into correlations1 without1 however1 any assumption whatever as to the character of these correlations4 ,it seems to me that such a thesis is the most general that can be made if nature is to be intelligible at all4 ,this thesis underlies all the considerations of this essay1 and we shall not try to find anything more general4 ,we shall1 however1 recognize that any assumption as to the character of the correlations constitutes a special hypothesis which may restrict the future1 and that therefore these special hypotheses are to be subjected to special examination4 ,we return to this matter in more detail in discussing the causality concept1 which is closely related to the concept of explanation4 ,in this view of explanation there is no implication that the element is either a smaller or a larger scale thing than the phenomenon being explained2 thus we may explain the properties of a gas in terms of its constituent molecules1 or perhaps some day we shall become so familiar with the idea of a non-,euclidean space that we shall explain 7instead of describe7 the gravitational attraction of a stone by the earth in terms of a space-time curvature imposed by all the rest of the matter in the universe4 ,if this is accepted as the true nature of explanation1 we see that an explanation is not an absolute sort of thing1 but what is satisfactory for one man will not be for another4 ,the savage is satisfied by explaining the thunderstorm as the capricious act of an angry god4 ,the physicist demands more1 and requires that the familiar elements to which we reduce a situation be such that we can intuitively predict their behavior4 ,thus even if the physicist believed in the existence of the angry god1 he would not be satisfied with this explanation of the thunderstorm because he is not so well acquainted with angry gods as to be able to predict when anger is followed by a storm4 ,he would have to know why the god had become angry1 and why making a thunderstorm eased his ire4 ,but even with this additional qualification1 scientific explanation is obviously still a relative affair--relative to the elements or axioms to which we make reduction and which we accept as ultimate4 ,these elements depend to a certain extent on the purpose in view1 and also on the range of our previous physical experience4 ,if we are explaining the action of a machine1 we are satisfied to reduce the action to the push and pull of the various members of the machine1 it being accepted as an ultimate that these members transmit pushes or pulls4 ,but the physicist who has extended his experimental knowledge further1 may want to explain how the members transmit pushes or pulls in terms of the action on each other of the electrons in their orbits in the atoms4 ,the character of our explanatory structure will depend on the character of our experimental knowledge1 and will change as this changes4 ,formally1 there is no limit to the process of explanation1 because we can always ask what is the explanation of the elements in terms of which we have given the last explanation4 ,but the point of view of operations shows that this is mere formalism which ends only in meaningless jargon1 for we soon arrive at the limit of our experimental knowledge1 and beyond this the operations involved in the concepts of our explanations become impossible and the concepts become meaningless4 ,as we extend experimental knowledge and push our explanations further and further1 we see that the explanatory sequence may be terminated in several possible ways4 ,in the first place1 we may never push our experiments beyond a stage into which the elements with which we are already familiar do not enter4 ,in this case explanation is very simple3 it involves nothing essentially new1 but merely the disentanglement of complexities4 ,the kinetic theory of gases1 in explaining the thermal properties of a gas in terms of ordinary mechanical properties of the molecules1 suggests such a situation4 ,or1 secondly1 our experiments may bring us into contact with situations novel to us1 in which we can recognize no familiar elements1 or at least must recognize that there is something in addition to the familiar elements4 ,such a situation constitutes an explanatory crisis and explanation has to stop by definition4 ,or thirdly1 we may try to force our explanations into a predetermined mold1 by formally erecting or inventing beyond the range of present experiment ultimates more or less like elements already familiar to us1 and seek to explain all present experience in terms of these chosen ultimates4 ,leaving for the present the third possibility1 which is within our control to accept or reject1 and is a formal matter1 it is merely a question of experimental fact which of the first two possibilities corresponds to the actual state of affairs4 ,the most perfunctory examination of the present state of physics shows that we are now facing the second of these possibilities1 and that in the new experimental facts of relativity1 and in all quantum phenomena1 we are confronted with an explanatory crisis4 ,it has often been emphasized that ,einstein's theory of gravitation does not seek at all to give an explanation of gravitational phenomena1 but merely describes and correlates these phenomena in comparatively simple mathematical language4 ,no more attempt is made to reduce the gravitational attraction between the earth and the sun to simple terms than was made by ,newton4 ,in the realm of quantum phenomena it is of course the merest commonplace that our old ideas of mechanics and electrodynamics have failed1 so that it is a matter of the greatest concern to find how many1 or indeed whether any1 elements of the old situations can be carried over into the new4 ,an examination of many of the so-called explanations of quantum theory constitutes at once a justification of the definition of explanation given above1 and of the statement that in quantum phenomena we are at an explanatory crisis4 ,for the endeavor of all these quantum explanations is to find in every new or more complicated situation the same elements which have already been met in simpler situations1 and which are therefore relatively more familiar4 ,for example1 many quantum phenomena are made to involve the emission of energy when an electron jumps from one orbit to another4 ,but always the elements to which reduction is made are themselves quantum phenomena1 and these are still so new and unfamiliar that we feel an instinctive need for explanation in other terms4 ,we seek to understand why the electron emits when it jumps4 ,the explanatory crisis which now confronts us in relativity and quantum phenomena is but a repetition of what has occurred many time in the past4 ,a similar crisis confronted ,prometheus when he discovered fire1 and the first man who observed a straw sticking to a piece of rubbed amber1 or a suspended lodestone seeking the north star4 ,every kitten is confronted with such a crisis at the end of nine days4 ,whenever experience takes us into new and unfamiliar realms1 we are to be at least prepared for a new crisis4 ,now what are we to do in such a crisis8 ,it seems to me that the only sensible course is to do exactly what the kitten does1 namely1 to wait until we have amassed so much experience of the new kind that it is perfectly familiar to us1 and then to resume the process of explanation with elements from our new experience included in our list of axioms4 ,not only will observation show that this is what is now actually being done with respect to quantum and gravitational phenomena1 but it is in harmony with the entire spirit of our outlook on nature4 ,all our knowledge is in terms of experience2 we should not expect or desire to erect an explanatory structure different in character from that of experience4 ,our experience is finite2 on the confines of the experimentally attainable it becomes hazy1 and the concepts in terms of which we describe it fuse together and lose independent meaning4 ,furthermore1 at every extension of our experimental range we must be prepared to find1 and as a matter of fact we have often found1 that we encounter phenomena of an entirely novel character for which previous experience has given us no preparation4 ,the explanatory structure proposed above has all these properties2 it is finite1 being terminated by the edge of experiment1 the final stages of our explanations are hazy in that it becomes more and more difficult to distinguish elements of familiar experience1 and every now and then we must admit new elements into our explanations4 ,the first step in resuming our explanatory progress1 after we have been confronted with such a crisis1 is to seek for various sorts of correlation between the elements of our new experience1 in the confident expectation that these elements will eventually become so familiar to us that they may be used as the ultimates of a new explanation4 ,this is exactly what is now happening in quantum theory4 ,diametrically opposed to the views above1 there is another ideal of the explanatory process which is held by many physicists1 and which has been mentioned above as the third possible way in which the explanatory sequence may be terminated1 namely1 the endeavor to devise beyond the limit of present experiment a structure built of elements like some of those of our present experience1 in the action of which we endeavor to find the explanation of phenomena in the present range4 ,now a program such as this1 as a serious program for the final correlation of nature1 is entirely opposed to the spirit of the considerations expounded here4 ,there is no warrant whatever in experience for the conviction that as we penetrate deeper and deeper we shall find the elements of previous experience repeated1 although sometimes we do find such repetitions1 as in the behavior of gases4 ,yet this has been the attitude of many eminent physicists1 for example1 ,faraday and ,maxwell1 in seeking to explain distant electrical action by the propagation through a medium of a mechanical push or pull1 or by ,hertz1 who sought in all phenomena the effect of concealed masses with ordinary mechanical inertia4 ,although as a general principle this program seems to be absolutely without justification1 nevertheless it may be justified if the specific character of the physical facts seems to indicate a repetition at lower levels of elements familiar higher up4 ,hertz undoubtedly had this justification1 as did also ,maxwell to a certain extent1 in the discovery that the general equations of electrodynamics are of the same form as the generalized ,lagrangean equations of mechanics4 ,for ,faraday1 however1 there seems no such justification2 the urge to this sort of thing in ,faraday came from an uncritical acceptance of his own temperamental reactions4 ,from a less serious point of view it may1 however1 be quite justified to make such a working hypothesis as that in the action of electrical forces may be discovered the same elements with which we are familiar in the everyday experiences of mechanics4 ,for such a hypothesis often enables us to make partial correlations which suggest new experimental tests1 and thus gives the stimulus to an extension of our experimental horizon4 ,many physicists recognize the tentative character of such attempted explanations1 but others apparently take them more seriously1 as for example ,lord ,kelvin in his continuous life-long attempts to find a mechanical explanation of all physical phenomena4 ,this quotation from ,kelvin is illuminating4 ,i never satisfy myself until ,i can make a mechanical model of a thing4 ,if ,i can make a mechanical model1 ,i can understand it4 ,as long as ,i cannot make a mechanical model all the way through1 ,i cannot understand it4444 ,but ,i want to understand light as well as ,i can without introducing things that we understand even less of4 ,so much for general considerations on the nature of explanation4 ,coming now to greater detail1 many explanations involve what may be described as a mechanism4 ,it is difficult to characterize exactly what we mean by mechanism1 but it seems to be associated with an attitude of mind that strives to realize the third of the possibilities mentioned above4 ,as a matter of fact1 the mechanism sought for is usually of a particular type1 in that the ultimate elements selected are mechanical elements4 ,this point of view is particularly characteristic of the ,english school of physicists4 ,although mechanism usually implies mechanical elements1 we may show by specific examples that we do actually use the word in a broader significance4 ,if1 for example1 we could devise within the core of an atom a revolving system of electrical charges1 acting on each other with the ordinary inverse square forces of electrostatics1 such that every now and then the system becomes unstable and breaks up1 we should doubtless say that we had found a mechanism for explaining radioactive disintegration4 ,however1 the formulation of a precise definition of mechanism is of secondary concern to us2 we are primarily interested in understanding the attitude of mind that feels a mechanism is necessary4 ,a typical example of such an urge to a mechanism is afforded by the gravitational action between distant bodies4 ,to many minds the concept of action at a distance is absolutely abhorrent1 not to be tolerated for an instant4 ,such an intolerable situation is avoided by the invention of a medium filling all space1 which transmits a force from one body to the other through the successive action on each other of its contiguous parts4 ,or the dilemma of action at a distance may be avoided in other ways1 as by ,boscovitch in the eighteenth century1 who1 in order to explain gravitation1 filled space with a triply infinite horde of infinitesimal projectiles4 ,now of course it is a matter for experiment to decide whether any physical reality can be ascribed to a medium which makes gravitation possible by the action of its adjacent parts1 but ,i can see no justification whatever for the attitude which refuses on purely a priori grounds to accept action at a distance as a possible axiom or ultimate of explanation4 ,it is difficult to conceive anything more scientifically bigoted than to postulate that all possible experience conforms to the same type as that with which we are already familiar1 and therefore to demand that explanation use only elements familiar in everyday experience4 ,such an attitude bespeaks an unimaginativeness1 a mental obtuseness and obstinacy1 which might be expected to have exhausted their pragmatic justification at a lower plane of mental activity4 ,although it will probably be fairly easy to give intellectual assent to the strictures of the last paragraph1 ,i believe many will discover in themselves a longing for mechanical explanation which has all the tenacity of original sin4 ,the discovery of such a desire need not occasion any particular alarm1 because it is easy to see how the demand for this sort of explanation has had its origin in the enormous preponderance of the mechanical in our physical experience4 ,but nevertheless1 just as the old monks struggled to subdue the flesh1 so must the physicist struggle to subdue this sometimes nearly irresistible1 but perfectly unjustifiable desire4 ,one of the large purposes of this exposition will be attained if it carries the conviction that this longing is unjustifiable1 and is worth making the effort to subdue4 ,the situation with respect to action at a distance is typical of the general situation4 ,i believe the essence of the explanatory process is such that we must be prepared to accept as an ultimate for our explanations the mere statement of a correlation between phenomena or situations with which we are sufficiently familiar4 ,thus1 in quantum theory1 there is no reason why we should not be willing to accept as an ultimate the fundamental fact that when an electron jumps radiation is emitted1 provided always that we can give independent meaning in terms of operations to the jumping of an electron4 ,if there is no experiment suggesting other and intermediate phenomena1 we ought to be able to rest intellectually satisfied with this4 ,of course it is quite a different matter1 and entirely justified1 to imagine what the assumption of finer details in the process would involve experimentally1 and then to seek for these possible new experimental facts4 ,it is a consequence of this view that any correlation is adapted to be an absolutely final element of explanation1 and can never be superseded by the discovery of new experimental facts1 if the correlation is by definition beyond the reach of further experiment4 ,such a possibility1 for example1 is contained in a correlation between the numerical magnitude of the gravitational constant and the total mass of the universe4 ,something of this sort may be well attempted by those who desire their explanations to take a formally final shape4 ,we shall return to this subject later4 ,the instinctive demand for a mechanism is fortified by observation of the many important cases in which mechanisms have been discovered or invented4 ,however1 the significance of such successful attempts must be subject to most careful scrutiny4 ,the matter has been discussed by ,poincar1#f who showed that not only is it always possible to find a mechanistic explanation of any phenomenon 7,hertz's program was a perfectly possible one71 but there are always an infinite number of such explanations4 ,this is very unsatisfactory4 ,footnote #f3 ,henri ,poincar4 ,wissenschaft and ,hypothese1 ,translated into ,german by ,f4 and ,l4 ,lindemann1 ,teubner1 ,leipzig1 #aijf4 ,see especially p4 #bag4 ,we want to be able to find the real mechanism4 ,now an examination of specific proposed mechanisms will show that most mechanisms are more complicated than the simple physical phenomenon which they are invented to explain1 in that they have more independently variable attributes than the phenomenon has been yet proved to have4 ,an example is afforded by the mechanical models invented to facilitate the study of the properties of simple inductive electrical circuits4 ,the great number of such models which have been proposed is sufficient indication of their possible infinite number4 ,but if the mechanism has more independently variable attributes than the original phenomenon1 it is obvious that the question is without meaning whether the mechanism is the real one or not1 for in the mechanism there must be simple motions or combinations of motions which have no counterpart in features of the original phenomenon as yet discovered4 ,obviously1 then1 the operations do not exist by which we may set up a one to one correspondence between the properties of the mechanism and the natural phenomenon1 and the question of reality has no meaning4 ,if1 then1 a mechanism is to be taken seriously as actually corresponding to reality1 we must demand that it have no more degrees of freedom than the original phenomenon1 and we must also be sure that the phenomenon has no undiscovered features4 ,physical experience shows that such conditions are most difficult to meet1 and indeed the probability is that they are impossible4 ,a mechanism with more independently variable attributes than the phenomenon may prove to be a very useful tool of thought1 and therefore worth inventing and studying1 but it is to be regarded no more seriously than is a mnemonic device1 or any other artifice by which a man forces his mind to give him better service4 ,there is another possible program of explanation1 the converse of that considered above1 namely1 to explain all familiar facts of ordinary experience in terms of less familiar facts found at a deeper level4 ,the most striking example of this is the recent attempt to give a complete electrical explanation of the universe4 ,the original attempt was to explain electrical effects in mechanical terms2 this attempt failed4 ,at about the same time the existence of the electron was experimentally established1 so that it was evident that electricity is a very fundamental constituent of matter4 ,the program of explanation was reversed1 and an electrical explanation sought for all mechanical phenomena1 including in particular mechanical mass4 ,but this attempt has also failed2 we recognize that part of mass may be non-electrical in character1 we postulate non-electrical forces inside the electron1 and further1 we usually postulate for electrons and protons the property of impenetrability1 a property derived from experience on a higher scale of magnitude4 ,a program of this general sort is likely to be regarded with considerable sympathy1 and indeed the chances of success seem much greater than do those of the converse program1 for in our experience large scale phenomena are more often built up from and analyzed into small scale phenomena than the converse4 ,but as a matter of principle we must again recognize that the only appeal is to experiment1 and that we have to ask just one question3 ,is it true1 as a matter of fact1 that all large scale phenomena can be built up of elements of small scale phenomena8 ,it seems to me that the experimental warrant for this conviction has not yet been given4 ,the failure of the attempted electrical explanation of the universe is a case in point4 ,however1 the failure to prove a proposition is no guarantee that some time it may not be proved1 and many physicists are convinced of the ultimate feasibility of this program4 ,personally ,i feel that the large may not always be analyzed into the smaller2 the subject will be discussed again4 ,a conviction of the significance of microscopic analysis has many features in common with the usual conviction of the ultimate simplicity of nature4 ,the thesis of simplicity involves in addition the assumption that the kinds of small scale elements are few in number1 but actually this involves no important difference between the two convictions1 because we have seen that the elements of which we build our structure become fewer in number as we approach the limit of the experimentally attainable4 ,we may properly grant to convictions of this sort pragmatic value in suggesting new correlations and experiments1 but a recognition of the empirical basis of all physics will not allow us to go further4 ,,models ,,and ,,constructs ,in discussing the concept of length1 we could find no meaning in questions such as3 ,is space on a scale of #aj-h cm4 ,euclidean8 ,nevertheless it will seem to many that they do attach a perfectly definite meaning to a question of this kind4 ,of course it must be agreed that magnitudes of #aj-h cm4 cannot be thought of in terms of immediate sensation4 ,when one thinks of an atom as a thing with any geometrical properties at all1 ,i believe he will find that what he essentially does is to imagine a model1 multiplying all the hypothetical dimensions by a factor large enough to bring it to a magnitude of ordinary experience4 ,this large scale model is given properties corresponding to those of the physical thing4 ,for example1 the model of the atom which was accepted in the fall of #aibe contains electrons rotating in orbits1 and every now and then an electron jumps from one orbit to another1 and simultaneously energy is radiated from the atom4 ,such a model is satisfactory if it offers the counterpart of all the phenomena of the original atom4 ,now ,i believe the only meaning that any one can find in his statement that the space of the atom is ,euclidean is that he believes that he can construct in ,euclidean space a model with all the observed properties of the atom4 ,this possibility may or may not be sufficient to give real physical significance to the statement that the space of the atom is ,euclidean4 ,the situation here is very much the same as it was with respect to mechanisms4 ,the model may have many more properties than correspond to measurable properties of the atom1 and in particular1 the operations by which the space of the model is tested for its ,euclidean character may and as a matter of fact ,i believe do not have any counterpart in operations which can be carried out on the atom4 ,further1 we cannot attach any real significance to the statement that the space of the atom is ,euclidean unless we can show that no model constructed in non-,euclidean space can reproduce the measurable properties of the atom4 ,in spite of all this1 ,i believe that the model is a useful and indeed unescapable tool of thought1 in that it enables us to think about the unfamiliar in terms of the familiar4 ,there are1 however1 dangers in its use3 it is the function of criticism to disclose these dangers1 so that the tool may be used with confidence4 ,closely related to the mental model are mental constructs1 of which physics is full4 ,there are many sorts of constructs3 those in which we are interested are made by us to enable us to deal with physical situations which we cannot directly experience through our senses1 but with which we have contact indirectly and by inference4 ,such constructs usually involve the element of invention to a greater or less degree4 ,a construct containing very little of invention is that of the inside of an opaque solid body4 ,we can never experience directly through our senses the inside of such a solid body1 because the instant we directly experience it1 it ceases by definition to be the inside4 ,we have here a construct1 but so natural a one as to be practically unavoidable4 ,an example of a construct involving a greater amount of invention is the stress in an elastic body4 ,a stress is by definition a property of the interior points of a body which is connected mathematically in a simple way with the forces acting across the free surface of the body4 ,a stress is then1 by its very nature1 forever beyond the reach of direct experience1 and it is therefore a construct4 ,the entire structure of a stress corresponds to nothing in direct experience2 it is related to force1 but is itself a six-fold magnitude1 whereas a force is only three-fold4 ,we have next to ask whether the stress1 which we have invented to meet the situation in a body exposed to forces1 is a good construct4 ,in the first place1 a stress has the same number of degrees of freedom as the observable phenomenon1 for it is one of the propositions of the mathematical theory of elasticity that the boundary conditions1 which are the experimental variables1 uniquely determine the stress in a given body i4e4 a body of given elastic constants2 and of course it is at once obvious1 by an inspection of the equations1 that conversely a possible stress system uniquely determines the boundary conditions to the significant amount4 ,there is1 therefore1 a unique one-to-one correspondence between a stress and the physical situation it was made to meet1 and so far a stress is a good construct4 ,up to this point a stress1 from the point of view of the operations in terms of which it is defined1 is a purely mathematical invention1 which is justified because it is convenient in describing the behavior of bodies subjected to the action of force4 ,but we wish now to go farther and ascribe physical reality to a stress1 meaning by this that a stress in a solid body shall correspond to some real physical state of the interior points4 ,let us examine1 from the point of view of operations1 what the meaning of a statement like this may be4 ,since we now wish to ascribe an additional physical meaning to a stress beyond that of the mathematical operations in terms of which the stress was determined1 there must exist additional physical operations corresponding to this meaning1 or else our statement is meaningless4 ,now of course it is a matter of the most elementary experience that physical phenomena do exist which allow these other independent operations4 ,a body under stress is also in a state of strain1 which may be determined from the external deformations1 or the strain at internal points may be made more vividly real by those optical effects of double refraction in transparent bodies which are now so extensively used in model experiments1 or if the stress is pushed beyond a certain point1 we have such new phenomena as permanent set or finally1 rupture4 ,we have1 then1 every reason to be satisfied with our construct of stress4 ,in the first place1 from the formal point of view1 it is a good construct because there is a unique correspondence between it and the physical data in terms of which it is defined2 and in the second place we have a right to ascribe physical reality to it because it is uniquely connected with other physical phenomena1 independent of those which entered its definition4 ,this last requirement1 in fact1 from the operational point of view1 amounts to nothing more than a definition of what we mean by the reality of things not given directly by experience4 ,since now in addition to satisfying the formal requirements1 experience shows that a stress is most useful in correlating phenomena1 we are justified in giving to this construct of stress a prominent place among our concepts4 ,consider now another construct1 one of the most important of physics1 that of the electric field4 ,in the first place1 an examination of the operations by which we determine the electric field at any point will show that it is a construct1 in that it is not a direct datum of experience4 ,to determine the electric field at a point1 we place an exploring charge at the point1 measure the force on it1 and then calculate the ratio of the force to the charge4 ,we then allow the exploring charge to become smaller and smaller1 repeating our measurement of force on each smaller charge1 and define the limit of the ratio of the force to the charge as the electric field intensity at the point in question1 and the limiting direction of the force on a small charge as the direction of the field4 ,we may extend this process to every point of space1 and so obtain the concept of a field of force1 by which every point of the space surrounding electric charges is tagged with the appropriate number and direction1 the exploring charge having completely disappeared4 ,the field is1 then1 clearly a construct4 ,next1 from the formal point of view of mathematics1 it is a good construct1 because there is a one to one correspondence between the electric field and the electric charges in terms of which it is defined1 the field being uniquely determined by the charges1 and conversely there being only one possible set of charges corresponding to a given field4 ,now nearly every physicist takes the next step1 and ascribes physical reality to the electric field1 in that he thinks that at every point of the field there is some real physical phenomenon taking place which is connected in a way not yet precisely determined with the number and direction which tag the point4 ,at first this view most naturally involved as a corollary the existence of a medium1 but lately it has become the fashion to say that the medium does not exist1 and that only the field is real4 ,the reality of the field is self-consciously inculcated in our elementary teaching1 often with considerable difficulty for the student4 ,this view is usually credited to ,faraday1 and is considered the most fundamental concept of all modern electrical theory4 ,yet in spite of this1 ,i believe that a critical examination will show that the ascription of physical reality to the electric field is entirely without justification4 ,i cannot find a single physical phenomenon or a single physical operation by which evidence of the existence of the field may be obtained independent of the operations which entered the definition4 ,the only physical evidence we ever have of the existence of a field is obtained by going there with an electric charge and observing the action on the charge when the charges are inside atoms we may have optical phenomena1 which is precisely the operation of the definition4 ,it is then either meaningless to say that a field has physical reality1 or we are guilty of adopting a definition of reality which is the crassest tautology4 ,there can be no question whatever of the tremendous importance of the concept of the electric field as a tool in thinking about1 describing1 correlating1 and predicting the properties of electrical systems2 electrical science is inconceivable without this or something equivalent4 ,but in addition to this aspect of the field concept1 the further tacit implication of physical reality is almost always present1 and has had the greatest influence on the character of physical thought and experiment4 ,yet ,i do not believe that the additional implication of physical reality has justified itself by bringing to light a single positive result1 or can offer more than the pragmatic plea of having stimulated a large number of experiments1 all with persistently negative results4 ,it is sufficient to mention the fate of the attempt of ,faraday and ,maxwell to ascribe a stress like that of ordinary matter to the ether1 which failed because1 among other reasons1 nothing can exist in the ether analogous to the strain of ordinary matter1 to indicate the unfruitfulness of the idea of physical reality4 ,it seems to me that any pragmatic justification in postulating reality for the electric field has now been exhausted1 and that we have reached a stage where we should attempt to get closer to the actual facts by ridding the field concept of the implications of reality4 ,another indispensable and most interesting construct is that of the atom4 ,this is evidently a construct1 because no one ever directly experienced an atom1 and its existence is entirely inferential4 ,the atom was invented to explain constant combining weights in chemistry4 ,for a long time there was no other experimental evidence of its existence1 and it remained a pure invention1 without physical reality1 useful in discussing a certain group of phenomena4 ,it is one of the most fascinating things in physics to trace the accumulation of independent new physical information all pointing to the atom1 until now we are as convinced of its physical reality as of our hands and feet4 ,a construct which had to be abandoned because it did not turn out to have physical reality1 and which furthermore was not sufficiently useful in the light of newly discovered phenomena1 was that of a caloric fluid4 ,the notion of physical reality is not of prime importance to this discussion of the character of our constructs2 our definition of the meaning of physical reality may not appeal to everyone4 ,the essential point is that our constructs fall into two classes3 those to which no physical operations correspond other than those which enter the definition of the construct1 and those which admit of other operations1 or which could be defined in several alternative ways in terms of physically distinct operations4 ,this difference in the character of constructs may be expected to correspond to essential physical differences1 and these physical differences are much too likely to be overlooked in the thinking of physicists4 ,we must always be on our guard not to forget the physical differences between a thing like a stress in an elastic body and an electromagnetic field4 ,the moral of all this is that constructs are most useful and even unavoidable things1 but that they may have great dangers1 and that a careful critique may be necessary to avoid reading into them implications for which there is no warrant in experience1 and which may most profoundly affect our physical outlook and course of action4 ,,the ,r,,le ,,of ,,mathematics ,,in ,,physics ,practically all the formulations of theoretical physics are made in mathematical terms2 in fact to obtain such formulations is generally felt to be the goal of theoretical physics4 ,it is then evidently pertinent to consider what the nature of the mathematics is to which we assign so prominent a rle4 ,we have in the first place to understand why it is possible to express physical relations in mathematical language at all4 ,i am not sure that there is much meaning in this question4 ,it is the merest truism1 evident at once to unsophisticated observation1 that mathematics is a human invention4 ,furthermore1 the mathematics in which the physicist is interested was developed for the explicit purpose of describing the behavior of the external world1 so that it is certainly no accident that there is a correspondence between mathematics and nature4 ,the correspondence is not by any means perfect1 however1 but there is always in mathematics a precise quality to which none of our information about nature ever attains4 ,the theorems of ,euclid's geometry illustrate this in a preeminent degree4 ,the statement that there is just one straight line between two points and that this is the shortest possible path between the points is entirely different in character from any information ever given by physical measurement1 for all our measurements are subject to error4 ,it is possible1 nevertheless1 to give a certain real physical meaning to the ideally precise statements of geometry1 because it is a result of everyday experience that as we refine the accuracy of our physical measurements the quantitative statements of geometry are verified within an ever