,,on ,,the ,,theory ,,of ,,the          
,,infinite ,,in ,,modern ,,thought      
,,on ,,the ,,theory ,,of ,,the          
,,infinite ,,in ,,modern                
,,thought                          ,,two
 ,,introductory ,,studies               
,,by                               ,e4  
,f4 ,,jourdain                          
,,doctor ,,of ,,the ,,university ,,of   
,,paris                                 
,,vice-,,principal1 ,,st4 ,,hugh',s     
,,hall1 ,,oxford                        
,,longmans1 ,,green ,,and ,,co4         
#ci ,,paternoster ,,row1 ,,london       
,,new ,,york1 ,,bombay1 ,,and ,,calcutta
                                  #aiaa 
,all rights reserved          ,of the   
two papers here reproduced1 the first   
was given in #aije to  a meeting of the 
women science students in ,oxford2 the  
second1 in  #aijh1 to the ,philosophical
 ,society of this ,college4 ,they are   
printed  by request1 with the author's  
apologies for their incompleteness4     
,the lecture form has been retained4 ,i 

am indebted to my brother1 ,mr4  ,p4    
,jourdain1 for help in preparing the    
first lecture1 and for his  revision of 
the text4                               
,e4 ,f4 ,,jourdain4         ,,st4       
,,hugh',s ,,hall1 ,,oxford1             
,january1 #aiaa4                        
,,contents                              
,i                                      
,,page                    ,,the         
,,problem ,,of ,,the ,,finite ,,and     
,,the ,,infinite                        
#a                                      
,,ii                    ,,pragmatism    
,,and ,a ,,theory ,,of                  
,,knowledge                         #ca 
,i                 ,,the ,,problem ,,of 
,,the ,,finite ,,and ,,the ,,infinite   
,the influence of mathematics on        
philosophy and vice vers can be         
inferred from the historical progress of
 both studies1 though it has  not been  
possible till about within the last     
fifteen years to give  a logical        
explanation for the relations between   

them4 ,as long as it  was believed1     
according to the ,kantian view1 that the
 science of  mathematics was based on   
intuitions of time and space1 the       
alliance  between philosophy and        
mathematics could not be proved to be   
closer  than that between philosophy and
 experimental science1 although the     
historical fact remained that philosophy
 and mathematics exercised a  mutual    
stimulus1 and developed at the same     
periods of history4    ,but mathematics1
 as now defined1 is independent of      
intuitions of  space and time1 and also 
of axioms and hypotheses4#a             
,mathematics1  as now understood1 is    
based1 like formal logic1 on the        
prerequisites  of thought1 not on the   
notions of space and time4 ,here there  
is no  definition of number or space1   
but the conception of number and        
space1#b which is more complicated1 can 
be derived from them4 ,all  other       
complicated mind processes can1 in the  
same way1 be reduced to  the simple     

elements of the prerequisites of        
thought4    ,such a science might exist 
out of conditions of time and space as  
we  know them4 ,it is a science of      
relations rather than of mere number4   
,founded1 then1 on the laws of symbolic 
logic1 it is a valuable aid  and        
illustration to philosophy2 philosophy1 
on the other hand1 can  imagine lines   
for the exercise of the constructive    
power involved in  mathematics4 ,it is  
the object of this paper to show that   
the close  though apparently accidental 
union of philosophy and mathematics     
throughout the history of thought can   
now be explained1 and that the  problems
 with which pure mathematics is now     
concerned are those which  lie at the   
core of philosophic thought and         
speculation4 7,symbolic  logic has      
developed to meet the new demands made  
upon it4 ,it does  not now reduce itself
 to the syllogism1 as ,aristotle thought
 it did2  the prerequisites of thought  
are shown to be manifold instead of     

single4#c7    ,the use of the word      
philosophic in this connection suggests 
a  necessity for further definition4    
,philosophy is held to include at  least
 two great branches--,metaphysics and   
,ethics4 ,the influence of  mathematics 
is most evident on the metaphysical side
 of philosophy2  in fact1 the grouping  
of mathematics and metaphysics as allied
  sciences tends to bring out the       
essential distinction between           
metaphysics and ethics1 and--though not 
by any means to imply a break  in their 
real relation--to show where this has   
been misunderstood4  ,no philosophy has 
been equally strong on both sides2 they 
represent  different forms of activity  
of the human mind2 but it is still true1
  and from the conditions always must   
be1 that an ethical system grows  out of
 metaphysics as practice follows precept
 and conduct implies  belief4 ,the new  
definition of mathematics does not touch
 these  consequences2 it merely marks   
the limits within which philosophy on   

the metaphysical side can submit to1 or 
rest upon1 the conclusions of           
mathematics4    ,as to the historical   
relation between metaphysics and        
mathematics1  the subject is so vast    
that we shall only attempt a very rapid 
generalisation of its results on the    
growth of the conception of  the ,finite
 and the ,infinite4 7,of course1 there  
are many other sides  of the relation   
which might be studied47 ,the general   
result of the  inquiry has been1 as far 
as we can judge1 that metaphysics has   
exercised an inspiring force on         
mathematics1 and mathematics has        
defined and strengthened the conceptions
 of metaphysics at every  critical stage
 in the history of philosophy4 ,but     
where metaphysics  has been treated as  
the proof of science1 where it has been 
laid  down as the foundation for exact  
knowledge1 the results have not         
corresponded with the truth of          
experience1 and the quality of thought  
has become degenerate4 ,progress depends

 on the right perception of  the        
relations between the sciences and parts
 of philosophy4    ,such progress is    
especially evident in the early ,greek  
and in the  modern periods1 while the   
large period from the ,christian era to 
the  ,renaissance gives examples of the 
unfortunate reversal of the parts  of   
metaphysics and science and consequent  
confusion of thought4#d    ,the problems
 of the metaphysician are no doubt in a 
sense always the  same2 but this is     
equally true of the problems of any     
other science4  ,the methods by which   
the problems are attacked and the       
adequacy  of the solutions they receive 
vary1 from age to age1 in close         
correspondence with the general         
development of science4 ,every great    
metaphysical conception has exercised   
its influence on the general  history of
 science1 and in return every important 
movement in science  has affected the   
development of metaphysics4 ,the        
metaphysician could  not if he would1   

and would not if he could1 escape the   
duty of  estimating the bearing of the  
great scientific theories of his time   
upon our ultimate conceptions of the    
nature of the world as a whole4  ,every 
fundamental advance in science thus     
calls for a restatement and             
reconsideration of the old metaphysical 
problems in the light of the  new       
discovery4           99       99        
99       99       99    ,during the     
,greek period mathematics was the only  
branch of science  which was at all     
developed1 and its development coincided
 with  the age of the philosophers4     
,thus when ,plato spoke of science he   
always meant mathematics4 ,and even     
later1 when the physical sciences  had  
begun to develop1 ,aristotle put        
mathematical ideas into close           
connection with metaphysical ones when  
he stated that they occupied  the middle
 term between the ideal and the         
sensible4 ,both ,plato and  ,aristotle  
referred to and depended upon           

mathematical proofs and  illustrations  
of philosophical questions4 ,during this
 ,greek period  the conception of       
,infinity took shape4 ,the pre-,platonic
 notion1  reproduced again later in the 
decline of ,socratic theory by the      
,stoics1 was that the ,infinite was the 
aggregate of the ,finite2 the  ,platonic
 and ,aristotelian theory1 that1 namely1
 of the most vigorous  moment of ,greek 
thought1 was that the ,infinite was more
 than the  aggregate of the ,finite2    
that it had a self-determined existence 
from  which the ,finite had been        
derived4 ,existence1 as known to man1   
was  treated as a compromise between the
 ,finite and the ,infinite4             
,neo-,platonism altogether separated the
 ,infinite from the ,finite4 ,in  the   
,alexandrine metaphysics1 which         
represented a decadent stage of         
philosophy and its deviation from the   
sciences1 the conception of  the        
,infinite became less clear and logical2
 it diverged from the  view which had   

