The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics by John L. Bell

Author:John L. Bell
Language: eng
Format: epub, pdf
ISBN: 9783030187071
Publisher: Springer International Publishing

But in the end archimedean systems are to be preferred on the grounds of simplicity: The validity of Veronese’s system has been criticized by Cantor and others on the ground that the definitions contained in it, relating to equality and inequality, lead to contradiction; it is however unnecessary for our purpose to enter into the controversy on this point. The straight line of geometry is an ideal object of which any properties whatever may be postulated, provided that they satisfy the conditions, (1), that they form a valid scheme, i.e. one that does not lead to contradiction, and (2), that the object defined is such that it is not in contradiction with empirical straightness and linearity. There is no a priori objection to the existence of two or more such adequate conceptual systems, each self-consistent even if they be incompatible with one another; but of such rival schemes the simplest will naturally be chosen for actual use. Assuming then the possibility of setting up a valid non-Archimedean system for the straight line, still the simpler system, in which the system of Archimedes is assumed, is to be preferred, because it gives a simpler conception of the nature of the straight line and is adequate for the purposes for which it was devised. 153

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