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Additional info for The Foundations of Mathematics; A Contribution to the Philosophy of Geometry
The concrete items of the whole remain, in their constitutional elements, the same. N o energy is lost; no particle of matter is annihilated; and the change that takes place is mere transformation. 3 The l a w of causation is otherwise in the same predicament as the norms of logic. I t can never be satisfactorily proved by experience. Experience justifies the a priori and verifies its tenets in single instances which prove true, but single instances can never demonstrate the universal and necessary va lidity of any a priori statement.
H e shows that the mathematician starts from definitions and then proceeds to show how the product of thought may originate either by the single act of creation, or by the double act of positing and combining. The former is the continuous form, or magnitude, i n the narrower sense of the term, the latter the discrete form or the method of combination. H e distin guishes between intensive and extensive magnitude and chooses as the best example of the latter the sect or limited straight line laid down i n some definite direction.
Being void of particularity, it is universal; i t is the same throughout, and i f we proceed to build our air-castles in the domain of anyness, we shall find that considering the absence of all particularity the same construction w i l l be the same, wherever and whenever i t may be conceived. " B u t there is no "assumption" about i t . The atmosphere in which our mathemat ical creations are begotten is sameness. Therefore the same construction is the same wherever and whenever i t may be made.