Peirce's Theory of the Dimensionality of Physical Space

Peirce's Theory of the Dimensionality of Physical Space RANDALL R. DIPERT C. S. PEIRCE WAS AN ICONOCLAST in several significant respects, one being his theory of physical space. Against the increasingly popular Leibnizian tradition in mechanics, and vehemently against one of the relationalist theory's most capable defenders, Ernst Mach, Peirce argued for Newton's view, that there is an absolute space and that it is "alone logically tenable."' Against the popular view that physical space was either "probably" Euclidean (Helmholtz and Schubert) or certainly Euclidean (Frege and Mach), Peirce maintained that space must be nonEuclidean . 2 The only question remaining a pure matter of fact for him was whether space is hyperbolic (Lobatchevskian) or elliptical (Riemannian). To determine which metric is applicable to physical space, Peirce used observations and many, sometimes rather complex, methods of analyzing them. He tentatively concluded that physical space is hyperbolic. Finally, and for some of the same reasons employed to argue that space is non-Euclidean, Peirce claimed that "no experiences, familiar or l For details, see my "Peirce on Mach and Absolute Space," Transactions of the C. S. Peirce Society IX (1973) :79-94. J. K. Feibleman in his Introduction to the Philosophy of Charles S. Peirce (Cambridge, Mass.: M.I.T. Press, 1946), p. 359, mistakenly creates the impression, by taking a remark of Peirce's out of context, that Peirce was "on the whole" a Leibnizian relationalist. 2 See my "Peirce on the Geometrical Structure of Physical Space," Isis 68 (1977):404-413. Murray Murphey in his Development of Peirce's Philosophy (Cambridge, Mass.: Harvard University Press, 1961), pp. 219-228, discusses Peirce's treatment of physical space and the influence on it of Riemann. Thomas Goudge in The Thought ofC. S. Peirce (New York: Dover, 1950), p. 70, unjustifiably confuses the issues of the Euclideanness of physical space and the dimensionality of physical space, an all too common error. Moreover, Peirce is probably free from Goudge's charge that he "sometimes confuses pure with applied geometry" (p. 72). For example, at R-94a, Peirce (probably C. S.--see n. 19) writes, "We all see, now, that geometry has two parts; the one deals with the facts about real space, the investigation of which is a physical, or perhaps a metaphysical problem, at any rate, outside of the Purview of the mathematician." There is little reason to think that, in the passage with which Goudge concerns himself, Peirce was talking about pure geometry in discussing the measurements of the angles of large celestial triangles. Both with regard to the metric and the dimensionality of physical space, Peirce was never confused about the pure/applied geometry distinction. However, when Peirce claims that physical space must be non-Euclidean, the "must" has to be distinguished from logical necessity, or pure and applied geometry will be interlocked. For if physical space must (in the logical sense) be non-Euclidean, then physical geometry will fall under (pure) mathematics, which is defined by Peirce as the science of what is and what is not logicallypossible (1.185). Peirce does indeed distinguish between metaphysical and logical necessity (see 1.489) but fails to be explicit about what kind of necessity is involved in the statement that physical space must be non-Euclidean. If we are to be generous, we may assume that the 'must' is metaphysical necessity, thus absolving Peirce from any implicit pure/applied geometry confusion. [61] 62 HISTORY OF PHILOSOPHY otherwise, are absolutely inconsistent with space having four dimensions ''3 and that "there is room for serious doubt whether the laws of mechanics hold good for single atoms, and it seems quite likely that they are capable of motion in more than three dimensions" (6.11). At least two arguments are used to defend this latter claim, and it is these arguments, along with their relations to other aspects of Peirce's later philosophy, that I will discuss and criticize. Few capable philosophers in the nineteenth century with a mathematical background were quite so liberal with regard to physical space's being four dimensional as was Peirce. Riemann, in his famous "Ober die Hypothesen welche der Geometrie zu Grunde liegen," admitted the possibility that...