decreasing margin of error4 ,from this arises that view of the nature of mathematics which apparently is most commonly held2 namely that if we could eliminate the imperfections of our measurements1 the relations of mathematics would be exactly verified4 ,abstract mathematical principles are supposed to be active in nature1 controlling natural phenomena1 as ,pythagoras long ago tried to express with his harmony of the spheres and the mystic relations of numbers4 ,this idealized view of the connection of mathematics with nature could be maintained only during that historical period when the accuracy of physical measurement was low1 and must now be abandoned4 ,for it is no longer true that the precise relations of ,euclid's geometry may be indefinitely approximated to by increasing the refinements of the measuring process1 but there are essential physical limitations to the very concepts of length1 etc41 which enter the geometrical formulations1 set by the discrete structure of matter and of radiation4 ,this is no academic matter1 but touches the essence of the situation4 ,there is no longer any basis for the idealization of mathematics1 and for the view that our imperfect knowledge of nature is responsible for failure to find in nature the precise relations of mathematics4 ,it is the mathematics made by us which is imperfect and not our knowledge of nature4 ,from the operational point of view it is meaningless to attempt to separate nature from knowledge of nature4 ,the concepts of mathematics are inventions made by us in the attempt to describe nature4 ,now we shall repeatedly see that it is the most difficult thing in the world to invent concepts which exactly correspond to what we know about nature1 and we apparently never achieve success4 ,mathematics is no exception2 we doubtless come closer to the ideal here than anywhere else1 but we have seen that even arithmetic does not completely reproduce the physical situation4 ,mathematics appears to fail to correspond exactly to the physical situation in at least two respects4 ,in the first place1 there is the matter of errors of measurement in the range of ordinary experience4 ,now mathematics can deal with this situation1 although somewhat clumsily1 and only approximately1 by specifically supplementing its equations by statements about the limit of error1 or replacing equations by inequalities--in short1 the sort of thing done in every discussion of the propagation of error of measurement4 ,in the second place1 and much more important1 mathematics does not recognize that as the physical range increases1 the fundamental concepts become hazy1 and eventually cease entirely to have physical meaning1 and therefore must be replaced by other concepts which are operationally quite different4 ,for instance1 the equations of motion make no distinction between the motion of a star into our galaxy from external space1 and the motion of an electron about the nucleus1 although physically the meaning in terms of operations of the quantities in the equations is entirely different in the two cases4 ,the structure of our mathematics is such that we are almost forced1 whether we want to or not1 to talk about the inside of an electron1 although physically we cannot assign any meaning to such statements4 ,as at present constructed1 mathematics reminds one of the loquacious and not always coherent orator1 who was said to be able to set his mouth going and go off and leave it4 ,what we would like is some development of mathematics by which the equations could be made to cease to have meaning outside the range of numerical magnitude in which the physical concepts themselves have meaning4 ,in other words1 the problem is to make our equations correspond more closely to the physical experience back of them2 it evidently needs some sort of new invention to accomplish this4 ,we return later1 in discussing ,lorentz's equations of electrodynamics1 to the disadvantages arising from the present undiscriminating character of mathematics4 ,in the meantime1 we must recognize that there are very important advantages here1 as well as disadvantages4 ,all experience justifies the expectation that the laws of nature with which we are already familiar hold at least approximately and without violent change in the unexplored regions immediately beyond our present reach4 ,by assuming an unlimited validity for the laws as we now know them1 mathematics enables us to penetrate the twilight zone1 and make predictions which may be later verified4 ,it is only when we are carried too far afield that we must deprecate this characteristic of our mathematics4 ,there is another aspect of the use of mathematics in describing nature that is often lost sight of2 namely1 that any system of equations can contain only a very small part of the actual physical situation2 there is behind the equations an enormous descriptive background through which the equations make connection with nature4 ,this background includes a description of all the physical operations by which the data are obtained which enter the equations4 ,for instance1 when ,einstein formulates the behavior of the universe in terms of the world lines of events1 the events as they enter the equations are entirely colorless things1 merely #c space and #a time cordinate4 ,to make connection with experience there must be a descriptive background giving the physical contents of the events2 for example1 there may be the statement that some of the events are light signals4 ,this descriptive background is supposed to remain fixed1 unaffected by any operations to which the equations themselves are subject4 ,if1 for example1 the frame of reference of the equations is altered by changing its velocity1 the physical significance of the descriptive background is supposed to remain unaltered1 or rather no mention is usually made of this question at all4 ,it would seem1 however1 that this matter needs some discussion4 ,the descriptive background gets its meaning only in terms of certain physical operations4 ,if the descriptive background remains unaltered when the uniform velocity of the frame of reference is changed1 for instance1 this means that the motion of the frame of reference does not at all affect the possibility of carrying out certain operations4 ,this is pretty close to the restricted principle of relativity itself1 which states that the form of natural laws is not affected by uniform velocity4 ,until a more careful analysis of the situation is made it would seem therefore that there is some ground for the suspicion that the principle of relativity is involved in the possibility of giving to physical phenomena a complete mathematical formulation1 understanding complete to mean including the descriptive background4 ,,chapter ,,iii ,,detailed ,,consideration ,,of ,,various ,,concepts ,,of ,,physics ,,we now begin our detailed consideration of the most important concepts of physics4 ,it is entirely beyond the scope of this essay to attempt more than an indication of some of the most important matters4 ,neither is it to be expected that the parts of this discussion will always have a very close connection with each other2 the purpose of the discussion is to aid in acquiring the greatest possible self-consciousness of the whole structure of physics4 ,,the ,,concept ,,of ,,space ,a logically satisfying definition of what we understand by the concept of space is doubtless difficult to give1 but we shall not be far from the mark if we think of it as the aggregate of all those concepts which have to do with position4 ,position means position of something4 ,the position of things is determined by some system of measurement2 perhaps the simplest is that implied in a ,cartesian cordinate system with its three measurements of length4 ,hence much of the essential discussion of space has already been given in connection with the concept of length4 ,we have seen that measurements of length are made with physical measuring rods applied to some physical object4 ,we cannot measure the distance between two points in empty space1 because if space were empty there would be nothing to identify the position of the ends of the measuring rod when we move it from one position to the next4 ,we see1 then1 from the point of view of operations that the framework of ,cartesian geometry1 often imagined in an ideal mathematical sense1 is really a physical framework1 and that what we mean by spatial properties is nothing but the properties of this framework4 ,when we say that space is ,euclidean1 we mean that the physical space of meter sticks is ,euclidean3 it is meaningless to ask whether empty space is ,euclidean4 ,geometry1 therefore1 in so far as its results are expected to apply to the external physical world1 and in as far as it is not a logical system built up from postulates1 is an experimental science4 ,this view is now well understood and accepted1 but there was a time when it was not accepted1 but vigorously attacked2 the change of attitude toward this question is symptomatic of a change of attitude toward many other similar questions4 ,we have already emphasized that the space of astronomy is not a physical space of meter sticks1 but is a space of light waves4 ,we may1 therefore1 have different kinds of space1 depending on the fundamental operations4 ,the space of meter sticks we have called tactual space1 and the space of light beams optical space4 ,if we ask whether astronomical space is ,euclidean1 we mean merely to ask whether those features of optical space which are within the reach of astronomical measurement are ,euclidean4 ,the only possible attitude with respect to this question1 or such related questions as whether the total volume of space is finite1 or whether space has curvature1 is that it is entirely for experiment to decide1 and that we have no right to form any preconceived notion whatever4 ,it is therefore beyond the scope of this discussion4 ,it is interesting to notice that the restricted theory of relativity virtually assumes1 although often without making the explicit statement1 that tactual and optical space are the same4 ,this equivalence results from the properties assumed for light beams4 ,the distance of a mirror may be found equally well by measuring it with meter sticks1 or by determining the time required by a light signal to travel there and back4 ,this situation is1 however1 logically unsatisfying1 because it must be assumed that the operations for measuring time are independently defined1 and we shall see that they are not4 ,it is a consequence of the assumed equivalence of tactual and optical space that the path of a beam of light is a straight line1 as a straight line is determined by operations with meter sticks4 ,when we come to astronomical phenomena1 the physical operations with meter sticks can no longer be carried out1 and it is meaningless to ascribe to beams of light on an astronomical scale the same geometrical properties that we do on a small scale4 ,,the ,,concept ,,of ,,time ,according to our viewpoint1 the concept of time is determined by the operations by which it is measured4 ,we have to distinguish two sorts of time2 the time of events taking place near each other in space1 or local time1 and the time of events taking place at considerably separated points in space1 or extended time4 ,as we now know1 the concept of extended time is inextricably mixed up with that of space4 ,this is not primarily a statement about nature at all1 and might have been made simply by the observation that the operations by which extended time is measured involve those for measuring space4 ,of course historically the doctrine of relativity was responsible for the critical attitude which led to an examination of the operations of measuring time1 but relativity was not necessary for a realization of the spatial implications of time1 any more than the discovery of ,planck's quantum unit h was necessary for the invention by ,planck of his absolute units of measurement1 although historically he was inspired to make this invention by discovering h1 and in his own mind seems to have thought of the connection as a necessary one4#g ,footnote #g3 ,max ,planck1 ,the ,theory of ,heat ,radiation1 translated by ,masius1 ,p4 ,blakiston's ,son ,co41 #aiad edition1 p4 #agd4 ,the physical operations at the basis of the measurement of time have never been subjected to the critical examination which seems to be required4 ,one method of measurement1 for instance1 involves the properties of light4 ,a meter stick is set up with mirrors at the two ends1 and a light beam travels back and forth between the two mirrors without absorption4 ,the time required for a single passage back and forth is defined as the unit of time1 and time is measured simply by counting these intervals4 ,but such a procedure is unsatisfactory if we are to permit ourselves all those operations which are demanded by even the simplest postulate of relativity1 for we must be able to move our clock from place to place1 transfer it from one system to another in relative motion1 and with it determine the properties of light beams in the stationary or moving system4 ,we recognize in principle that the length of the meter stick may be different when it is in motion1 that it may change also during the acceleration incident to moving it from one place to another1 and that until proved to the contrary the velocity of light may be a function of velocity or acceleration4 ,the complicated interplay of all these possibilities leaves us in much doubt as to the physical significance of such postulates as1 for example1 that the velocity of light is the same in the moving system and the stationary system4 ,in order to ascribe any simple significance to postulates about the velocity of light1 it would seem that we must have an instrument for measuring this velocity1 and therefore for measuring time1 which does not itself involve the properties of light4 ,to do this we might seek to specify the measurement of time in purely mechanical terms1 as for instance in terms of the vibration of a tuning fork1 or the rotation of a flywheel4 ,but here again we encounter great difficulties1 because we recognize that the dimensions of our mechanical clock may change when it is set in motion1 and that the mass of its parts may also change4 ,we want to use the clock as a physical instrument in determining the laws of mechanics1 which of course are not determined until we can measure time1 and we find that the laws of mechanics enter into the operation of the clock4 ,the dilemma which confronts us here is not an impossible one1 and is in fact of the same nature as that which confronted the first physicist who had to discover simultaneously the approximate laws of mechanics and geometry with a string which stretched when he pulled it4 ,we must first guess at what the laws are approximately1 then design an experiment so that1 in accordance with this guess1 the effect of motion on some phenomenon is much greater than the expected effect on the clock1 then from measurements with uncorrected clock time find an approximate expression for the effect of motion on mass or length1 with which we correct the clock1 and so on ad infinitum4 ,however1 so far as ,i know1 the possibility of such a procedure has not been analyzed1 and until the analysis is given1 our complacency is troubled by a real disquietude1 the intensity of which depends on the natural skepticism of our temperament4 ,in practice1 the difficulties of such a logical treatment are so great that the matter has been entirely glossed over4 ,it is convenient to postulate a clock1 of unknown construction1 but such that the velocity of light1 when measured in terms of it1 has certain properties4 ,such1 for example1 is the point of view in ,birkhoff's recent book4#h ,footnote #h3 ,g4 ,d4 ,birkhoff4 ,relativity and ,modern ,physics1 ,harvard ,university ,press1 #aibc4 ,the difficulty with this method is that the resulting edifice is as divorced from physical reality as is the logical geometry of postulates4 ,we cannot be at all sure that the properties of light as measured with our physical clocks are the same as the theoretical properties4 ,the difficulty is particularly important and fundamental in the general theory of relativity2 the basis of the whole theory is the infinitesimal interval ds1 which is supposed to be given4 ,once given1 the mathematics follows4 ,but in a physical world1 ds is not given1 but must be found by physical operations1 and these operations involve measurements of length and of time with clocks whose construction is not specified4 ,in any actual physical application the question must be answered whether the physical instrument used in measuring the temporal part of ds is really a clock or not4 ,there is at present no criterion by which this question can be answered4 ,if the vibrating atom is a clock1 then the light of the sun is shifted toward the infra-red1 but how do we know that the atom is a clock 7some say yes1 others no78 ,if we find the displacement physically have we thereby proved that general relativity is physically true1 or have we proved that the atom is a clock1 or have we merely proved that there is a particular kind of connection between the atom and the rest of nature1 leaving the possibility open that neither is the atom a clock nor general relativity true8 ,in practice1 of course1 we shall adopt the solution which is simplest and most satisfying aesthetically1 and doubtless shall say that the atom is a clock and relativity true4 ,but if we adopt this simple view1 we must also cultivate the abiding consciousness that at some time in the future troubles may have their origin here4 ,it seems to me that the logical position of general relativity theory is merely this3 ,given any physical system1 then it is possible to assign values to ds such that relations mathematically deduced by the principle of relativity correspond to relations between measurable quantities in the physical system2 but that the things that we physically call ds are anything more than approximately connected with the ds's required to give the mathematical relations1 is at present no more than a pious faith4 ,to return to the concept of time1 we have already stated that there are two main problems1 that of measuring time at a single point of space1 and that of spreading a time system over all space4 ,the second aspect of the problem is that to which attention has been directed by relativity theory2 the following detailed examination shows how the operations of relativity for setting and synchronizing clocks at distant places involve the measurement of space4 ,it is a fundamental postulate that the adjustment of the clocks is to be accomplished by light signals4 ,the synchronization of the clocks is now simple enough4 ,we merely demand that light signals sent from the master clock at intervals of one second arrive at any distant clock at intervals of one second as measured by it1 and we change the rate of the distant clock until it measures these intervals as one second4 ,after its rate has been adjusted1 the distant clock is to be so set that when a light signal is despatched from the master clock at its indicated zero of time the time of arrival recorded at the distant clock shall be such that the distance of the clock from the master clock divided by the time of arrival shall give the velocity of light1 assumed already known4 ,this operation involves a measurement of the distance of the distant clock1 so that in spreading the time coordinates over space the measurement of space is involved by definition1 and the measurement of time is1 therefore1 not a self-contained thing4 ,this is the physical basis for the treatment of space and time as a four-dimensional manifold4 ,although mathematically the numbers measuring space and time enter the formulas symmetrically1 nevertheless the physical operations by which these numbers are obtained are entirely distinct and never fuse1 and ,i believe it can lead only to confusion to see in the possibility of a four dimensional treatment anything more than a purely formal matter4 ,the notion of extended time1 therefore1 involves the measurement of space4 ,it is an interesting question whether the notion of local time also involves the measurement of space4 ,a rigorous answer to this question involves giving the specifications for the construction of a clock1 which we have seen has not yet been done4 ,it seems to me probable1 however1 that the construction of even a single local clock involves in some way the measurement of space4 ,if1 for example1 we use a vibrating tuning fork1 we must find how the time of vibration depends on the amplitude of vibration1 and this involves space measurement1 or if we use a rotating flywheel1 we have to correct for the change of moment of inertia due to the change of dimensions when it is set into motion or brought into a gravitational field1 and all this involves space measurement4 ,however1 these considerations are not certain1 and perhaps the question is not important4 ,there is now the further consideration that actually in practice the concept of local time is not entirely divorced from that of extended time1 for two bodies cannot occupy the same space at the same time1 and the time of any event is actually measured on an instrument at some distance1 communication being maintained by light or elastic signals4 ,but experience convinces us that in the limit1 as the phenomenon to be measured gets closer to the clock1 there is no measurable difference1 whether communication with the clock is maintained by light1 or acoustical or tactual signals1 so that we have come in physical practice to accept measurement of the time of events in the immediate neighborhood of the clock 7local time7 as one of the ultimately simple things behind which we do not attempt to go4 ,local time is1 therefore1 a concept treated by the physicist even now as simple and unanalyzable4 ,this concept is what most people have in mind when they think of time4 ,time1 according to this concept1 is something with the properties of local time2 it was something of this kind that ,newton must have meant by his absolute time1 and it is the tacit retention of this sort of concept that is responsible for the difficulty so often found in grasping the idea of the relativity of simultaneity1 which is of course entirely foreign to our experience of simultaneity in local time4 ,an examination of the operations involved in extending time has shown how the concept of extended time is different from that of simple local time2 this difference leads to appreciably different numerical relations when we are dealing with high velocities or great distances4 ,local time is proved by experience not to be a satisfactory concept for dealing with events separated by great distances in space or with phenomena involving high velocities4 ,for instance1 we must not talk about the age of a beam of light1 although the concept of age is one of the simplest derivatives of the concept of local time4 ,neither must we allow ourselves to think of events taking place in ,arcturus now with all the connotations attached to events taking place here now4 ,it is difficult to inhibit this habit of thought1 but we must learn to do it4 ,the nave feeling is very strong that it does mean something to talk about the entire present state of the universe independent of the process by which news of the condition of distant parts is determined by us4 ,i believe that an examination of this feeling will show that it is psychological in character2 what we mean by the totality of the present is merely the entire present content of our consciousness4 ,this is apparently a simple direct thing2 we do not appreciate until we make further analysis that our present consciousness of the existence of the moon or a star is due to light signals1 and that therefore the apparently simple immediate consciousness of events distant in space involves complicated physical operations4 ,similarly1 if we continue to use local time1 we get into trouble1 when we go to high velocities1 with our simple concept of velocity1 which may be defined in terms of a combination of space and time concepts4 ,the concept of local time thus loses its value and becomes merely a blunted tool when we try to carry it out of its original range4 ,but the concept of extended time1 with which we have to replace local time1 is a complicated thing1 to which we have not yet got ourselves accustomed2 it may perhaps prove to be so complicated as never to be a very useful intuitive tool of thought4 ,all these considerations about time have been concerned only with intervals of such an order of magnitude that they are readily experienced by any individual4 ,if we have to deal with intervals either very long or very short1 it is obvious that our entire procedure changes1 and consequently the concept changes4 ,in extending the time concept to eras remote in the past1 for example1 we try as always1 to choose the new operations so as to piece on continuously with those of ordinary experience4 ,a precise analysis of the change in the concept of time when applied to the remote past does not seem to be of great significance for our present physical purpose1 and will not be attempted here4 ,it is perhaps worth while to point out1 however1 that all our other concepts1 as well as that of time1 must be modified when applied to the remote past2 an example is the concept of truth4 ,it is amusing to try to discover what is the precise meaning in terms of operations of a statement like this3 ,it is true that ,darius the ,mede arose at #f3cj on the morning of his thirtieth birthday4 ,of more concern for our physical purposes is the modification which the time concept undergoes when applied to very short intervals4 ,what is the meaning1 for example1 in saying that an electron when colliding with a certain atom is brought to rest in #aj-ah seconds8 ,here ,i believe the situation is very similar to that with regard to short lengths4 ,the nature of the physical operations changes entirely1 and as before1 comes to contain operations of an electrical and optical character4 ,the immediate significance of #aj-ah is that of a number1 which when substituted into the equations of optics1 produces agreement with observed facts4 ,thus short intervals of time acquire meaning only in connection with the equations of electrodynamics1 whose validity is doubtful and which can be tested only in terms of the space and time coordinates which enter them4 ,here is the same vicious circle that we found before4 ,once again we find that concepts fuse together on the limit of the experimentally attainable4 ,this discussion of the concept of time will doubtless be felt by some to be superficial in that it makes no mention of the properties of the physical time to which the concept is designed to apply4 ,for instance1 we do not discuss the one dimensional flow of time1 or the irrevocability of the past4 ,such a discussion1 however1 is beyond our present purpose1 and would take us deeper than ,i feel competent to go1 and perhaps beyond the verge of meaning itself4 ,our discussion here is from the point of view of operations3 we assume the operations to be given1 and do not attempt to ask why precisely these operations were chosen1 or whether others might not be more suitable4 ,such properties of time as its irrevocability are implicitly contained in the operations themselves1 and the physical essence of time is buried in that long physical experience that taught us what operations are adapted to describing and correlating nature4 ,we may digress1 however1 to consider one question4 ,it is quite common to talk about a reversal of the direction of flow of time4 ,particularly1 for example1 in discussing the equations of mechanics1 it is shown that if the direction of flow of time is reversed1 the whole history of the system is retraced4 ,the statement is sometimes added that such a reversal is actually impossible1 because it is one of the properties of physical time to flow always forward4 ,if this last statement is subjected to an operational analysis1 ,i believe that it will be found not to be a statement about nature at all1 but merely a statement about operations4 ,it is meaningless to talk about time moving backward3 by definition1 forward is the direction in which time flows4 ,,the ,,causality ,,concept ,the causality concept is unquestionably one of the most fundamental1 perhaps as fundamental as that of space and time1 and therefore at least equally entitled to a first place in the discussion4 ,but as ordinarily understood1 there are certain spatial and temporal implications in the causality concept1 so that it can best be discussed in this order after our examination of space and time4 ,there is an aspect of the causality concept that in many respects is closely related to the question of explanation1 for to find the causes of an event usually involves at the same time finding its explanation4 ,but there are nevertheless sufficient differences to warrant a separate discussion4 ,it seems fairly evident that there was originally in the causality concept an animistic element much like that in the concept of force to be discussed later4 ,the physical essence of the concept as we now have it1 freed as much as possible from the animistic element1 seems to be somewhat as follows4 ,we assume in the first place an isolated system on which we can perform unlimited identical experiments1 that is1 the system may be started over again from a definite initial condition as often as desired4#i ,footnote #i3 ,we must include in general in the concept of initial condition the past history of the system4 ,in order not to make this condition so broad as to defeat itself1 we have to add the observation that actually identity of past history is necessary over only a comparatively short interval of time4 ,logical precision seems unattainable here--the physical concepts themselves have not the necessary precision4 ,we assume further that when so started1 the system always runs through exactly the same sequence of events in all its parts4 ,this contains the assumption that the course of events runs independent of the absolute time at which they occur--there is no change with time of the properties of the universe4#aj ,it is a result of experience that systems with these properties actually exist4 ,an alternative way of stating our fundamental hypothesis is that two or more isolated similar systems started from the same initial condition run through the same future course of events4 ,upon the system given in this way1 which by itself runs a definite course of events1 we assume that we can superpose from the outside certain changes1 which have no connection with the previous history of the system1 and are completely arbitrary4 ,now of course in nature1 as we observe it1 there is no such thing as an arbitrary change1 without connection with past history1 so that strictly our assumption is a pure fiction4 ,it is here that the animistic element still seems to persist1 although perhaps not necessarily4 ,footnote #aj3 ,as so often in physics1 we appear to be doing two things at once here4 ,it is doubtful whether we can give a meaning to definite initial condition apart from the future behavior of the system1 so that we have no real right to infer from uniform future behavior both a constancy of the laws of nature1 independent of time1 and a constancy of initial condition4 ,i very much question whether a thoroughgoing operational analysis would show that there are really two independent concepts here1 and whether the use of two formally quite different concepts is anything more than a convenience in expression4 ,it seems to me that it may be just as meaningless to ask whether the laws of nature are independent of time as it was to ask with ,clifford whether the absolute scale of magnitude may not be changing as the solar system travels through space4 ,we regard our acts as not determined by the external world1 so that changes produced in the external world by acts of our wills are1 to a certain degree of approximation1 arbitrary4 ,the system1 then1 on which we are experimenting1 is one capable of isolation from us in that we may regard ourselves as outside the system1 and having no connection with it4 ,the system1 furthermore1 is capable of isolation from the rest of the physical universe1 in that events taking place outside the system have no connection with those taking place inside4#aa ,experience gives the justification for assuming that physical isolation of this sort is possible4 ,actually1 of course1 isolation is never complete1 but only partial1 up to presumably any desired degree of approximation4 ,footnote #aa3 ,here again1 the concept of isolation or connection is defined only in terms of the behavior of the system1 and it is not clear whether this is really an operationally independent concept or not4 ,the statement that two exactly similar isolated systems1 starting from the same initial conditions 7including past history in the general idea of initial condition7 will run through the same future course of events involves as a corollary that if differences develop in the behavior of two such apparently similar systems these differences are evidence of other previous differences4 ,the thesis that this corresponds to experience may be called the thesis of essential connectivity and is perhaps the broadest we have3 it is the thesis that differences between the behavior of systems do not occur isolated but are associated with other differences4 ,it is essentially the same thesis as that already mentioned in connection with explanation1 namely that it is possible to correlate any of the phenomena of nature with other phenomena4 ,if now the connectivity or correlation between phenomena is of a special kind1 we have a causal connection2 namely1 if whenever we arbitrarily impress event ,a on a system we find that event ,b1 always occurs1 whereas if we had not impressed ,a1 ,b would not have occurred1 then we say that ,a is the cause of ,b1 and ,b the effect of ,a4 ,by suitably choosing the event ,a1 we may find the effect of any event of which the system is susceptible4 ,the relation between ,a and ,b is an unsymmetrical one1 by the very nature of the definition1 the cause being the arbitrary variable element1 and the effect that which accompanies it4 ,furthermore1 ,a may obviously be the cause of more than one event ,b1 and may cause a whole train of events4 ,the causal concept analyzed in this way is not simple by any means4 ,we do not have a simple event ,a causally connected with a simple event ,b1 but the whole background of the system in which the events occur is included in the concept1 and is a vital part of it4 ,if the system1 including its past history1 were different1 the nature of the relation between ,a and ,b might change entirely4 ,the causality concept is therefore a relative one1 in that it involves the whole system in which the events take place4 ,in practice we now take an exceedingly pregnant step and seek to extend the concept1 and rid ourselves as much as possible of its relativity4 ,it is a matter of experience that there are often a great number of systems in which ,a is the cause of ,b4 ,in many cases the causal relation persists through such a very wide range of systems that we lose sight entirely of the system1 and come to assume that we have an absolute causal connection between ,a and ,b4 ,for instance1 when ,i strike a bell1 and hear the sound1 the causal connection persists through such a great number of different kinds of system that ,i might think that here is an absolute causal connection4 ,such an absolute causal connection would mean that always under all circumstances1 the striking of the bell is accompanied by a sound4 ,but all conditions means only all those conditions covered by experiment4 ,thus in the case of the bell1 all our experiments were made in the presence of the atmosphere4 ,the causal connection between the striking of the bell and the sound should have been always recognized in principle as relative to the presence of the atmosphere4 ,indeed1 later experiments in the absence of the atmosphere show that the atmosphere does play an essential part4 ,now as a matter of fact1 the atmosphere is so comparatively easy to remove that we very readily include the atmosphere in the chain of causal connection4 ,but if the atmosphere had been impossible to remove1 like the old ether of space1 our idea of the causal connections between the striking of the bell and its sound might have been quite different4 ,in actual physical applications of the causality concept1 the constant background which is maintained during all the variations by which the causal connection is established usually has to be inferred from the context4 ,it is a matter of perhaps universal experience that the event ,a is accompanied by not only one event1 which is the effect of ,a by definition1 but ,a entails a whole causal train of events4 ,it seems to be a generalization from experience that the causally connected train of events started by ,a is a never ending train1 provided the system is large enough4 ,this is perhaps not necessary in the general case1 