RoGEKS — The Logical Basis of Mathematics. 191 



VII. 



I must add that the evidence for the logical theory has become 

 overwhelmingly strong in the last few years or so, owing to the large amount 

 of accurate and careful work that has been done in the subject. The 

 logicians have taken to constructive measures ; and they can only be refuted 

 now by those who have taken the trouble to learn some of their methods. 

 Thus, for example, Peano shows that finite integers can be defined by the 

 use of three fundamental ideas or indefinables. Infinite series of the kind 

 required in geometry and elsewhere may be defined and classified by certain 

 universal, and therefore logical, properties. To Dedekind and George Cantor 

 the beginnings of this work are due. 



The Euclidean treatment of irrationals and incommensurables (based on 

 Euclid's theory of ratio) has been shown by Professor F. Purser to lead to the 

 ordinary symbolism of simple algebra. This way of treating the subject is 

 both interesting and pleasing; and we are not troubled with explicit 

 statements of the axioms used. The Dedekind-Cantor method of treating 

 the subject explicitly states the axioms, and shows that the laws of 

 irrationals (including algebraic and transcendental numbers) are laws of one- 

 dimensional series of a definable nature, and that spatial figures are not 

 required to establish them. Euclid's theory really assumes a particular 

 form of Dedekind's axiom. 



VIII. 



The Logical analysis of mathematics, besides throwing light on the 

 foundations of the science, and giving promise of extending our knowledge 

 of Functions generally, has a value for metaphysics as well as for 

 mathematics, in that it has succeeded in clearing up the once troublesome 

 question of infinite number. Every mathematician knows that no finite 

 integer is the greatest, that no fraction is the smallest, that between any two 

 points in a line there is always another, that there is no largest and no 

 smallest possible figure, volume, or line, no first and no last moment of time. 

 Thus mathematicians use the conception of a class containing an infinite 

 (transfinite) number of members. This conception is not indefinite, but quite 

 definite and precise, because, for example, every point of a straight line is 

 sharply distinguishable from every other point. The Kantian difficulty 

 about this as implied in the Dialectic is self-created ; the actual infinite, it is 

 said, does not exist, because it cannot be imagined ; and the explanation then 

 put forward is that points, past Time, outer Space, and so forth, do not exist 



