Why Metaphysics Needs Logic and Mathematics Doesn't: Mathematics, Logic, and Metaphysics in Peirce's Classification of the Sciences

Cornells de Waal Why Metaphysics Needs Logic and Mathematics Doesn't: Mathematics, Logic, and Metaphysics in Peirce's Classification of the Sciences The view I defend here,1 and which I take to be Peirce's, is that while metaphysics needs to be grounded in logic, mathematics does not. The metaphysician who is ignorant of the rules of logic is bound to go astray, as he is too easily charmed by the splendor of his speculations. Interestingly, it is precisely the need to retain this splendor of speculations that motivated Peirce to take the exact opposite view for mathematics. Mathematics should not be grounded in logic, as doing so would unduly restrict the mathematician and hence block the road of inquiry. Admittedly, logic provides the mathematician with interesting material to work with, but so do the rules of chess, quantum mechanics, and the seven bridges of Königsberg, but mathematics does not rely on logic to make sure that its inferences are correct. As will be shown, the difference between mathematics and metaphysics regarding their relation to logic is a direct result of a difference in the kind of mistakes that are made in either discipline, which is in turn caused by the different nature of the disciplines in question. Metaphysics is a positive science, as is logic, but mathematics is not. To substantiate and further explicate this, a wider exploration of the relations between mathematics, metaphysics, and logic is needed. To get a good sense of how Peirce understood the relation between metaphysics, logic, and mathematics, and how this differs from the received view, I will begin by briefly examining Auguste Comte's classification of the sciences, which can be considered representative of the standard view in Peirce's day, if not its paradigm. Next, I run quickly through Peirce's own classification, pointing in very general terms to some of the main differences with Comte. Having thus set the scene, I give a more detailed account of mathematics, logic, and metaphysics as they were understood by Peirce, together with their interrelations. This will show why metaphysics needs logic whereas Transactions of the Charles S. Peirce Society Spring, 2005, Vol. XLI, No. 2 284 Cornells de Waal mathematics does not, and will also show the different ways in which mathematics and logic each relate to metaphysics. Comte's Classification Peirce's classification of the sciences is in part a reaction to a division developed by Comte in his six-volume Cours de philosophie positive (1830-42). Weary of explanations in terms of unobserved and unverifiable causes, Comte restricted positive philosophy (or positive science) strictly to general descriptions of phenomena. As he put it in the Cours, "In the final, the positive state, the mind has given over the vain search after Absolute notions, the origin and destination of the universe, and the causes of phenomena, and applies itself to the study of their laws, — that is their invariable relations of succession and resemblance."2 Having thus defined positive philosophy, Comte next divides it into the abstract and the concrete sciences. Abstract sciences aim to discover the regularities (or laws) in the phenomena we encounter; concrete sciences inquire how these regularities can be applied to special cases. In Comte's scheme, the abstract sciences are mathematics, astronomy, physics, chemistry, biology, and sociology, with each subsequent science relying on the principles of those preceding it. Sociology, since it concerns relationships among biological entities, relies on the findings of biology; biology, since its objects are physical objects, relies on the findings of physics; and physics, dealing with objects that can be counted, sequenced, and measured, relies on the findings of mathematics. By taking this approach, mathematics becomes a positive science for Comte. Like physics and biology, it too concerns itself purely with the description of phenomena. Geometry and mechanics, Comte writes, must "be regarded as true natural Sciences, founded, like all others, on observation, though, by the extreme simplicity of their phenomena, they can be systematized to much greater perfection" (PP 1:33). The phenomena studied by geometry and mechanics, Comte continues, "are the most general, the most simple, the most abstract of all, — the most irreducible to others, the most independent...