238 SCIENCE PROGRESS 



" If \, /I, p are distinct members of k, and any two of them 



possess a common member but there is no member 



common to all three, and if the same is true for X, fM, a, 



then p and a- possess a common member." 



In all about ten or a dozen such " axioms " are required to 



produce a subject interesting from its apparent relevance to the 



space of experience. 



Then the subject consists of the hypothetical propositions 

 which can be proved concerning k, ivhatever class of classes k 

 may be, where the hypotheses of the propositions consist of all 

 or some of these ** axioms " concerning k. These hypothetical 

 are true for any value of k, whether the "axioms" which from 

 the hypotheses be false or true for that value. And further if 

 for some special determination of k, we know that the " axioms " 

 are true, then the conclusions of the propositions are also true 

 for this value of k. These further propositions, arising from a 

 special determination of k for which the "axioms" are true, 

 belong in general to applied mathematics. 



Accordingly the modern mathematician withdraws, or should 

 withdraw, from the apodictic certainty of his results the whole 

 question of fact involved in any application of his reasoning. 

 What then are the properties of space as known to us in' our 

 common human experience ? The mathematician should leave 

 the answer to the physicist, or to the psychologist, or to the 

 plain man in the street, or to the metaphysician, or to whomever 

 is the proper person to deal with this question. It is true that 

 mathematics forms an essential element in the inductive proof 

 of the properties of space, if the proof is inductive ; an 

 interesting chapter could be written upon this point. But the 

 result arrived at has not the shadow of a right to masquerade 

 as possessing the certainty of mathematics ; it may be as certain 

 as you like but it is not mathematical certainty. This rapid 

 account, taking geometry as an example, must suffice to show 

 how the discovery of the doctrine of the variable has enabled 

 mathematical philosophy to be reconstructed. 



Another distinction, which the book under consideration 

 ignores in practice, is that the problem of the presentation of 

 mathematical ideas to students in the initial stages of know- 

 ledge must be entirely separated from the discussion of the true 

 " principia " of the subject. Elementary mathematics and the 

 elements of mathematics are widely different subjects, though 



