An essay on the foundations of geometry

An essay on the foundations of geometry

1. Geometry, throughout the 17th and 18th centuries, remained, in the war against empiricism, an impregnable fortress of the idealists. Those who heldas was generally held on the Continentthat certain knowledge, independent of experience, was possible about the real world, had only to point to Geometry: none but a madman, they said, would throw doubt on its validity, and none but a fool would deny its objective reference. The English Empiricists, in this matter, had, therefore, a somewhat difficult task; either they had to ignore the problem, or if, like Hume and Mill, they ventured on the assault, they were driven into the apparently paradoxical assertion that Geometry, at bottom, had no certainty of a different kind from that of Mechanicsonly the perpetual presence of spatial impressions, they said, made our experience of the truth of the axioms so wide as to seem absolute certainty. Here, however, as in many other instances, merciless logic drove these philosophers, whether they would or no, into glaring opposition to the common sense of their day. It was only through Kant, the creator of modern Epistemology, that the geometrical problem received a modern form. He reduced the question to the following hypotheticals: If Geometry has apodeictic certainty, its matter, i.e. space, must be priori, and as such must be purely subjective; and conversely, if space is purely subjective, Geometry must have apodeictic certainty. The latter hypothetical has more weight with Kant, indeed it is ineradicably bound up with his whole Epistemology; nevertheless it has, I think, much less force than the former. Let us[2] accept, however, for the moment, the Kantian formulation, and endeavour to give precision to the terms priori and subjective. 2. One of the great difficulties, throughout this controversy, is the extremely variable use to which these words, as well as the word empirical, are put by different authors. To Kant, who was nothing of a psychologist, priori and subjective were almost interchangeable terms[1]; in modern usage there is, on the whole, a tendency to confine the word subjective to Psychology, leaving priori to do duty for Epistemology. If we accept this differentiation, we may set up, corresponding to the problems of these two sciences, the following provisional definitions: priori applies to any piece of knowledge which, though perhaps elicited by experience, is logically presupposed in experience: subjective applies to any mental state whose immediate cause lies, not in the external world, but within the limits of the subject. The latter definition, of course, is framed exclusively for Psychology: from the point of view of physical Science all mental states are subjective. But for a Science whose matter, strictly speaking, is only mental states, we require, if we are to use the word to any purpose, some differentia among mental states, as a mark of a more special subjectivity on the part of those to which this term is applied. ......Buy Now (To Read More)

Product details

Ebook Number: 52091
Author: Russell, Bertrand
Release Date: May 17, 2016
Format: eBook
Language: English