been affected by mathematical thought1  
and tended  to assimilate to itself the 
ideas of perfection and universality1   
which1 philosophically speaking1 are    
conceptions distinct from that  of      
,infinity--universality referring to a  
common principle of unity1  and         
perfection involving the moral ideal4   
,real progress was deferred  by the too 
rapid coherence of ideas only partially 
analysed and  understood4 ,thinkers     
passed quickly from the exclusive       
contemplation  of subject to that of    
object and back again1#e each new period
  negativing all previous experience1   
till the result was the exclusion  of an
 imperfectly analysed ,relative and     
,finite from an insufficiently          
apprehended ,absolute and ,infinite4    
,after the ,christian era ,greek        
philosophy drifted off into             
scholasticism and lost touch of reality1
 the grammar of ,aristotelian  logic    
replacing the vital connection of ideas4
 ,st4 ,anselm1 it is  true1 attempted to

 find a rational proof of the existence 
of ,god1 and  identified ,him with the  
,infinite of ,greek thought2 but ,st4   
,thomas  ,aquinas led away the argument 
to a discussion as to how far form and  
matter1 separately considered1 shared in
 the quality of ,infinity4 7,he  thought
 form did1 but not matter47 ,an         
overpowering sense of mystery1  joined  
to a premature desire for definition    
without scientific  analysis1 sapped the
 vigour of medival thought4             
,throughout the middle ages1 then1 we   
see the conditions of the ,greek  period
 reversed3 philosophy during the second 
period is not1 as  in the first1 engaged
 in giving a stimulus to the efforts of 
pure  reason2 rather the intuitions of  
philosophy are treated as axiomatic1    
and a false superstructure of knowledge1
 alien to experience and  reality1 is   
erected upon these foundations4         
,philosophy1 in fact1 is  used as a     
general basis for science4 ,the parts of
 philosophy and  mathematics1 correctly 

though imperfectly seen by the ,greeks1 
are in  the second period exchanged1 and
 the result is confusion of idea4  ,the 
notion of the ,infinite1 as in the      
,alexandrine metaphysics1 is  held to   
include perfection and universality1 and
 does not exist as a  conception apart  
from these4    ,after the ,renaissance1 
the scholastic philosophy falling into  
disuse1  the attempt to find an         
explanation of the ,cosmos1 a synthesis 
of the  universe1 was abandoned1 and    
replaced by the ,cartesian idea--the    
inference of existence from thought1 and
 the limitation of the sphere  of       
inquiry to that which could be known by 
the ego4 ,new scientific  and           
mathematical discoveries kept pace with 
this new analysis and  development of   
thought1#f and the surer ground in      
philosophy was  definitely allied with  
the work of the mathematical mind4 ,the 
philosophical thesis developed from     
8,the ,infinite is the negation  of the 
,finite10 to 8,the ,infinite presupposes

 the ,finite and does  not exclude it40 
,the problem of the ,finite and the     
,infinite became  the great idea of the 
age1 and there was a reversion to the   
,greek  notion of existence as a        
compromise between the two1 and almost  
the  hint of a coming explanation of    
them4 ,in the decline of ,cartesian     
philosophy1 when it drifted off into    
,pantheism1 there was only a  vague     
conception of the ,infinite1 and we     
trace a tendency to identify  the notion
 of ,infinity with that of the ,cosmos4 
,in medival thought  the idea of the    
,infinite had become confused with that 
of the ,perfect  and ,universal2 in the 
modern period the effort to give a      
concrete  expression to the notions of  
,infinity1 ,perfection1 and             
,universality  diverted the ideas from  
their relation to the ,creator and      
applied  them to the ,creation4         
,kant1 who gave a new impulse to some   
parts of the ,cartesian idea1  neglected
 both mathematical proofs and the search

 for a metaphysical  ,absolute4 ,in     
avoiding the subject he helped to       
perpetuate the  vague descriptions of   
the ,finite and the ,infinite1          
uncorrected by  mathematical thought1   
which had been the currency of the      
philosophy  of his age1 and which       
corrupted the philosophy of the         
succeeding  century4 ,the nineteenth    
century produced nothing more than      
guesses  at truth1 which were1 perhaps1 
not very far wrong1 and which the       
present century is engaged in correcting
 and substantiating4 ,the  same         
vagueness afflicted both mathematics    
7theory of functions7  and philosophy4  
,fichte1 ,schelling1 and ,hegel1        
particularly ,hegel1  identify the      
metaphysical ,absolute with reality1    
infinity1 and the  universal4 ,the ideas
 of continuity and infinity are not     
separated by  them from those of        
perfection and universality1 nor from   
one another1  and their nature is not   
understood4    ,leaving aside the       

,french neo-critical ,school            
7,renouvier7 and the  ,english ,school  
7,spencer7--the first of whom deny the  
,infinite1  thus acting in opposition to
 mathematical reasoning1 while the      
second perpetuate ,kant's error of      
considering the ,infinite1 though       
thinkable1 as unknowable 7,dr4 ,caird   
has pointed out that this  position is  
illogical7--we arrive at a moment in    
history which is  more fruitful in      
result on the mathematical side1 and    
will1 no doubt1  have an effect on      
metaphysics4 ,for1 owing to recent      
discoveries  in ,germany and ,england1  
mathematics is now in a position to give
  greater support than before to the    
intuitions of philosophy4  ,hitherto1   
philosophers have been reluctant to     
allow full value to  the mathematical   
conceptions of ,infinity1 and with some 
justice1 as  the notion had not been    
sufficiently analysed4 ,philosophers1   
who  never attempted the analysis1 have 
been inclined to accept certain         

contradictions in their conception as   
inherent in the nature of  ,infinity4   
,within the last twenty-five years      
,cantor and ,dedekind have  cleared up  
the notion of continuity1 and ,russell  
has given greater  precision to the     
idea1 and has applied this reasoning to 
philosophy4    ,present-day             
metaphysicians seem to be divided into  
two groups2 on  the one side1 those who 
consider in philosophy the value of a   
theory  of being1 and1 on the other1    
those who chiefly consider the value  of
 a theory of knowledge1 i4e4 the        
,epistemologists4 ,the first  group1    
devoting themselves to psychology1      
evolution1 and history1  have no        
necessary belief in the ,infinite4 ,the 
,epistemologists1  whose work is founded
 on ,kant1 discuss the theory of        
knowledge and  enumerate the conditions 
of knowledge4 ,their argument may not   
touch1  but does not exclude1 the notion
 of the ,infinite4 ,the position of  the
 ,epistemologist has been made          

infinitely more secure by recent        
mathematical work4 ,that of the         
psychologist remains almost untouched4  
,it is necessary now to examine more    
closely the mathematical results  to    
which reference has been made4    ,in   
general terms it may be said that       
mathematics has1 as a study1  led       
immediately from the nature of the      
subject to the perception  of the       
,infinite1 and to a knowledge of the    
connection between the  ,infinite and   
the ,finite4 ,the simplest form in which
 the idea can be  put is stated by ,st4 
,augustine1 who said that numbers       
considered  individually were finite1   
but considered as an aggregate were     
infinite4#g ,before ,st4 ,augustine1 and
 after him down the long  stream of     
philosophic thought1 the theologian and 
the philosopher  have turned to         
mathematics for illustrations of the    
infinitely great  and infinitely little1
 as developed from the concrete         
processes of  arithmetic and geometry4  