but if the event ,a involves imparting external energy to the system1 or the action of external force 7momentum change71 there can be no question4 ,that there is a causal train started by ,a is particularly evident if ,a and ,b are separated in space4 ,thus in the case of the bell1 the impulse given to the air by the vibration of the bell is propagated through the air as an elastic wave1 which thus constitutes the causal train of events4 ,the phenomenon of propagation is characteristic of causal connections of a mechanical character1 and is the justification for the introduction of the time concept in connection with the causality concept1 where it now appears for the first time4 ,it is evident that when a disturbance is propagated to a distant point1 the effect follows the cause in time1 as time is usually measured4 ,we extend this result1 and usually think that the effect necessarily follows the cause4 ,we now examine whether this is a necessary result of the causality concept4 ,if we are to talk about the time of events at different places1 we must have some way of setting clocks all over space4 ,if this is done arbitrarily1 there is no necessary connection between the local clock times of a cause and its effect1 but nevertheless the causality concept involves a certain temporal relation even in this most general case4 ,suppose that event ,a takes place at point #a and its effect1 event ,b1 at point #b4 ,we station a confederate at #b who sends a light signal 7or any other sort of signal7 to #a1 as soon as the event ,b occurs at #b4 ,then it is a consequence of the nature of the causality concept that the signal cannot arrive at #a before event ,a occurs4 ,for if it did arrive before ,a1 we should merely omit to perform ,a1 which by hypothesis is arbitrary1 and entirely in our control1 and then our assumption would be violated that the system is such that the event ,b occurs only when ,a also occurs4 ,the same argument shows a fortiori that if the effect ,b occurs at the same place as its cause ,a1 it cannot precede it in time4 ,i cannot see that the nature of the causality concept imposes any further restriction on the time of ,b4 ,the restricted principle of relativity1 however1 in postulating that no signal can be propagated faster than a light signal1 virtually makes a further assumption about the temporal connection of causally connected events1 namely1 that the event ,b at #b cannot occur before the arrival at #b of a light signal which started from #a at the instant that ,a occurred at #a4 ,for if ,b did occur earlier1 we could use events ,a and ,b as a signaling code1 thus violating our hypothesis4 ,there is thus a closest connection in time1 when time is extended over space as the theory of relativity directs1 between cause and effect1 depending on their separation in space2 from this arises the relativity concept of the causal cone1 which in the four dimensional manifold of space-time divides the aggregate of all those events which may be causally related from the aggregate of those which are separated by such a small interval in time and such a large interval in space that communication by light signals and therefore a causal connection is not possible4 ,given now two events ,a and ,b which are related as cause to effect in one system of reference1 then they must be causally related also in every other system of reference4 ,for if they were not1 we could by definition of causality suppress the event ,a in one of the systems in which the causal relation does not hold1 and this1 because of the nature of the concept of event1 involves suppressing ,a in all the systems1 thus violating our hypothesis of a causal connection in the original system4 ,the concept of event involved in this argument will be examined later4 ,it appears then1 that the fundamental postulate of relativity 7that the form of natural laws is the same in all reference systems7 demands that the temporal order of events causally connected be the same in all reference systems4 ,the whole universe at this present moment is often supposed to be causally connected with all succeeding states4 ,this means that if we could repeat experience1 starting from the same initial conditions1 the future course of events would always be found to be the same4 ,the truth of this conviction can never be tested by direct experiment1 but it is something at which we arrive by the usual physical process of successive approximation4 ,it is difficult to formulate precisely what we mean by present state of the universe1 and there is every reason to think that such a formulation is not unique1 but the concept contains the necessary implication that none of the events constituting the present can be causally connected4 ,the events in distant places which constitute the present must be separated by an interval of time less than time required by light to travel between the two places4 ,the conviction1 arising from experience1 that the future is determined by the present and correspondingly the present by the past1 is often phrased differently by saying that the present causally determines the future4 ,this is in a certain sense a generalization of the causality concept4 ,it is one of the principal jobs of physics to analyze this complex causal connection into components1 representing as far as possible the future state of the system as the sum of independent trains of events started by each individual event of the present4 ,how far such an analysis is possible must be decided by experiment4 ,it is certainly possible to a very large extent in most cases1 but there seems to be no reason to expect that a complete analysis is possible4 ,so far as the system is describable in terms of linear differential equations1 the causal trains started by different events propagate themselves in space and time without interference and with simple addition of effects1 and conversely the present may be analyzed back into the simple sum of elementary events in the past1 but if the equations governing the motion of the system are not linear1 effects are not additive1 and such a causal analysis into elements is not possible4 ,no emphasis is to be laid here on the differential aspect of the equations3 it is quite possible that finite difference equations may have the same property of additivity4 ,although there can be no question that linear equations enormously preponderate1 neither can there be any doubt that some phenomena cannot be described in terms of linear equations 7e4g41 ferro-magnetism71 so that there seems no reason to think that a causal analysis is always possible4 ,i believe1 however1 that the assumption that such an analysis into small scale elements is possible is tacitly made in the thought of many physicists4 ,if the analysis is not possible1 we may expect to find results following the cooperation of several events which cannot be built up from the results of the events occurring individually4 ,when a causal analysis is possible1 finding the simplest events which act as the origin of independent causal trains is equivalent to finding the ultimate elements in a scheme of explanation1 so that here we merge with the concept of explanation1 as already mentioned4 ,as was true of the explanatory sequence1 so here there can be no formal end of the causal sequence1 because we can always ask for the cause of the last member4 ,but it may be physically meaningless to extend the causal sequence beyond a certain point4 ,we have seen from the point of view of operations that the causal concept demands the possibility of variation in the system4 ,it is therefore meaningless to say that ,a is the cause of ,b unless we can experience systems in which ,a does not occur4 ,now if in extending the causal sequence1 we eventually arrive at a condition so broad that physically no further variation can be made1 our causal sequence has to stop4 ,corresponding to this property of the causality concept1 the causal sequence may be terminated either formally1 by postulate1 or naturally1 by the intrinsic physical nature of the elements of the sequence4 ,thus if we say that light gets from point to point because it is propagated by a medium of unalterable properties1 which fills all space1 which is always present and can never be eliminated physically1 we have by the postulated properties of the medium brought the possibility of further inquiry to a close1 because to take the next step and ask the cause of the properties of the ether1 demands that we be able to perform experiments with the ether altered or absent4 ,such an ending of the sequence is evidently pure formalism1 without physical significance4 ,but other considerations may give physical significance4 ,thus if there are other sorts of experiment that can be explained by assuming a universal medium of the same properties1 the concept proves not only to be useful1 but to have a certain degree of physical significance4 ,an example of an inevitable termination of the causal sequence is afforded by the possibility1 already mentioned1 that the value of the gravitational constant may be determined by the total quantity of matter in the universe4 ,without further qualification1 this is an entirely sterile statement1 but if it can be shown that there is a simple numerical connection1 the matter takes on interest1 and we may seek further for a correlation between the numerical relation and other things4 ,this analysis of the causality concept does not pretend to be complete and leaves many interesting questions untouched4 ,perhaps one of the most interesting of these questions is whether we can separate into cause and effect two phenomena which always accompany each other1 and whether therefore the classification of phenomena into causally connected groups is an exhaustive classification4 ,but the discussion is broad enough for our purpose here2 the most important points of view to acquire are that the causality concept is relative to the whole background of the system which contains the causally connected events1 and that we must assume the possibility of an unlimited number of identical experiments1 so that the causality concept applies only to sub-groups of events separated out from the aggregate of all events4 ,,the ,,concept ,,of ,,identity ,one of the most fundamental of all the concepts with which we describe the external world is that of identity2 in fact1 thinking would be almost inconceivable without such a concept4 ,by this concept we bridge the passage of time2 it enables us to say that a particular object in our present experience is the same as an object of our past experience4 ,from the point of view of operations1 the meaning of identity is determined by the operations by which we make the judgment that this object is the same as that one of my past experience4 ,in practice there are many indirect ways of making this judgment1 but ,i believe the essence of the situation lies in the possibility of continuous connection between the object of the present and the past by continuous observation 7either direct or indirect7 through all intermediate time4 ,we must1 for example1 be able to look continuously at the object1 and state that while we look at it1 it remains the same4 ,this involves the possession by the object of certain characteristics--it must be a discrete thing1 separated from its surroundings by physical discontinuities which persist4 ,the concept of identifiability applies1 therefore1 only to certain classes of physical objects2 no one thinks of trying to identify the wind of to-day with the wind of yesterday4 ,it is somewhat easier to identify a liquid such as water in its flow in a stream1 because we can make the motion of the water visible by solid particles suspended in it1 but even here it is not easy to prove to a captious critic that it is really the water and not the suspended particles of solid that we are identifying4 ,even solids1 when our measurements are sufficiently refined1 seem to lose their discontinuous edges1 as has been suggested in the discussion of the approximate character of experimental arithmetic1 and the identity concept becomes hazy4 ,there can be no question that the concept of identity is a tool perfectly well adapted to deal approximately with nature in the region of our ordinary experience1 but we have to ask a more serious question4 ,does not the apparent demand of our thinking apparatus to be furnished with discrete and identifiable things to think about impose a very essential restriction on any picture of the physical universe which we are able to form8 ,we are continually surprising ourselves in the invention of discrete structure further and further down in the scale of things1 whole sole raison d'tre is to be found entirely within ourselves4 ,thus ,osborne ,reynolds#ab has speculated seriously and most elaborately about an atomic structure in the ether1 and we find ,eddington#ac hinting at the existence of structure of an order of magnitude of #aj-dj cm4 ,on a much larger scale of magnitude we also think in the same terms1 and conceive positive and negative elementary charges with hard and impenetrable cores1 which involves a complete change in the law of force at points sufficiently close4 ,what physical assurance have we that an electron in jumping about in an atom preserves its identifiability in anything like the way that we suppose1 or that the identity concept applies here at all8 ,footnote #ab3 ,osborne ,reynolds1 ,the ,sub-,mechanics of the ,universe1 #bed pp41 ,cambridge ,university ,press1 #aijc4 ,footnote #ac3 ,a4 ,s4 ,eddington1 ,report on ,gravitation1 ,lon4 ,phys4 ,soc41 #aiah1 p4 #ia4 ,in fact1 the identity concept seems to lose all meaning in terms of actual operations on this level of experience4 ,the mind seems essentially incapable of dealing with continuity as a property of physical things2 it is not even able to talk about continuity except in negative terms4 ,to each attempted description of the properties of a truly continuous substance1 it can say ,no1 it is not that1 but cannot imagine experience which corresponds to what it conceives a really continuous thing ought to feel like4 ,in terms of operations1 continuity has only a sort of negative meaning4 ,now certain implications of this inability of the mind can be removed by appropriate postulates1 as1 for example1 we can postulate the complete annihilation of a negative by a positive charge1 as is now being done in certain speculations4#ad ,there is point in doing this1 because the annihilation of two charges has physical meaning4 ,but it is a question whether all implications of this habit of thought can be removed1 and whether any picture that we can form of nature will not be tinged--sickbed o'er with the pale cast of thought4 ,footnote #ad3 ,for example1 ,j4 ,h4 ,jeans1 ,nat4 #aad1 #hbh-hbi1 #aibd4 ,the operational view suggests that in this last we are coming perilously close to a meaningless question1 although there is a certain sense in which there is meaning here4 ,it may turn out as a matter of fact that we shall not be able to carry our delving into small-scale phenomena deeper than a certain point1 and that nature will appear to be finite downward1 so that we shall bring up against a wall of some kind4 ,but to ask in such a situation whether we have come to the end because nature is really finite1 or whether we only appear to be at an end because of some property of our minds1 such as inability to deal with continuity1 is1 ,i believe1 a meaningless question4 ,in actual use the identity concept is extended1 and identity is used in other senses than the fundamental one examined above4 ,for instance1 we speak of two observers seeing the same object1 or if the object moves or does something1 we may speak of two observers perceiving the same happening4 ,a happening about which the judgment of sameness is possible when perceived by different observers 7or mathematically expressed when observed in two reference frames7 is what we mean by an event1 which is one of the fundamental concepts of relativity theory4 ,what now is involved in this concept of event1 or what do we mean when we say that two observers experience the same event8 ,a first crude attempt might say that the event is the same if it is described in the same way by the two observers4 ,but this leads us into all the complicated questions of the meaning of language1 which we would gladly avoid1 and is furthermore not true1 because the whistle of a locomotive1 for example1 does not have the same pitch for two observers moving with different velocities4 ,a satisfactory analysis of the situation is difficult to give1 but ,i believe the essence lies in the discrete character of the event1 just as the identity concept when applied only to objects involved discreteness4 ,the event is bounded on all sides by discontinuities1 both in space and time4 ,now it seems to be a result of experience that discontinuities have a certain absolute significance1 in that there is a one-to-one correspondence between the discontinuities observed in any one reference system and those observed in any other4 ,corresponding discontinuities in two reference systems are by definition the same4 ,an event is by definition the aggregate of all phenomena bounded by certain discontinuities1 and two reference systems are by definition describing the same event if the discontinuous boundaries of the event are the same1 irrespective of the appearance of the event in the two systems4 ,the emission of a light signal1 for example1 is an event according to this definition1 although it may appear as red light in one reference system and green in another4 ,we now see that the concept of event is only an approximate concept1 as was also that of identity1 and for the same reason1 namely1 there are no such things in experience as sharp discontinuities1 but as our measurements become more refined1 the edges of supposed discontinuities become blurred4 ,as we go to smaller scales of magnitude this blurring becomes more important1 until the physical possibility of performing those operations by which the discontinuities are detected entirely disappears1 and the concept of event acquires1 in terms of operations1 an entirely different meaning4 ,we continue to think of the event in the same way as before in terms of a mental model1 but the true operational significance now depends on the particular phenomenon under consideration4 ,the concept of event is really not the same sort of thing when applied to the emission of a quantum of radiation from an atom1 or the emission of gamma radiation from a radioactive disintegration1 or the flashing of a signal from a dark lantern by opening and closing a shutter4 ,here as always1 when our range of experience is extended1 we must be prepared at some future time to find that1 by extending the ordinary concept of event to small-scale phenomena by the device of the mental model1 we have by implication smuggled into our picture phenomena which do not exist1 so that it will be necessary to revise our thinking1 casting it into terms corresponding to direct experience4 ,,the ,,concept ,,of ,,velocity ,the concept of velocity1 as ordinarily defined1 involves the two concepts of space and time4 ,the operations by which we measure the velocity of an object are these3 we first observe the time at which the object is at one position1 and then later observe the time at which it is at a second position1 divide the distance between the two positions by the time interval1 and if necessary1 when the velocity is variable1 take the limit4 ,as long as we deal with fairly low velocities we do not have to inquire carefully as to the kind of time we use in these operations1 but when the velocities become high1 we do have to take care to use the local times at the two positions of the body1 which means that we must have a time system spread over space1 or1 in other words1 the extended time system4 ,this velocity concept1 defined in this way1 may be used as a tool in describing nature1 and it will be found that nature has certain properties2 for example1 the velocity of light is #c x #ajaj cm4_/sec4 ,further1 no material thing can be given a velocity as high as this1 but as its velocity is made to approach this value1 increments of energy increasing without limit are required4 ,but now it is very much a question for examination whether the velocity concept defined in this particular way has been chosen wisely as a tool for describing natural phenomena4 ,it is quite possible to modify the velocity concept1 that is1 to set up other operations which correspond to our instinctive feeling of what velocity is in terms of immediate sensation and such that all numerical measures are unmodified at low velocities4#ae ,for example1 a traveller in an automobile measures his velocity by observing the clock on his instrument board and the mile stones which he passes on the road4 ,this operation differs from that of the definition above in that the time is no longer extended time1 but is the local time of the moving object4 ,the space coordinates used in this alternative operation at first seem a hybrid sort of thing1 but they are what the observer would actually most naturally use3 they are what he would measure with a tape measure fixed to a point of the road and allowed to unwind as he proceeds1 or what is measured by a vessel at sea with a log line let out behind4 ,footnote #ae3 ,it is an interesting question for the psychologist whether the velocity concept is not a more primitive thing in order of apprehension than that of time1 and whether the concept of time is not derived from observing things in motion1 or whether indeed there is any necessary connection at all between velocity and time in terms of untutored experience4 ,or there is still another most interesting way of defining velocity1 in which the analysis into space and time is not made at all1 but velocity is directly measured by building up the given velocity by physical addition of a unit velocity selected arbitrarily4 ,this matter is discussed at some length in my book ,dimensional ,analysis1#af but is of sufficient pertinence here to describe briefly4 ,we may in the first place construct a concrete standard for velocity1 as1 for example1 by stretching a string between two pegs on a board with a fixed weight4 ,if we strike the string1 a disturbance travels along the string which we can follow with the eye1 and we define unit velocity as the velocity of this disturbance4 ,an object has greater than unit velocity if it outruns the disturbance1 and less if it lags behind4 ,we may now duplicate our standard1 making another board with pegs and stretched string1 and check the equality of the two velocities by observing that the two disturbances run together4 ,we now define two units of velocity as the velocity of anything which runs with the disturbance of the string of the second board1 when the second board is made to move bodily with such a velocity that it runs with the disturbance of the first string4 ,the process may be extended indefinitely1 and any velocity measured4 ,footnote #af3 ,yale ,university ,press1 #aibb4 ,if either of these two alternative definitions of velocity were adopted1 it would be found that the velocity of light is infinite4 ,further1 there would be no limit to the velocity which can be imparted to material bodies on giving them unlimited energy1 which is what we are prepared to regard from ordinary experience is natural and simple4 ,the infinite velocity for light1 on the other hand1 is most unnatural1 particularly if we favor a medium point of view4 ,we are here faced with a dilemma--all sorts of phenomena cannot at the same time be treated simply4 ,if we attach the most fundamental significance to the behavior of material bodies1 we shall do well to adopt one of the alternative definitions of velocity4 ,if1 on the other hand1 we regard the phenomena of light as the most fundamental1 we shall endeavor to form our definition so that the properties of light are simple4 ,this was precisely the point of view of ,einstein2 it is characteristic of his entire scheme of restricted relativity that light is the fundamental thing1 and this influenced him in adopting the first definition of velocity4 ,now one can have no quarrel with this desire to make light fundamental 7the wisdom of doing this is to be justified by the results71 and if the properties of light are to be treated mathematically1 one can easily see the desirability of getting rid of infinite attributes1 and so admit the desirability of making the velocity of light finite4 ,but all this involves another very insidious assumption which we ourselves have tacitly used in all our preceding discussion1 namely1 that the notion of velocity properly pertains to light at all4 ,einstein has very definitely adopted this point of view1 and so determined the character of the entire structure of relativity4 ,i believe1 on the contrary1 that it is very gravely to be questioned whether the identification of light with a thing travelling1 which is involved in applying the velocity concept1 should be made4 ,this discussion must be postponed1 however1 until we deal with the properties of light4 ,the important points for us to notice at present are that the definition of velocity actually used involves the concept of extended time1 and that it would be possible to define velocity in different ways1 which would give quite a different complexion to phenomena at high velocities1 but which would leave untouched our ordinary experience4 ,the velocities at which the precise form of definition becomes important are higher than can be reached in ordinary mechanical experiments4 ,such velocities can be attained in terrestrial laboratories only with electrified particles1 as in experiments in high vacua or with radioactive disintegrations4 ,it is interesting to notice that we very seldom attempt a direct measurement of velocity in such experiments by following a discrete particle in its flight and finding the time required to pass over a measured distance1 but the velocities are measured indirectly1 by calculation from the equations of electrodynamics and in terms of such immediately observed things as curvature of path4 ,it is true that one or two experiments have attempted a more direct measure of velocity1 but it seems there is room for more work here4 ,,the ,,concepts ,,of ,,force ,,and ,,mass ,another concept of great importance is that of force4 ,since the usual analysis finds a connection between force and acceleration1 and acceleration involves velocity1 this is a natural place for the discussion of force4 ,this concept has been subjected to much analysis by various writers4 ,in origin the concept doubtless arises from the muscular sensations of resistance experienced from external bodies4 ,this crude concept may at once be put on a quantitative basis by substituting a spring balance for our muscles1 or instead of the spring balance we may use any elastic body1 and measure the force exerted by it in terms of its deformation4 ,of course1 the various precautions which must be taken in carrying out this idea physically are complex2 the matter of precautions against temperature changes1 for example1 is one of the most easily understood4 ,the concept of force so defined is limited to static systems2 it is the task of statics to find the relation between the forces in systems at rest4 ,we next extend the force concept to systems not in equilibrium1 in which there are accelerations1 and we must conceive that at first all our experiments are made in an isolated laboratory far out in empty space1 where there is no gravitational field4 ,we here encounter a new concept1 that of mass1 which as it is originally met is entangled with the force concept1 but may later be disentangled by a process of successive approximations4 ,the details of the various steps in the process of approximation are very instructive as typical of all methods in physics1 but need not be elaborated here4 ,suffice it to say that we are eventually able to give to each rigid material body a numerical tag characteristic of the body1 such that the product of this number and the acceleration it receives under the action of any given force applied to it by a spring balance is numerically equal to the force1 the force being defined1 except for a correction1 in terms of the deformation of the balance1 exactly as it was in the static case4 ,in particularly1 the relation found between mass1 force1 and acceleration applies to the spring balance itself by which the force is applied1 so that a correction has to be applied for a diminution of the force exerted by the balance arising from its own acceleration4 ,we now extend the scope of our measurements by bringing our laboratory into the gravitational field of the earth1 and immediately our experience is extended1 in that we continually see bodies accelerated with no spring balance 7that is1 no force7 acting on them4 ,we extend the concept of force1 and say that any body accelerated is acted on by a force1 and the magnitude of this force is defined as that which would have been necessary to produce in the same body the same acceleration with a spring balance in empty space4 ,there is physical justification for this extension in that we find we can remove the acceleration which a body acquires in a gravitational field by exerting on it with a spring balance a force of exactly the specified amount in the opposite direction4 ,this extended idea of force may also be applied to systems in which there are electrical actions4 ,we thus see that in extending the notion of force from bodies in rest to bodies in motion1 the character of the concept has changed1 because the operations by which force is measured change--the force acting on a body is now measured in terms of its acceleration4 ,but in determining the force from the acceleration1 we have to know the mass4 ,this mass has to be independently measured with the original concept of force2 otherwise we have no basis for such simple statements as that the force of gravity on a body is proportional to its mass4 ,all this applies to the ordinary range of experiments with low velocities4 ,if now we extend the range of measurements1 we find phenomena which we had not expected2 for example1 there seem to be difficulties in the way of indefinitely increasing the velocity of a material body1 as of a charged atom4 ,we begin to ask searching questions3 is the force of gravity independent of velocity at high velocities1 or is the mass independent of velocity under the same conditions or independent of the gravitational field1 etc48 ,in attempting to answer these new questions1 we find difficulty with the concepts in terms of which they are formulated4 ,there are no operations by which we can find whether force is independent of velocity unless we first know the mass1 or any operations by which a mass can be measured unless we know a force4 ,the purely mechanical systems with the highest velocities of which we have any experimental knowledge are the heavenly bodies4 ,the motion of these is1 with the important exception of ,mercury1 that predicted by the ordinary laws of mechanics1 so that at first it might appear that we have here confirmation of the laws of mechanics for bodies with comparatively high velocities4 ,but it must be remembered that all we can observe of the heavenly bodies is their positions1 and that we cannot perform on these bodies all the operations by which we can check the laws of mechanics for terrestrial phenomena4 ,if1 for example1 mass and the force with which gravity acts on mass were both equally affected by velocity1 the motion of the heavenly bodies would be exactly the same as that observed now4 ,hence as we increase the range of velocity1 the concepts of force and mass simultaneously lose their definiteness1 and become partially fused4 ,this is typical of what we have now come always to expect near the limit of the experimentally attainable2 experience becomes less rich1 the choice of physical operations more restricted1 concepts change and become fewer in number4 ,if we are to retain the same formal number of concepts1 we must introduce arbitrary conventions or definitions4 ,these definitions are to be determined largely by convenience4 ,in the case of mechanical systems1 this motive of convenience is supplied by considerations from outside the domain of mechanical phenomena4 ,the highest velocities of practice are not reached in mechanical1 but in electrical systems1 in experiments with vacuum tubes1 etc4 ,considerations of convenience are therefore dictated from the electrical point of view4 ,these considerations will be gone into in much more detail later2 the conclusion is all that we need here1 which is that it is convenient to assume for the charge of the electron a constant number1 independent of the velocity1 and this involves making its mass variable in a definite way with velocity4 ,now if the principle of relativity is accepted1 the mass of mechanical objects must vary with velocity in the same way as the mass of electrical charges4 ,since the variability of this latter is fixed1 mechanical mass becomes a definite function of velocity1 and the force is therefore also fixed in any specific physical case4 ,the fundamental definition of force given above is highly academic1 involving as it does hypothetical experiments in laboratories situated far out in empty space4 ,some sort of procedure like this seems to correspond to more or less explicit statements to be found in the literature of mechanics4 ,the meaning in terms of actual operations to be given to such definitions involves complicated inferential reasoning4 ,we would make much closer connection with the conditions of actual experiment if in the definition we substituted for the hypothetical operations in empty space more or less approximately realizable operations on bodies sliding on level table tops without friction4 ,i suppose our instinctive feeling for the laws of mechanics is such that we are convinced that definitions in terms of an interstellar space laboratory or a level table top are actually the same4 ,but in principle we must recognize that when the operations are different1 the concepts are different1 and if we adopt something equivalent to the table top definition1 as it seems we are physically forced to do1 we must leave open in our thinking the possibility of finding in the present penumbra1 when our accuracy is sufficiently increased1 such phenomena perhaps as directional attributes of mass in a gravitational field4 ,we have just considered the sort of problem that we encounter on ordinary scales of magnitude on going from low to high velocities2 what becomes of the concepts of force and mass when we go to a very small scale8 ,down to the atomic scale we may at least slur over the new physical difficulties1 for although we cannot of course experiment with actual atoms1 we can nevertheless make measurements of the ,brownian#ag movement of suspensions in liquids settling in a gravitational field1 for example1 and the extrapolation to the atom is not a very great one4 ,the mass of each individual atom is obtained by what is equivalent to a process of counting1 assuming the law of conservation of mass on an atomic scale4 ,this is justified by all chemical experience4 ,the mass of the component parts of the atoms1 the electrons1 may also perhaps be given a unique significance after we have decided on the laws of the electrical field1 by experiments on acceleration in electrical fields4 ,the question which interests in principle here is what meaning1 if any1 shall be attached to the mass of the elements of the electron4 ,footnote #ag3 ,this phenomenon is discussed at length in the book by ,j4 ,perrin1 ,brownian ,movement and ,molecular ,reality1 translated by ,f4 ,soddy1 ,taylor and ,francis1 ,london1 #aiji4 ,it is evident that we here go beyond any possible experience1 at least for the present1 and that experience has again become poorer and our concepts fewer in number4 ,all that we can now demand is that certain combinations of numbers1 some of which represent mechanical mass and others electrical charge1 have proper relations to each other when integrated throughout the entire body of the electron4 ,similar questions confront us when we ask what are the forces which the parts of the electron exert on each other4 ,we return to this question in considering the nature of the electrical concepts4 ,in any event1 the concepts of both force and mass are entirely altered in this domain4 ,it is interesting to note1 in passing1 that present electrical theory gives no meaning to the mass of the elements of the electron1 since the total electromagnetic mass of the electron is built up from the mutual terms in the action of the elements--the total mass is not a linear resultant of the action of the elements4 ,,the ,,concept ,,of ,,energy ,in examining the concept of energy1 we start with purely mechanical energy4 ,in isolated mechanical systems1 in which there are only conservative forces1 the sum of kinetic and potential energy is constant4 ,the kinetic energy may be defined as mv#b1 formed for all parts of the body4 ,the potential energy is determined by the position of the parts