Abstract
Kant’s view of geometry has been widely criticised and generally rejected by twentieth-century analytic philosophers. In this paper I shall try to examine what I believe has come to be the most widespread form of their criticism, and I shall suggest that it is inconclusive. Although I have no wish to accept Kant’s view of geometry in its entirety, I nevertheless shall try to suggest a partial defense of it. My argument will be that those of its theses which have been most vigorously condemned by analytic philosophers are less indefensible than they have usually supposed.
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Notes
Michael Friedman in his Foundations of Space-Time Theories (Princeton, 1983), p. 3, discusses the strong linkage between logical positivism and the theory of relativity.
Moritz Schlick, Allgemeine erkenntnislehre (Berlin, 1925), translated as General Theory of Knowledge (New York, 1974), especially section 29.
Hans Reichenbach’s view of geometry developed through his close contact with Einstein from 1919 onwards. It was expressed especially in his Relativitätstheorie and erkenntnis apriori (Berlin, 1920), translated as The Theory of Relativity and A Priori Knowledge (Berkeley, 1965), and in his Philosophie der raum-zeit-lehre (Berlin, 1928), translated as The Philosophy of Space and Time (New York, 1958).
Rudolf Camap’s mature views receive a lively formulation in his Philosophical Foundations of Physics (New York, 1966), Part III.
One expression by Einstein of philosophical views about geometry is in his “Reply to Criticisms”, especially pp. 676–679, in P. A. Schillip (ed.), Albert Einstein: Philosopher-Scientist (Evanston, 1949).
Russell’s best known writings relating to the philosophy of mathematics have little to say specifically about geometry, but advocate an over-all view of mathematics according to which Euclidean geometry could not have the kind of preeminence Kant ascribes to it. See A. N. Whitehead and B. Russell, Principia Mathematica (Cambridge, vol. I, 1910; vol. 2, 1912; vol. 3, 1913). See also Russell’s more popularly written An Introduction to Mathematical Philosophy (London, 1919), especially chapter 18.
Carl G. Hempel wrote a pair of widely read articles which came to be regarded as summing up the outlook of logical empiricists concerning the status of mathematics: “Geometry and Empirical Science” and “On the Nature of Mathematical Truth”, both in American Mathematical Monthly, vol. 52 (1945).
Ernest Nagel, The Structure of Science (New York, 1961), chapters 8, 9.
Hilary Putnam, “The Logic of Quantum Mechanics”, in his Philosophical Papers (Cambridge, second edition, 1979), vol. I, pp. 174–177.
Putnam, op. cit., p. 174.
Nor is Putnam alone in allowing his presentation to fall into disorder in this way. His mentor Reichenbach had become entangled in a similar difficulty in The Theory of Relativity and A Priori Knowledge. There Reichenbach had declared that modern physics shows Euclidean geometry to be wrong (pp. 3–4); presumably he was thinking of astronomical observations which show that the paths of two light rays can inclose an area, and was thinking that we are to identify straight lines with the paths of light rays, yielding the result that two straight lines can inclose an area, contrary to Euclidean geometry. However, elsewhere in his presentation Reichenbach says of such a situation that the rays are deflected and curved as they pass through the gravitational field (p. I1, p. 18). Thus Reichenbach says both that the paths of light rays are straight lines and that these paths are curved. But one and the same line cannot be both straight and curved.
Immanuel Kant, Inaugural Dissertation and Early Writings on Space, translated by John Handyside (Chicago, 1929), pp, 5–11.
Henry E. Allison in his Kant’s Transcedental Idealism (New Haven, 1983) presents one interpretation of transcendental idealism.
Kant’s theses (I) and (2) would have seemed completely obvious to eighteenth-century readers. Thesis (3) would have seemed more controversial, as Leibniz and Hume had denied it. But the interest of thesis (3) would have lain in what it was thought to imply concerning the nature of mathematical knowledge, and what that in turn would imply concerning metaphysical views such as transcendental idealism.
For example, P. F. Strawson in The Bounds of Sense (London, 1966) treats transcendental idealism as a misguided part of Kant’s philosophy.
Russell expresses this view in various places, for example in his Principles of Mathematics, 2nd ed. (London, 1937), p. vii.
Camap discusses this approach, for example in his Foundations of Logic and Mathematics (Chicago, 1939). This is Volume 1, number 3, of the International Encyclopedia of Unified Science.
The Encyclopedia Britanica (15th edition, Chicago, 1986), p. 699, describes Gauss as “one of the first to doubt that Euclidean geometry was inherent in nature and thought.”