,the recurring decimal in arithmetic1   
the  properties of the circle and       
ellipse in geometry1 of the cone in     
conic sections1 and of the surd in      
algebra1 all touch the problem of       
number and space on the side of         
,infinity4    ,in higher mathematics it 
is possible to start from the idea of   
the  ,finite and reach the conception of
 the ,infinite2 or to reverse the       
process1 and from the ,infinite to      
deduce the ,finite4 ,thus in the        
familiar puzzle of the subdivision of   
the parts of a straight line  by halving
 the remainder1 there will be a crowding
 and a coalescing  of the points of     
division towards one end of the line1   
the points of  division getting         
infinitely nearer1 but the steps will   
never meet4  ,here in the centre of a   
straight line--a limited straight       
line--we  are confronted with the       
problem of ,infinity4    ,again1 from a 
series of finite numbers we can gain the
 notion of an  infinite series4 ,take   

two series which have a correspondence  
with one  another4 ,if for every element
 of the one we can choose an element of 
the other1 and of the other there is an 
element for the one1 when at  any point 
we cut off its progress to infinity1    
this happens3--    ,one series1 if      
summed up1 will give a larger numerical 
result than  the other1 and therefore   
can be said to be greater than the      
second4  ,let us call the first series  
,a1 and the second ,b4 ,let us now      
imagine  the two series1 though starting
 at a definite point1 are never cut  at 
the further end4 ,then to all infinity  
series ,b is without certain  numbers   
which series ,a possesses1 and as an    
infinite series is  smaller than series 
,a4 ,but1 on the other hand1 when       
neither series  is cut1 series ,b       
retains its correspondence with series  
,a4 ,thus we  attain a definition of an 
infinite series4 ,it is such that the   
part1  while being less than the whole1 
has yet a complete correspondence  with 

the whole4 ,the whole is greater than   
the part1 but take away  the part from  
the whole and that which remains        
corresponds to it in  infinity1 because 
the test of summing the series 7which   
would give  a contrary result7 involves 
limitation1 and thus cannot be applied4 
,subtraction can take place in ,infinity
 without loss4    ,by reversing this    
process1 and by starting from the theory
 of the  ,infinite1 we may gain some    
idea of the discovery of the ,finite4   
,so  ,dedekind and ,russell define      
finite numbers not only in the usual way
  as those which can be reached by      
mathematical induction1 starting  from  
#j and increasing by #a at each step1   
but also as those of classes  which are 
not similar to the parts of themselves  
obtained by taking  away single terms4  
,that is the reversal of the process    
applied just  now4 ,dedekind also has   
deduced the ,finite from the ,infinite  
by a  novel process4 ,he predicates a   
world of thought which we each and all  

possess1 filled with thoughts and       
things1 to each thing corresponding  a  
thought4 ,there are thus two            
8trans-finite0 series in the minds of   
each and all of us2 we cannot say when  
the series of thoughts and  things will 
end2 but they have number1 though it is 
infinite number4  7,number exists       
wherever there is a correspondence1 one 
to one1  between two aggregates47 ,but  
in this ,gedankenwelt1 says ,dedekind1  
there is one thing to which there is no 
corresponding thought3 that  is the ego4
 ,each man is part of his own world of  
thought1 but  there is no thought of    
himself in his mind corresponding       
exactly to  himself1 as a thought in his
 mind corresponds to another object4#h  
,two important results follow from      
,dedekind's theory3 first1 the          
existence of a finite number one1 the   
number of the ego1 as deduced  from the 
,gedankenwelt of two infinite systems2  
second1 by putting  together all the    
,gedankenwelts there are or may be1 we  

get the  notion of series of series1    
which seems to transcend ,infinity1 and 
it gives us the conditions which are    
possibly gathered up in the  ,absolute4 
,now the argument from the ,finite to   
the ,infinite and the  converse process 
may both be employed in mathematics 7or 
both may  be neglected1 as in the       
elementary methods of calculation used  
in  arithmetic74 ,a discussion has taken
 place in the ,hibbert ,journal on  the 
relative value of the two methods4      
,keyser1 in an article called  the      
,axiom of ,infinity1 argued that one    
method1 that of ,dedekind1  should be   
exclusively developed4 ,russell answered
 him1 stating that  it was not necessary
 to hold exclusively to either4 ,if the 
,finite  and the ,infinite can in turn  
be deduced from one another1 neither    
conception can be truly called an axiom4
 ,the real axiom is existence1  which   
includes both1 and which is defined by  
mathematicians as that  which is not    
self-contradictory4    ,now the problem 

of ,infinity includes also that of      
continuity2 in  other words1 the problem
 of number includes that of cardinal and
  ordinal number4 ,it is time to get to 
the mathematical definition of  number1 
which we have found as a conception can 
be attached both to  the ,finite and to 
the ,infinite4 ,what is number in       
mathematics8    ,take any collection of 
things--we call that an aggregate4 ,if  
an  aggregate corresponds one to one    
with another aggregate1 they are  both  
said to have a number1 and the same     
number4 ,subtract from the  idea of an  
aggregate the idea of quality or kind1  
and order or  arrangement1 what is left 
is its cardinal number4 ,if you subtract
  quality and not order1 the result is  
an ordinal number4 ,this  reasoning     
applies both to finite and infinite     
aggregates2 in fact1  the ,infinite may 
be said to possess most of the          
properties which  we attach to the      
,finite4 ,two infinite aggregates1 for  
example1 can  have an ordinal           

correspondence1 and infinite aggregates 
submit1 like  finite ones1 to           
arithmetical processes4    ,the         
mathematician analyses still more       
closely the relation between  the       
,finite and the ,infinite1 as follows3--
    ,he starts from the aggregate1 which
 he analyses into the ,finite  and the  
,infinite1 and the latter he analyses   
into the ,transfinite  and the          
,absolute4 ,of these two elements1 one  
only has till just  lately been the     
subject of mathematical treatment--the  
one called  the ,transfinite4 ,it is the
 transfinite subdivision of the         
,infinite  to which the idea of number  
is applicable1 and which is1 therefore1 
in a sense inseparable from the ,finite4
 ,infinite numbers or series  ought then
 to be more correctly described as      
,transfinite4 ,but the  processes of    
mathematics do not end here2 they reach 
up to the idea  of the ,absolute        
,infinite1 the conception of which has  
been attained  in recent years by       

mathematical work4 ,the results of this 
work may  now be briefly summarised4    
,i4 ,the ,absolute appears to have the  
same relation to the ,transfinite  as   
the ,transfinite to the ,finite4 ,if the
 ,finite deals with numbers1  and the   
,transfinite with series of numbers1 the
 ,absolute deals  with series of series4
 ,thus there are at least two examples  
of the  ,infinite within our grasp which
 lead up to the idea of the ,absolute4  
,one is the class of all classes of     
propositions2 the other is the  series  
of all worlds of thought1 in ,dedekind's
 sense4    ,,ii4 ,the ,finite1          
,transfinite1 and ,absolute can be      
further defined in  this way4 ,there is 
no greatest finite number1 but there is 
a least  transfinite number1 which has  
been called ,aleph #j1 and which can  be
 proved to be greater than any possible 
finite number1 however  large1 because  
if there were a last number it must be  
smaller than  the sum of the whole      
series4 ,there are unending series of   

