190 Proceedings of the Royal Irish Academy. 



The question, Is geometry hypothetical ? includes the question, Do 

 indivisible points, lines, and surfaces actually exist ? On this point expert 

 thinkers have different views ; and this proves that the ideal theory is the safer. 

 If we knew what ' exist ' meant, the question might be answered definitely. 

 That they have a logical existence there can be no doubt, as terms whether 

 definable or indefinable. 



Similarly we have no right to assert dogmatically the physical existence 

 of indivisible moments of Time ; psychology shows that experience of such a 

 moment is impossible. But if we assume their existence, we must also 

 assume the existence of indivisible surfaces in Space, because motion implies 

 a correlation between elements of Space and the elements of Time. From the 

 logical point of view, however, the units of Space and Time may be regarded 

 as referring to definite divisible portions of each. 



The concepts of Euclidean geometry are thus, I hold, logically real, and 

 practically useful ; but the question of the existence of exact extra-mental 

 correlatives may be put on one side as being metaphysical.* This proves that 

 the non-Euclidean view is actually the only intelligible way of explaining 

 the reasoning in Euclidean geometry. Thus the term ' point ' is only a 

 logical name for the material property of position, which, however, in rcriim 

 natv/ra, always involves filling Space; the Logical or Hypothetical view of 

 mathematics saves us from all metaphysical questions about the extra-mental 

 existence of points. The physical correlatives of the logical points may be 

 Space-filling volumes, if we please. 



The Space of geometry, whether Euclidean or otherwise, is not given by 

 intuition or by experience. To speak figuratively, it is an ideal logical 

 structure, the properties of which — that is, axioms, terms, and definitions — 

 are only suggested by images given by experience. But the properties of 

 geometrical Space are never given in immediate experience; nor can we say 

 strictly that the Space of experience forms even a part of the denotation of 

 the logical concept, for it is incomplete. 



To take only one example. We assume that between any two points 

 on a line there exists another point. Imagination or intuition (pure or 

 otherwise) can never give an image satisfying this logical axiom of a compact 

 series, because this would imply the intuition of an infinite number. 



Those who assert that Imagination actually gives the ideal logical 

 structure will have to decide whether such images are given by sight or by 

 what senses, and will find themselves, whatever answer they give, in a 

 variety of difficulties escaped by the logical view. 



* Kant adopts this view ia the Dialectic (Bk. ii, Ch. iii, § 4), but his followers seem to have 

 disowned it, and cling to the Aesthetic, 



