H. Jerome Keisler
Vilas Professor of Mathematics, Emeritus
As a mathematician, I will instead write about the proper role of mathematics, and in particular of formal methods in mathematics, in relation to other disciplines. In order to answer this question, it seems necessary to take a stance on the question of mathematical existence. According to the classification in P. Maddy's recent paper "Mathematical Existence" [M], my position is somewhere between arealism and thin realism. Briefly, mathematical intuition about infinite objects exists, but I reserve judgment on whether or not the infinite objects themselves exist.
Gödel [G] wrote in 1964 that "the question of the objective existence of the objects of mathematical intuition ... is an exact replica of the question of the objective existence of the outer world". "The mere psychological fact of the existence of an intuition which is sufficiently clear to produce the axioms of set theory and an open series of extensions of them suffices to give meaning to the question of the truth or falsity of propositions like Cantor's continuum hypothesis."
I agree with Gödel that a mathematical intuition exists which is sufficiently clear to produce the axioms of set theory and an open series of extensions of them. I also agree that mathematical intuitions exist independently of the observer. However, I reserve judgment on whether the intuition is or can be clear enough to give meaning to the question of the truth or falsity of propositions like Cantor's continuum hypothesis.
I view mathematical research as exploring mathematical intuitions. This is ordinarily done by a community of mathematicians collectively building a formal system and proving a large number of theorems. Formal systems are used to clarify, sharpen, and communicate intuitive observations. We use mathematical intuition to find new axioms, make conjectures, and find proofs of theorems which follow from the axioms. This provides evidence of the existence of mathematical intuition. As Gödel pointed out, the consequences of the axioms, in turn, can be checked against intuition, and provide evidence for or against the axioms, in a manner analogous to the scientific method of using observations to test a theory. Computers give mathematicians another tool for testing axioms, as well as for experimentation which can stimulate mathematical intuition.
Mathematical intuition occurs in a variety of settings, for example: the cumulative hierarchy in set theory; constructive mathematics; computational complexity; modal logic; nonstandard universes; category theory; probability theory; string theory; biological and social sciences. One should expect that completely different intuitive viewpoints are possible, and that some of them will be discovered in the future.
A successful example which is often cited is the use of the intuitive concept of the cumulative hierarchy of sets to find the basic axioms of set theory, add axioms of infinity, and draw conclusions about sets at the lower levels. One can formally represent most of mathematics within set theory. However, some caution is needed because it is often hard to transfer mathematical intuition from one setting to another, or even from one subarea of mathematics to another. For instance, when one is immersed in the usual set-theoretic hierarchy, it is hard to think intuitively in terms of functors in category theory, or hyperfinite sets in a nonstandard universe, or constructive existence. For this reason, I am in favor of a pluralistic approach which encourages the unrestricted exploration of mathematical intuition in different settings. Fruitful ideas can be discovered in one setting but obscured in others.
Robinson’s nonstandard analysis is one of many examples where different mathematical intuitions have led to new concepts and results. Constructive mathematics is another example of this kind. It is a good idea to find out what can be done by constructive methods, but a bad idea to limit mathematics to such methods.
I see two roles for mathematics in relation to other disciplines. One role is to provide algorithms to solve practical problems. A second, more interesting role, where mathematical intuition comes into play, is to create models or theories which shed light on some phenomenon that one wishes to understand. From this standpoint, pure mathematics can be viewed as the study of a mathematical intuition for its own sake, rather than for some outside phenomenon.
One can get some insight into the role of mathematics in relation to other disciplines by looking at a particular area. For example, in the area of mathematical economics there is a very substantial literature on exchange economies. A real-life exchange economy has a large but finite set of small agents who interact in some way, and one would like to understand phenomena such as coalitions, equilibria, the movement of prices, and behavior under uncertainty. In the literature, three different ways of representing exchange economies have been developed and used extensively. These are: an increasing sequence of finite economies where one studies the asymptotic behavior, an economy with a continuum of agents, and an economy with a hyperfinite set of infinitesimal agents. Even though one knows that there are only finitely many agents in a real-life economy, many phenomena are best understood intuitively by using infinite structures. Moreover, the three different approaches involve different mathematical intuitions which are helpful in understanding different phenomena.
What is going on here is that well-behaved infinite objects are intuitively simpler than large complicated finite objects. In all real situations (even in physics) one wishes to explain a large but finite set of observations about the world. The role of mathematics is to use mathematical intuition, often about infinite objects, to help understand the observations. One does not try to give an exact description, but instead searches for an idea which is simple enough to be grasped intuitively and which somehow captures the essential features of the phenomenon. For this reason, mathematics can serve its role best if one has the flexibility to exploit different mathematical intuitions.