,alephs or  infinite numbers1 which are 
as distinct from one another in idea  as
 #a is from #j1 and which can no more be
 derived from one another  by a         
mathematical process than #a can be     
derived from #j1 but can  be reached in 
the same way by induction4 ,beyond the  
,transfinite we  cannot discover in the 
,absolute the idea of least or of       
greatest4    ,,iii4 ,the relation of    
cardinal and ordinal number also throws 
some  light on the ,finite1             
,transfinite1 and ,absolute4 ,in the    
,finite1  cardinals and ordinals are    
parallel to one another2 in the         
,transfinite they strikingly diverge2 in
 the ,absolute we cannot trace  any     
connection between cardinals and        
ordinals1 i4e41 it is possible  to have 
an ordinal series to which there can be 
no corresponding  cardinal number or    
type4#i    ,,iv4 ,if arithmetical       
processes are applied to the ,finite1   
,transfinite1  or ,absolute1 we get     
interesting results4 ,we know the effect

 of  addition1 multiplication1 and      
raising to a power1 on the ,finite4     
,the first two processes have been      
applied to the ,alephs2 the last  has   
been formulated1 but the mathematical   
results have not yet been  brought to a 
satisfactory conclusion4 ,broadly       
speaking1 we may say  that the raising  
of an ,aleph to a power may make it     
transcend the  ,finite and the          
,transfinite and melt into the          
,absolute4 ,thus all  mathematical      
processes which find their goal in the  
,absolute would  find their annihilation
 there4 ,no finite mathematical         
conception  would be applicable to it4  
,now the conception of this ,absolute   
,infinite1 of which the aggregate  of   
all ordinal numbers is perhaps a        
symbol1#aj has been subjected to        
criticism4 ,some mathematicians#aa think
 that it exists1 but has no  number4 ,it
 is discovered by a logical process1 but
 defies analysis  and the application to
 it of the notion of number4 ,all       

mathematical  conceptions find in it    
their aim and conclusion4 ,the          
importance of  this theory1 its         
practical importance1 lies in the very  
much simpler  mathematical formul that  
can be produced now that the logical    
process is shown to extend from the     
,finite to the ,absolute ,infinite  7in 
the same way that the labour of summing 
a series arithmetically  by statement   
and addition is shortened by the        
application of  algebraical principles  
which depend on larger knowledge74 ,its 
philosophical importance is great3 the  
,absolute is here1 as  elsewhere1 the   
goal of human thought1 and is the       
mathematician's name  for the highest   
power discoverable by human reason4     
,it would be very interesting to discuss
 the probable attitude of a  ,pascal or 
a ,hegel to these mathematical          
conceptions1 if they had been  aware of 
them4 ,take ,pascal's puzzle of the     
,finite and the ,infinite4  ,he thought 
that if the ,finite could be subtracted 

from the ,infinite1  the ,infinite would
 thereby lose some of its quality of    
infinity4 ,how  differently would it    
have appeared to him had he realised    
that an  aggregate infinite cardinal can
 have subtracted from it either finite  
or transfinite terms3 if transfinite    
terms1 many different answers  result1  
giving different degrees of transfinity3
 if only finite terms  are taken away1  
the ,infinite remains in its entirety4  
,how1 again1 would ,hegel have rejoiced 
in a definition of thought and          
existence which would bridge over the   
logical gulf in his system6  ,hegel     
asserted that thought and existence were
 one4 ,he is objected  to by many       
philosophers1 who ask where is the      
tertium quid which  makes it possible to
 reach from one to the other1 or        
predicate their  essential unity8 ,but  
the mathematician defines existence as  
something  which is not                 
self-contradictory4 ,thought1 then1 to  
him is a form  of existence1 for thought

 is not self-contradictory2 but the two1
  thought and existence1 are not        
necessarily conterminous4#ab ,hence1  to
 say that non-contradiction is a        
fundamental condition of true  thinking 
is as much as to say that it is a       
fundamental characteristic  of real     
existence1 and he identifies thought    
with reality4    ,dr4 ,caird remarks    
that the secular conscience conceives of
 the  ,infinite as opposed to the       
,finite2 the religious conscience treats
  the ,infinite as real1 presupposed by 
the illusory ,finite4 ,where does  the  
truth lie8 ,mathematics does not admit  
the necessity of adopting  either view  
at the expense of the other4            
,metaphysics standing alone produces    
results that may be disproved1  but     
cannot be proved4 ,mathematics standing 
alone produces results  that are        
susceptible of proof4 ,both are based on
 logic1 and rest on  the prerequisites  
of thought4 ,together they are a field  
for the best  powers of human reason3   

metaphysics supplies insight1 intuition1
  imagination2 mathematics offers the   
indubitable proof and translates  the   
ideal into the actual4    ,but the      
element in philosophical thought which1 
employing the  psychological method1    
tends to the discussion of a theory of  
being  rather than that of knowledge1   
and thus to the realisation of an       
ethical system rather than to           
metaphysical discovery1 is averse from  
accepting these conclusions4 ,it        
remains1 therefore1 for us to examine   
the criticism offered by the            
psychological school on what they call  
the mathematising of philosophy2 and it 
will be found that the attack  deals    
both with the ground of the alliance and
 its results4    ,a typical exponent of 
this school is ,moisant1 who1 in the    
,revue  ,philosophique for ,january     
#aije1 attacked what he considered to be
  the characteristic of modern          
philosophy and also its vice4 ,it will  
be observed that at the outset he       

reverses the rles of philosophy  and    
mathematics as we have apprehended them4
 ,philosophy1 he says1  should expect to
 be inspired by mathematics1 but should 
avoid its  method4 ,next1 he connects   
the modern movement with the theories of
  ,leibniz1 who aimed at substituting   
general formul for elementary  forms of 
reason and calculation4 ,these short    
cuts1 which seem to the  mathematician  
to liberate the mind from a burden which
 prevents it  from employing its full   
activity1 seem to the psychologist to   
tend to  a mechanical method1 in which  
the thinker is only aware of premises   
and results1 and in which the           
mathematical concept tends to replace   
the real idea4 ,then he attacks the new 
definition of mathematics as  the       
science of relations1 asserting that it 
still contains notions of  space4#ac    
,finally1 he comes to the real question 
at issue1 and enters into  the          
comparison of a metaphysical and a      
mathematical problem4 ,he  takes as his 

subject the argument from the known to  
the unknown4  ,descartes had said that  
argument should lead from the known to  
the unknown1 simple to complex1 and had 
defined the first as that  which could  
be known without the help of the second4
 ,this logical  order of reasoning has  
been attributed to mathematics1 but has 
been  considered to be inapplicable to  
philosophy4 ,mathematics1 in its  recent
 development1 by the argument from the  
,finite to the ,infinite  and back      
again1 starts from two propositions1    
neither of which can  be said to be     
axiomatic1 because each in turn can be  
proved from  the other1 but in the      
course of argument from either          
mathematics  makes use of the logical   
process4 ,the real axiom1 as has been   
shown1  is that of existence or being4  
,a metaphysical argument has the  same  
root--that of existence--but a          
metaphysical problem deals with         
paradoxes1 with questions which are     
sometimes defined as having two         

answers1 each equally correct1 and      
sometimes as yielding no answer  at all4
 ,the method of thesis1 antithesis1 and 
synthesis is in the  ,hegelian logic    
applied to their solution4    ,a        
mathematical and a metaphysical problem 
are not1 then1 problems of  the same    
kind to be solved by the same method2   
nor is the conception  of the           
mathematical ,absolute reached in the   
same way as that of  the metaphysical   
,absolute4 ,we are even unable to say   
how far they  correspond except in      
respect of their absoluteness4#ad ,but  
the  contention of the mathematician    
to-day and of the epistemologist  school
 of philosophy is not the identity of   
methods and results in  the two         
sciences4 ,it is the axiom of existence 
on which they both  depend3 the law of  
thought by which all methods are        
developed1 and1  above all1 the         
correlative value of each science to the
 other1  which allows us1 in developing 
our knowledge from the standpoint  of   

