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Kant and the Synthetic Nature of Geometry*

Published online by Cambridge University Press:  09 June 2010

Colwyn Williamson
Colwyn Williamson
Affiliation:
University College of Swansea
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Extract

The purpose of this paper is to explore the significance of Kant's claim that geometry is synthetic. I begin by outlining certain criticisms of the Kantian position, criticisms selected with an eye to their popularity, rather than their importance in the abstract. I am no expert on the textual exegesis of Kant, and serious Kantian scholars would not, perhaps, be much troubled by the criticisms I propose to discuss: indeed, they might properly maintain that some of these problems (for example, the significance of non-Euclidean systems) were, for them, resolved long ago. But within the prevailing tradition of English-speaking philosophy certain sorts of criticism of Kant do seem to have sunk deep into our attitudes. It is with these criticisms that I hope to settle accounts. This small aim leads to the more difficult one of trying to understand what Kant means when he says that geometry is synthetic: about this larger task I will make only some preliminary remarks.

Type
Articles
Information
Dialogue: Canadian Philosophical Review / Revue canadienne de philosophie , Volume 6 , Issue 4 , March 1968 , pp. 497 - 515
Copyright
Copyright © Canadian Philosophical Association 1968

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References

1 Ayer, A. J., Language, Truth and Logic, 82.Google Scholar

2 G. Frege, The Foundations of Arithmetic, 102–103: “In calling the truths of geometry synthetic a priori, he revealed their true nature. And this is still worth repeating, since even today it is often not recognized.”

3 Ayer, op. cit., 80.

4 Ibid., 78.

5 Bertrand Russell, Introduction to Mathematical Philosophy, 145.

6 S. Körner, Kant, 35. And see Ayer, op. cit., 82.

7 Immanuel Kant's Critique of Pure Reason, translated by Norman Kemp Smith (Kant), 48.

8 Kant, 190.

9 For this, and for some pertinent remarks about “contained in the meaning of”, vide: J. L. Austin, “The Meaning of a Word”, Philosophical Papers.

10 Vide: A. N. Prior, Formal Logic, 16.

11 Vide: Carnap, Rudolf, “The Logicist Foundations of Mathematics”, Philosophy of Mathematics (Ed. Benacerraf, and Putnam, ).Google Scholar

12 Vide: “Critical Philosophy and Mathematical Axiomatics”, Socratic Method and Critical Philosophy.

13 Russell, op. cit.; Ayer, op. cit., 83.

14 Kant, 52

15 Kant, 54 (my emphasis).

16 E.g. “… it has to be noted that mathematical propositions, strictly so called, are always judgements a priori, not empirical … If this be demurred to, I am willing to limit my statement to pure mathematics, the very concept of which implies that it does not contain empirical, but only pure a priori knowledge.” (Kant, 52).

17 Kant, 68.

18 Kant, 200.

19 Kant, 95.

20 Kant, 17–18.

21 Gottfried Martin, Kant's Metaphysic and Theory of Science, 18.

22 Vide: H.-J. de Vleeschauwer, La Déduction transcendentale dans l'auvre de Kant, Vol. I, Chap. III. The Lambert referred to is Johann Heinrich, distinguished Berlin mathematician.

23 G. Saccheri, Euclides ob omni naevo vindicatus.

24 This work of 1747 is translated in Gabriele Rabel, Kant. The specific quotation is from page 5 of Rabel's book.

25 Kant, 52.

26 Kant himself sometimes calls them axioms: e.g. Kant, 589.

27 Vide: Martin, op. cit., 17. & Vide: Die philosophischen Schriften von G. W. Leibniz (Ed. Gerhardt), I, 188, for the view that there are no real axioms except for the principle of identity.

28 Vide: Martin, ibid., 207, #2, 2, and especially Meinecke, W., “Die Bedeutung der nichteuklidischen Geometrie”, Kantstudien XI, 1906.Google Scholar

29 W. Kneale & M. Kneale, The Development of Logic, 385–386.

30 Martin, op. cit., e.g. 19.

31 Ibid., 88.

32 Nelson, op. cit., 164.

33 Ayer, op. cit., 84. Ayer, who in this regard is much worse than Kant, uses several quite different criteria for analyticity/syntheticity: the one I have cited is merely one of these criteria.

34 Vide: Martin, op. cit., 19.

35 Ayer, op. cit., 81.

36 For better or worse, and for reasons partly indicated, I want to say that, e.g., having three angles is not a property of a triangle in the sense in which having angles whose sum is 180 degrees is a property.

37 Kant, 577.

38 Kant, 590.