of the system1 and has physical significance only with reference to a datum position1 that is1 only changes of potential energy have meaning in terms of operations4 ,the total energy ascribed to the system has therefore an element of arbitrariness in that the datum position may be chosen at random1 and energy acquires meaning only on tracing the history back to the epoch of the datum position4 ,the concept of energy may be extended from mechanical systems to all systems with which we are acquainted2 the operations by which meaning is given to the extended energy concept involve the generalized conservation principle1 or the first law of thermodynamics4 ,the extension to thermal systems is immediate2 the inclusion of optical and electrical systems in the scheme was a most important physical step1 which of course required careful experimental justification4 ,because of its wide range of application1 the energy concept has now come to be regarded as one of the most important in physics2 this idea was held by ,ostwald#ah twenty and more years ago1 and is now much to the front because of the connection between mass and energy indicated by the theory of relativity1 and the important rle assigned to energy levels in spectrum analysis4 ,footnote #ah3 ,w4 ,ostwald1 ,die ,energie1 ,barth1 ,leipzig1 #aijh4 ,what now is the precise nature and significance of the general energy concept8 ,in the first place the conservation property of energy is one of the simplest and most obvious of the properties of matter1 so that in this property of energy is seen a reason for ascribing to it certain of the properties of matter1 in particular and most important1 that of localization in space4 ,we must recognize1 however1 that this idea of a location in space is injected into the situation entirely by ourselves1 and corresponds to nothing directly given by the operations of experiment4 ,the idea has had a most important effect1 however4 ,witness1 for instance1 the importance ascribed to the discovery by ,kelvin of a function by which the total energy of an electric field can be represented as distributed through space2#ai this was one of the most important props of the medium point of view4 ,footnote #ai3 ,this function is times the scalar product of electric force and displacement4 ,if ,maxwell's definition of displacement is adopted1 the factor is replaced by 1 and an accurate analogy results between the energy stored in the ether and the elastic energy stored in a bent spring4 ,a more critical examination is likely to diminish considerably our satisfaction with this naive analogy drawn between matter and energy4 ,with regard to matter1 we may still be tolerably satisfied with our ascription to matter of location in space1 but it is quite different with regard to conservation of matter4 ,in just what sense is matter conserved8 ,certainly not in terms of mass1 as we at one time thought4 ,nevertheless we undeniably have a feeling that there is some sort of conservation property here1 and are driven to formulate it badly in terms of a hypothetically constant number of protons and electrons4 ,i have long thought that ,newton was groping after some very similar idea when he so far forgot himself as to define mass as quantity of matter1 a definition perfectly meaningless to a rigorous and unsympathetic interpretation4 ,on the other hand1 whatever meaning may reside in our idea of conservation of matter1 it certainly is not1 in at least one important respect1 like the conservation of energy4 ,for the energy of an isolated mechanical system is a function of the frame of reference in which it is described2 merely by giving velocity to the reference frame and altering in no way the mechanical system we may change its kinetic1 and so its total1 energy by any direct amount4 ,this does not even remotely resemble ordinary matter4 ,i cannot see that the operations which are equivalent to the energy concept justify us in saying more than that energy is a property of a material system2 the operations do not seem to give any unique meaning to a location associated with energy4 ,we now ask what significance is to be ascribed to the sort of conservation that energy does have4 ,we restrict ourselves first to mechanical systems4 ,the motions of a mechanical system satisfy certain differential equations of the second order1 and the actual motion is to be found by an integration of the equations4 ,in the integral of a differential equation certain constants appear which are determined by the initial conditions1 and are therefore the same during all the future motion of the system2 obviously these constants of motion correspond to conservative properties4 ,this reasoning can of course be at once extended4 ,any system1 mechanical or not1 whose motion is determined by differential equations1 will have certain conservative properties4 ,for the systems of mechanics energy is one of the conservative functions2 others are momentum and moment of momentum4 ,energy is particularly simple1 in that it is connected with measurable properties of the system by a simple formula 7 mv#b71 and is furthermore scalar1 which is also a property of quantity of matter4 ,but to go further and ascribe to energy other properties of matter1 such as localization in space1 is entirely overlooking the essential difference in the character of the operation by which matter and quantity of energy are measured1 that is1 overlooking the essential difference in their physical character4 ,the possible extension of the energy concept from mechanics to thermodynamics receives a sufficient physical explanation in terms of our views of the essentially mechanical character of thermal phenomena4 ,that the idea can be extended also to simple electrical or magnetic systems1 in which the effect of velocity of propagation is neglected1 is a consequence of the fact that in these systems the equations of motion remain of the same general mechanical type1 it having been shown by ,maxwell that the equations of such systems may be written in the generalized ,lagrangean form4 ,when1 however1 we extend our formulas to systems in which the velocity of propagation is important 7that is1 when we consider the field equations in their general form7 we find that the ,lagrangean equations no longer apply to matter taken by itself1 and energy is no longer conserved in the original sense4 ,a new function appears1 however1 which behaves mathematically in the same way that the energy did before4 ,the equations of motion of the system remain ,lagrangean in form if the mechanical parts of the system are supplemented by the electric and magnetic fields in space4 ,in this extended form we have1 therefore1 a conservative function as before1 and the energy concept may be retained in this enlarged aspect4 ,the physical operations by which energy is determined are entirely altered1 however1 and the physical character of the concept is changed4 ,no more than before is there justification for localizing energy in space1 or ascribing to it other properties of matter4 ,yet the materialization of the energy concept1 and the consequent desire that energy be localized in space1 is one of the strongest arguments in many minds for the existence of a medium4 ,as far as ,i can see1 therefore1 the existence of conservative functions is involved in the possibility of describing natural phenomena with differential equations4 ,that further there is a conservative function of the precise form found in mechanics is a consequence of the particular form of the equations and the nature of the forces4 ,the question of the significance of the fact that the forces of nature appear to be conservative1 with respect to this particular function of mechanics1 is of much interest1 but it is not our immediate concern now4 ,we are interested rather to ask under what general conditions we shall have conservative functions4 ,quantum theory strongly suggests that when we pass to phenomena on a small enough scale1 we may no longer be able to employ differential equations in our descriptions1 and hence the previous reason for the existence of constants connected with the motion disappears4 ,now there is one obvious remark to be made about this more general situation4 ,whenever the future history of a system is so connected with its present condition that we can retrace our way to the present from any future configuration1 we shall always have conservative functions4 ,for any future configuration contains certain fixed 7or conservative7 features1 in that we can reconstruct the unique present from any future state4 ,there is no reason to expect that the operations by which we find the fixed features will always be simple1 as in the mechanical case4 ,now the determination of the future by the present1 and conversely the possibility of reconstructing the present from the future 7or the past from the present71 is1 we are convinced1 a property which is at least approximately true of phenomena down to a smaller scale of magnitude than we have yet reached1 and so we expect to find these conservative functions in systems whose ultimate laws of motion are much more general than any with which we are yet familiar4 ,the particular form of the conservative function depends on the character of the system4 ,that there is a scalar conservative function for ordinary systems depends of course on particular properties of the system1 but we are at least prepared to find that a scalar conservative function does not necessarily mean a differential equation of the second order4 ,the potential energy of a system has a particular significance with respect to this point of view4 ,in an ordinary mechanical system1 the potential energy simply measures the work done by the applied forces in being displaced from the initial to the final positions2 that is1 the potential energy is a measure of the deviation from the initial position1 and so measures a certain feature of the history of the system4 ,in the more general system1 in which we may not have differential equations1 we may look for something analogous to the potential energy which shall measure the displacement of the system from its initial configuration4 ,such a measure is always possible as long as the past can be reconstructed from the present 7or the present from the future74 ,we recall a remark of ,poincar's#bj to the effect that we of necessity must always have conservation1 because if we have a system in which conservation apparently fails1 we merely have to invent a new form of potential energy4 ,this remark is obviously not of entire generality1 but applies only to such systems as those considered here1 in which the past may be reconstructed from the present4 ,footnote #bj3 ,henri ,poincar1 ,wissenschaft und ,hypothese1 translated into ,german by ,f4 and ,l4 ,lindemann1 ,teubner1 ,leipzig1 #aijf1 ,chap4 ,,viii4 ,of late there has been much discussion of the advisability1 on the basis of certain quantum phenomena1 of giving up conservation as a principle applied to the details of the emission and absorption of light1 retaining it only in a statistical sense4 ,it seems to me that the question here in the minds of physicists was always merely one of convenience1 and that few1 if any1 doubted the ultimate applicability of the principle of ,poincar1 or thought that we were here concerned with a system of such great generality that the past could not be reconstructed from the present4 ,the question was merely whether those variables in terms of which the potential energy is defined make close enough connection with other things of immediate experimental significance1 or whether on the whole the retention of a potential energy is not more trouble than is justified by its convenience1 making a treatment from the statistical point of view preferable4 ,however1 this is all now a matter of more or less past history1 because with the recent extension of the experiments of ,compton1#ba we seem to have experimental evidence for the validity of the conservation law in detail for elementary quantum processes1 with a corresponding simple potential energy4 ,footnote #ba3 ,w4 ,bothe and ,h4 ,geiger1 #b,s4 f4 ,phys4 #cb1 #fci-ffc1 #aibe4 ,a4 ,h4 ,compton1 ,proc4 ,nat4 ,acad4 ,soc41 ,,ii1 #cjc-cjf1 #aibe4 ,going still deeper1 however1 there are quantum phenomena which still may have to be treated by statistical methods1 and this may mean giving up conservation in detail4 ,we have no experimental evidence1 for example1 of what the electron is doing while jumping from one quantum orbit to another4 ,a situation like this merely means that those details which determine the future in terms of the past may lie so deep in the structure that at present we have no immediate experimental knowledge of them1 and we may for the present be compelled to give a treatment from a statistical point of view based on considerations of probability4 ,but ,i suppose that no one1 except perhaps ,norman ,campbell1#bb will maintain that such a situation is more than temporary1 or will cease to search for consequences of these details of structure which may be open to experimental verification4 ,footnote #bb3 ,norman ,campbell1 ,time and ,chance1 ,phil4 ,mag4 ,i1 #aajf-aaag1 #aibf4 ,similarly1 we cannot permanently be satisfied with a picture of radioactive phenomena which represents radioactive disintegration as a matter of chance4 ,the general conclusion to which all this discussion leads is that energy is probably not entitled to the fundamental position that physical thought is inclined to give it1 but that it is a more or less incidental consequence of more deep-seated properties1 and that the character of these deep-seated properties is subject to only the most general restrictions1 so that very little about the nature of the details can be inferred from the existence of any energy function4 ,,the ,,concepts ,,of ,,thermodynamics ,we shall not be concerned here with the many technical questions which are the proper subject of treatises on thermodynamics1 but shall attempt an examination only of some fundamental concepts4 ,the most fundamental of these1 which sets thermodynamics off apart from the simpler subjects1 is probably that of temperature4 ,in origin this concept was without question physiological1 in much the same way as the mechanical concept of force was physiological4 ,but just as the force concept was made more precise1 so the temperature concept may be more or less divorced from its crude significance in terms of immediate sensation and be given a more precise meaning4 ,this precision may be obtained through the notion of equilibrium states4 ,we have in the first place the fundamental experimental fact that when a small body is placed inside a large system1 which we recognize by crude means as comparatively invariable in temperature as time goes on1 the small body very soon acquires a steady condition1 that is1 it comes to equilibrium with its surroundings4 ,we now have the further experimental fact that if the small body ,a is in equilibrium with its environment1 and body ,b is also in equilibrium with the same environment1 there will be no change of condition of ,a and ,b when they are brought into contact with each other--that is1 ,a and ,b are each in equilibrium with the other and also with the environment and therefore1 by definition1 at the same temperature as the environment4 ,the temperature of the environment is now measured in terms of some of the properties of ,a and ,b which crude experience has shown change with the physiological temperature of ,a and ,b4 ,the physiological notion of temperature is thus made more precise by being connected with the physical phenomenon of equilibrium4 ,now it is at once evident that stated in this way without qualification we have said things that are not true4 ,it is not true in general that1 when ,a is in equilibrium with an environment and ,b is in equilibrium with the same environment1 ,a will be in equilibrium with ,b4 ,suppose1 for example1 that the environment is a stream of water and ,a is a tiny water wheel moving freely in its bearings1 and that ,b is a similar wheel with much friction4 ,then we know that ,b will become warm1 and will not be in equilibrium with ,a when brought into contact with it4 ,or we may choose for ,a a mercury thermometer with bulb covered with putty1 and for ,b a similar thermometer with bulb sheathed in platinum1 and we know that the two thermometers will not register the same temperature in the water stream4 ,or still more simply1 we may try to read the temperature of the air in our garden on any bright day with a silvered and with a blackened bulb thermometer2 we know that the two thermometers will read different temperatures4 ,it is evident1 therefore1 that we shall have to specify much more carefully the conditions under which equilibrium holds if we are to give precise significance to the temperature concept4 ,it seems fairly evident in the first place that we shall have to rule out systems in which there is large scale mechanical motion2 the simple notion of temperature does not apply to a system moving with respect to us4 ,only when the two thermometers ,a and ,b move with the same velocity as the stream do we have three-fold equilibrium between the stream1 ,a and ,b4 ,we may state this in another way by saying that the temperature of a moving body must be measured on a thermometer stationary with respect to the body1 but this is only a matter of words1 and properly speaking the temperature concept applies only to a certain aspect of the relation between two bodies mutually at rest4 ,we here entirely neglect relativity questions such1 for example1 as the proper way of correcting for the change of dimensions of moving thermometers4 ,if now the body whose temperature we are measuring does not move with the same velocity in all its parts1 we may still give a meaning to local temperature by dividing the body into parts so small that the velocity of each part is essentially uniform1 and measuring the temperature of each part with a thermometer stationary with respect to it4 ,we are now confronted with the question of how far to carry the process of subdivision4 ,suppose we have a fluid whose motion is completely turbulent when measured with instruments of the ordinary scale of magnitude4 ,for such a fluid the fundamental equilibrium proportions hold between two measuring bodies ,a and ,b and the fluid1 provided that the bodies ,a and ,b are so large that the motion is completely turbulent on their scale of magnitude4 ,we may then define the temperature of the turbulent fluid from the standpoint of these large scale bodies4 ,but we may also define the temperature from the small scale point of view as the average of the temperatures recorded by sufficiently small thermometers1 each moving with the velocity of a local bit of the fluid4 ,these two temperatures will in general be different1 and we must more or less arbitrarily select one which we define as the true temperature4 ,it would seem that the small scale temperature is the better one to choose1 because there is a certain degree of arbitrariness in specifying the scale from which the motion shall be judged completely turbulent ,but on the other hand1 there are difficulties in the small scale definition1 because the turbulence may become more and more fine grained1 until we end with the motion of the molecules themselves1 when the operations certainly fail which give meaning to the temperature concept4 ,in this case of molecular turbulence1 we are driven back to the large scale definition1 which obviously corresponds to ordinary physical practice4 ,it appears then that the temperature concept is not a clean cut thing1 which can be made to apply to all experience1 but that it is more or less arbitrary1 involving the scale of our measuring instruments4 ,in any special case1 the meaning of the temperature concept must be set by special convention4 ,in practice this does not often make difficulty1 because in the majority of cases there is no large scale motion with respect to the thermometers4 ,consider now the other aspect of the equilibrium problem suggested by the thermometers with blackened and silvered bulbs in the sunshine4 ,our common experience tells us how to deal with this situation effectively enough for ordinary purposes4 ,we recognize that the possibility of temperature equilibrium is disturbed by the radiation1 and we protect the bulbs of the thermometers from the sun's radiation by appropriate shields4 ,but this only minimizes the difficulty4 ,for the shield is warmed by the sun1 and in turn warms to a less degree by its radiation the bulb of the thermometer within4 ,we must recognize that every body1 no matter what its temperature1 is always emitting radiation1 so that the bulb of our thermometer is always in a radiation field4 ,at first this puts us in a serious quandary as to the whole question of equilibrium and the meaning of temperature4 ,the situation is saved by the experimental observation that there is a particular radiation field which affects all thermometers equally1 namely1 the field inside an infinite body all at the same temperature4 ,logically this looks like the vicious circle again1 for we have not yet defined what we mean by the same temperature4 ,but actually we avoid the circle here1 as in so many other physical cases1 by a process of asymptotic approximation4 ,the procedure is perhaps something like this3 we find that if we experiment with larger and larger bodies1 isolated and at great distances from other bodies at approximately the same temperature as judged by crude physiological sensations1 two thermometers1 identical except that the bulb of one is blackened and that of the other is silvered1 record more nearly the same temperature as time goes on and as the thermometers are sunk to greater depths in the body4 ,in actual practice1 of course1 the radiational opacity of most bodies is so high that these precautions against the effects of external radiation can usually be entirely ignored4 ,at high temperatures1 on the other hand1 radiation has to be explicitly dealt with4 ,the conclusion for us from these considerations is that operationally the concept of temperature is tied up with that of radiation--the equilibrium concept of temperature is strictly never exactly applicable2 it is only a limiting sort of concept applicable when the radiation field is of a special sort1 namely1 that of a black body4 ,in spite of the explicit recognition which we have to give radiation in defining temperature1 we usually entirely lose sight of it in thinking about the mechanism of ordinary physical processes1 as for instance when we picture the temperature of a gas as determined by the kinetic energy of its molecules4 ,now ,i have no doubt that negligence of this sort can be justified1 but the necessary logical analysis is apparently complicated1 and involves a great many different sorts of experiment by methods of asymptotic approximation1 by which we establish the existence of various sorts of physical constants1 such as constants of emission and absorption and reflection and scattering and fluorescence and thermal conductivity4 ,we do not need to make the analysis here1 but ,i believe that some time it would be worth while to attempt it4 ,such an analysis will justify the principle so often used3 that if a body is in thermal equilibrium the various processes involved1 such as radiation or thermal conductivity1 must when taken separately also be in equilibrium4 ,doubtless1 if our experience had been confined to higher temperatures1 like that of the sun1 this notion of different mechanisms acting independently would have been more difficult to acquire4 ,we next consider another fundamental concept of thermodynamics1 that of quantity of heat4 ,we are at first perhaps inclined to think of this as a comparatively straightforward concept1 given immediately in terms of experience1 but an analysis of the operations by which we measure quantity of heat will show that the situation is really most complicated4 ,consider1 for example1 ,joule's experiment in which the mechanical equivalent of heat was measured by determining the rise of temperature of the water in a container when stirred by paddles driven by a falling weight4 ,we do not question that the rise of temperature of the water has its origin in the mechanical work done on it by the paddles4 ,but what about the rise of temperature of the container8 ,we shall doubtless say that part of this rise comes from heat communicated to it by the warmer water in contact with it1 and part from mechanical work done on it by turbulent impact of the water4 ,but by what operations shall we measure what part of the energy communicated to the container is heat and what part mechanical work8 ,we try to give an idealized answer to this question in terms of ,maxwell demons stationed at all parts of the boundary of the containing vessel with small scale measuring instruments4 ,to measure the heat entering at any point ,i can see nothing else for the ,lilliputian observers to do but to determine the temperature gradient at every point of the boundary from temperature observations at two different levels1 and calculate the heat inflow from the gradient and the thermal conductivity of the material of the walls--there seems no way of measuring a flow of heat as such4 ,the inflow of mechanical energy must be calculated from a detailed knowledge of the elastic waves and other large scale deformations of the walls4 ,here again there is an arbitrary element in our procedure2 if our mechanical measuring instruments are on too gross a scale1 we may miss mechanical energy which we would catch with finer instruments4 ,this situation which we have just submitted to detailed analysis is1 ,i believe1 typical of the general situation2 it is not possible in the general case to find anything which we can call heat as such4 ,without further explicit examination1 we can unambiguously speak of a body losing or gaining heat only when there has been no energy interchange of any other sort with other bodies4 ,in such a case the heat is measured in terms of the temperature change of the body4 ,the heat concept is in the general case a sort of wastebasket concept1 defined negatively in terms of the energy left over when all other forms of energy have been allowed for4 ,the essential fact that a quantity of heat can by itself be defined only in terms of a drop of temperature is somewhat obscured by the usual method of thermodynamic analysis4 ,in describing a ,carnot engine1 for example1 it is specified that the engine shall work between a source and a sink so large that their temperature is not changed by the heat given out or absorbed by them1 so that the impression is natural that heat may in some way be measured apart from temperature changes4 ,this of course is not the case2 we merely require that the source and sink be so large that their temperature changes are of a different order of magnitude from those in the working substance itself1 so that with respect to the working substance1 source and sink may be considered to be at constant temperature4 ,assuming now that we are able to measure quantity of heat in those cases in which the concept has meaning1 let us examine the first law of thermodynamics1 which we write in the form3 d,q d,w d,e ,here d,q is the heat imparted to a given body by other bodies1 d,w is the work of all kinds done on it from outside1 and d,e is the increase of internal energy4 ,now if this equation says what appears at a nave first glance1 it should say that we find experimentally that the relation written always holds between the measured quantities d,q1 d,w1 and d,e4 ,we have seen that in the general case it is not possible to assign a unique operational significance to d,q and d,w1 and presumably not to their sum4 ,we ignore for the present difficulties of this kind and confine attention on d,e2 how shall we measure it8 ,i believe it does not take much examination to convince us that there are no physical operations for measuring d,e as such1 and that therefore the equation expressing the first law must have a different significance from that which appears on the surface4 ,this is often recognized in the statement that the essence of the first law is that d,e is an exact differential determined only by the variables which fix the internal condition of the body1 and not a function of the path by which the body is carried from one condition to another4 ,but what shall we mean by internal condition1 and how shall we be sure that we have found all the variables required to specify it completely8 ,internal condition may be a most complicated thing and require many variables1 as shown by a piece of iron with a complicated magnetic history or by a piece of aluminum about to undergo recrystallization after overstrain4 ,here again ,i believe there is no physical procedure by which general meaning can be given to this concept of internal condition4 ,in specific cases we can state what the variables are which determine internal condition1 and the criterion that we have found the correct internal variables is that d,e shall be a complete differential in terms of them4 ,the first law of thermodynamics properly understood is not at all a statement that energy is conserved1 for the energy concept without conservation is meaningless4 ,the essence of the first law is contained in the statement that the energy concept exists 7or has meaning in terms of operations74 ,the first law is often thought to be one of the most general of physics1 but in a paradoxical sense it is the most special of all laws1 because no general meaning can be given to the energy concept1 but only specific meaning in special cases4 ,the first law owes its complete generality to the fact that no specific case has yet been found of so broad a character that it cannot be included under one or another special case4 ,examination will at once justify this view4 ,thus we find a great many systems which are adequately described in terms of two variables1 pressure and temperature1 in that a function of p and t can be found such that its differential equals d,q d,w4 ,there are other systems in which the six components of stress and t completely fix the internal condition in the sense that they determine a d,e4 ,in other systems the specification of a magnetic field may be necessary1 or an electric1 or a gravitational field4 ,no case is known which cannot be handled in terms of the action of external forces of the proper kind1 but there is no general procedure1 and the first law owes its generality to the exhaustive cataloging of special cases4 ,we may now return to the question left in abeyance above of the ambiguity in d,q d,w4 ,in all the cases in which the specific variables can be found which define d,e1 d,q and d,w also have meaning4 ,consider1 for example1 a gas1 the internal condition of which may be characterized in terms of t and p4 ,the mere fact that the internal condition can be specified in terms of two variables1 one a mechanical variable1 shows that the substance is mechanically homogeneous4 ,being mechanically homogeneous1 we do not have the possibility of ambiguous values of d,w varying with the scale of the measuring instruments1 and in fact we know that d,w p dv4 ,similarly the gas being homogeneous and at rest as a whole allows unique values for d,q4 ,of course this cannot obscure the physical fact that even in such a gas1 when we go to a small enough scale1 we find inhomogeneities arising from the ,brownian movement1 etc4 ,practically our statement means that the inhomogeneities are so fine grained that over a very wide range of scale of the measuring instruments we find the same definite results4 ,the same sort of considerations apply to more complicated systems4 ,if d,e is a complete differential in terms of t and six stress components1 this means again that the body is homogeneous1 its condition is determined by temperature and stress1 which are the same throughout the body1 and again there is no possible ambiguity from the scale of the instruments which measure d,w and d,q4 ,it seems in general1 then1 that if the body allows operations by which d,e acquires meaning1 at the same time d,q and d,w are provided for4 ,in working out this idea in full detail1 some care must be given to the question of order of differentials4 d,q1 for unit time and unit volume1 is strictly equal to k#b t1 where k is thermal conductivity1 so that in determining d,q the second derivatives of temperature are involved4 ,if the body is obviously not homogeneous1 it is still a matter of experience that it can be divided into small pieces1 each of which are by themselves sufficiently homogeneous1 and the first law in its usual form may be applied to each of the pieces4 ,finally1 we emphasize a fact already implicitly mentioned1 namely1 that no physical significance can be directly given to flow of heat1 and there are no operations for measuring it4 ,all we can measure are temperature distributions and rates of rise of temperature4 ,as at present defined1 a heat current is a pure invention1 without physical reality1 for any determined heat flow may always be modified by the addition of a solenoidal vector1 with change in no measurable quantity4 ,if someone states that throughout all space there is a uniform heat current of #ajf cal4_/cm4#b sec4 in the direction of ,polaris1 no disproof can be given1 for such a stream is solenoidal1 and as much heat flows out of every closed surface in unit time as flows in4 ,such a solenoidal flow is meaningless in terms of operations2 we could give meaning to such a flow only in terms of some slight modification of the solenoidal condition introduced by the measuring instrument4 ,in all ordinary conditions the flow of heat given by the simple relation q k ,grad t corresponds exactly to what our atomic pictures lead to expect in those cases where the details of the picture can be worked out4 ,but there may be cases where it is advantageous to supplement the ordinary heat flow 7 k ,grad t7 by the pure fiction of a solenoidal flow1 because in this way it may be possible to account for new phenomena which appear when the solenoidal conditions are slightly departed from4 ,thus if in a conductor at uniform temperature carrying a steady electric current we say that a heat current is also flowing proportional to the electric current and therefore solenoidal1 we may provide the possibility for a simple correlation of phenomena found under those more complicated conditions when an electric current flows in a conductor of non-uniform temperature in a magnetic field4 ,if it should turn out that the heat current is uniquely determined by considerations of this character1 then we have taken the first step away from the pure formalism which this sort of thing otherwise is in the direction of giving physical reality to the invention of heat current4 ,there are other interesting questions of a fundamental thermodynamic character1 such for example1 as whether the entropy concept has any general significance apart from the scale of our measuring instruments1 and what is the operational significance of applying thermodynamic concepts to radiation1 but we shall not consider these questions here4 ,,electrical ,,concepts ,we now set ourselves the problem of finding the meaning of the various concepts in terms of which we describe the behavior of electrical systems1 assuming that we understand what we mean by electrical4 ,we start with the simplest electrical systems1 namely1 those in which we deal with static phenomena on a large scale4 ,in such systems there are independent physical operations by which we may find the magnitude of any charge1 provided that it is effectively concentrated in a geometrical point4 ,the measurements involved in these operations are measurements of ordinary mechanical forces2 we assume that our knowledge of mechanics has already taught us how to make such measurements4 ,an electrically charged body experiences forces1 which may be measured by tying a string to it and pulling on the string with a spring balance hard enough to keep the body in equilibrium4 ,three charges are numerically equal if when each is placed at unit distance from another1 in the absence of the third 7or other charge71 the forces are always the same4 ,if furthermore the forces are of unit magnitude1 the charges are defined as unit charge4 ,having obtained unit charge1 we define the magnitude of any other charge as equal to the force which it experiences when placed at unit distance from unit charge4 ,this of course is all very trite2 the important thing for us is merely that magnitude of charge1 or quantity of electricity1 is an independent physical concept1 and that unique operations exist for determining it4 ,these operations presuppose the ability to perform certain operations of mechanics4 ,having now learned how to measure electrical quantities1 we discover experimentally the inverse square law of force1 and later arrive at the concept of the electric field4 ,as we have seen1 the field is an invention2 here we shall use this concept only for the purpose for which the invention was made1 and shall not involve ourselves in any of the implications of