the two sciences1 to recognise something
 of the greatness of the  ,absolute     
principle to which they both reach up1  
and in which their  being consists4     
,,footnotes3          #a ,of course1 if 
we comprehend in our view only          
elementary            geometrical and   
algebraical science1 it is easy to      
show that they do demand both axioms and
 intuitions4            ,take1 e4g4     
,euclid ,i4 ,i41 where in the           
construction            it is necessary 
to employ intuition for the assertion   
that the arcs really cut one another4   
,there is no            logical         
certainty that they do2 in fact1 in some
 other            conditions1 e4g4 in   
those of other space dimensions1        
they might not4          #b ,this is1 of
 course1 not the space of experience4   
,logic            and mathematics deal  
with implications of thought4 ,see      
,b4 ,russell 7,hibbert ,journal1 #aijd1 
pp4 #hji-ab71 who            has shown  
that in all pure mathematics it is only 

the            implications that are    
asserted1 not the premiss or the        
consequence1 as mathematicians used     
formerly to assume4          #c ,de     
,morgan1 ,peirce1 ,schrder1 and ,b4     
,russell have worked            out the 
logic of relations as well as the       
syllogism4          #d ,see ,taylor1    
8,elements of ,metaphysics10 p4 #ac4    
#e ,see ,dr4 ,caird1 8,evolution of     
,theology in the ,greek                 
,philosophers40          #f ,so         
,galileo1 ,newton1 ,huygens were        
philosophers in            science4     
,descartes1 ,pascal1 ,leibniz were      
mathematicians            as well as    
philosophers4          #g ,see ,s4      
,augustine1 ,de ,civitate ,dei1 ,book   
,,xii4 ch4            xix43 8,ita vero  
suis quisque numerus proprietatibus     
terminatur1 ut nullus eorum par esse    
cuicumque alteri            possit4     
,ergo et dispares inter se atque diversi
 sunt1            et singuli quique     
finiti sunt1 et omnes infiniti sunt40   

#h ,see ,r4 ,dedekind1 ,was sind und was
 sollen die ,zahlen8            #ahic4  
#i ,two transfinite aggregates can have 
an ordinal            correspondence    
with one another4         #aj ,see ,g4  
,cantor1 ,zur ,lehre vom ,transfiniten4 
#ahij4         #aa e4g4 ,mr4 ,p4        
,jourdain1 ,philosophical ,magazine4    
#aijd4         #ab ,the same result is  
hinted at by ,mr4 ,taylor4              
,taylor1 8,elements of ,metaphysics10 p4
 #bb4         #ac ,linear order1 #a1 #b1
 #c1 c4 ,circular4 ,a ,,cd ,b1          
,a ,,cd ,b4''' ,the latter1 it is true1 
involves the idea            of         
separation4 ,but this idea can be       
developed from            those of      
inclusion and exclusion1 which belong to
 the            fundamental laws of     
thought4              #ad ,the          
,absolute1 according to a recent        
metaphysical            thinker1 is 8a  
conscious life which embraces the       
totality of existence1 all at once1 and 
in a            perfect systematic      

unity1 as the content of its            
experience40--,taylor1 8,elements of    
,metaphysics10 p4 #fj4                  
,,ii                    ,,pragmatism    
,,and ,a ,,theory ,,of ,,knowledge      
,the question before us is the relation 
of ,pragmatism to a body of  knowledge4 
7a7 ,one question at issue between the  
,idealist#ae and the  ,pragmatist has to
 do with the way in which each defines  
knowledge  and gauges its ultimate aim4 
,both say that knowledge is relative1   
but one school asserts that the human   
mind slowly and laboriously  uncovers or
 discovers what ,goethe calls the       
8,living garment of  ,deity10 i4e4 the  
world of nature1 and comes into a       
heritage of  scientific truth which     
increasingly corresponds to the subject 
of his  faith2 the other claims that we 
live in a self-evolving universe in     
which in the course of long ages a new  
heaven and a new earth may be  created  
which are not foreseen or implied in    
present conditions4 ,in  other words1   

the ,idealist finds the ,divine in human
 life2 he finds  in his own small corner
 of the universe the microcosm and      
symbol of  ,infinity3 the ,pragmatist   
considers that nothing is which is not a
  result of human action1 and lowers the
 ,divine element to the result  of      
individual human activity4 ,a compromise
 between the two ideas on  new and      
interesting lines has recently been made
 by ,bergson4 ,the  ,christian doctrine 
of ,immanence and ,transcendence also   
combines them4    ,now the increase of a
 body of knowledge would seem to depend 
on  the comparison of the successful    
working out of hypotheses with  the     
discrepancies from theory that from time
 to time appear4 ,taken  together1      
proofs and discrepancies point to the   
evidence of a larger  law4 ,this is     
,hegel's logic1 and the principle1 so   
far as it is  here implied1 is not      
denied in modern times1 for no one      
wishes to  found a logic on a study of  
discrepancies as such4 ,even ,w4 ,james 

says1 8,whenever you once place yourself
 at the point of view of any  higher    
synthesis you see exactly how it does1  
in a fashion1 take up  opposites into   
itself40#af ,in fact1 without the notion
 of unity1  that of discrepancy could   
not exist3 there must be a background on
  which the differences appear4 ,the    
ultimate unity is symbolised in the     
,idealist doctrine of an ,absolute4     
,the ,absolute of ,idealistic thought is
 not1 however1 now conceived  of 7as the
 ,pragmatist would have us believe7 as  
an abstract unity1  but as one involving
 a social bond1 and hence relations     
which can  be described as personal1 if 
we remember that the ,personality of    
the ,absolute transcends our notion of  
human personality4 ,such a  conception  
of the term ,absolute1 a new reading of 
the theory of  the ,one and the ,many1  
has been led up to by ,bradley and      
,royce by  methods of logic1 and without
 any reference to dogma4 ,it has been   
conveniently expressed by ,taylor4 ,the 

argument is briefly that  ultimate      
,reality must be ,one1 ,many1 and       
,personal4    8,for our conclusion that 
mere truth cannot be the same thing as  
ultimate reality was itself based upon  
the principle that only  harmonious     
individuality is finally real1 and this 
is the very  principle employed by the  
intellect itself whenever it judges one 
thought-construction relatively higher  
or truer than another40#ag    ,and      
again3--    8,if we speak of existence  
as a society1 then we must be careful to
  remember that the individual unity of 
a society is just as real a  fact of    
experience as the individual unity of   
the members which  compose it1 and that 
when we call the ,absolute a society    
rather than  a self1 we do not do so    
with any intention of casting doubt upon
 its  complete spiritual unity as an    
individual experience40#ah    ,the      
,absolute has been stated in modern     
thought to be ,one1 ,many1 ,real1  and  
,personal or ,social1 and these terms of

 its qualification have  been           
successively arrived at4    ,w4 ,james's
 words ring hollow when he attempts to  
dissociate such a  conception from the  
reality of which it is the crown and    
inclusive  symbol1 and type and essence4
 8,i personally10 he says1 8give up the 
,absolute4 ,i find it entangles me in   
metaphysical paradoxes that are         
inacceptable40 ,he allows that there may
 be a ,god1 though limited  in power and
 goodness1 8one helper amongst others1  
primus inter  pares in the midst of all 
the shapers of the great world's fate40 
,in such a system1 as ,h4 ,jones has    
pointed out1 8there is neither in  the  
universe nor in ,god any principle to   
inspire or guide1 or in any  way to     
bring about the amelioration desired4   
,the process is guided  by no end4 ,the 
universe begins by being an aggregate of
 accidents1  pluralistic1 discontinuous1
 irrational1 and1 of itself1 cannot     
become  otherwise4 ,there is nothing    
actual within to change its             

