164: Proceedings of the Royal Irish Academy. 



Mathematics in both its great departments is therefore competent to 

 give us new truths. Are these truths of any vakie ? That great philosopher 

 John Locke, by his inaccurate expressions and hazy modes of thinking, led 

 many to believe that Mathematics is concerned only with what Locke's 

 followers called abstract ideas, but not with anything else. 



Locke does not appear to have grasped the true state of the case, viz. that 

 Geometry has to do directly, not with material objects, but with figures 

 in pure space, and Algebra with series, discrete or continuous, in time. 

 Mathematics is, indeed, the science of space and time ; and consequently its 

 truths condition all our knowledge of material objects, wdiich cannot be 

 cognized except in space and time, and in conformity with their laws. When 

 Locke said that no perfect circle exists, if he meant by a circle an object of 

 sense, he was no doubt right ; but if he meant a figure in pure space, he was 

 entirely wrong, for perfect circles exist as much as space itself. 



The boundary between two contiguous portions of space is a perfect 

 mathematical surface, the boundary between two portions of a surface a 

 perfect line, and the boundary between two portions of a line a pouit. The 

 existence of mathematical figures is therefore as real as that of objects of 

 experience — indeed more real, for space and time are necessary conditions 

 uf our consciousness. To Kant we owe the complete explanation of the 

 nature of jMathematics : and his theory seems to be the only one which fully 

 accounts for all the characteristics of mathematical truths. 



There is, however, a School of I'hilosophy different from his, which, 

 though not falling into the mistake of supposing mathematical theorems to 

 be analytical propositions, and mathematical processes to be nothing but 

 pure reasoning, yet fails to appreciate their true character, and looks upon 

 them as merely experimental truths, and modes of obtaining results by the 

 aid of experience. Among philosophical writers, one of the ablest advocates 

 of this theory was John Stuart Mill, who, following Bain, tried to reduce 

 space to a series of muscular sensations in time. He failed, however, to 

 account for the fact that the intuitions by means of which new truths are 

 arrived at in JMathematics can be obtained from the representations of the 

 imagination without any appeal to experience, and that yet these truths apply 

 to objects of experience. 



The theory that geometrical axioms are simply experimental truths was 

 held by the^illustrious Helmholtz. Intimately coimected with this theory is 

 the ISTon-Euclidean Geometry of which Helmholtz was an upholder, and 

 which is looked upon with favour by many distinguished mathematicians. 



The Non-Euclidean Geometry, originally started by the Eussian 

 mathcDiiatician Lobatschewskv, asserts that there is no evidence for the 