ascribing physical reality to the field4 ,notice that as long as we deal only with point charges we do not have to define field strength in terms of the limiting procedure of making the exploring charge smaller1 for the limiting small charge is necessary only to avoid the reaction of the exploring charge on the positions of the charges which generate the field4 ,all this again is trite2 the important point is that the operations by which the inverse square law and the concept of the field are established presuppose that the charge is given as an independent concept1 since the operations involve a knowledge of charges4 ,the operations also involve the measurement of forces by the ordinary static procedure of mechanics with spring balances4 ,with the means now at our command we establish one very important property of electric charges1 namely that the total amount of charge on an isolated body of finite size is conserved1 no matter how the charge is forced to rearrange itself by the motion of charges on adjacent bodies4 ,by procedures exactly like those outlined above1 we may treat all the corresponding magnetic quantities2 there is formal parallelism between the two sets of phenomena1 but there is the physical difference that we have to realize a single magnetic pole by the device of using a very long slender magnet4 ,we now give our electrical system more freedom1 in that we allow the charges to be in motion with respect to each other4 ,perhaps the most immediate question which we now have to ask is whether charge continues to be conserved when set in motion1 or whether the total charge on an isolated body is a function of its velocity8 ,to answer this question we must generalize the procedure by which we assigned a numerical value to a stationary charge4 ,perhaps the simplest way is to allow two unit charges each to move with constant velocity1 remaining at unit distance apart1 and measure with a spring balance the force required to keep them at constant distance apart4 ,now we immediately find that the force is altered under these conditions1 so that our first impulse is to say that the charge is a function of the velocity4 ,but as we experiment further1 we find that the state of affairs is very complicated2 the force between the two charges at any moment of their motion depends not only on the charges1 their distance apart1 and their velocities1 but also on the angle between the line joining them and the direction of motion in the lines4 ,further experiment of other kinds yields other information2 it requires a force to maintain a charge in uniform motion in a magnetic field1 or to maintain a magnetic pole in motion in an electric field4 ,a moving electric charge exerts a force on a stationary magnetic pole1 so that by definition the moving charge is surrounded by a magnetic field1 and similarly a moving magnetic pole is surrounded by an electric field4 ,returning to our two moving electric charges1 we are impelled to ask whether1 if all these complications are possible1 the numerical constant 7unity for static charges7 in the inverse square law of force is a function of velocity as well as the magnitude of the charges themselves8 ,if we broaden the question in this way1 as we apparently must1 our problem becomes indeterminate1 for we are trying to answer two different questions with a single kind of measurement1 namely of the force between moving charges4 ,i have had no better luck on trying other methods of measurement4 ,apparently the operations do not exist by which unique meaning can be given to the question of whether the magnitude of a charge is a function of its velocity4 ,on realizing this situation1 we are at first embarrassed to know how to proceed1 but we reflect that the embarrassment is not of our own making1 but corresponds to a physical fact4 ,the concept of charge as a unique and independent thing essentially pertains only to static systems4 ,we may extend the concept to moving systems if we wish1 as a matter of convenience to ourselves1 but must recognize that such an extension is an invention of ours and not a reality of nature4 ,now we do make such an extension1 and we make it in the simplest possible way1 that is1 we define the charge on an isolated body in motion as that which we should find on it if we reduced it to rest and made measurements according to the regular static procedure4 ,that this is a convenient thing to do depends on the experimental result that the charge so found is independent of the way in which velocity is imparted to or removed from the body2 in other words1 whenever the body is reduced to rest1 the same charge is always found on it4 ,although this is pure definition on our part1 it turns out to have a most simple and convenient connection with experimental facts which were discovered after the decision to treat a moving charge in this way was made2 the discovery is of the atomic structure of electricity4 ,if then we agree to call each elementary charge a constant independent of the velocity1 the total charge on a body becomes merely proportional to the count of the total number of atomic charges on the body1 which is certainly highly convenient and suggestive4 ,having now fixed what we mean by the magnitude of a moving charge1 we are ready to turn to the general problem of the behavior of any system of charged bodies in motion4 ,for the present we consider only phenomena of the scale of everyday experience4 ,the most general problem that has meaning here is to determine all measurable properties of the system in terms of those data which experiment shows can be arbitrarily specified4 ,now we have already emphasized that the electromagnetic field itself is an invention1 and is never subject to direct observation4 ,what we observe are material bodies1 with or without charges 7including eventually in this category electrons71 their positions1 motions1 and the forces to which they are subject4 ,the forces are to be measured according to definition in mechanical terms1 either by the strains in members of a framework if the system is in equilibrium1 or in terms of accelerations and masses if it is not in equilibrium4 ,the electromagnetic field as such is not the final object of our calculations1 but the calculation of it is only an intermediate auxiliary step1 convenient to make because our mathematical formulation gives so simple a connection between electromagnetic field1 charges1 and mechanical action that the latter can be calculated at once in terms of the former4 ,in fact the connection is so simple that in many cases we have come to regard our problem as solved if we can compute the electromagnetic field1 overlooking the fact that the field has no immediate meaning in terms of experience4 ,electromagnetic theory now presents us with a solution of the general problem2 this solution is contained in the four-field equations of ,maxwell1 the constitutive equations1 and those additional equations 7quite often lost sight of7 which give the forces exerted by the field on electric charges1 or currents1 or dielectrics4 ,let us inquire how we may set about testing the physical correctness of these equations4 ,we may begin with one of the simplest possible tests1 and inquire whether the equations are correct in stating that the force acting on a charge moving in an electric field is simply the product of the charge and the field strength4 ,this1 on the face of it1 is a surprising statement4 ,the field itself is affected by the motion of the charges which generate it1 and it is natural to expect a converse effect4 ,if1 furthermore1 we have sympathy with the medium point of view1 it is easy to think that whatever it is in the medium that gets hold of a charge and exerts a force on it will find it harder to take hold when the charge is in motion4 ,in attempting to check our statement experimentally1 the only additional complication1 as compared with the static case which we have already checked1 is afforded by the motion of the charge1 for we have defined the magnitude of a charge in motion1 so there is no difficulty here1 and we may furthermore suppose that the field is generated by stationary charges1 so that we need not trouble to inquire whether the procedure by which the field was originally defined is here applicable4 ,the task of checking the equation then reduces to the simple physical task of measuring the force on the moving charge4 ,how shall we do this8 ,if the velocity is low1 we may tie a string to the charge and measure the force with a spring balance 7or its equivalent74 ,but now an examination of the equations shows that in more complicated phenomena perceptible deviations from the static behavior are to be expected only at much higher velocities than can be attained by towing charges with a string and a spring balance1 so that it is evidently necessary to check the simple equation for the force on a moving charge also at high velocity4 ,since at high velocity the spring balance method for measuring forces fails1 we are driven to the only procedure that we have1 namely a measurement in terms of the resultant acceleration1 calculating the force by ,newton's first law of mechanics4 ,but this involves a knowledge of the mass of the moving body1 which we recognize in general may be a function of the velocity4 ,now we have already seen1 in discussing the concepts of mechanics1 that the operations by which mechanical mass is defined cannot be carried out at high velocities1 so that either the concept of mechanical mass becomes meaningless at high velocities1 or we must adopt another definition4 ,in attempting to give this new definition of mass at high velocities1 we are driven to a result of special relativity theory1 namely that all mass1 mechanical or electrical1 must be the same function of velocity4 ,if now electrical mass can be found in terms of velocity1 our immediate problem is solved and we shall be in a position to complete the experimental check of the equation4 ,but as a matter of fact1 in order to determine electrical mass1 we have to use that equation which we are now engaged in trying to establish4 ,logically we have again the vicious circle1 the physical significance of which is that independent operations do not exist for giving unique meaning to the concept of force on a charge at high velocity4 ,we seemed so close to our goal a minute ago2 that we may allow ourselves to jump the logical chasm1 and assume that the equation is correct4 ,electrical mass now becomes a definite function of velocity1 mechanical mass the same function1 and we are in a position to compare the actual acceleration received by a charge in a field with that calculated by the equation4 ,our conviction1 on the basis of all experience up to the present1 is that the two accelerations will be found to agree4 ,the equation then does somehow make correct connection with experience in that a consequence of the equation can be verified experimentally1 in spite of the fact that as the equation stands it is meaningless1 because the operations do not exist by which meaning can be given to the individual terms4 ,at low velocities the equation really says what it seems to say1 because the individual terms have meaning in terms of operations2 and1 what is more1 what the equation says agrees with experiment4 ,at high velocities the equation does not mean at all what appears on the surface2 by itself it has no meaning2 it has meaning only when considered as a member of a system of equations1 and only in so far as the system of equations makes by implication statements about nature that have meaning in terms of operations that can be carried out physically4 ,the individual terms of the equation of the system do not have meaning at high velocities1 and in fact there are more terms than there are independent physical operations4 ,an exact analysis from the operational point of view of the significance of the equations at high velocities has perhaps never been made1 and is not necessary for our immediate purpose4 ,the discussion has brought out1 however1 that the number of physically independent concepts has been cut down by two at least1 in that we have made purely formal definitions of the meaning of quantity of electricity1 and of the force exerted by a field on a charge at high velocity4 ,there is no reason to think that there is anything unique about this analysis1 or that formal definitions might not have been given to other concepts than charge and force4 ,we can only state that as far as physical content goes the equations have at least two degrees of freedom4 ,it should then be possible to find quite different sorts of equations which agree equally well with experience4 ,in particular1 since we have seen that the force on a moving charge has no meaning in terms of independent operations1 it should be possible by arbitrary definition to make this force any function of velocity that we please 7of course reducing to the proper value at low velocities71 and then to determine the other equations so that the entire group of equations is consistent with experiment4 ,so far as ,i know1 no one has tried to give such a modified set of equations1 and indeed there is no particular reason why anyone should bother to do this1 because the present equations are simple enough1 and the modified equations1 although perhaps differing greatly in appearance from the present ones1 would have no advantage in any greater or different physical content4 ,but there is no reason to think that the present state of affairs will always continue4 ,we have seen that the decrease in the number of concepts corresponds to our inability to measure as many sorts of physical things at high velocities as at low4 ,now it is the task of the future experimenter so to refine the possibilities of measurement at high velocities as to restore these two degrees of freedom4 ,in particular1 mass should be made measurable in mechanical terms at high velocities4 ,when this restoration has been made1 and all the quantities in our equations receive independent physical meaning1 the significance of the equations in terms of operations will be quite altered1 although the formal appearance will be unchanged4 ,we must then be prepared to find1 as always when we change the range of phenomena1 that the equations in their present form do not correspond to the facts at all1 and that one of the alternative forms allowed by our present two degrees of freedom is the correct form4 ,but until the new experimental facts have been obtained1 it seems hardly worth while to attempt to specify the doubly infinite variety of forms which the equations might have consistently with present experiment4#bc ,footnote #bc3 ,since this was written1 a paper has appeared by ,v4 ,bush1 ,jour4 of ,math4 and ,phys41 vol4 ,v41 ,no4 #c1 #aibf1 in which it is shown that there are advantages in supposing the charge of an electron to change when it is set in motion4 ,so far we have discussed the extension of ordinary electric phenomena in only one direction1 to high velocities4 ,there is another extension which is much more important physically1 namely to very small scales of magnitude4 ,this extension is necessary to an understanding of the properties of matter in bulk1 the electrical nature of the atom having been once established4 ,our problem is to show how the statistical average of the behavior of a large number of electrons gives the large scale effects which are within the reach of observation1 and which are described by the equations we have just discussed4 ,to get this statistical average we must be able to calculate at least certain features of the behavior of the individual electrons1 which means that we must know the form of the equations down to dimensions of the order of those of an electron1 or smaller4 ,now if one contrasts the scale of the supposed dimensions of the electron with the smallest dimensions on which we can make independent experimental verification of those equations1 he must admit that there is an enormous chance for change in the type of equation beyond the limit that we can reach by direct experiment1 and the chances of guessing the correct extension of the equation to small dimensions are correspondingly almost vanishingly small4 7,we may perhaps say that experiments on the ,brownian motion on a scale a good many atoms in diameter bring us the closest possible directly1 which means that we are #ajf or #ajg fold away from electronic dimensions47 ,in spite1 however1 of the apparently enormous chances against it1 this program of extending the field equations to small dimensions and following out the consequences was exactly the program which ,lorentz set himself4#bd ,that ,lorentz saw that such a program might be carried through must be recognized as a vision of extraordinary genius1 and that he was willing to devote to it the years of arduous and detailed calculation that he did is evidence of a pertinacity of purpose of the highest moral order4 ,footnote #bd3 ,see for example1 ,h4 ,a4 ,lorentz1 ,the ,theory of ,electrons1 ,b4 ,g4 ,teubner1 #aiaf4 ,we now have to examine critically this program and to inquire what is the significance of the measure of success that ,lorentz attained4 ,the precise extension of the equations that he made was very simple1 for the large scale equations of ,maxwell were taken over with as little change as possible4 ,the equations are so familiar that it is not necessary for us to write them in detail2 they express relations between the electric and magnetic force vectors 7force and induction now becoming the same thing1 the difference between them in ponderable bodies being one of the things that is to be explained in terms of the electrons71 the space density of electric charge1 its velocity1 and the force acting on elementary charge4 ,we have to notice that although formally the equations have changed little in appearance1 nevertheless the physical content1 as judged by the operations1 has changed a great deal4 ,consider1 for instance1 the meaning of charge density4 ,in the ,maxwell equations1 was merely the number of discrete elementary charges per unit volume1 the distances between these charges being supposed so small compared with the scale of the phenomena involved that their average effect could be fairly represented in terms of their numbers4 ,in the ,lorentz equations1 on the other hand1 has a value different from #j only inside the electron2 everywhere else #j4 ,now an examination of the previous discussion1 in which we questioned whether the magnitude of the charge might be a function of its velocity1 will show that there are no physical operations whatever by which meaning can be given to at individual points inside an electron4 ,there is a single condition on this 1 namely1 that its integral throughout the total volume assigned to the electron shall equal the total static charge of the electron4 ,obviously a single scalar condition is a pretty blunt tool with which to attempt to determine a point function throughout a volume4 ,again1 the equations talk about the velocity of the charge at interior points of the electron2 what possible physical operations are there by which meaning can be assigned to the velocity of an amorphous structureless substance in regions inaccessible to experiment8 ,here again1 the concept as a detailed description of the behavior at a point has become meaningless1 and again there is a single integral condition1 namely1 that the v associated with every must be such that when integrated over the volume of the electron it will give a total transport of charge equal to that carried by the electron in its motion4 ,this again is a single condition on a function distributed through space4 ,still again1 the equations contain the electric and magnetic vectors at points inside the electron4 ,what is the possible meaning of these field vectors in terms of operations8 ,our procedure for finding the field at a point involves by definition finding the force on an electric charge placed at that point4 ,but there is no charge smaller than an electron1 and the procedure degenerates into a fiction4 ,again there is a single integral condition on the field vectors2 the integral of the force on the assumed charge density when taken over the total volume of the electron must give a value corresponding to experiment4 ,except for this single condition1 the concept of the field at points inside the electron is an invention without physical reality4 ,not only is the field concept meaningless at points inside the electron1 but it is meaningless at points outside within a certain distance1 because the exploring charge can never be made smaller than the electron itself1 and so can never come closer than a certain distance4 ,the actual state of affairs is much worse than has already appeared4 ,it was shown in the discussion of space and time that no independent physical meaning can be attached to lengths and times as small as have to be assumed in describing the behavior of the individual electrons4 ,the operations ,div1 ,curl1 d_/dt which enter the field equations are1 therefore1 physically meaningless as they stand2 they have only a mathematical meaning which begins to acquire physical complexion in a most complicated way when the equations are integrated over large enough volumes4 ,it is evident1 therefore1 that the concepts which enter the field equations have entirely lost their large scale significance2 they have become blurred1 fused together1 and fewer in number4 ,a precise analysis of this situation has probably never been attempted and would obviously be difficult3 it would be interesting to know at least how many really independent concepts there are at this order of phenomena4 ,an attempt at an analysis would probably be worth while from a physical point of view in suggesting possible experiments by which the number of physically independent concepts could be extended4 ,since the quantities in the field equations are meaningless in the naked form in which they enter the equations1 it is meaningless to inquire whether the equations as they stand are true or not4 ,in our present state of experimental knowledge it is also meaningless to ask whether1 for example1 the inverse square law between electric charges continues to hold1 or whether an accelerated charge radiates4 ,these questions have meaning only when applied to phenomena on a scale large enough to correspond to possible experiment4 ,there is a rather interesting obverse to the statement that it is meaningless to ask whether the field equations are true1 namely1 that it may not be meaningless to state that they are false4 ,a statement is not true unless it is true in every particular1 but it is false if it is false in a single particular4 ,if we can show that a single consequence of the field equations of ,lorentz1 when integrated or averaged in such a way as to correspond to experimental possibilities1 is false1 then the equations must be false4 ,it seems that1 regarded as a complete description of physical behavior on a small scale1 the equations must be judged false1 because they contain no suggestion of quantum phenomena4 ,even if we have to recognize that the equations are false1 there can be no question that they correspond to an important part of reality1 and that they have been of the greatest service to physics4 ,what is the significance of the success that they have attained8 ,it is to be noticed that all the phenomena to which the ,lorentz equations have been successfully applied1 although not large scale phenomena in the ordinary sense of the word1 are nevertheless phenomena involving the coperation of a number of atoms1 and that the equations unquestionably fail when applied to phenomena involving single electrons4 ,it appears from our best present evidence that on a small scale the behavior of nature is governed by quantum principles and is therefore quite different from large scale behavior1 ,which we have seen is governed by the ,maxwell equations4 ,there must of course be a transition zone in which the character of phenomena changes from quantum to ,maxwell4 ,now any program like that of ,lorentz is almost inevitably bound to begin to give correct results when we get up as far as the transition zone1 for the simple reason that the relations of ,maxwell have been put into the equations and are always there ready to appear as soon as the quantum relations begin to give way4 ,the physical significance of the success of the ,lorentz program seems to be that the transition from ,maxwell to quantum takes place at a stage pretty far down toward the individual atoms4 ,to find the precise details of the transition from ,maxwell to quantum phenomena constitutes a large part of the program of the immediate future4 ,all this skepticism about the classical work of ,lorentz is likely to be rather irritating or depressing1 particularly if one attempts to imagine what other course could have been adopted4 ,indeed it does seem that we find ourselves in a real quandary2 ,lorentz was practically forced1 because of the character of the mathematical tools at his command1 to take the course that he did1 in spite of any recognition of the physical meaninglessness of the mathematical operations4 ,we have already seen that conventional mathematics does not correspond to the physical reality2 it cannot easily make a qualified statement subject to limitations1 and it recognizes no difference between the physically big and the physically little and the corresponding change in the operational meaning of its symbols4 ,it begins by being a most useful servant when dealing with phenomena of the ordinary scale of magnitude1 but ends by dragging us by the scruff of the neck willy nilly into the inside of the electron where it forces us to repeat meaningless gibberish4 ,larmor recognized this1 and in his electron theory1 developed practically contemporaneously with that of ,lorentz1 endeavored to treat electrons as wholes1 and not to make meaningless statements about their insides4#be ,but he was much less successful than ,lorentz in making his analysis give physical results1 and one may suspect that it was at least in part due to difficulty with his tools4 ,footnote #be3 ,joseph ,larmor1 ther and ,matter1 ,cambridge ,university1 ,press1 #aijj4 ,in this book the electron is treated as a point singularity in the ether4 ,what we should like to be able to do is easy to see4 ,the things that go into our equations must have independent physical meaning1 and the character of our mathematical formulation should change to keep pace with the change in the physical operations which give meaning to the terms4 ,for example1 electrical density has a meaning for large scale phenomena1 but means nothing on a small scale4 ,our ultimate electric unit is the electron2 when we get down to this scale of magnitude1 our mathematics ought to be making statements about the relative behavior of discrete electrons1 and not mention so much as by implication the density at points inside an electron4 ,but this sort of thing we apparently cannot yet do2 the proper mathematical language has not been developed4 ,such a language1 when developed1 must not only be able to resist the temptation to burrow inside the electron1 but must also try to get along without the field concept1 which we have seen is liable to so much physical abuse1 and must reduce effects in complicated electrical systems to the ultimate elements that have physical meaning1 namely1 a dual action between pairs of electrical charges1 with no implications about physical action where the charges are not4 ,,the ,,nature ,,of ,,light ,,and ,,the ,,concepts ,,of ,,relativity ,we have already discussed several aspects of the theory of relativity in connection with the relation to it of some of our fundamental concepts4 ,there are still other topics connected with relativity which demand attention2 most of these involve the properties of light4 ,it will now be convenient to discuss together the properties of light and these concepts of relativity4 ,we restrict our discussion of light to those simple properties which bear on the theory of relativity4 ,practically all our thinking about optical phenomena is done in terms of an invention1 by means of which these phenomena are assimilated to those of ordinary mechanical experience1 and so made easier to think about4 ,to realize that invention has been active here1 we must think ourselves back into that naive frame of mind in which experience is given directly in terms of sensation4 ,the most elementary examination of what light means in terms of direct experience shows that we never experience light itself1 but our experience deals only with things lighted4 ,this fundamental fact is never modified by the most complicated or refined physical experiments that have ever been devised2 from the point of view of operations1 light means nothing more than things lighted4 ,now experience shows that these things lighted may stand to each other in varied relations2 in attempting to reduce these relations to order and understandability we make a certain invention4 ,this is prompted by several cardinal experimental facts3 in the first place1 things lighted have a simple geometrical relation to each other1 in that screens placed on straight lines between the lighted objects may suppress the illumination of one or the other and themselves become illuminated4 ,this leads to the concept of rectilinear beams of light1 which is no more than a description of the geometrical relation between lighted objects4 ,then we have the experimental fact of the asymmetrical relation of the lighted objects1 described in terms of sources and sinks4 ,finally1 we have the discovery made at a much later stage1 and not possible until physical measurements had reached a high refinement1 that light has properties analogous to the velocity of material things4 ,this was first discovered in connection with astronomical phenomena in the shift of the time of eclipse of ,jupiter's satellites and in aberration1 but was later found to hold for purely terrestrial phenomena1 in that a beam of light reflected from a distant mirror does not return to the source until after the lapse of a time interval that can be measured with means sufficiently refined4 ,this property of return after the lapse of time is precisely like that of material things1 such as a messenger despatched for an answer1 or a ball or a water wave bouncing from a wall4 ,these various properties of light lead quite naturally and almost inevitably to the invention of light as a thing that travels1 thing not necessarily connoting material thing4 ,the question now for us is whether we shall regard this as a mere invention1 made for convenience in thinking1 or shall go further and ascribe physical reality to it1 that is1 shall we think of light as capable of independent physical existence in the space between the matter that constitutes the source and the mirror8 ,now in spite of the resemblances pointed out above1 there is at least one universal and fundamental difference between a thing that travels and light4 ,we have independent physical evidence of the continued existence of the ball1 for example1 at all intermediate points of space2 we can see it1 or hear it1 or feel the wind in the air as it passes1 or even touch it4 ,all these phenomena are independent of the initial and terminal phenomena1 and hence by our criterion for the physical reality of an invention1 we are justified in ascribing physical reality to the ball in transit4 ,but with the beam of light it is entirely different2 the only way by which we can obtain physical evidence of the intermediate existence of the beam is by interposing some sort of a screen1 and this act destroys just that part of the beam whose existence we have thereby detected4 ,there is no physical phenomenon1 whatever by which light may be detected apart from the phenomena of the source and the sink 7understanding that a mirror is included in the idea of1 sink72 that is1 no phenomenon exists independent of the phenomenon which led us to the invention of a thing travelling4 ,hence from the point of view of operations it is meaningless or trivial to ascribe physical reality to light in intermediate space1 and light as a thing travelling must be recognized to be a pure invention4 ,the status of light is exactly the same as that of an electric field2 there is not the slightest warrant for ascribing physical reality to either at points of empty space--light and field-at-a-point have no meaning until we go there and make experiments with some material thing4 ,of course the electromagnetic theory of light makes this resemblance inevitable1 provided the theory and our views of the nature of light and the field are correct4 ,it cannot be denied that there are some phenomena which uncritically considered appear to justify thinking of light as a thing that travels2 these will now be discussed4 ,probably the argument to which most significance is usually ascribed is derived from the phenomena of energy4 ,the passing of light from source to sink is accompanied by the transfer of energy4 ,but energy is conserved1 so that we have to ask where the energy is in the time interval between the emission of light from the source and its absorption by the sink4 ,there is an obvious answer3 the energy is in transit1 of course1 somewhere in the intermediate space between source and sink4 ,if we think of light as propagated through a medium1 then the medium is such that energy may reside in it1 as in the electromagnetic theory of light1 or if light is more material and ballistic in character1 the thing that travels has itself energy4 ,we notice in the first place that the conservation principle involves the time concept1 because what we mean by conservation is that the total energy of the universe1 at a fixed instant of time1 is constant4 ,that is1 we have to integrate over all space the local energy at a definite instant of time1 and this involves spreading the time concept over all space4 ,it is further evident that unless we spread the time concept over space in the right way we shall not get conservation4 ,the proof that it is possible to spread the time concept over space in such a way as to give conservation involves a knowledge of the properties of light4 ,it would seem1 then1 that we ought not to assume conservation in deducing the properties of light1 when a knowledge of the properties of light is necessary to establish conservation4 ,these considerations cannot be accepted as final1 however1 until a detailed analysis has been made1 and this would be most complicated4 ,but there is a more important consideration derived from our previous critique of the energy concept1 namely1 that there is no basis for asserting that energy is localized in space at all2 energy is not a physical thing1 but rather what we would call a property of a system as a whole4 ,if this view of energy be granted1 the whole energy argument for light as a thing travelling1 and also for the existence of a medium1 falls4 ,i believe that similar considerations apply to any arguments from the conservation of momentum4 ,the possibility of detecting light in apparently empty space by a screen constitutes perhaps the most immediate reason for considering light as a thing that travels4 ,this point of view ,i believe is characteristic of the entire attitude of ,einstein in deducing the theorems of the special theory of relativity4 ,einstein's light signal is for the purposes of the deduction thought of as a simple spherical wave spreading from the source and capable of being watched as it spreads by an observer outside the system1 in much the same way that a water wave can be watched4 ,of course the light signal cannot actually be watched in transit1 but we can come fairly close to this ideal by placing screens at any point we please to make the wave visible4 ,it is true that the mere act of showing the existence of the light destroys that part of the beam whose existence is detected1 but the screen needs only an infinitesimal amount of light to make it visible1 and so by the usual physical argument we may suppose that the detecting screen produces only an infinitesimal modification of the total original light4 ,our satisfaction with this picture evaporates if our present quantum views of the nature of light are correct4 ,we can no longer think of the spherical light pulse as of irreducible simplicity1 but it is an exceedingly complicated thing1 perhaps more complicated than a gas from the point of view of kinetic theory1 and simulates simplicity by some sort of averaging of the effects of the elementary quantum processes of which it is composed4 ,if the principles of relativity are to continue to be regarded as fundamental1 or even if they are to remain intelligible1 we must apply our reasoning1 not to spherical wavelets1 but to the elementary process of which these wavelets are composed4 ,now the elementary quantum act is essentially a twofold thing3 there is a discrete act of emission at some discrete material particle1 and the act is consummated by another discrete act 7absorption or scattering7 at some other discrete particle4 ,we cannot yet fully characterize the details of this twofold process1 but have to connect