192 Proceedings of the Royal Irish Academif. 



until some one makes use of them. This is a relapse to the Berkeley-Hume 

 theory, that Space and Time are composed of a finite number of points and 

 instants {minima sensihilia), and finally ends with the absurdity that the 

 number of points on a straight line is, say, the greatest number that anyone 

 perceives at the present moment; that Space is essentially non-Euclidean, 

 since all straight lines come to an end on its boundary ; and that the Past 

 began when the oldest person now living began to have conscious experience. 

 All these absurdities are due to the refusal to go beyond imagination or 

 intuition, and to the fallacious distinction between phenomenon and thing 

 2Jer se. 



The simple solution of the difficulty is that the conception of actually 

 infinite number is not self-contradictory ; it is quite conceivable, though not 

 imaginable. G-eorge Cantor has placed this fact beyond doubt. There is no 

 longer an Antinomy. Thus the logical or hypothetical or ideal theory of 

 mathematics is necessary in order to justify the application to actual entities 

 of the conception of Infinity, whereas the purely intuitional theory defeats 

 its own end in its haste to grasp reality in a single image. 



IX. 



If the utility of such abstract investigations is questioned, it is enough to 

 reply (without tracing the ethical problem any further) that not only Logic 

 and Philosophy, but Pure Mathematics itself, is moving in the direction 

 pointed out by this kind of logical analysis ; that such analysis has a directly 

 practical value, because it tends to satisfy an intellectual need felt by many 

 thinkers ; and that, in the course of time, it is likely to influence the more 

 abstract parts of Applied Mathematics, by checking romanticism, and by 

 assisting in the formation of new conceptions of Nature, suggested and 

 perhaps mentally retained by the imagination, but not representable except 

 by precise definition. 



X. 



There is a continuous logical order connecting all branches of this subject. 

 Integers are defined by three fundamental ideas; next, rationals (a class 

 quite distinct from integers) are defined. Eeal numbers (including irrationals 

 and algebraic numbers) are most satisfactorily treated as transfinite sets 

 of rationals, and provide us with the conception of one-dimensional 

 series of the kind required in Euclidean, Cartesian, and all forms of non- 

 Euclidean Geometry. The doctrine of Transfinite numbers, cardinal and 

 ordinal, leads, or will lead, to a clearer classification of infinite series and of 



