#axiomatization

Incommensurable Formalizations.—Given that mathematics wears its mind on its sleeve, making its presuppositions explicit as axioms, how are the axioms to be understood in their turn? Gödel was among the few to perceive this problem, and he devoted his later life to the pursuit of principles underlying the axioms of set theory, rather than to a revision of the axioms themselves. Below the level of foundations there is a conceptual substructure that we scarcely know, and which is the basis of our understanding. These conceptual substructures could be called the regulatory principles mathematics. There are analogous principles in natural science. Indeed, with the natural sciences we have a number of explicitly formulated regulatory principles—the principle of parsimony (Ockham’s razor), the uniformity of nature, thermodynamics—many of them of long philosophical provenance. Cosmology is particularly rich with regulatory principles, such as the cosmological principle, the anthropic principle, the Copernican principle, and the principle of mediocrity. Such principles regulate our reasoning without being directly implicated in the derivation of particular results. These principles, even if disputed, are as familiar to us as the regulatory principles of mathematical thought are unfamiliar to us. It is as though natural science skipped over the stage of axiomatization and went directly to the principles underlying the axioms—if only the axioms had been formulated in the first place. This asymmetry between natural science and mathematics gives each discipline a fundamentally different relationship to formalization, and, as Hilbert noted, “Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.” What he meant is that a mature science is formalized, but what he did not say was that natural science and mathematics involve incommensurable forms of formalization.

Hilbert’s Paradise.—Mathematics differs starkly from the natural sciences in that axiomatics emerged early in the history of mathematics, even if its promise was not fully realized until much later in history, but it is by way of axiomatization that presuppositions are rendered explicit. With this mechanism for ferreting out presuppositions, mathematics wears its mind on its sleeve. Today there are axiomatized expositions of the natural sciences—J. H. Woodger even formulated an axiomatic biology—but it is unlikely that such methods will ever gain traction among those pushing the discipline forward. One might suppose, given this degree of epistemic transparency, that mathematics was in a better place than the natural sciences, but, strangely, mathematics has become among the most strongly selective of disciplines in terms of native understanding. Either you get it or you don’t. But if you don’t have a feeling for mathematics, you still need to learn the basics. As a result, mathematics is the discipline most vulnerable to being taught as a grab bag of rules to be applied as the occasion presents itself, presented without regard to any mathematical intuition, except those intuitions that come prepackaged with the hard-won formalisms that have proved themselves over time. Some become quite clever in the manipulation of meaningless tokens according to specified rules, obtaining the desired result and acquiring a practical facility, but that strategy contributes nothing to the growth of the discipline. In an unexpected way, the formalist conception of mathematics has been realized in practice. Hilbert is supposed to have said of axioms, “One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs.” The terms aren’t to be meaningful on their own account, but only counters in a game; the point is to learn the game well. It may as well be so. No one will expel us from the paradise that Hilbert has left us.