the place at which absorption takes place with the place of emission by statistical considerations4 ,it is evident1 however1 that to think of emission as starting some process like a spherical wavelet travelling like a thing through space presents an entirely incorrect view1 because in the wave there is no hint of the discrete place which is to terminate it4 ,we may say crudely that there is no way by which the wave can know what discrete material particle is to complete the emission process4 ,we may perhaps try to save the situation by remembering that a spherical wave is polarized and so has a unique direction associated with it2 but further examination shows that this does not help1 because the unique direction is that of no energy flow1 and absorption can take place in any direction except this4 ,it appears then that instead of being a help1 the thing travelling point of view is a positive embarrassment when we try to picture by means of it the essentially twofold nature of the elementary quantum act4 ,another plausible argument for light as a thing travelling may be deduced from our principle of connectivity4 ,imagine a dark lantern with a shutter that can be opened or closed so as to emit a momentary flash of light1 a distant mirror1 and a receiving instrument near the source4 ,one of the properties of light that we always assume is that no permanent trace of the act of emission is left behind in the source4 ,the most minute examination of all the details of the lantern and its surroundings at some time after the emission of the flash has not yet shown any phenomenon that betrays a remembrance of the emission of the flash1 unless perhaps we measure the total energy or momentum and have some way of knowing what the energy or momentum would have been if the flash had not been emitted1 and in any event we cannot specify the moment in past time when the signal was emitted4 ,in the same way we cannot tell from an examination of a mirror whether it has at any time in the past reflected a beam of light4 ,consider now two systems1 each consisting of a source and a mirror distant #c x #ajaj cm41 identical in all respects1 except that in one a light signal was flashed from the source #a4e seconds ago1 and in the other only #j4e seconds ago4 ,according to our hypothesis1 the most complete examination of source and mirror in either system fails to show the slightest difference1 but nevertheless there is something essentially different about the two systems1 for in one a light signal arrives at the screen in #j4e seconds1 and in the other not until #a4e seconds4 ,this violates what we have suggested might be regarded as the cardinal and most general principle of all physics1 the principle of essential connectivity1 which states that differences between two systems must be associated with other differences4 ,a most obvious and simple way of maintaining our principle is merely to point out that the system really included more than we investigated3 the system properly consists of source1 mirror1 screen1 and all intermediate space1 so that if we had examined intermediate space we should have found light there in transit in different positions in the two systems1 thus correlating with the differences in subsequent history4 ,this argument appeals to me as perhaps the strongest that can be advanced for the view of light as a thing travelling4 ,but it seems in no way conclusive4 ,our principle of essential connectivity made no mention of the time concept1 but we have somehow smuggled it in making the application above4 ,we sought to give a complete description of our system at some one instant of time1 and this involved spreading the time concept over space4 ,this itself is a questionable operation and may be done in different ways4 ,but1 more important1 what is the justification for supposing that the system can be completely described by giving a complete description of all the measurable parts of it at some one epoch8 ,we have seen that in the most general case the principle of essential connectivity must recognize that the concept of initial condition of a system involves all the past history1 and we may have here a case in point4 ,the answer can be given only by experiment4 ,in dealing with ordinary experience1 when we do not have to distinguish between local and extended time1 and are not dealing with optical phenomena1 there can be little question that experience at least approximately justifies the expectation that future behavior is determined in terms of the present condition and that present condition may be specified in terms of the results of present operations performed in the system4 ,but before extending this principle to phenomena in which we have to distinguish between local and extended time1 we have to answer just that question which we are now considering2 namely1 whether there are physical phenomena taking place in apparently empty space1 and whether therefore empty space has to be included in the system4 ,we find ourselves again treading the vicious circle4 ,perhaps experience will show that the extension of the principle of connectivity to optical phenomena involves something like this3 namely1 the future at any point in a material system is determined by a complete description of the present state of the system in the immediate vicinity1 and by a history of the behavior at more distant points1 this history extending over longer and longer intervals of time as the point becomes more remote4 ,i believe1 however1 that these possibilities will not seem very satisfactory1 and that most physicists will discover in themselves a very strong disposition to feel that the future is determined in terms of a complete description of some sort of instantaneous configuration1 time being extended in some suitable way over space4 ,this instinctive demand that the future be determined in terms of the present may easily be consistent with the optical phenomena in our two systems consisting of source1 mirror1 and screen1 without involving the material existence of light in empty space1 provided that our assumption that the emission of light leaves behind it no permanent record in the source was incorrect4 ,it may be that detailed examination of a source after emission will disclose permanent traces1 from which the instant of emission may be found by extrapolation4 ,if the conviction of determinism of the future is strong1 the physicist may well be impelled to search here for new phenomena indicating such a memory of emission4 ,let us now inquire how our physical structure might be affected if we should give up the identification of light with a thing travelling4 ,one consequence is that light need no longer be thought of as having the property of velocity1 since velocity1 in terms of immediate experience1 is a property of things moving from place to place4 ,giving up the concept of light as a thing travelling would enable us1 then1 to adopt an alternative method of describing nature with a different concept of velocity2 we have seen that it is possible to define velocity in terms of operations different from the usual ones1 in such a way as to give the usual numerical results at small velocities1 but different results at high velocities1 and in particular to give an infinite velocity for light4#bf ,footnote #bf3 ,no difficulty arises from the asymmetric character of light in assigning an infinite velocity to light because those physical operations by which we discover which is the source and which the sink are entirely distinct from the operations by which a velocity is measured2 or in other words1 even in the limit1 it still has meaning to say that an infinite velocity has a direction associated with it4 ,there is now no objection to an infinite number associated with light1 if we no longer think of light as having physical velocity4 ,we may1 if we like1 continue for the sake of convenience to talk of the velocity of light1 clearly understanding that the infinite value which must be ascribed to this velocity corresponds to the fact that the physical concept of velocity does not apply in all respects to light4 ,we should now have to revise our process for extending the time concept over space1 because this was formerly so done as to give light a finite velocity4 ,we are now to make this velocity infinite1 which is obviously to be done simply by setting a distant clock on zero at the instant it receives a light signal flashed from our clock at its zero4 ,the behavior of material things now takes on a simple aspect--there is no longer a finite upper limit to the velocity that can be given a material thing1 and light has no longer the paradoxical property of bearing the same finite relation to each of two material systems which differ from each other by a finite amount 7that is1 the first postulate of relativity that the velocity of light is #c x #ajaj in all reference systems74 ,light instead now bears the relation of infinitude to each of two systems which differ from each other by only a finite amount1 and this is natural from the mathematical point of view4 ,however1 all is not simplified by this change in the method of setting clocks1 but a price has to be paid4 ,the price is that we have to give up the simple connection between the velocity of a thing and its go and come time4 ,our changes have not affected local time2 the time of passage of light to a distant mirror and return to the source is not changed1 and is therefore still finite1 although we describe the velocity of light as infinite4 ,now examination shows at once that there is no immediate connection between the concepts of velocity and of go and come time1 because the operations involved are different4 ,a measurement of a linear velocity according to our definition involves two clocks at two different places or else a clock travelling with the object1 while go and come time demands only a single clock at a single place1 and also involves necessarily a reversal of direction of motion in the object under measurement4 ,we see then that1 according to the definition adopted for velocity1 we have the choice either of doing as ,einstein did in the restricted theory of relativity and making go and come time very simply related to velocity2 or we may say that refined physical measurements show that something of significance happens when the direction of motion is reversed1 and that phenomena are not symmetrical with respect to a reversal of direction4 ,the asymmetry which results from reversing the direction of motion we may visualize as a sort of curvature in space and time1 as of a small piece of an arc of a circle bent back on itself1 with the two ends diverging4 ,this alternative way of treating velocity would mean that velocity can be measured simply only by a specially situated observer2 this need not be considered disturbing1 because in fact the operations have been defined only with respect to such an observer4 ,which of these two possible treatments of velocity shall be adopted is to a certain extent a matter of convenience1 determined by the sort of phenomena in which we are most interested and wish most to simplify4 ,einstein's chief concern was with optical phenomena1 so that the motive for his choice is evident4 ,in this choice of ,einstein it is not very evident that the desire to make go and come time simply connected with velocity played a very prominent part1 but it seems rather that the desire to think of light as a thing travelling1 with a finite velocity1 was much more influential4 ,this way of thinking of light is fundamental to all the treatment of restricted relativity2 without this sort of picture all the mathematical deductions would lose their simplicity and convincingness1 for in all the deductions we inevitably think of ourselves as an observer from outside1 watching a thing that we call light travelling back and forth like any physical thing4 ,now there can be no doubt that1 when choice is possible1 convenience and simplicity are important considerations2 but ,i believe that there is another much more important consideration1 namely1 the most perfect reproduction possible of the physical situation4 ,it seems to me that it is very questionable whether ,einstein1 and all the rest of modern physics1 for that matter1 have not paid too high a price for simplicity and mathematical tractability in choosing to treat light as a thing that travels4 ,physically it is the essence of light that it is not a thing that travels1 and in choosing to treat it as a thing that does1 ,i do not see how we can expect to avoid the most serious difficulties4 ,of course the whole problem of the nature of light is now giving the most acute difficulty4 ,the thing-travelling point of view1 even as treated by ,einstein1 does not land us in a situation which is at all satisfying logically4 ,we are familiar with only two kinds of thing travelling1 a disturbance in a medium1 and a ballistic thing like a projectile4 ,but light is not like a disturbance in a medium1 for otherwise we should find a different velocity when we move with respect to the medium1 and no such phenomenon exists2 neither is light like a projectile1 because the velocity of light with respect to the observer is independent of the velocity of the source4 ,on the other hand1 in aberration we have a phenomenon similar to that shown by projectiles4 ,the properties of light are more like those of a projectile than is perhaps commonly realized1 as is shown in the papers of ,la ,rosa#bg on the ballistic theory of light4 ,the properties of light remain incongruous and inconsistent when we try to think of them in terms of material things4 ,footnote #bg3 ,m4 ,la ,rosa1 ,scientia1 ,july-,august1 #aibd4 ,einstein's restricted relativity has made a great contribution in so grouping and coordinating the phenomena that they can all be embraced in a simple mathematical formula1 but he does not seem to have presented them in such a light that they are simple or easy to grasp physically4 ,the explanatory aspect is completely absent from ,einstein's work4 ,in view of all our present difficulties it would seem that we ought at least to try to start over again from the beginning and devise concepts for the treatment of all optical phenomena which come closer to physical reality4 ,no one realizes more vividly than ,i that this is a most difficult thing to do4 ,if we are ever successful in carrying through such a modified treatment1 it is evident that not only will the structure of most of our physics be altered1 but in particular the formal approach to those phenomena now treated by relativity theory must be changed1 and therefore the appearance of the entire theory altered4 ,i believe that it is a very serious question whether we shall not ultimately see such a change1 and whether ,einstein's whole formal structure is not a more or less temporary affair4 ,although it is exceedingly difficult to forsee what the treatment of the future will be like1 it is easy to surmise certain of its features4 ,in essence the elementary process of all radiation perceived as radiation is twofold4 ,there is some process at the source and some accompanying process at the sink1 and nothing else1 as far as we have any physical evidence2 furthermore1 the elementary act is unsymmetrical1 in that the source and the sink are physically differentiated from each other4 ,this is the most complete expression of the physical facts2 there is nowhere any physical evidence for the inclusion of a third element 7the ether74 ,therefore all the phenomena apprehended by an observer 7and this embraces all physical phenomena7 can be determined only by the source and the sink and the relation to each other of source and sink1 for there is nothing else that has physical meaning in terms of operations4 ,this formula covers not only the possibility of such first order phenomena as aberration and the ,dppler effect1 but also shows that such second order effects as that looked for by ,michelson and ,morley must be non-existent4 ,it will thus be seen that some of the consequences of relativity theory are implicitly contained in certain very broad points of view4 ,one interesting question that must be answered before we can get very far with a new treatment is whether the elementary optical process is of necessity twofold1 or whether we may have emission without absorption1 that is1 radiation into empty space4 ,lewis seems to imply in recent papers that this is not possible4#bh ,the astronomers have already pointed out difficulties in explaining phenomena like the temperature equilibrium of the planets if we suppose this is the case4 ,footnote #bh3 ,for example1 in the book3 ,g4 ,n4 ,lewis1 ,the ,anatomy of ,science1 ,yale ,university ,press1 #aibf1 p4 #abi4 ,,other ,,relativity ,,concepts ,we now turn to some of the other concepts of relativity4 ,one of the most important of these is the event2 in fact this concept is made fundamental by ,whitehead4#bi ,we have already discussed the concept of event under the identity concept with which it is closely involved4 ,the event is usually thought of by ,einstein as merely an aggregate of four coordinates1 three of space and one of time4 ,the principle of general relativity1 namely1 that the laws of nature shall be of invariant form1 when formulated mathematically1 involves the assumption that nature may be analyzed into events1 and is expressed by the requirement that the mathematical relations between the coordinates of a chain of events shall be invariant4 ,the same idea is also expressed by ,einstein in another form1 namely1 that nature may be completely characterized in terms of space-time coincidences4 ,in elaborating this idea1 ,einstein assumes that the results of all measurement may be given in terms of such coincidences4 ,now it appears to me most questionable whether the analysis of nature into events is possible or sufficient4 ,with regard to the coincidence point of view1 it seems perfectly obvious that the world of our immediate sensation cannot be described in terms of coincidences2 how1 for example1 shall we describe in terms of space-time coincidence the photometric comparison of the intensity of two sources of illumination1 or the comparison of the pitch of two sounds1 or the location of a sound by the binaural effect8 ,footnote #bi3 ,a4 ,n4 ,whitehead1 ,an ,enquiry ,concerning the ,principles of ,natural ,knowledge1 ,cambridge ,university ,press1 #aiai1 ,chap4 ,v4 ,to justify the coincidence point of view we apparently have to analyze down to the colorless elements beyond our sense perception4 ,it does not seem unreasonable1 perhaps1 to expect that the universe is completely determined in terms of the positions as a function of time of all the positive and negative electrons2 but to introduce such a thesis now certainly goes beyond present experimental warrant1 and is contrary to the general spirit of relativity1 which nowhere else involves any reference to the small scale structure of things4 ,even if we were willing to overlook all these objections1 we would still have the fact that the difference between a positive and negative electron is not contained in any specification of the mere coordinates4 ,a further very important doubt in principle as to the possibility of the analysis of nature into events is afforded by the character of the concept of event itself4 ,we have seen that the idea of event involves the existence of discontinuities1 and that this can correspond only approximately to the physical fact1 because discontinuities apparently lose their abruptness as we make our measurements more refined4 ,the thesis that nature can be described in terms of discontinuities of a very small scale seems much too special to be made a fundamental part of a theory of the general pretensions of that of relativity4 ,in fact this1 as well as a consideration to be mentioned later1 suggests that the argument and result of general relativity may be intrinsically restricted to large scale phenomena4 ,we now pass from these somewhat special questions to ask why it is that ,einstein was able in the general theory of relativity to obtain new and physically correct results from general reasoning of an apparently purely mathematical character4 ,we are convinced that purely mathematical reasoning never can yield physical results--that if anything physical comes out of mathematics it must have been put in another form4 ,our problem is to find where the physics got into the general theory4 ,there are two questions to be disentangled here3 we have to consider in the first place the significance of the fact that ,einstein has been able to describe relations in nature in mathematical form1 and in the second place of the fact that he was able to arrive at the mathematical formulation of these physical relations by reasoning of apparently a purely mathematical character1 from postulates of merely formal mathematical content 7invariance of natural laws in generalized coordinates74 ,now the theory of relativity does not seem to differ in the first respect from any other branch of mathematical physics1 such as the classical mathematical theory of electricity and magnetism1 for instance1 and this matter has already been touched in an earlier chapter4 ,it is a fact that the behavior of nature can in many cases be expressed to a high degree of precision in mathematical language1 and relativity is not unique in this respect4 ,in any event1 we must not allow this possibility of mathematical formulation to obscure the essential fact that all physical knowledge is by its nature only approximate1 so that we may expect at any time to find1 when we have carried our measurements to a higher degree of precision1 that our mathematical expression of the laws was not quite exact1 as seems now to be the case with ,newton's law of gravitation1 for example4 ,i do not suppose that ,einstein would claim that the statements of relativity differ in this respect from any of our other statements about nature1 although apparently some of his followers see something more here4 7,from the operational viewpoint the meaning to be attached to something more is somewhat obscure47 ,with respect to the second question1 we may stop to notice that the special theory stands in quite a different position from the general theory4 ,the special theory is much more physical throughout2 its postulates are physical in character1 and it is obvious that the physics got into the results through the postulates4 ,it seems to me without question that ,einstein showed the intuitive insight of a great genius in recognizing that there are mutual relations between physical phenomena which can be described in very much simplified language in terms of concepts slightly modified from those already in common use4 ,in view of the remarks made on the nature of light1 it is legitimate to wonder1 however1 whether the formulations of even the special theory will always stand4 ,it seems to be true that all the facts of nature1 even in the absence of a gravitational field1 cannot be connected by the simple formulations of the special theory2 that the physical relations are simple only in a sub-group2 and that if we wish to deal with all optical phenomena1 we have carried our simplifications too far1 for the emission of a light signal is not a simple event1 and light is not in nature like a thing travelling4 ,just the sorts of physical thing which are ignored in treating light as the special theory does are coming to be more and more important in the minds of physicists1 and this is a reason for wondering whether ultimately ,einstein's special theory may not be regarded merely as a very convenient way of tying together a large group of important physical phenomena1 but not as being by any means a full or complete statement of natural relations4 ,with respect to the general theory1 however1 ,i believe the situation is quite different4 ,the fundamental postulate that the laws of nature are of invariant form in all cordinate systems is highly mathematical1 and of an entirely man-made character4 ,of what concern of nature's is it how man may choose to describe her phenomena1 and how can we expect the limitations of our descriptive process to limit the thing described8 ,furthermore1 ,einstein's method of connecting his mathematical formulation and nature by way of coincidences of #d-;events 7three space1 one time coordinates7 seems to be very far removed from reality1 since it entirely leaves out the descriptive background in terms only of which the #d-;event takes on physical significance4 ,nevertheless1 three definite conclusions about the physical universe have been taken out of the hat by the conjuror ,einstein 7shift of the perihelion of ,mercury1 displacement of apparent position of stars at the edge of the sun's disk1 and the shift toward the infra-red of spectrum lines from a source in a gravitational field71 and the problem for us as physicists is to discover by what process these results were obtained4 ,an examination of what ,einstein actually did in deriving his results will show1 ,i believe1 that the situation is really different from that suggested above4 ,in the first place1 the requirement that the laws of nature be of invariant form actually places no restriction1 as any one can see by setting himself the task of expressing1 for example1 an inverse cube law for gravitation in terms of generalized coordinates4 ,the work of expressing such a law can be attacked in a perfectly routine way4 7,the essential difference between the invariability requirements of the special and general theories is to be noted2 the special theory requires that the velocity of light1 for instance1 have the same numerical value in all allowed systems3 the general theory merely that all laws have the same literal form1 but with variable numerical coefficients47 ,but1 as ,einstein says1 if any one actually attempts to carry through the work of expressing an inverse cube law in generalized coordinates1 he will find the task prohibitively complicated1 and will seek for some simpler formulation4 ,what ,einstein actually did1 therefore1 was to require that the laws of nature be simple in generalized form4 ,now we know that the law of gravitation as formerly expressed in ordinary coordinates as an inverse square law was approximately exact1 and was also simple4 ,any deviations from this law are small1 and all experience leads us to expect that to the first order of small quantities the deviations can be taken care of mathematically in the form of small correction terms to this law4 ,this by itself gives nothing1 however1 because a small correction term can be added to our equations in an infinite number of ways4 ,if1 however1 we know that the equation must be of a certain type after the correction terms have been added1 the possibilities are so much restricted that the form of the correction term may be determined4 ,in arguing as to the probable type of the equation1 ,einstein advanced the considerations by which physics gets into the situation4 ,in the first place1 the special theory had prepared us for the possibility of finding that our measuring instruments might be modified in a gravitational field1 analogously to the shortening of a meter stick when set into motion4 ,in fact1 special theory had indicated that in an accelerated system the modifications might be too complicated to be treated by that theory4 ,in the absence1 then1 of specific information we must be prepared for the most general possible alteration in space-time in a gravitational field4 ,in describing space-time we must therefore use cordinates adapted to handling the most general possible relations1 and these are the generalized cordinates of ,riemann1 which had been already discussed by mathematicians4 ,going back now to ,einstein's criterion that the equations are to be simple1 we have the demand that the equations be simple in generalized cordinates1 and of course they must also reduce to the ordinary equations 7that is1 the equations of special relativity7 in space where there is no gravitational field4 ,in deciding the further question as to what the type of equation probably is1 we are influenced by considerations of convenience as well as by physical considerations4 ,practically the only type of equation that can be handled mathematically is linear1 so that we shall certainly try first whether this type of equation may not continue to hold4 ,now the ,newtonian law of the inverse square may be expressed in terms of a linear differential equation of the second order in the old ,cartesian cordinates 7,poisson's equation71 so that our most immediate suggestion is that the equations remain linear and of the second order in generalized cordinates4 ,as a matter of fact1 this requirement turns out to be sufficient to determine the small correction term by which the ordinary equations can be generalized2 ,einstein's papers must be studied to see how this works out in detail4 ,all this looks pretty mathematical1 but as a matter of fact there is much physical content1 because systems which can be described by linear equations of the second order have definite physical properties4 ,the requirement that the equations be linear corresponds to one of the most fundamental properties of our universe--the causality concept would not be possible or would be much modified in a universe governed by non-linear equations1 for the joint effect of two causes acting together would not be the sum of their effects acting separately1 so that the analysis of a situation into simple elements would be impossible and the causality concept probably would not have arisen4 ,furthermore1 an equation of the type of ,poisson of the second order means that there are propagation phenomena1 and equations of mechanics of the second order involve the existence of a scalar energy function4 ,if1 then1 the behavior of the universe can be described by differential equations at all1 these equations must be linear of the second order if the universe is to have the broadest physical characteristics of our own universe4 ,what ,einstein really did1 therefore1 was to demand that even when space-time is warped by the presence of a gravitational field1 those physical phenomena which can be described in terms of differential equations continue to be described by linear differential equations of the second order2 that is1 that nature continues to be describable in terms of a causality concept1 with propagation phenomena1 and a simple energy function4 ,the consequences of a guess like this about the properties of nature appeal to our physical intuition as being worth following out1 and of course we know the experimental justification4 ,several general comments may now be made on the structure reached in this way4 ,in the first place1 the whole structure is only descriptive in character2 we find certain correlations in nature which we describe with considerable completeness in mathematical equations1 without introducing any new element of explanation or of mechanism4 ,we have seen that as we increase our range from the realm of ordinary phenomena to phenomena of different character we arrive at a stage where for a time the process of explanation apparently halts1 and we have to be satisfied with a statement of mere correlation between elements2 later1 however1 these elements may be accepted as the ultimates in a broadened scheme of explanation1 and the explanatory process resumed4 ,are we at such a stage now with the general relativity theory1 and may later a new scheme of explanation be established based on the correlations of ,einstein8 ,this is of course a matter of individual judgment2 ,i personally question whether the elements of ,einstein's formulation1 such as curvature of space-time1 are closely enough connected with immediate physical experience ever to be accepted as an ultimate in a scheme of explanation1 and ,i very much feel the need for a formulation in more intimate physical terms4 ,in the second place1 we must repeat the comment already made in discussing time1 namely1 that there is still a very wide gap between the theory and its physical application1 in that we have no way of identifying our physical clocks and our physical measures of time with the thing called time in the formulas4 ,this gap must be filled by a specification of the physical structure of a clock4 ,it has always been very puzzling to understand why ,einstein has so strenuously insisted that the shift toward the infra-red is an integral part of the general theory1 and that if the shift is not found1 the theory must fall4 ,in other words1 ,einstein insists that the assumption that an atom is a clock is an integral part of his theory4 ,i believe that this attitude may be due to a realization by ,einstein of that very flaw in the logical structure which we are now emphasizing4 ,in the absence of any method of specifying the details of construction of at least one clock1 relativity becomes a purely academic affair1 unless there exist in nature concrete things which may serve as clocks4 ,einstein must either be able to tell how to construct a clock1 or else be able to point to a specific example of a clock4 ,he chose the atom as the specific thing4 ,doubtless the reason was the apparent simplicity of the vibrating mechanism of an atom1 as shown by the precise equality of the frequencies emitted by all atoms of the same element4 ,if the atom is not a clock1 where in nature can one be found8 ,but in the last few years we have come to appreciate the exceedingly complicated quantum structure of an atom1 and ,einstein's thesis loses much of its instinctive appeal4 ,since ,einstein created the theory of relativity1 it is perhaps ungracious to question his right to stipulate that the assumption that the atom is a clock is an integral part of the theory4 ,this1 however1 degenerates to a mere matter of language1 and does not touch the arbitrary nature of the procedure4 ,it does not prevent us from having a second brand of relativity theory1 that of ,x instead of ,einstein1 exactly like that of ,einstein except that perhaps now the clock is constructed in terms of the life period of a radioactively disintegrating element4 ,the only way to eliminate the arbitrariness seems to be to postulate that all natural processes1 which run naturally of themselves independently of what we may do1 may equally well serve as clocks and give the same results4 ,but in answering the question of the operational meaning of independently of what we may do we shall effectively have to answer the question of what is a clock4 ,this point of view may possibly1 however1 get us a little nearer to our goal of finding how to specify the structure of a clock4 ,finally1 the general theory is not completely general1 but applies only to a certain range of phenomena1 just as we saw that the special theory does not embrace all optical phenomena4 ,the general theory applies only to those phenomena which can be described in terms of differential equations1 that is1 par excellence1 to large scale phenomena4 ,if quantum phenomena cannot be described by differential equations1#cj as apparently now they cannot1 general relativity cannot by its very nature be applicable4 ,general relativity does not give us a comprehensive formulation of the behavior of all nature1 and as far as we can see1 we are still as far as ever from such a general formulation4 ,footnote #cj3 ,this statement now takes on a very questionable aspect in view of the new quantum wave mechanics 7,march1 #aibg74 ,,rotational ,,motion ,,and ,,relativity ,physically there is a great difference between the behavior of systems in uniform relative rectilinear motion and those in uniform relative rotation4 ,the special theory of relativity states that there is a triply infinite number of systems with all possible uniform rectilinear velocities with respect to each other1 in all of which physical phenomena have exactly the same mutual relations1 that is1 natural laws are the same4 ,now the mere formulation of the principle suggests the sense in which system is here used4 ,it is obvious that system refers only to a part of the universe2 we are not making a hopelessly academic statement about what would happen if we had an infinite number of universes to experiment with1 but are talking about operations that may be approximately realized in our own universe4 ,the system of the formulation we may think of as a completely equipped laboratory1 out in empty space1 so far from the heavenly bodies that they can have no effect4 ,the different systems of the formulation are different laboratories1 all built to exactly the same architectural blue prints4 ,the phenomenon to which the postulates of relativity apply are phenomena which pertain entirely only to one or another of these laboratories4 ,the meaning of this restriction is not completely definite and has1 in any special case1 to be judged partly by the context4 ,obviously1 to see from the window of a laboratory another laboratory passing with a certain velocity cannot be counted as one of the allowed phenomena4 ,still less is it one of the allowed phenomena to observe that the center of gravity of the entire stellar universe has a certain velocity of translation with respect to the laboratory4 ,the