character4'''  ,god is himself finite1  
helpless to bring about this great      
change1 a  part1 and no more1 of a      
universe broken in fragments40          
,another form1 and a very scholarly one1
 of the argument against  the existence 
of an ,absolute has been stated by ,bax 
in the 8,roots  of ,reality40 ,he       
appears to have reached the conclusion  
that the  telos1 the goal of human      
thought1 is not an ,absolute involving  
any notion of fixity1 but that it may be
 conceived of as a 8moving  synthesis40 
,he argues that everything of which we  
are conscious in  the universe is seen  
against a background which itself moves1
 and  is only realisable or             
distinguishable if it shifts upon       
something  relatively motionless behind 
it4 ,he concludes1 therefore1 that by   
analogy there is no ,absolute1 since    
what we perceive always implies         
something against which we perceive it2 
thus that there is no goal by  which and
 at which the spirit of man can find    

rest4 ,on his theory we  could never    
claim to reach the conception of an     
,absolute1 though he  admits the        
progressive character of human thought1 
and the increasing  reach1 lucidity1 and
 depth of the human mind4 ,the true     
answer to this  argument is that it     
proves exactly what it sets out to      
disprove4  ,as it is acknowledged that  
only the permanent or the relatively    
permanent can produce the phenomena of  
change1 so the appearance  of the goal  
of thought as a moving synthesis would  
presuppose an  ,absolute as a ground    
reality4#ai    ,if in truth we were able
 to apprehend entirely the source of all
  life and the background of all        
experience1 we might say that it did    
not exist for us as an ,absolute1 but   
the fact that whatever we  perceive     
postulates an unending series behind it1
 carries with it  the proof of an       
,absolute ,infinite4 7,this conclusion  
is led up to by  the mathematician's    
idea of the series of all finite and    

transfinite  ordinal numbers47 ,some    
part of this argument has been already  
suggested in ,ormond's 8,foundations of 
,knowledge10 and so far was used  by    
,mr4 ,illingworth in the 8,doctrine of  
the ,trinity40#bj    8,from a deeper    
metaphysical point of view it is the    
concept of  evolution itself that must  
submit to the determination of          
knowledge1  for it will be found that in
 so far as it becomes epistemologically 
necessary to ground relative processes  
in an ,absolute experience1  just so far
 will it become necessary also to       
connect the evolutionary  aspect of the 
world itself with a ground reality that 
is stable1  and involves the flux of    
change only as transcending and         
including  it40#ba    ,the further      
answer that any judgment1 even the      
,pragmatist's 8judgment  of value10     
implies an ,absolute has been stated in 
his ,oxford  ,lectures#bb by ,professor 
,h4 ,jones4    7b7 ,the next point we   
should like to work out is the relation 

of  fact to law4 ,the ,pragmatist denies
 scientific law and also logic1  and    
makes his appeal to facts4 ,no          
conclusion can be drawn from that       
denial except by the use of logic       
itself4 ,if he consistently denied      
logic1 his position would be            
unassailable by logic1 but he uses the  
method he denies1 and is thus open to   
attack4 ,on the subject of the  ,laws of
 ,science the ,pragmatist points out    
truly that there is no  actual          
continuity between a fact and a law4    
,but laws are concepts1  the result of  
mental activities2 they are themselves  
subject to the  laws of logic4 8,they   
were means1 and you make them ends10    
complains  the ,pragmatist4 ,that is    
just what nature herself does4 ,she     
perfects  means1 such as the means of   
supporting life1 and then these become  
ends4 ,language1 again1 is at first a   
means1 and then becomes an end4  ,so    
does any science change its character to
 the onlooker4 ,a law1 too1  though it  

generalises facts1 is a limit on        
absolute generalisation4  ,it thus      
stands midway between the abstraction   
and the fact4 ,the  ,pragmatist1        
however1 opposes to law what he calls a 
new fact--what  should rather be called 
a hypothesis4 ,he asserts that in every 
event1  action1 experiment1 there is a  
margin unseen and unrecognised by  us2  
that at every moment1 therefore1 the    
unknown1 the unexpected1  may take shape
 and voice and denounce all our careful 
and reasoned  conclusions4 8,why should 
the sun rise to-morrow because he has   
risen to-day and yesterday80 asks the   
,pragmatist4 8,we are making  an        
enormous assumption10 he says1 8in      
claiming the uniformity of  ,nature and 
the principle of causality40 ,the       
,idealist answers that the  ,pragmatist 
makes a larger assumption in doubting   
the truth of the  principles1 which     
though relative and not absolute1 still 
do work  out in practice1 than the      
,idealist does in his act of faith4 ,in 

fact1  the act of faith is rational as  
well as natural2 it is the act of       
doubting that is in this case due to a  
mere scholastic quibble4 ,it is  the    
,idealist and not the ,pragmatist who   
makes his appeal to the truth  of facts4
 ,each day that the sun goes on rising  
finds the ,idealist in  a better        
philosophical position and the          
,pragmatist in a worse1 except  on the  
assumption that the link between man and
 the external world  is a false         
imagination4 ,let us emphasise3--,it is 
the ,pragmatist who  quibbles with      
logic1 and the ,idealist who appeals to 
facts4    7c7 ,now there are certain    
facts and certain deductions from facts1
  well known to mathematicians1 which we
 should like to quote here as  having a 
bearing on the theory of the ,absolute1 
because they deal  with aspects of      
,infinity1 and mark a connection between
 the world as  we know it and the       
concepts of the philosopher4 ,all have  
the support  of science1 and furnish the

 ,idealist philosopher with examples    
which  support his theories1 and        
strengthen his position in the face of  
the  ,pragmatist attack4 ,they have to  
do with the theory of ,infinity as      
shown in3--    ,i4 ,the ,indefinite     
,regress4    ,,ii4 ,infinite series4    
,,iii4 ,dimensions in space and time4   
,before entering upon them we must      
repeat that the question of number  and 
series in mathematics is independent of 
the assumptions of space  and time4 ,as 
a science1 mathematics could exist      
outside them3 order  is not necessarily 
spatial or temporal4 ,our conclusions1  
therefore1  cannot be attacked on the   
ground that they are based on ,euclidean
  conceptions of space3 they are based  
on the laws of logic4                   
,i4 ,,the ,,indefinite ,,regress        
,hume and1 later1 ,kant argued that by  
the principle of association  when we   
think of one quality of a thing the     
others are naturally  brought before our
 minds1 and thus that we get into the   

habit of  attributing to the notion of  
the thing a certain group of qualities4 
,and it is true that we do attend to a  
thing all at once1 including  in the    
notion of it all the qualities which we 
know belong to it4    ,now experience1  
according to ,leibniz1 gives us an      
example of a  unity which embraces a    
multiplicity of detail4 ,thus a thing is
 one  substance as embodying an         
individual experience1 and its qualities
  belong to it in the same sense as the 
constituents of experience  belong to   
the single experience4 ,these qualities 
are in relation4  7,the ,pragmatist     
denies the existence of relations as    
part of a  higher unity4#bc7 ,but they  
are not only relation1 since relation   
always implies something more than      
itself4 ,let us take the example  of    
number4 ,numbers could never have been  
counted if there had not  been things to
 count4 ,now suppose each quality could 
be analysed  into a new relation1 we    
should still not get rid of the quality4

