The Historical and Conceptual Relations between Kant's Metaphysics of Space and Philosophy of Geometry

The Historical and Conceptual Relations between Kant's Metaphysics of Space and Philosophy of Geometry" TED B. HUMPHREY IN THE "TRANSCENDENTAL AESTHETIC" Of the Kritik der reinen Vernun]t, Kant formulates and argues for three related doctrines concerning space and geometry: 9(1) Space is the pure a priori form of outer intuition. (2) Geometrical truth is a priori and synthetic. (3) The metric of humanly intuited space is Euclidean, and therefore the propositions of Euclid's geometry are a priori synthetic truths. The generally accepted view that the third doctrine somehow follows deductively from the first two, indeed, that Kant used it in their initial justification, has been the source of several apparently serious criticisms not only of the Aesthetic but also of Kant's entire epistemology and metaphysics,x In particular, three objections having their source in recent research in mathematics and psychology seem to cast considerable doubt on the defensibility of the second and third doctrines. The two criticisms stemming from mathematics, namely, that geometries other than the Euclidean are formulable with complete logical consistency, a fact many believe Kant could not have been aware of, and that some persons can actually visualize non-Euclidean spatial images, were most powerfully advanced by Hans Reichenbach .2 The second of Reichenbach's criticisms is effectively generalized by the psychological research of R. K. Luneberg and A. A. Blank, the conclusion to be drawn from which, according to Adolf Griinbaum, is that "although the physical space in which sensory depth perception by binocular vision is effective is Euclidean , the binocular r/suM space resulting from psychometric coordination possesses a Lobatchevskian hyperbolic geometry of constant curvature. ''s Because * I wish to thank the Arizona State University Grants Committee for a faculty grant-inaid supporting the research for this paper. TbAnk~ are also due Professor Peter Fuss, University of Missouri, SL Louis, Dr. Frank D. Farmer, Arizona State University, and an unknown reader for this Yournal for extensive, helpful comments. x Much of what I have said thus far about the doctrines concerning space is equally true, mutatis mutandis, of the Aesthetic's doctrines concerning time. Consequently, most of the paper's results regarding the historical and logical relations obtaining among the doctrines pertaining to space will apply mutatis mutandis to time. For the most part, however, time will not be a topic of explicit concern in what follows. 2 Hans Reichenbach, The Philosophy of Space and Time, trans. Maria Reichenbaeh and John Freund, (New York: Dover Publications, Inc., 195"I), pp. 30-58 passim. s Adolf Griinbaum, Philosophical Problems of Space and Time (New York: Alfred A. Knopf, 1963), p. 154. [483] 484 HISTORY OF PHILOSOPHY most persons believe, either implicitly or explicitly, that in the development of the Aesthetic's doctrines, the third is either essential to the defense of the first two or a necessary conclusion from them, 4 they have been dissuaded by these developments from accepting any portion of the position Kant propounds in the Aesthetic . This is a mistake. The Aesthetic contains no argument for the view that humanly intuited space is Euclidean, nor do any of its arguments concerning the origin and metaphysical status of space depend on that view. In this paper I shall argue that critics have misunderstood the historical and conceptual relations obtaining among the Aesthetie's three main doctrines concerning space and therefore have tended to reject the first two on the basis of arguments against which Kant, or at least a Kantian, can defend. The argument will develop in three stages: (1) Kant's theory of mathematical, especially geometrical, truth, particularly with respect to the precise nature of its syntheticity, and his belief that Euclidean geometry is true of humanly intuited space did not serve as premises in the development of his metaphysics of space. (2) The conceptual relations among the three doctrines is different from the one most believe to obtain. (3) Objections from contemporary mathematics to Kant's belief in Euclidean geometry cannot serve as grounds for rejecting his philosophy of mathematics and metaphysics of space. By way of conclusion I will point out some of the ways in which Kant's philosophy of mathematics and beliefs about...