See, for example, P. A. M. Dirac, “Development of the physicist’s conception of nature”, pp. 2–3, in Jagdish Mehra (ed.), The Physicist’s Conception of Nature (Dordrecht, 1973).
’Visual space’ is mentioned by Hans Reichenbach in “The Philosophical Significance of the Theory of Relativity”, in Albert Einstein: Philosopher-Scientist, edited by Paul A. Schillp (Evanston, 1949), p. 299. See also Adolf Grünbaum, “Catnap’s Views on the Foundations of Geometry”, in The Philosophy of Rudolf Carnap, edited by Paul A. Schillp (La Salle, 1963), p. 666.
Here I do not intend to suggest that it is at all clear what it would mean to see according to one type of geometry rather than another. Patrick Heelan in his Space-Perception and the Philosophy of Science (Berkeley, 1983) suggests that many people do visually experience the world as nonEuclidean, and in support of this he refers to the nonstandard type of perspective found in the paintings of Cezanne and others. However, Heelan’s account seems to entail, implausibly, that when Cezanne looked at the landscape he saw it according to one scheme of projection, while when he looked at his canvas he saw it according to a different scheme.
Of course there will be some line descriptions which pick out lines that must necessarily be straight (e.g., “The straight line between London and Paris”, or “The shortest distance between London and Paris”). However, with a vast range of typical line descriptions (e.g., “The route I lately took from London to Paris”) it will be an empirical question whether the line referred to is straight.
Reichenbach, in his The Philosophy of Space and Time, distinguishes between inductive simplicity and descriptive simplicity. For him, inductive simplicity has to do with the nature of the phenomena being described; where two opposing hypotheses are both consistent with the observed evidence, the inductively simpler hypothesis is more probably true. Descriptive simplicity has to do merely with ease of description; a descriptively simpler hypothesis is more convenient than a descriptively complex alternative, but they describe the same phenomenon and cannot differ in probability. Reichenbach wishes to show that Euclidean geometry is unacceptable, and to do so he would need to show that some nonEuclidean theory of the world is inductively simpler than any Euclidean theory. His arguments do not succeed in showing that this is so, however.
Russell, in Our Knowledge of the External World, lecture IV, stimulated discussion of this idea through his emphasis on Occam’s razor as a principle of scientific philosophizing.
Michael Friedman in his Foundations of Space-Time Theories (Princeton, 1983) makes an impressive attempt to justify this claim.
This is pretty much the view of Henri Poincaré, who writes, “What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true… One geometry cannot be more true than another; it can only be more convenient.” Science and Hypothesis (New York, 1952), p. 50 (the original French edition was 1902).
Gary Rosenkrantz, in “The Nature of Geometry”, American Philosophical Quarterly, vol. 18, no. 2 (April, 1981), argues, as I want to do, that Euclidean geometry can be interpreted so as to be necessary and a priori. However, on p. 108 he says that even if Euclidean geometry is necessarily true, space still may be nonEuclidean; with this I disagree.
It seems to be Plato’s view that phenomena in the realm of becoming do not have any exact geometrical character, so he could be said to hold that entities of the type discussed in geometry do not exist in the physical world. He does not conclude, however, that geometrical knowledge is unimportant. For Plato, some things in the physical world do come close to having geometrical character; moreover, there is another world containing the geometrical Forms that have it to perfection.
P. F. Strawson in his Individuals (London, 1959), chap. 2, discusses alternative patterns of experience, such as an “auditory world”. But it is only when these begin to exhibit something like spatial order that they can be experience of a physical world.
For discussion of this point, see Michael Friedman, Foundations of Space-Time Theories (Princeton, 1983), pp. 27–28.
I thank Athanasse Raftopoulos and Mauro Dorato for drawing my attention to this objection.
I argued to this effect in my Philosophy of Mathematics (Englewood Cliffs, 1965), ch. 2. More recently Gary Rosenkrantz in the paper cited above has argued that geometrical principles can be regarded as necessary, a priori, and synthetic.
This is the conception of the analytic-synthetic distinction which Gottlob Frege adopts and clarifies in The Foundations of Arithmetic (Oxford, 1953), Section 3. Frege held that geometry is synthetic in this sense.
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Barker, S. (1992). Kant’s View of Geometry: A Partial Defense. In: Posy, C.J. (eds) Kant’s Philosophy of Mathematics. Synthese Library, vol 219. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8046-5_9
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