 ,at  each stage there remains a quality
 in relation1 and this goes on  to      
,infinity4 ,such a constant subdivision 
perhaps results from our  finite        
experience seizing facts in a disjointed
 way4 ,when we analyse  a law in its    
working1 we always do seem to come to   
this ,indefinite  ,regress4 ,now it has 
been the reproach against metaphysics1  
as uttered  by the ,pragmatist1 that    
there is no correspondence in scientific
 fact  to this road into ,infinity4     
,w4 ,james asserts3 8,but in point of   
fact1 nature doesn't make eggs by       
making first half an egg1 then a        
quarter1 then an eighth1 c41 and  adding
 them together4 ,she either makes a     
whole egg at once or none  at all1 and  
so of all her other units4 ,it is only  
in the sphere of  change1 then1 where   
one phase of a thing must needs come    
into being  before another phase can    
come1 that ,zeno's paradox gives        
trouble4  ,and it gives trouble then    
only if the successive steps of change  

be  infinitely divisible40#bd    ,the   
sphere of change1 however1 one would    
answer1 includes all nature1  and       
science in its discoveries acts on the  
hypothesis that these  steps of change  
may be infinitely divisible4 ,royce held
 to it firmly  that any consistent      
attempt to make an orderly arrangement  
of the  terms of an infinite whole must 
lead to the ,indefinite ,regress4 ,and  
he further shows the connection with the
 fact that an infinite series  can be   
adequately represented by a part of     
itself4    ,in the ,boyle ,lecture1     
delivered in ,oxford in #aijh1 on the   
properties  of radium1 two facts emerged
 which show that the ,indefinite        
,regress  is now recognised in science4 
,first1 that in the region of experiment
 we become aware of groups of  elements 
allied to radium1 which seem1 in the    
number of individuals  in their groups1 
to follow a simple arithmetical         
progression4    ,secondly1 that         
radio-active elements lose in activity  

at a certain  rate1 which always        
represents an exact proportion of the   
mass which  remains4 ,the tremendous    
disintegrating force slackens in exact  
relation to the time which passes1 so   
that the smaller the morsel  the less   
the relative loss of mass4 ,here1 then1 
is the ,indefinite  ,regress4 ,in the   
world of fact as well as of ideas we are
 dealing with  aspects of ,infinity4#be 
,,ii4 ,,infinite ,,series    ,there are 
other aspects of ,infinity which we can 
get at by studying  series1 and which in
 the conception of series of series give
 strength  and point to the philosophic 
conception of an ,absolute4    ,prof4   
,c4 ,keyser develops this thought1 and  
shows 7in two recent  articles1 ,january
 and ,april #aiji1 in the ,hibbert      
,journal7 that  certain theological     
dogmas1 such as the doctrine of the     
,trinity1  and certain attributes of the
 ,divine ,being1 such as ,omniscience   
and  ,omnipresence1 are entirely        
conceivable by the human mind if        

regarded  without the paralysing        
limitations of the ,finite4 ,he shows   
that in  our mathematical formul which  
have to do with infinite series we  have
 the exact replica of what to the lay1  
non-mathematical mind seem  to be the   
paradoxes of the ,athanasian ,creed4 ,he
 first shows that in  a mathematical    
analogy points of view about an         
,infinite ,being1 even  if partially    
discordant1 may all be true if regard is
 had to ,his  ,infinity4#bf    ,further1
 he shows that certain assumptions1 such
 as the whole  is greater than its part1
 are inapplicable to ,infinite ,being4  
,the conception of a ,trinity in ,unity 
in which 8none is afore or  after other1
 none is greater or less than another1  
but the whole  three persons are        
co-eternal together and co-equal0 is    
rationally  conceivable by the          
mathematician who is familiar with the  
theory of  manifolds4#bg    ,we have1 he
 shows1 three infinite manifolds3--     
,e of the even integers4        ,o of   

the odd ones4        ,f of the fractions
 having integers for their terms4    ,no
 two of these have a single element in  
common1 yet the three  together         
constitute one manifold ,m1 that is     
exactly equal in wealth  of elements to 
each of its infinite components4        
,again1 there is the apparent opposition
 between the ,omniscience of  ,god and  
the freedom of man4 ,the antithesis     
disappears if we realise  that from the 
point of view of ,infinites the dignity 
and power of  ,omniscience remain the   
same1 even if some part of experience is
 not  yet drawn into the sphere of      
,omniscience4#bh    ,here we have the   
present conceived of as a moving plane  
separating  the unknown from the known4 
,the 8past0 can be said to be known1    
though its content changes every        
instant4 ,this is the real answer  to   
,w4 ,james's cry that he could accept an
 ,absolute if it had even  the fragment 
of an 8other40 ,there can be this       
8other10 and yet the  ,absolute still   

remains an ,absolute4    ,the doctrine  
of ,omnipresence follows from the       
argument of the  ,continuum 7which is   
the aggregate of all real numbers74     
,thus the  number of points in space of 
infinite dimension is no greater than   
the number of points in any part of     
space as known to us4 ,the whole  is    
incarnate in every part1 because to each
 part1 in however small an  atom1       
corresponds a point in the universal    
whole1 and the number of  points in a   
space of infinite dimensions is equal to
 the number of  points in a straight    
line however small4    ,and this is true
 not merely of points but also of       
forces4 8,the  ,universe is dynamic1    
charged throughout with innumerable     
modes of  motion4 ,each point1 however1 
of any moving thing--an ion of gas1 a   
vibrating fibre of brain--is represented
 by a corresponding point  in ,s 7a     
small typical atom71 and so within the  
tiny sphere--indeed1  in every room1    
however small--the whole dynamics of the

 universe  is depicted completely and   
co-enacted by motion of points and      
transformation of point configurations4 
,there in miniature proceed at  once the
 countless play and interplay of every  
kind of motion1 small  and large1 simple
 and complex1 the quivering dance of the
 molecule1  the wave and swing of       
universal ther40#bi                     
,,iii4 ,,dimensions ,,in ,,space        
,there is another argument1 one relating
 to the theories of time  and space1    
which greatly affects the conception of 
omnipresence4  ,this is the argument of 
the many dimensions1 called by ,keyser  
the  8radiant concept of hyper-space1   
which only a generation ago was         
regarded1 even by mathematicians--most  
adventurous of men--as being            
purposeless and vain1 but which         
meanwhile has advanced so rapidly to    
commanding position that even the       
following statement by ,poincar1  in his
 recent address before the              
,international ,mathematical ,congress  

at ,rome on #8,l',avenir des            
,mathmatiques1' is well within the      
limits  of conservatism3 #8,nous sommes 
aujourd'hui tellement familiariss  avec 
cette notion que nous pouvons en parler1
 mme dans un cours  d'universit1 sans   
provoquer trop d'tonnement4' ,the fact  
is that the  doctrine already exists in 
a vast and rapidly growing literature1  
flourishes in all the scientific        
languages of the world1 and in its      
essential principles has become for     
mathematics as orthodox as the          
multiplication table40    ,the present  
position of the theory is briefly this3 
,if there did  not exist a fourth       
dimension1 we could not be aware of a   
third as  such1 and so on4 ,are we then 
looking out upon a third dimensional    
world1 and realising it as such because 
we are mentally capable of  conceiving  
dimensions beyond it8 ,our world        
sensibly contains one  dimensional and  
two dimensional facts--the first such as
 a time  series1 for which one number is

 sufficient to fix a point1 and the     
second such as a plane where position   
can be fixed by two numbers4  ,does our 
world contain facts of other dimensions8
    8,all particles of air are          
four-dimensional in magnitude when1  in 
addition to their position in space1 we 
also consider the  variable densities   
which they assume1 as they are expressed
  by the different heights of the       
barometer in the different  parts of the
 atmosphere4 ,similarly all conceivable 
spheres in  space are four-dimensional  
magnitudes1 for their centres form  a   
three-dimensional point-aggregate1 and  
around each centre a  one-dimensional   
totality of spheres1 the radii of which 
can be  expressed by every numerical    
magnitude from zero to infinity4        
,further1 if we imagine a               
measuring-stick of invariable length to 
assume every conceivable position in    
space1 the positions so obtained  will  
constitute a five-dimensional aggregate4
 ,for in the first place  one of the    

