[ 182 J 



IX. 



THE LOGICAL BASIS OF MATHEMATICS. 

 By E. a. p. EOGERS, F.T.C.D. 



Eead Januaky 27. Ordered for Publication JA^'UAEY 29. Published March 11, 1908. 



The object of this paper is to show generally that there are reasons for 

 believing that, bj the use of a limited number of mutually consistent axioms, 

 definitions, and premisses involving indefinables, it is possible to deduce all 

 the conclusions of mathematics by means of logical reasoning alone. The 

 full proof of this thesis would be to actually state these premisses. This 

 has to a large extent been done (some references are given below). Here 

 the subject is discussed from a general point of view without entering into 

 details, and the criticisms are necessarily brief. 



That the premisses of mathematics, if it is to be useful in increasing our 

 knowledge of the laws of nature, must be suggested by some experiences of 

 objective reality, no one can deny. I have thus no quarrel with the intuitional 

 or empirical "sdews, provided it is understood that intuition or perception is 

 only to suggest the premisses ; to logic exclusively belongs the demonstration. 

 Immediate experience or intuition has always given the start to mathematical 

 investigation ; but if mathematics never went beyond immediate experience, 

 by the aid of logic, it would be absolutely useless. The hyper-practical view 

 of mathematics — the theory which insists on actualizing in the material world 

 every step of the reasoning — is thus suicidal, because the practical value of 

 this — as of every other Deductive Science — is due to the fact that it leaves 

 the world of immediate experience behind, and, by leaving it, obtains new 

 results which, in many cases, may be applied and verified in direct experience, 

 whether by intuition or by measurement. Logical principles, including the 

 Law of Contradiction, are the correlatives in this ideal process of the 

 L'niformity of Nature and of the Permanence of certain real physical relations. 



"Without entering into a criticism of the well-known Kantian distinction 

 between ' pure ' intuition and ' empirical ' perception, I shall assume that 

 Kantians and the mere empiricists agree in regarding the objects of 

 mathematics as being immediately given images. And though, as just stated, 

 intuition is always used in mathematics, the term ' intuitionism ' will be 

 understood to connote the extreme ^-iew that all mathematical reasoning 

 consists in experimenting with particular images. 