special principle of relativity contains by implication therefore the statement that certain very large and important classes of physical phenomena may be isolated and treated as taking place unaffected by the rest of the universe4 ,granted now the possibility of isolation1 we have a second statement1 which is usually treated as if it were the entire statement of the restricted principle1 namely1 that there is a triply infinite set of systems in which these phenomena run in the same way independent of the relative motion of the systems with respect to each other4 ,when once the significance of the observation is grasped that absolute motion has no meaning in terms of operations1 we see that this last statement takes immediately a most simple and satisfying aspect1 in fact1 so simple and inevitable that we are inclined to see in this the complete essence of the situation and regard the meaninglessness of absolute motion as affording peremptory proof of the restricted principle4 ,with this bias we now turn to examine the facts of rotary motion1 and are disconcerted to find them quite different4 ,no meaning in terms of measuring operations can be given to absolute rotary motion any more than to absolute translation1 but nevertheless phenomena are obviously entirely different in different systems in relative rotary motion 7phenomena of rupture1 for example71 so that apparently there are physical phenomena by which the concept of absolute rotary motion might be given a certain physical significance4 ,given two worlds like our own in empty space1 but surrounded by impenetrable clouds1 and each provided with a ,foucault pendulum1 then we believe that it is physically possible that we may find on one of these worlds the plane of rotation of the pendulum gradually changing in direction1 while on the other it remains stationary4 ,this difference we regard as possible without other accompanying physical phenomena which are causally related to the rotation of the pendulum 7of course we have to make the two worlds of infinitely rigid material and eliminate other phenomena which we regard as purely incidental71 so that we apparently have here a contradiction of our cardinal physical principle of essential connectivity4 ,we are certainly not inclined to give up our principle1 and we believe that as a physical fact1 if the clouds could be evaporated1 an observer in one world would find that he was rotating with respect to the system of the fixed stars1 whereas the corresponding observer on the other world would find that he was stationary4 ,our principle of essential connectivity is therefore maintained1 in that the rotation of the plane of the pendulum is connected with a rotation with respect to the rest of the universe of the entire world in which the pendulum is mounted4 ,as far as ,i am aware1 no other way of maintaining our principle has ever been suggested4 ,but this demands that we give up our physical hypothesis of the possibility of isolating a system4 ,there is here no question of limiting behavior2 we believe that no matter how far our rotating world gets from the rest of the universe the ,foucault pendulum would always behave in the same way2 the system can never be isolated1 but such local phenomena as the invariance of the plane of the pendulum are always essentially determined by the rest of the universe4 ,if now our system cannot be isolated1 we must return to the phenomena of translational motion4 ,in principle the act of isolation cannot be performed1 the rest of the universe cannot be disregarded1 and we should expect that different states of translational motion as well as different states of rotational motion with respect to the rest of the universe would have an effect on phenomena4 ,we set ourselves the problem of understanding this apparent enormous difference between phenomena of translation and rotation4 ,we remark that what apparently is a difference in principle may1 in virtue of the approximate character of all measurement1 be only a difference in magnitude1 and that translational effects may exist too small to detect4 ,a physical basis for such a difference may be found in the enormously different numerical values of translational and rotational velocities with respect to the rest of the universe attainable in practice4 ,in describing phenomena of cosmic magnitude1 we may plausibly measure the phenomena in units commensurable with the scale of the phenomena4 ,thus in measuring linear distances1 we may perhaps choose as the unit of length the diameter of the stellar universe1 and in measuring rotation1 a complete reversal of direction with respect to the entire universe4 ,this last means a change of angular orientation of #b 1 the first means a length of the order of #ajf light years4 ,measured in such cosmic units the angular velocities attainable in practice are incomparably greater than linear velocities4 ,we now see that it is possible that the real state of affairs is as follows3 namely1 phenomena in any system are affected by motion with respect to the entire universe1 whether that motion is of translation or of rotation1 and the magnitude of the effect is connected with the velocity of the motion by a factor which is of the general order of unity when velocity is measured in cosmic units4 ,this last is merely an application of the argument so often made in physics as to the order of magnitude of unknown numerical factors1 and will be found expanded on page #hh of my book on ,dimensional ,analysis4 ,the linear velocities attainable in practice are now so exceedingly low that their effect has not yet been detected experimentally1 but angular velocities are high1 and the effect is easily demonstrable4 ,in this light the special principle of relativity is no different in character from any other physical law2 it is only approximate1 and some day our measurements may become refined enough to detect its limitations4 ,we have made a hypothesis here1 which we may call the hypothesis of the immanence of the entire universe1 namely1 that isolation is impossible1 or that the rest of the universe1 no matter how distant1 always has a local effect on at least some phenomena4 ,this is essentially the hypothesis of ,mach1#ca and leads to a situation which can1 ,i think1 be contemplated with logical equanimity1 although it has always seemed to many physicists most highly antiphysical in character4 ,footnote #ca3 ,e4 ,mach1 ,the ,science of ,mechanics1 translated by ,mc,cormack1 ,the ,open ,court ,publishing ,co41 ,chicago1 #ahic4 ,see especially p4 #bce4 ,it must certainly be admitted that most physical experience justifies us in thinking that effects may be made as small as we please by getting far enough away from the cause of the effect4 ,but if we accept the considerations of the preceding pages1 we must be prepared to admit that as phenomena change in range their character may change1 and that in these new realms we must1 at first at least1 be satisfied with a mere statement of correlations4 ,certainly we have very strong physical evidence of a formal correlation between the ,foucault pendulum and the rest of the universe4 ,but a correlation of this sort may be without significance because of its very breadth2 we never can prove the significance of the correlation by performing an experiment with the rest of the universe absent4 ,have we really done anything more than merely get things into such a formal situation that they cannot be assailed1 a possibility which the mere laws of our thinking seem always to leave open1 as has been suggested1 or is there any physical content to what we have done8 ,we have seen that if our correlation is also suggested by other phenomena1 then we may accept it as having physical content4 ,now there is just a glimmer of a suggestion that our hypothesis of the immanence of the universe may be needed in other ways4 ,the gravitational constant and the velocity of light are always treated as arbitrary magnitudes thrust on the universe from outside with no connection with other phenomena4 ,nevertheless1 ,i suppose that no one regards this situation as ultimately satisfactory and does not entertain the hope that some day we may be able to give some sort of account of the numerical magnitude of these constants4 ,we have not hitherto succeeded in finding any connection between these constants and small scale phenomena such as the charge on the electron1 its mass1 etc41 so that there is some plausibility in expecting that a connection may be sometime found with cosmic things2 indeed general relativity theory already prepares us for exactly this possibility4 ,now the velocity of light and the gravitational constant control small scale experiments1 for of course these two constants can be measured by local experiments1 so that if the cosmic connection is found1 we should have a control of local behavior by cosmic things1 and therefore another example of the immanence of the entire universe4 ,there is no need for me to waste time in apologizing for the highly speculative character of all this4 ,it is worth while to emphasize1 however1 that our general considerations on the meaning of explanation have prepared us to admit as reasonable just the sort of explanation contained in the hypothesis of the immanence of the universe1 and therefore to reserve a place in our physical thinking for possibilities of this sort1 in spite of the fact that such considerations are not usually entertained1 and may seem to many opposed to the spirit of physics4 ,,quantum ,,concepts#cb ,the history of quantum theory up to the present is a repetition in many respects of that of the early theories of electricity1 in that all our thinking has been in mechanical terms4 ,as far as we now know1 quantum phenomena are always associated with atoms4 ,we make for the atom a mental model with all the properties of the mechanisms of the ordinary scale of magnitude and with a few impressed properties in addition which represent the new quantum relations4 ,as we now think of it1 the atom has a massive core about which electrons revolve under an inverse square law1 the connection between the mass of the electron1 its acceleration1 and the force acting on it being that usual in ,newtonian mechanics4 ,footnote #cb3 ,this section was written early in #aibf without access to recent literature4 ,our attitude toward quantum phenomena has been so much changed since then by the new quantum mechanics1 that a number of the following statements are superseded as a statement of present opinion4 ,however it has seemed worth while to let the section stand as written1 because many of the developments actually taken in the new mechanics follow the lines that it is here urged they ought to take1 and in so far afford interesting confirmation of the point of view of this essay4 ,the space in which the electron circulates is thought of as ,euclidean1 and the motion is described in time1 which may be measured with clocks in the usual way4 ,the general equations of electrodynamics do not apply2 there are no propagation effects inside the atom1 the motion of the electrons does not produce a magnetic field1 and there is no radiation when the electron is in one of its possible stable states1 in spite of the acceleration4 ,we may1 if we please1 in working out the character of the motion1 entirely neglect the electrical origin of the inverse square law1 and treat this merely as an impressed force without further implications4 ,superposed on the ordinary spatial1 temporal1 and mechanical characteristics of the model are additional quantum properties1 one which determines the particular orbit in which the electron moves pdq nh1 and another which determines the frequency of the radiation emitted when the electron passes from one allowed orbit to another4 ,no mechanism is suggested to account for these quantum conditions1 although the conditions are formulated in mechanical terms4 ,we now have to ask what is the meaning in terms of operations of our usual concepts of space-time and mechanics when applied to phenomena of this order4 ,it is of course evident1 as has already been emphasized1 that the concepts have entirely changed in character1 because we do not measure an electron orbit1 for example1 by stepping off the diameter with meter sticks1 or by measuring the time required for light to travel across the diameter4 ,the particular feature of immediate interest in this changed situation is the change in number of our concepts on the atomic level4 ,i shall not attempt to find by an exact analysis the number of independent concepts at this level2 probably such an analysis is not possible4 ,we may1 however1 make an approximate suggestion4 ,apparently the most important concept in describing relations inside a quantum system corresponds to that of energy on the ordinary scale4 ,changes of energy determine the frequency of emitted radiation1 as well as the relations during collisions of atoms and electrons2 these collisional relations make direct connection with experiment through the voltages applied to electrons in collision experiments4 ,the analogue of the momentum concept also seems to have independent significance1 as shown by the ,compton effect4 ,the frequency of emitted radiation is also something with independent experimental significance4 ,i believe that these three things are all that have direct significance for quantum experiments made up to the present time4 ,in any event1 it is perfectly evident that on the quantum level the concepts which at present have operational significance are considerably fewer than on the level of ordinary experience4 ,apart from the question of convenience1 there may be justification in continuing to use our old mechanical forms of thought if new experimental relations are thereby suggested4 ,that a very large number of such as yet undiscovered relations may be suggested in some such way is at once evident4 ,thus we have no present knowledge of any phenomenon associated with what the electron does when passing from one energy level to another4 ,how long does it take to make the passage8 ,what is its path during passage8 ,is it subject to the ordinary laws of electrodynamics during passage8 ,when and where is the radiation emitted that corresponds to passage8 ,when the electron leaves one stable orbit is the orbit on which it will eventually land already determined8 ,does the radiation train emitted during a change from one energy level to another have a definite length in space1 or may it have a variable length and correspondingly something that corresponds to variable amplitude8 ,what happens to the radiation when the electron passages are interfered with before the emission of a quantum has been completed8 ,what is the mechanism by which the quantum conditions are imposed8 ,is it not possible that part of the clew to the riddle of the manner of transition from purely quantum behavior to the behavior of classical mechanics may be found in the behavior of the electron during passage from one energy level to another8 ,certainly we have a tendency to the classical behavior under those conditions1 such as at high temperature or in strongly condensed systems1 in which the time occupied in passage might be expected to become a more important part of the total time4 ,corresponding to these questions there should be many as yet undiscovered phenomena1 and the mechanical point of view therefore has its value in suggesting experiments to detect such effects4 ,it is of course too early to see what the final result will be here2 we cannot tell whether eventually enough new experimental kinds of behavior will be found to restore the number of independent concepts to that of the level of ordinary experience or not1 or whether indeed it will turn out that a greater number of concepts is required4 ,it is contrary to our instincts to expect a greater number1 and a smaller number now seems to us not unnatural1 but the considerations of this essay should prepare us for either possibility4 ,it is often said that quantum phenomena are inconsistent with ordinary mechanics1 and proofs of this assertion are often offered4 ,i believe that no such proof1 in the spirit in which the attempt is usually made1 can be correct1 for it seems to me that the remark of ,poincar applies1 namely1 that any sort of behavior can be imitated by a mechanical system1 provided it is only complicated enough4 ,a peremptory proof of this can be given to any one who is not a believer in vitalism4 ,if a sentient being can be regarded as a mechanical system1 we merely have to station inside each atom a ,maxwell demon1 with instructions to make the atom react according to quantum rules4 ,opposed to the spirit of this sort of reduction of quantum phenomena to mechanical terms1 we have to remember that it makes sense to talk about the character of our conceptual structure only when the number of concepts is reduced to the number that have independent operational significance1 that is1 to the minimum number4 ,in the meantime let us examine what may be the significance in the light of present experiment of statements like those ascribed to ,bohr that our usual concepts of space and time may be inapplicable in dealing with quantum phenomena4 ,this idea is often given the more explicit form that space and time may be essentially discontinuous at the quantum level4 ,from the operational point of view1 it is most difficult to see exactly what this more explicit statement means1 at least in terms of those operations by which length and time were originally defined4 ,thus if space were discontinuous1 it might mean that a point exists which may be reached by laying off a meter stick fourteen times1 for example1 and another point by laying off sixteen times1 but that no point can be found with fifteen applications4 ,such a state of affairs seems to be inconsistent with our definition of the counting operation and to have no concern with any properties of space2 for what shall we mean by laying off a meter stick sixteen times if it cannot be laid off fifteen times8 ,it is conceivable that space might end1 in the sense that beyond a certain limit there might be some irremovable physical hindrance to the continued laying off of distances with a meter stick 7although ,i think that we should be inclined to describe such a state of affairs in terms of matter enclosing empty space rather than as the end of space71 but to say that space may be discontinuous seems to be meaningless4 ,in the same way1 ,i believe it meaningless to speak of discontinuous time4 ,we may have phenomena discontinuous in space and time1 but not discontinuous space or time4 ,it seems then that we must give up the idea that in the quantum domain the usual concepts of space and time may fail1 in the specific sense that they may become discontinuous4 ,what may we understand by the failure of these concepts in a more general sense8 ,no one of course would expect that even eventually the concepts will have the same operational significance for the inside of an atom that they have on the ordinary scale2 it must be a modified sort of concept with which we are concerned1 such as we have already seen is given by the field equations of electrodynamics4 ,if now the number of operationally independent concepts on the quantum level turns out to be the same as on the level of ordinary experience1 and if there is also the possibility of continuous transition from the operations of the quantum domain to those of ordinary experience1 then it seems to me that we should say that our usual concepts of space and time still apply in the quantum domain4 ,but if the number of operationally independent concepts is either greater or less than on the ordinary level1 then ,i believe we must say that the ordinary concepts of space and time cannot apply4 ,one might still look for the possibility of separating out from the complex of concepts on the quantum level a group which might change continuously to those of space and time on the ordinary level1 but ,i think that such a possibility is very remote when one considers that the total number of concepts changes1 and that in the zone where the number changes the definitions are not unique by which one extrapolates a concept from one domain to another4 ,if ,bohr's idea is true that space and time cannot be used in describing ultimate quantum phenomena1 one of the most immediate implications in terms of experiment might be that phenomena corresponding to intermediate positions of the electron between stable orbits do not exist4 ,finally1 we must comment on the general tactics of the quantum situation4 ,it would seem that there have already been a sufficient number of unsuccessful attempts to formulate quantum behavior in terms of ordinary mechanics to justify the expectation that ultimately something quite different must evolve4 ,the difficulties of an unmodified carrying over of ordinary mechanical notions to quantum phenomena may be illustrated by a simple example4 ,consider a particle of mass m rotating in a frictionless circular track of radius r4 ,then according to quantum conditions it can move stably on this track only with certain definite velocities1 such that pdq mv #br nh4 ,suppose now the particle rotating with one of the allowed velocities1 and a tangential force applied4 ,if the usual mechanical notions of force are still valid1 the particle must respond by moving in its track with continually increasing velocity4 ,after the velocity has been increased by a small amount1 we remove the force4 ,the motion is now no longer one of the allowed ones1 and the particle must in some way change its velocity2 it must either slow down or speed up4 ,in the first case it must either radiate energy1 which a system of the simple mechanical properties we have supposed is not capable of doing1 or else the law of conservation of energy fails1 and also ,newton's first law of motion during the process of acquiring the steady condition4 ,if1 on the other hand1 the particle speeds up1 it must increase its energy from nowhere1 and again ordinary mechanics does not apply4 ,it seems then a mistake to attempt to formulate the quantum conditions in terms of the notions of ordinary mechanics 7momentum1 and position cordinates in either the ordinary or the generalized ,lagrangean sense74 ,it would seem1 on the other hand1 plausible to expect that mechanics is not a fundamental thing1 but is in some way an effect produced by the aggregate action of a great many elementary quantum processes4 ,amplitude of radiational vibration1 for example1 may be such a statistical aspect of a great many processes1 in some such way as on the ordinary level of experience temperature is a statistical aspect of the average kinetic energy of the atoms4 ,one possibility of this kind has already been more explicitly indicated2 in the elementary process of emission of radiation1 frequency and energy are not two independently assignable variables1 but are connected ,e h4 ,that is1 on the quantum level radiation has only a single property1 which is properly neither energy or frequency4 ,we are now neglecting the polarization aspect of radiation4 ,on a higher level1 that of ordinary radiation1 the single elementary property has expanded itself into two 7energy and frequency7 through the additional variable of the number of elementary quantum processes in the complex radiation4 ,the program of the immediate future should be an extension of something of this sort1 namely1 to invent new concepts corresponding to the experimentally independent things on the quantum level 7such perhaps as the resultant of the fusion of the energy and frequency concepts for radiation71 and then to show how the ordinary concepts of mechanics 7and very likely those also of space and time7 are generated by statistical effects in aggregates of great numbers4 ,perhaps it is yet too early for an attempt of this sort1 because it may seem that there are still too many possibilities of new experimental discoveries which might upset the results of elaborate theoretical speculation4 ,if this should really be felt to be the case1 ,i believe that physics ought for the present to hold in partial abeyance its theoretical activities in this field1 and devote itself to acquiring as rapidly as possible the necessary experimental facts4 ,we may emphasize again that the possibility of carrying out this plausible program can be proved only by experiment2 it may be that more concepts will be required on the quantum level than for ordinary experience4 ,the invention of new concepts is certainly not an easy thing1 and is something which physics has always deliberately1 and perhaps justifiably1 shirked1 as shown by the persistent attempts to carry the notions of mechanics down into the finest structure of things4 ,this shirking has not had bad results1 but on the contrary good results1 as long as physics has been primarily concerned with phenomena near the range of ordinary experience1 but ,i believe that as we get farther and farther away from ordinary experience1 the invention of new concepts will become an increasing necessity4 ,,chapter ,,iv ,,special ,,views ,,of ,,nature ,,in this last chapter we propose to discuss certain special hypotheses about the structure of nature1 and certain other matters that could best be left until we had examined our fundamental concepts4 ,we have seen that in setting up the general rules which are to guide us in describing and correlating nature1 we have to take extreme care to allow no special hypotheses to creep in1 as otherwise we might be restraining possible future experience4 ,even here there is no hard and fast line of separation of the general from the special1 and one might entangle himself in inextricable difficulties if his ideals were too meticulous4 ,how for example is the critic to be answered who says3 ,your very endeavor to formulate principles so broad as not to restrict future experience means1 when examined in the light of operations1 that you are seeking for principles which past experience suggests will not limit the future4 ,it is in the very nature of things impossible to escape all the implications of past experience and therefore to find any completely general principle4 ,i believe that we must admit the critic is right1 and that rigorously our goal is impossible of attainment4 ,we may say in partial self-defense that all the discussion of this essay has been subject to one explicit assumption1 namely1 that the working of our minds is understood1 which of course involves the assumption that our minds continue to function in the future in the same way as in the past4 ,even with this proviso we can not rigorously avoid the implications of the past1 but there can be no practical question that we recognize certain assumptions about the behavior of nature to be so special as to limit seriously the physical possibilities1 and other assumptions to be less restricting4 ,in the previous discussion we had to make assumptions1 but ,i hope these assumptions will be recognized by all with physical experience to be so broad as not to restrict us seriously4 ,more special assumptions or hypotheses have their very great use1 however1 when we attempt to push forward the domains of experimental knowledge1 because they may suggest new experiments or aid in correlating information already obtained4 ,these special hypotheses may cover a very wide range of generality2 some of them are general enough in character to be discussed here4 ,among these special hypotheses there is a group which play an important part in the speculation of most physicists1 and which have features in common4 ,these are3 the hypotheses of the simplicity of nature1 of the finiteness of nature in the direction of the very small1 and of the determinateness of the future in terms of the present4 ,that these views have points of similarity is obvious if we consider a hypothetical special case4 ,suppose that no physical structure beyond the electrons and protons can be discovered1 or is even suggested by any known phenomenon1 so that the entire future behavior of a system can be determined by a specification of the present relations of all its protons and electrons2 in this case nature would be both simple and finite and the future determined by the present4 ,,the ,,simplicity ,,of ,,nature ,of these hypotheses1 perhaps the most important is that of the simplicity of nature1 because of its wide spread diffusion and the effect it has had on physical thought4 ,the hypothesis of simplicity assumes several forms2 some physicists are convinced that the laws which govern nature are simple1 others that the ultimate stuff of which nature is composed is simple 7perhaps protons and electrons and energy71 or there may be a combination of both views into the belief that ultimately we shall find simple ultimate elements behaving according to simple laws4 ,in one respect it is obvious that nature is not simple1 namely numerically--try counting the electrons or atoms or stars6 ,consider now the first of these aspects of the thesis of simplicity1 which may be expressed as the conviction that the behavior of the entire universe can be comprehended in a few principles of great breadth and simplicity1 such as the inverse square law of force1 or the second law of thermodynamics1 or perhaps still better the equality of the elementary positive and negative charges1 which apparently holds to an enormous degree of precision4 ,in explanation of a view like this there is in the first place the mental urge1 because we can take a satisfaction almost sthetic in contemplating such a universe1 and there is in the second place a strong suggestion from experience4 ,practically all the history of physics is a history of the reduction of the complicated to the simpler4 ,for example1 the behavior of a large part of the world of immediate experience can be reduced to the simple laws of mechanics4 ,the behavior of another very large group of natural phenomena can be reduced to thermodynamics4 ,the behavior of the heavenly bodies1 which at first was described in a rather complicated way in the ,ptolemaic system of astronomy1 can be reduced to those same laws of mechanics which we find in our immediate neighborhood1 with the one addition of the universal law of gravitation1 which later refined experiment discloses is really active in our immediate surroundings4 ,similarly the laws of thermodynamics 7except that part dealing with radiation7 are reduced to the ordinary laws of mechanics through the additional assumption of the atomic structure of matter4 ,truly a stupendous accomplishment that may well color our whole future outlook4 ,one may find great justification here for the belief that all nature will ultimately be reduced to a similar simplicity1 and1 in particular1 justification for the attempt to find the explanation of all nature in the action of mechanical laws4 ,now1 of course1 as a matter of physical and historical fact1 this program could not be carried through1 but obdurate physical phenomena were discovered4 ,electric phenomena1 which at first seemed so promising1 refused to fit into the scheme1 and the converse attempt1 to explain mechanical effects in terms of electrical effects1 also failed4 ,we still carry our ordinary mechanical notions down into the realm of small electric effects1 and still talk1 for instance1 about non-electrical forces which hold an electron together4 ,nor are there experiments affording sufficient basis for believing that all the mass of a positive nucleus is electrical in character4 ,we also think of electrical charges as having the property of identifiability1 which involves the possession of sharp edges and a change in the law of force at small distances1 and this is certainly a property carried over from our large scale experience4 ,it seems fairly evident then that the laws of nature cannot be reduced to either those of mechanics or of electricity1 nor probably1 as is suggested by quantum phenomena1 to a combination of both4 ,this of course does not preclude the possibility that the laws still may be simple when expressed in other forms4 ,an example of such a broad general law that goes deeper than mechanics or electrodynamics is probably afforded by the second law of thermodynamics when extended to include radiation phenomena4 ,examples of attempts to find other such simple laws are ,tolman's ,principle of ,similitude1#cc and ,lewis's theory of ,ultimate ,rational ,units1#cd and his recently enunciated principle of ,complete ,reversibility4#ce ,the first two of these attempts ,i do not believe are successful1 for reasons ,i have stated elsewhere1#cf the third also seems somewhat doubtful4 ,footnote #cc3 ,r4 ,c4 ,tolman1 ,phys4 ,rev4 #c1 #bdd-bee1 #aiad2 #f1 #bai-bcc1 #aiae2 #ae1 #eba1 #aibj2 ,jour4 ,amer4 ,chem4 ,soc41 #dc1 #hff-hge1 #aiba4 ,footnote #cd3 ,g4 ,n4 ,lewis4 ,vol4 #ae1 #aiba of the ,contributions from the ,jefferson ,physical ,laboratory1 dedicated to ,professor ,hall1 ,cambridge1 ,mass42 ,phil4 ,mag41 #di1 #gci-gej1 #aibe4 ,footnote #ce3 ,g4 ,n4 ,lewis1 ,proc4 ,nat4 ,acad4 ,sci41 #aa1 #agi-ahc1 #dbb-dbh1 #aibe4 ,footnote #cf3 ,p4 ,w4 ,bridgman1 ,phys4 ,rev4 #h1 #dbc-dca1 #aiaf2 ,dimensional ,analysis1 p4 #aje4 ,with regard to the general question of simple laws1 there are at least two attitudes2 one is that there are probably simple general laws still undiscovered1 the other is that nature has a predilection for simple laws4 ,i do not see how there can be any quarrel with the first of these attitudes4 ,let us examine the second4 ,we have in the first place to notice that simple means simple to us1 when stated in terms of our concepts4 ,this is in itself sufficient to raise a presumption against this general attitude4 ,it is evident that our thinking must follow those lines imposed by the nature of our thinking mechanism3 does it seem likely that all nature accepts these same limitations8 ,if this were the case1 our conceptions ought to stand in certain simple and definite relations to nature4 ,now if our discussion has brought out any one thing1 it is that our concepts are not well defined things1 but they are hazy and do not fit nature exactly1 and many of them fit even approximately only within restricted range4 ,the task of finding concepts which shall adequately describe nature and at the same time be easily handled by us1 that is1 be simple1 is the most important and difficult of physics1 and we never achieve more than approximate and temporary success4 ,consider the example of time4 ,the original concept of local time1 which for long seemed satisfactory1 turns out to be inadequate1 and has to be replaced by extended time1 which is so complicated that it is questionable whether we shall ever be able to grasp it with the confidence that we must demand in a useful concept 7by grasp ,i mean intuitive command of all the implications of the operations which are involved74 ,the concept has not yet been found which describes simply the temporal relations of the universe4 ,not only are concepts hazy around the edges and so incapable of fitting nature exactly1 but there is always the chance that there are concepts other than those which we have adopted which would fit our present phenomena4 ,finding concepts to fit nature is much like solving a cross-word puzzle4 ,in the puzzle there may be some parts of the pattern which we fill completely and easily1 but sometimes we find parts in which we can fill in everything except one or two obstinate definitions1 so that we are sure we are on the right track1 and rack our brains for the missing words1 when with a flash of inspiration we see that the obstinate words can be fitted in by a complete change in those which we had already accepted4 ,it may be that we are soon to witness a similar change in our concept of the nature of light4 ,an important difference between the cross-word puzzle and nature is that we can never tell when we have filled in all the squares in any of the parts of nature's puzzle2 there is always the possibility of new phenomena which our present scheme does not touch4 ,considering1 then1 the nature of our conceptual material1 it seems to me that the overwhelming presumption is against the laws of nature having any predisposition to simplicity as formulated in terms of our concepts 7which is of course all that simplicity means71 and the wonder is that there are apparently so many simple laws4 ,there is this observation to be made about all the simple laws of nature that have hitherto been formulated2 they apply only over a certain range4 ,we have not extended the laws of gravitation to small bodies1 nor have we found that our electrical laws will work on a cosmic scale4 ,it does not seem so very surprising that over a limited domain1 in which the most important phenomena are of a restricted type1 the conduct of nature should follow comparatively simple rules4 ,a tempting question is whether there may not be some laws of nature that are really simple1 without relation to our mode of formulation1 such as the law of the inverse square4 ,i leave it to the reader to decide whether this question has meaning4 ,in this connection it is possibly significant that the average physicist is strangely reluctant to tamper with the inverse