BERTRAND RUSSELL’S MATHEMATICS AND GEOMETRY: BRITISH NEO–KANTIANISM
Bertrand Arthur William Russell (1897)
Geometry, throughout the 17th and 18th centuries, remained, in the war against empiricism, an impregnable fortress of the idealists. Those who held as was generally held on the Continent that 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 … 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 hypothetical: If Geometry has apodeictic certainty, its matter, i.e. space, must be à 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 …  we can no longer affirm, on purely geometrical grounds, the apodeictic certainty of Euclid. But unless Metageometry has done more than this — unless it has proved, what I believe it alone cannot prove, that Euclid has not apodeictic certainty — then Kant’s other line of argument retains what force it may ever have had …  I shall contend, as a result of these conclusions, that those axioms, which Euclid and Metageometry have in common, coincide with those properties of any form of externality which are deducible, by the principle of contradiction, from the possibility of experience of an external world. These properties, then, may be said, though not quite in the Kantian sense, to be à priori properties of space, and as to these, I think, a modified Kantian position may be maintained … The next step, in defining a form of externality, is obtained from the idea of dimensions. Positions, we have seen, are defined solely by their relations to other positions. But in order that such definition may be possible, a finite number of relations must suffice, since infinite numbers are philosophically inadmissible. A position must be definable, therefore, if knowledge of our form is to be possible at all, by some finite integral  number of relations to other positions. Every relation thus necessary for definition we call a dimension. Hence we obtain the proposition: Any form of externality must have a finite integral number of dimensions …The above argument, it may be urged, has overlooked a possibility. It has used a transcendental argument, so an opponent may contend, without sufficiently proving that knowledge about externality must be possible without reference to the matters external to each other. The definition of a position may be impossible, so long as we neglect the matter which fills the form, but may become possible when this matter is taken into account. Such an objection can, I think, be successfully met, by a reference to the passivity and homogeneity of our form. For any dependence of the definition of a position on the particular matter filling that position, would involve some kind of interaction between the matter and its position, some effect of the diverse content on the homogeneous form. But since the form is totally destitute of thinghood, perfectly impassive, and perfectly void of differences between its parts, any such effect is inconceivable …  The Kantian argument — which was correct, if our reasoning has been sound, in asserting that real diversity, in our actual world, could only be known by the help of space — was only mistaken, so far as its purely logical scope extends, in overlooking the possibility of other forms of externality, which could, if they existed, perform the same task with equal efficiency. In so far as space differs, therefore, from these other conceptions of possible intuitional forms, it is a mere experienced fact, while in so far as its properties are those which all such forms must have, it is à priori necessary to the possibility of experience.¹
1. Bertrand Arthur William Russell (1872–1970), An Essay on the Foundations of Geometry, Cambridge, Cambridge University Press, 1897, 1–56–57–140–186.
See: “It is the Transcendental Deduction that has played the most important part in the arguments of the English Kantio–Hegelians.”
Andrew Seth Pringle–Pattison in Hiralal Haldar, Essays in Philosophy,Calcutta, University of Calcutta, 1920, 6.
See also: “[Hegel] was great, on the one hand by his metaphysical results, on the other by his logical method; on the one hand as the crown of dogmatic philosophy, on the other as the founder of the dialectic, with its then revolutionary doctrine of historical development. Both these aspects of Hegel’s work revolutionized thought … the practical tendency of his metaphysics was, and is, to glorify existing institutions, to see in Church and State the objective embodiment of the Absolute Idea, his dialectic method tended to exhibit no proposition as unqualified truth, no state of things as final perfection … The validity of this view we need not here examine; it is sufficient to point out that Hegel, in his ‘Philosophy of History,’ endeavored to exhibit the actual course of the world as following the same necessary chain of development which, as it exists in thought, forms the subject of his logic … the development of the world therefore proceeds by action and reaction, or, in technical language, by thesis and antithesis, and these become reconciled in a higher unity, the synthesis of both …  we might live to see another French Revolution, perhaps even more glorious than the first, leaving Social Democracy to try one of the greatest and most crucial experiments in political history.”
Bertrand Arthur William Russell, German Social Democracy: Six Lectures, With an Appendix on Social Democracy and the Woman Question in Germany by Alys Russell, London/New York, Longmans, Green and Company, 1896, 2–163.
See finally: “No logical absurdity results from the hypothesis that the world consists of myself and my thoughts and feelings and sensations, and that everything else is mere fancy … Philosophy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true.”
Bertrand Arthur William Russell, The Problems of Philosophy, London, Williams & Norgate, 1912, 34–249.