extremities of the measuring-stick may  
be conceived to  assume a position at   
every point of space1 and this          
determines for  one extremity alone of  
the stick a three-dimensional totality  
of  position1 and1 secondly1 as we have 
seen above1 there proceeds from  every  
such position of this extremity a       
two-dimensional totality  of directions1
 and by conceiving the measuring-stick  
to be placed  lengthwise in every one of
 these directions1 we shall obtain all  
the  conceivable positions which the    
second extremity can assume1 and        
consequently the dimensions must be #c  
#b or #e '''0 c41 c4#cj                 
,mathematicians have for long done      
problems in the seventh and eighth      
dimensions4 ,they have told us that you 
cannot tie a knot in the  second        
dimension1 because there is no up or    
down1 and the threads  would not        
cross--nor in the fourth1 because the   
knot would pull out  in a new direction 
and would not hold4 ,but it has only    

lately been  realised that fourth and   
other dimensions may be actual fact in  
the  world round us4 ,of course1 from   
the point of view of a point there  are 
only three dimensions to be known1 but  
to a line in the same  space there are  
five1 to the surface probably six4 ,our 
intelligence  at present does not go    
beyond the point2 but if we could think 
of  space from the point of view of a   
solid1 worlds upon worlds would  rise   
before our view4    ,of the fourth      
dimension we can discover some facts by 
analogy4 ,we  can count the edges of its
 typical figure1 and apply thought to   
determining some of its conditions4 ,but
 a more interesting subject  of research
 is the inquiry into the light thrown by
 the theory of  four dimensions on the  
determination of certain atoms in       
chemistry1  that are known to be        
distinct elements1 but could only be    
determined  actually in another         
dimension4#ca    8,in chemistry1 the    
molecules of a compound body are said to

 consist  of the atoms of the elements  
which are contained in the body1 and    
these are supposed to be situated at    
certain distances from one  another and 
to be held in their relative positions  
by certain  forces4 ,at first the       
centres of the atoms were conceived to  
lie in  one and the same plane4 ,but    
,wislicenus was led by researches in    
paralactic acid to explain the          
differences of isomeric molecules of    
the same structural formul by the       
different positions of the atoms  in    
space4 ,in fact1 four points can always 
be so arranged in space  that every two 
of them may have any distance from each 
other2 and the  change of one of the six
 distances does not necessarily involve 
the  alteration of any other4    8,but  
suppose our molecule consists of five   
atoms8 ,four of these  may be so placed 
that the distance between any two of    
them can be  made what we please4 ,but  
it is no longer possible to give the    
fifth  atom a position such that each of

 the four distances by which it  is     
separated from the other atoms may be   
what we please4 ,on the  contrary1 the  
fourth distance is dependent on the     
three remaining  distances1 for the     
space of experience has only three      
dimensions4 ,if1  therefore1 ,i have a  
molecule which consists of five atoms1  
,i cannot  alter the distance between   
two of them without at least altering   
some  second distance4 ,but if we       
imagine the centres of the atoms placed 
in a four-dimensioned space1 this can be
 done2 all the ten distances  which may 
be conceived to exist between the five  
points will then be  independent of one 
another4 ,to reach the same result in   
the case of  six atoms we must assume a 
five-dimensional space1 and so on40#cb  
,here we see that if chemistry as a     
science is bound to take account  of all
 its facts1 the scientist is confronted 
with a problem of  dimensions that is   
really a problem of ,infinity applied   
not1 as in  the other cases quoted1 to  

number1 but to space4    ,and there is a
 reason which explains why the same     
problem tends  to appear in these       
different ways4 ,both time and space can
 be  most correctly thought of as       
series3 the former known to us as       
possessing one direction1 though        
possibly involving more1 and the  latter
 three1 though possibly involving more4 
,time is not a thing nor  a condition1  
but it is the way in which we are       
enabled to apprehend  the relations of  
actions to one another4 ,the assumption 
of the  ,pragmatist1 that a different   
date in history is a new condition which
  might affect a chemical experiment1 is
 meaningless1 unless by that he  intends
 to say that at the different date new  
conditions prevailed4    ,the general   
conclusion of recent thought is then to 
establish  the ,idealist position more  
strongly by an appeal to mathematical   
argument4 ,this argument is strengthened
 by finding at the present  time some   
support in scientific fact and          

experiment4 ,the ,idealist  therefore   
appeals to fact1 and his position rests 
ultimately on a  truth which has its    
aspects of conformity with scientific   
experiment  and with logical or         
mathematical proof4      ,,footnotes3   
#ae ,this word is used here in the most 
general and            inclusive sense  
as applying to all thinkers who accept  
the reality of relations as part of a   
higher ,unity4         #af 8,a          
,pluralistic ,universe10 p4 #ii4        
#ag ,taylor1 8,elements of              
,metaphysics10 p4 #cab4         #ah     
,ibid41 p4 #cej4         #ai ,a         
succession of what is disconnected is   
not            change4 ,change is a     
succession within an identity3          
if not within the identity1 there is no 
change1 only            analysis and    
re-grouping4 ,the closer our knowledge  
is            of ourselves or anything  
else1 the more we see that              
change is the expression in time of an  
identity4         #bj ,illingworth1     

8,the ,doctrine of the ,trinity10 p4 #f4
         #ba ,ormond1 8,foundations of  
,knowledge10 p4 #ai4         #bb        
#aijh-aj4         #bc ,james1 8,a       
,pluralistic ,universe10 p4 #hj4        
#bd 8,a ,pluralistic ,universe10 p4     
#bcj4         #be ,see ,a4 ,t4 ,cameron1
 8,radio-,chemistry10 p4 #ag3 8,the     
curves illustrate two further points4   
,they approach            constant value
 towards the end of a month1 but it is  
seen that they reach a final value only 
at infinite            time4 ,this      
property is common to all such curves2  
it            illustrates the fact that 
the life of a radio-active              
element is infinite40 ,it is explained  
in the same book            7p4 #ca7    
that 8infinity is only a relative term2 
in            this connection it only   
means a longer time than we             
can measure40         #bf ,his theology 
is not so good as his mathematics2      
he seems to think that in the ,creed we 
assert our            belief in the     

,incomprehensible1 in the sense of      
that which is 8not capable of being     
seized by the            mind10 instead 
of in that which is 8untrammelled by    
limitations40 ,the word is immensus1    
best translated            infinite4    
#bg ,hibbert ,journal1 #aiji1 pp4       
#fbf-bh4         #bh ,hibbert ,journal1 
#aiji1 p4 #fbi4         #bi ,hibbert    
,journal1 #aiji1 p4 #fcb4         #cj   
,schubert1 8,mathematical ,essays and   
,recreations10            pp4 #gj1 #ga4 
#ca ,see ,van t' ,hoff1 ,la ,chimie dans
 l'espace1 and            ,schubert1    
8,mathematical ,essays and              
,recreations10 pp4            #hh-hi4   
#cb ,schubert1 8,mathematical ,essays   
and ,recreations10            pp4       
#hh-hi4 ,see also ,mach1 8,conservation 
of ,energy0            7trans4 ,open    
,court ,publishing ,co474               
,printed by ,,ballantyne1 ,,hanson      
,,co4                                   
,edinburgh  ,london                     
,transcriber's ,note3      ,words and   

phrases in italics are surrounded by    
underscores1 like  this4 ,footnotes were
 renumbered sequentially and were moved 
to the  end of the chapter4 ,words may  
have inconsistent hyphenation in the    
text4 ,one misspelled word was          
corrected4