square law4 ,i find in myself a lack of sympathy1 which ,i cannot justify by any of the considerations of this essay1 with attempts like the recent one of ,swann1#cg for example1 to explain a wide variety of hitherto obstinate effects by the assumption of slightly unequal departures from the inverse square law by the electrons and protons4 ,of course ,i hope that this feeling will turn out not to be prejudice1 but will perhaps be justified by some such general observation as that a departure from the inverse square law so slight as by definition to be forever beyond detection by direct experiment is meaningless2 but of this ,i am not at all sure4 ,footnote #cg3 ,w4 ,f4 ,g4 ,swann1 ,phys4 ,rev41 #be1 #bec1 #aibe4 ,we are now ready to consider the second respect in which nature may be simple1 namely1 because the material of which it is built may reduce to a few sorts of elements4 ,in this discussion it will be convenient to consider also at the same time the more inclusive simplicity arising from simple laws acting on simple elements4 ,the immediate question for us here is one of fact3 does nature seem to be getting intrinsically simpler as we get toward small scale phenomena8 ,there is much room for difference of opinion here2 personally ,i feel that this expected simplicity is not in evidence1 at least to the extent that we could desire4 ,for instance1 the fact that the electrons must have both electrical and mechanical properties is a straw in this direction4 ,it must also be remembered that a certain simulation of simplicity is inevitable as we approach the limits of experimental knowledge1 whatever the actual structure of nature1 for the mere reason that near the limit our possible experimental operations become fewer in number1 and our concepts fewer also4 ,the question which we are trying to answer has1 therefore1 its real meaning only in terms of the possible future4 ,do we believe that if we drive in our stakes at a certain point on our present frontiers1 this point will gradually1 as physics advances1 become possessed of a continually richer experience1 so that nature at this point will appear increasingly complicated8 ,or do we expect a termination of this process of expansion fairly soon8 ,it seems to me that as a matter of experimental fact there is no doubt that the universe at any definite level is on the average becoming increasingly complicated1 and that the region of apparent simplicity continually recedes4 ,this1 however1 is not the opinion of all observers4 ,thus ,bertrand ,russell1 in ,what ,i ,believe1 page #aj1 writes1 ,physical ,science is then approaching the stage where it will be complete1 and therefore uninteresting4 ,this is perhaps a particularly favorable epoch in the history of physics to urge the essential complexity of nature1 because all our new quantum phenomena indicate a vast wealth of hitherto unsuspected relations on the very edge of the attainable4 ,there is one aspect of quantum relations1 as also of our ideas of the nature of the structure of the nucleus of the atom1 which is particularly significant in this respect1 namely1 that we have to describe phenomena by statistical methods4 ,now a statistical method is used either to conceal a vast amount of actual ignorance1 or else to smooth out the details of a vast amount of actual physical complication1 most of which is unessential for our purposes4 ,there can be no doubt of the amount of ignorance that the statistical method conceals when applied to these phenomena1 but there are also strong indications1 particularly when applied to the nucleus1 that it covers a vast amount of actual physical complications4 ,the nucleus of a radium atom becomes unstable on the average every #ajd years1 which may be plausibly taken to indicate that every #ajd years the radium nucleus gets itself into some particular configuration4 ,considering the time scale on which we suppose events in the atom to take place1 and also considering the fact that radioactive disintegration seems unaffected by outside agencies1 this would indicate a perfectly appalling amount of structure4 ,we are similarly driven to statistical methods in quantum theory1 as for example1 in ,einstein's analysis of the details of equilibrium between emitting and absorbing atoms and radiation4 ,in general1 we cannot admit for a minute that a statistical method1 unless used to smooth out irrelevant details1 can ever mark more than a temporary stage in our progress1 because the assumption of events taking place according to pure chance constitutes the complete negation of our fundamental assumption of connectivity2 such statistical methods always indicate the presence of physical complications which it must be our aim to disentangle eventually4 ,it appears then that present experimental evidence makes very probable structures beyond the electron and the quantum2 we may go even further and say that there is no experimental evidence that the sequence of phenomena in nature as we go to ever smaller scales is a terminated sequence1 or that a drop of water is not in itself essentially infinite4 7,this statement contains by implications the meaning that we attach to infinite47 ,all the more1 then1 there is no evidence that nature reduces to simplicity as we burrow down into the small scale4 ,whatever may be one's opinion as to the simplicity of either the laws or the material structure of nature1 there can be no question that the possessors of some such conviction have a real advantage in the race for physical discovery4 ,doubtless there are many simple connections still to be discovered1 and he who has a strong conviction of the existence of these connections is much more likely to find them than he who is not at all sure they are there1 and is merely hunting for anything that may turn up4 ,it is largely a matter of psychology4 ,everyone knows that the mere suggestion that a problem has a solution1 or the knowledge that someone has already solved it1 is often sufficient to suggest a relation that otherwise might not have been noticed4 ,the chances are1 therefore1 that the relations between phenomena will be found by those who are previously convinced that the relations exist4 ,the observation that most of the discoveries are made by men with particular sorts of conviction naturally strengthens the belief that their convictions are true4 ,but this picture has an obverse side4 ,the man who is convinced that there is a relation where none exists may waste all his time in vain seeking for it4 ,granted that nature has no particular predisposition to simple relations1 the conviction that there are such relations is1 from the point of view of any one individual1 as likely to be a hindrance as a help4 ,from the point of view of physical society1 on the other hand1 it is desirable that there be such convictions1 for in such a society there will be more discoveries than in a society without such convictions4 ,we have here again the old conflict between the individual and society4 ,as in all other similar conflicts1 society will not be able to demand permanently from the individual the acceptance of any conviction or creed which is not true1 no matter what the gain in other ways to society4 ,if nature is not simple1 physicists will not continue to believe that it is1 even if such a conviction does increase the total number of discoveries4 ,it is an impossible attitude to expect that one can maintain4 ,does this then mean that physics is to face a drab future1 becoming continually more prosaic1 with new discoveries ever rarer1 made by a continually decreasing number of misguided but fortunate enthusiasts8 ,there may be such a danger1 but the greatest part of the danger is avoided if its nature is clearly recognized4 ,one of the problems of the future is the self-conscious development of a more powerful technique for the discovery of new relations without the necessity for preconceived opinions on the part of the observer4 ,there is an aspect here of our physical research that is often lost sight of1 namely1 the small proportion of successful discoveries compared with the number of investigators4 ,certainly the number of unsuccessful attempts1 even in the case of those fortunate individuals who make the great discoveries1 is very much greater than the number of their successful attempts4 7,faraday's reputed satisfaction with a return comes to mind47 ,this must always be taken into account in estimating the probable chances of correctness of any new theory4 ,with so many physicists working to devise new theories1 the chances are high that many false theories will be found1 in which a number of phenomena may apparently fit together into a new relation1 but which eventually prove to be inconsistent with other phenomena1 so that the proposed theory has to be abandoned4 ,as physics advances and the number of investigators and the amount of physical material increases1 one has to be more and more exacting in one's requirements of a new theory4 ,one must be particularly on guard against numerical coincidences4 ,an interesting chapter might be written on numerical relations which have been hopefully published1 but later had to be abandoned as without significance4 ,,determinism ,if we are right in supposing that physical evidence gives no warrant for the idea that nature is finite downward1 we have not only repudiated the thesis of simplicity but we have also made a very important observation on the other general thesis mentioned at the beginning of this chapter1 namely1 the thesis of physical determinism4 ,by determinism we understand the belief that the future of the whole universe1 or of an isolated part of it1 is determined in terms of a complete description of its present condition4 ,what we mean by present condition will be discussed later4 ,it is popularly assumed that every physicist subscribes to some such thesis as this4 ,but now if there is infinite structure even in a small isolated part of the universe1 a complete description of it is impossible1 and the doctrine as stated must be abandoned4 ,it seems to me that all present physical evidence prepares us to admit this possibility4 ,i suppose1 however1 that most physicists would subscribe to some modification of the original thesis1 perhaps along the following lines4 ,given a description of an isolated part of the physical universe in the most complete terms that have physical meaning1 that is1 down to the smallest elements of which our physical operations give us cognizance1 then the future history of the system is determined within a certain penumbra of uncertainty1 this penumbra growing broader as we penetrate to finer details of the structure of the system or as times goes on1 until eventually all but certain very general properties of the original system1 such as its total energy1 are forever lost in the haze1 and we have a system which was unpredictable4 ,i suppose that it is a further conviction of at least many physicists that by sufficiently refining our measurements1 the amount of haze at any fixed point in the future may be made indefinitely small1 and many might even go further and hope by studying the haze 7perhaps statistically7 to obtain some inferential evidence of structure beyond that yet experienced4 ,in fact it may be that this last contains the germs of the ultimate method of investigation1 if we ever reach a stage when we can no longer refine our methods of measurement4 ,determinism to the physicist is simply a way of stating certain implications of his conviction of the connectivity of nature4 ,we have seen that the broadest possible statement of the thesis of connectivity is3 ,given two isolated systems with identical past histories up to a certain epoch1 then the future histories will also be identical4 ,the thesis of the determinism of the future by the present constitutes a specialization of this general thesis in that we suppose that identity of all past history is not necessary for identity of future behavior1 but only identity of present condition4 ,the general and the special thesis are not equivalent by any means3 if past histories are identical then present conditions are also identical1 but the converse does not necessarily hold at all4 ,now ,i believe that the general thesis 7which ,i suppose all physicists will admit1 but whose truth is nevertheless subject to the verification of experience7 gets turned into the special thesis by a feeling of somewhat metaphysical content1 which we may perhaps state by saying that we can see no way by which the past can affect the future except through the present4 ,we do not like to think of the effect of a cause distant in the past jumping over the present and affecting the future without touching the present at all4 ,it is the analogue of that attitude of mind to which action at a distance in space is inconceivable2 just as it is difficult to conceive of a body here affecting a body there without in some way an action propagated through intermediate space1 so we do not like to think of a past cause jumping over time and producing a future effect without some sort of continuity in the causal chain through all intermediate time4 ,so far our discussion has been purposely loose3 it is evident that what we mean by present state is crying for definition4 ,what is meant by this may depend somewhat on the specific hypothesis that one adopts about the structure of nature4 ,historically the conviction of future determinism has been most intimately associated with a mechanical picture of the structure of the universe1 so that it may be well to begin from this point of view4 ,suppose the simplest possible system composed of point masses without structure1 as in the kinetic theory of gases4 ,what sort of specifications do we believe necessary to fix the present state of such a system8 ,the mechanical view of nature gives a definite answer4 ,by present state we mean the positions and velocities of all the masses4 ,this is sufficient for the complete determination of any purely mechanical system1 in which the forces between the elements are known functions of only their relative positions4 ,by a sort of extension of these ideas valid for mechanical systems1 it seems to be often thought that the present state of any system is determined by a complete specification of the positions and velocities of all the ultimate elements of the system 7provided always of course that this number is finite74 ,this principle1 however1 does not appear to bear the check of experiment when applied to electrical systems with radiation4 ,the theorems of the retarded potential show that such systems are determined by the present position and velocities of the charges in the immediate vicinity1 and by the corresponding data at remote points given for proper epochs in the past2 in this case1 therefore1 past and present history are necessary to determine the future4 ,but if we consider the electrical field as part of the system1 we may fix the future in terms of the present positions of the charges1 their velocities1 and the values of the field vectors all over space1 thus returning to a certain formal resemblance to mechanical systems1 and suggesting a reason for ascribing physical reality to the electric field4 ,this analogy with a mechanical system is1 however1 loose2 complete analogy would allow the instantaneous values of the time derivatives of the field to be given also1 and this is not possible4 ,how is it that velocity can strictly be regarded as characteristic of the present state of the system8 ,certainly the usual operations for measuring velocity demand that we know the configuration of the system at two different times1 and calculate the velocity from certain differences of the system at these two times4 ,the velocity is defined as a limiting result1 but even in the limit the essential physical fact does not disappear that we must know the positions of the system at two times4 ,we may now go further2 if the velocity is properly included in the present attributes of the system1 we can see no reason for not including a specification of all the higher time derivatives also4 ,in the case of the simple gaseous system under present consideration we can answer this question by examining the operations by which we actually go to work to determine the future of such a system4 ,the problem of determining the future condition of such a system reduces to the problem of writing the differential equations of motion of all its parts4 ,if the system is a mechanical system1 as in this case1 these equations are of the second order in the time derivatives of the position coordinates1 and also involve the forces1 which we suppose are known in terms of the relative positions of the parts of the system4 ,given1 then1 the positions and the way in which the forces depend on the relative positions of the parts1 the equations of motion can be written down for any configuration of the system1 and these equations may be integrated 7at least approximately7 in terms of the proper initial conditions4 ,now the only boundary conditions on a second order equation are the initial positions and velocities4 ,this is the reason that velocities have to be specified in giving the present condition of the system1 and that it is not necessary to give the higher derivatives4 ,apparently the reason why we instinctively include velocity among the present properties of the system is not because velocity is by its nature strictly a present property of the elements of the system1 but rather because our wide experience with mechanical systems has shown that as a matter of fact velocity is necessary in such systems to determine future motion4 ,but now if the equations of motion of the parts of the system are not those of mechanics1 they will in general be much more complicated in appearance and will involve higher derivatives of the time than the second4 ,suppose for the moment that the equations contain only derivatives and the mutual positions of the parts of the system4 ,then to integrate the equations and determine the motion we have to know the initial positions and the initial values of all the derivatives up to an order one lower than the highest which occurs in the equations4 ,the equations of motion of an electron are even more complicated than this1 in that the positions of distant parts of the system have to be given throughout an interval of time instead of merely an instant4 ,it would seem that the feeling that the present state of a system may be determined in terms of positions and velocities does not as a matter of fact apply to all the systems of our experience4 ,the discussion up to this point has been subject to the fundamental assumption that the behavior of the system is entirely determined if we can give the position of each part as a function of time4 ,this assumption is implicitly contained in ,einstein's formulation of the general principle of relativity1 namely1 that there is nothing more to a physical system than a set of space-time coincidences1 and that the system is fixed in terms of the space time cordinates of all its parts4 ,already in discussing the assumption of relativity we have indicated reasons for dissatisfaction with this as a means of reproducing all experience1 because in giving only the space-time coordinates of events we have entirely omitted the descriptive background of the equations1 which gives physical color to the system in question4 ,this discussion also assumes that a specification of the positions1 velocities1 and higher derivatives 7if necessary7 of the elements of the system is possible1 which amounts essentially to the assumption that the system contains only a finite number of elements4 ,now in view of the experimental fact that there is no reason for supposing that the structure of the universe is finite1 this conclusion must be modified1 but ,i do not believe that the necessary modification affects the essential argument4 ,in view of the possible infinite structure it would seem that we cannot expect more than that the future is determined by the present within a certain penumbra of uncertainty1 and this penumbra may be made less important by digging down deeper into the structure when specifying the present condition4 ,we have also slurred over the ambiguities in present condition when the system is spread over space4 ,probably a unique ascription of meaning to present is not possible for an extended system1 but at least one possibility is indicated by relativity theory4 ,imagine a staff of assistants distributed throughout space1 each equipped with clocks synchronized and set with the master clock by light signals in the conventional manner1 and each fully equipped with the necessary measuring instruments4 ,then what we mean at this point of the argument by present state of the system is the aggregate of all the information about the positions and velocities of the ultimate elements which ,i determine in my immediate vicinity at my origin of time plus the reports of similar observations made by all the assistants1 each local observation being made at the time origin of each local clock4 ,going back now to the main argument1 we have shown that the feeling that the present condition of the universe may be specified in terms of positions and velocities arose from experience with purely mechanical systems1 and that the more general formulation1 in which we add to the velocities the higher time derivatives1 applies only to systems in which the ultimate elements move according to differential equations of higher order than the second4 ,furthermore1 our analysis seems to have shown that systems in which there is radiation do not allow a determination of the future in terms of a present condition specified in terms such as these4 ,it seems1 however1 that the general principle of the determinism of the future by the present may be saved by a change in the definition of what we mean by the present condition of the system1 ridding it of its mechanical and other special implications1 and making more immediate connection with direct experiment4 ,let us understand by present condition of a system the aggregate of all information that can be obtained by any physical means whatever1 with any sort of physical instrument1 not attempting to get out of this analysis information about hypothetical ultimate physical elements1 with the proviso that the measurements are to be made now1 extending the concept of now to points distant in space in the way intimated above4 ,with such a general definition of the meaning of present we can now deal with systems in which there is radiation1 noticing that our assistant observers must be stationed throughout apparently empty space as well as in the neighborhood of matter4 ,that this does adequately cover the case of radiation is suggested by considering again the two systems of dark lanterns with screens and distant mirrors which we have previously considered1 in one system a light signal having been despatched #j4e second ago and in the other #a4e seconds ago4 ,our thesis demands that there be some present difference in these two systems1 because their future history is different1 in one of them a light signal arriving after the lapse of #a4e seconds1 and in the other after only #j4e second4 ,now there is a present difference as reported by our assistants1 for the assistant stationed half way between lantern and mirror reports in one system a flash of light on the side of a screen which is turned toward the lantern1 and in the other system on the side of the screen turned toward the mirror4 ,this more general point of view answers the question whether velocity may be regarded as a present attribute of the system1 for the parts of a system which are in motion have momentum1 and momentum may be detected by placing against such parts comparatively rigid members which will receive a minute deformation1 so that velocity has a meaning in terms of physical measurements made at a single instant of time4 ,there is a subtle and difficult question here1 namely1 whether in talking about operations of measurement we can ever get rid of temporal implications1 and therefore1 whether a condition of the system in which temporal implications remain can properly be described as present4 ,i shall not attempt to answer this question3 there must be some practically satisfying answer1 involving perhaps the physical analogue of differentials of different orders in mathematics1 short of carrying the analysis to such a degree of refinement that the concept of present becomes meaningless1 as we can see might easily happen4 ,with this enlarged understanding of what we mean by present state of the system1 it seems to me that physical evidence is now rather favorable to the view that the present determines the future1 subject to qualification about the penumbra1 at least as far as large scale phenomena are concerned4 ,it appears much more doubtful when we come to small scale phenomena1 and in particular it is doubtful whether the principle can be applied to the details of the quantum process1 and in fact it is not certain that it has meaning4 ,it is certain that if it is true an enormous amount of structure beyond any that has yet been detected is implied4 ,,on ,,the ,,possibility ,,of ,,describing ,,nature ,,completely ,,in ,,terms ,,of ,,analysis ,there is a certain thesis that is loosely related to the view that nature is finite downward1 namely1 that an explanation of the universe is possible in which we start with small scale things1 and explain large scale phenomena in terms of their small scale constituents1 the thesis1 in other words1 that all the properties of the large are contained in the properties of the small and that the large may be constructed out of the small4 ,some such thesis as this seems implied in the general attitude of many physicists4 ,let us examine the physical basis for this4 ,to maintain this thesis would demand that aggregates of things never acquire properties in virtue of their numbers which they do not already possess as individuals4 ,is this true8 ,consider1 for example1 the two-dimensional geometry on the surface of a sphere4 ,this is non-,euclidean4 ,is the geometry of the individual elements of the surface of the sphere non-,euclidean1 or do they acquire this property in changing scale8 ,is the kinetic energy of a number of electrons all moving together in such a way as to constitute an electric current the sum of the kinetic energies of the individual electrons1 or is there an additional term8 ,is the mass of an electron the sum of the masses of its elements8 ,a mathematical consideration is suggestive here4 ,those properties of a system which can be described in terms of linear differential equations have the property of additivity2 the effect of a number of elements is the sum of the effects separately1 and no new properties appear in the aggregate which were not present in the individual elements4 ,but if there are combination terms 7as in the electrical energy1 which contains the square of the field71 then the sum is more than 7or different from7 its parts1 and new effects may appear in the aggregate4 ,now of course the linear equation is of enormous importance in describing nature1 but many examples of systems with other types of equation can be found1 as that above for electromagnetic mass4 ,in expecting to find in nature such non-additive effects1 we need not commit ourselves at all to the view that nature is governed by differential equations1 but by analogy may expect similar effects if difference equations1 for instance1 should prove to be fundamental1 or even something beyond present mathematical formulation4 ,it is certainly very much easier to handle a system physically if the total action can be built up from that of its parts1 because the analysis which establishes the connection between the elements is easier to perform4 ,it is obviously easier to show that an explanation in such terms is correct1 because we have seen that explanation involves making experiments with representative elements absent or altered1 and it is easier to vary the small things than the large things4 ,those explanations which involve working from the small up will therefore be made first1 and will appear to be of disproportionate importance4 ,places where ,i look for an explanation from the large to the small are perhaps in accounting for the values of the gravitational constant and the velocity of light and in those phenomena which general relativity theory indicates may depend on all the matter in the universe1 as the ,foucault pendulum experiment4 ,we must1 of course1 also be prepared for such non-linear effects in the domain of unexplored quantum phenomena4 ,a ,,glimpse ,,ahead ,some of the general considerations of this essay may1 with considerable plausibility1 be expected to play a part in the future of both speculative and experimental physics4 ,the most important effect may be expected from the clearer recognition of the operational character of our physical concepts4 ,indeed during the writing of this essay there has been a very marked increase in emphasis on the necessity of understanding in terms of physical operations such fundamental concepts as that of the electron1 by the new quantum mechanics the mechanics of ,heisenberg-,born and ,schrdinger of #aibe-bf4 ,we are to expect then in the first place a more self-conscious and detailed analysis of the operational structure of all our physical concepts4 ,it has been beyond the scope of this essay even to begin to attempt a systematic and thoroughgoing analysis of this character4 ,this future analysis will show precisely how1 as we extend the range of experience1 the physical character of the operations changes by which we define our concepts1 as1 for example1 in mechanics the notion of force disappears at high velocity and is replaced perhaps by the notion of momentum4 ,in the region of change in the nature of our concepts1 special study will be made of the accuracy of our physical measurements1 and new experiments devised of greater accuracy1 in order that we may know precisely to what extent the new concepts are equivalent to the old4 ,past experience suggests that we may perhaps expect to find new phenomena especially in those regions where the difficulty of carrying out the usual operation forces us to change the operational character of our concepts4 ,there will be questions of a more or less formal nature to answer1 as for example the best way of extending concepts when there are several possible courses open to us4 ,we may expect more interesting results1 however1 when we get so far beyond ordinary experience that the character of the possible physical operations has become so restricted as to result in an apparent decrease in the number of independent concepts4 ,it seems plausible to expect that the structure of nature is more fundamentally connected with the number of independent concepts necessary for a complete description than with the precise details of the structure of the individual concepts1 such1 for example1 as whether space is measured optically or tactually4 ,in those regions where the number of concepts decreases1 we must make the most thoroughgoing experimental examination to discover if possible new sorts of operations by which the number of concepts may be brought back to normal4 ,in searching for such new experimental operations it seems to me that by far the greatest promise for the immediate future is offered by improvements in our powers of dealing with individual atomic and electronic processes1 such as we now have to a limited extent in the various spinthariscope methods of counting radioactive disintegrations1 or ,wilson's -track experiments4#ch ,in this self-conscious search for phenomena which increase the number of operationally independent concepts1 we may expect to find a powerful systematic method directing the discovery of new and essentially important physical facts4 ,footnote #ch3 ,c4 ,t4 ,r4 ,wilson1 ,proc1 ,roy4 ,soc41 #hg1 #bgg1 #aiab4 ,we can only conjecture whether the number of fundamental concepts will prove impossible of further increase or not1 but present experience seems to give greater probability to the view that as we penetrate deeper the number of fundamental concepts will always tend to become fewer4 ,we have already certainly one example in that the temperature concept disappears when we get to the atomic scale of magnitude1 and possibly a second example in the building up of separate concepts for energy and frequency by the combination of great numbers of that one operationally simple thing which characterizes the elementary quantum process in ordinary radiation4 ,different sorts of relations between concepts are conceivable in the transition zone where the number changes4 ,we may find that other examples are like that of temperature1 which is simply a statistical effect of a great many phenomena which may be described individually in terms of the ordinary concepts of mechanics1 so that in this case the number of concepts changes merely by temperature dropping out1 leaving the others more or less unaffected4 ,or all the concepts may be more closely interwoven1 so that when the total number of concepts changes it may not be possible to separate out a group of concepts whose defining operations are unchanged4 ,in such a case we must say that the original concepts are not applicable on the new level4 ,the most immediate application of this idea has been already mentioned1 namely1 to the concepts of space and time4 ,if the operations by which space and time are measured on the ordinary scale of magnitude cannot be carried down as a whole into the region of quantum phenomena1 then we must say that the ordinary concepts of space and time are not applicable to these phenomena4 ,closely connected with the sharper analysis of the operational structure of our concepts1 we may expect in the future also a closer analysis of our inventions4 ,this will take the form of a search for new physical facts which shall give to our inventions the character of physical reality4 ,in case prolonged search fails to disclose such phenomena 7as is probably now the case with the field concept of electrodynamics71 we must then find some way of embodying explicitly in our thinking the fact that we are dealing with pure inventions and not realities4 ,,index ,absolute1 #bf ,absolute time1 #d ,action at a distance1 #df ,analysis of large into small1 #ea1 #bbj ,arithmetic1 #ce ,atom1 #ei ,bell1 #hd ,birkhoff1 #gb ,black body1 #aab ,bohr1 #aij1 #aib ,born1 #bbb ,boscovitch1 #df ,bothe and ,geiger1 #aaf ,bridgman1 #bja ,brownian movement1 #ajg1 #abi1 #adc ,bush1 #adb ,caloric fluid1 #ei ,campbell1 #aag ,carnot engine1 #abe ,causality1 #hj ff ,causal train of events1 #he ,clifford1 #bh ,clock1 #gj ff1 #agf ,compton1 #aaf1 #ahh ,continuity1 #id ,conservation of charge1 #ace1 #acf ,conservative functions1 #aac ,constructs1 #ec ,correlation1 #cg ,cosmic units1 #ahb ,cross word puzzle1 #bjb ,descriptive background1 #fd ,determinism1 #aad1 #bji ff ,discontinuous space1 #aia ,dppler effect1 #aff ds1 #gb ,eddington1 #ic ,einstein1 vii1 #a1 #b1 #c1 #d1 #g1 #h1 #i1 #ab1 #ac1 #ad1 #fd1 #ajj1 #aee1 #afb1 #afc1 #afd1 #afg1 #afi1 #agj1 #aga1 #agb1 #agc1 #age1 #agf1 #agg1 #bjf ,electrical concepts1 #aca ff ,electrical explanation of universe1 #ej ,electrical mass1 #aci ,electric field1 #ef1 #acc ,empiricism1 #c ,energy1 #ajh ff1 #abf ff ,euclidean space1 #ad1 #ae1 #af1 #ah1 #bc1 #eb1 #fa1 #fg ,event1 #ie1 #afg ,explanation1 #cg ,explanatory crisis1 #da ,extended time1 #gg ,faraday1 #dd1 #eg1 #eh1 #bji ,final explanations1 #dh ,first law of thermodynamics1 #abf ff ,force1 #ajb ff ,foucault pendulum1 #ahj1 #ahd ,fourth dimension1 #gd ,future1 #bbb ,gauss1 #ae1 #cd ,go and come time1 #aab ,gravitational constant1 #ia ,haldane1 #be ,heat flow1 #acj ,heisenberg1 #bbb ,hertz1 #dd ,hoernl1 vi ,identity1 #ia ff ,isolation1 #hb ,joule1 #abd ,kelvin1 #de1 #aaj ,kinetic theory of gases1 #dj ,lagrangean equations1 #aab ,larmor1 #adi ,la ,rosa1 #afd ,length1 #i ff ,lewis1 #aff1 #bja ,light1 #aej ff ,local time1 #ge ,lorentz1 #adc1 #adg1 #adh1 #adi ,mach1 #ahc ,mass1 #ajb ff ,mathematics1 #fj ff ,maxwell1 #dd1 #eh1 #aab1 #acg1 #adh ,meaningless questions1 #bh ff ,measurement approximate1 #cc ,mechanism1 #de ,mercury1 #aje ,michelson1 #ae1 #bf ,michelson and ,morley1 #ff ,models1 #de1 #eb ,newton1 #d1 #aaj ,operational character of concepts1 #e ,operational thinking1 #cb ,optical space1 #fg ,ostwald1 #aji ,penumbra1 #cd ,perrin1 #ajg ,physical reality1 #ei ,planck1 #fi ,poincar1 #dh1 #aae1 #aaf1 #aij ,pythagoras1 #fa ,quantum act1 #aef ,quantum theory1 #dj1 #dg1 #ahf ,radiation and temperature1 #abc ,relative character of knowledge1 #be ,relativity1 #aej ff ,reynolds1 #ic ,rotational motion1 #agh ,russell1 #bje ,schrdinger1 #bbb ,space1 #ff ,silberstein1 #aa ,simplicity of nature1 #aih ,simultaneity1 #g1 #h ,spring balance1 #ajc ,statistical methods1 #aae1 #aag ,stress1 #ed ,swann1 #bjd ,table top1 #ajf ,tactual space1 #fg ,temperature1 #aah ff ,thermodynamics1 #aag ,thing traveling1 #aja1 #aeb1 #aeg1 #afd ,time1 #fi ,tolman1 #bja ,truth1 #gh ,turbulent motion1 #abj1 #abd ,velocity1 #ig ff1 #bac ff ,velocity of light1 #ajj ,whitehead1 #afg ,wilson1 #bbd