![[CS Dept., U Chicago]](http://www.cs.uchicago.edu/i/header_background.jpg)
This section is taken with very little change from my article on ``The Sources of Certainty in Computation and Formal Systems.''
Granting that formal systems play some sort of essential role in mathematics, there is still some controversy about the nature of that role. Many thinkers, especially working mathematicians, appear to accept the view that mathematics is a game played out by the rules of some formal system, or perhaps a collection of such games. That is, they accept the notion that mathematicians at work are carrying out derivations in a formal system. A variation on this view holds that, while mathematicians at work may employ mental steps that do not follow the rules of any well identified formal system, the correctness of that work depends on the possibility of recasting it as such a formal derivation. The actual practice of mathematics is viewed as a slightly risky, but more efficient, prediction of the results of particular formal derivations. Mac Lane calls the notion of mathematics as an arbitrary formal game ``vulgar formalism.'' I cannot find an explicit profession of ``vulgar formalism,'' even with a more dignified title, in print. But, I have heard mathematicians and users of mathematics express such views in casual conversation, and they sounded serious.
If mathematics is indeed the playing out of a game whose rules have only to do with the superficial forms of symbols, and nothing to do with their useful meanings, we are naturally puzzled why mathematics appears to be useful as a tool for practical work. R.~W. Hamming and E. P. Wigner expressed this puzzlement in papers called ``The unreasonable effectiveness of mathematics'' and ``The unreasonable effectiveness of mathematics in the natural sciences,'' respectively, whose titles are quoted widely by mathematicians who worry that a mere formal system should not capture scientific ideas with real-world content. I think that it is much more sensible to view mathematics as a study with real objective content, whose effectiveness is a result of the widespread applicability of that content. (Hamming and Wigner do not seem to hold a formalist view of mathematics themselves, and their papers contain a lot of interesting discussion of the actual practice of mathematics and science.)
Rather than mathematics itself being a formal game, I support the view that mathematics is essentially a rigorous study of the behavior of formal systems. Confusion about the relation of mathematics to formal systems probably derives partly from confusion between the ceremonies with which mathematical results are often presented, and the formality of the systems that they describe, and partly from the way in which mathematical tools are particularly effective for studying technical methodological issues in the practice of mathematics itself. For example, in mathematical logic we study formal systems that are designed to illuminate mathematical reasoning. It is not hard to misconstrue this application of formal tools to the study of mathematics as an a priori embedding of mathematics into a formal system.
Saunders Mac Lane explains the role of formal systems in mathematics very thoroughly in Mathematics, Form and Function, including the convolutions induced by the reflexive uses of mathematics. Haskell Curry covered similar ground earlier in Outline of a Formalist Philosophy of Mathematics. I find Curry's discussion less satisfying in its treatment of actual mathematical activity, but he contributes a slogan which I find extremely useful as long as it is used to stimulate thought, rather than as a final conclusion:
Mathematics is the science of formal systems.
That is, mathematics is not just the playing of a formally defined game; it has objective content---the qualities of formal systems. I propose another slogan to stimulate similar thought from a slightly different point of view:
The content of mathematics is form.
For present purposes, it is important that formal systems are real, objective, but not physical, things, that we can be highly certain about their behavior, and that the objective study of formal systems is at least a large part of the content of mathematics. But, it is not important to maintain Curry's slogan, or mine, as a precise characterization of everything in the practice of mathematics. Read Mathematics, Form and Function for a careful discussion of the rich variety of activities involved in the actual practice of mathematics.
The reflexivity of formal systems that we observe in a subsequent section may easily lead the unwary from a view of mathematics as a study heavily concerned with formal systems, back to the ``vulgar formalist'' view of mathematics as a particular formal system. If patterns of derivations in formal systems, and in classes of formal systems, can be modeled successfully by other formal systems, perhaps there is no real difference between the vulgar and refined versions of formalism. But, the vulgar view leads to an infinite regress of modeling systems in one another, explained very nicely by Lewis Carroll in ``What the tortoise said to Achilles.'' As a last defense, the vulgar formalist might point out that we may choose an individual formal system, such as the Combinator Calculus, that is capable of modeling the behavior of \emph{every} other formal system. Perhaps this cuts off the infinite regress, by taking the Combinator Calculus as the single source of all formal description (I can't resist comparing it to the bottom turtle in the famous story of the world being supported by ``turtles all the way down.''). No, there is still an infinite regress of \emph{modeling} steps. The Combinator Calculus provides a single language in which each of the modeling steps may be described, but it does not cure the infinitude of steps.
David Hilbert, one of the most productive mathematicians of the 20th century, proposed a program to put all of mathematics on an unshakably solid foundation. He divided mathematical notation into two categories. He regarded basic facts about computation with the integers, such as ``2+3=5'', as contentual statements about real live integer numbers. On the other hand, he regarded general statements about infinite numbers of integers, such as ``for all integers n there exists an integer m such that m>n and m is prime'', as purely formal ``ideal'' formulae, with no inherent quality of truth or falsity. To Hilbert, ideal formulae are acceptable in mathematics if and only if all of the basic facts about the integers that follow from them are true. Hilbert appeared to insist on a very fundamental distinction between the contentual and formal portions of mathematics, although both were critical to the power of mathematics.
Following the view of mathematics as the science of formal systems, we can understand a much larger portion of mathematics to be contentual. Basic integer arithmetic is just one particularly popular formal system. Mathematical statements about computations with the integers, such as ``2+3=5'', have content---they assert something about that formal system. Other more sophisticated looking mathematical statements may also have content as statements about other formal systems. From this point of view, there are still contentual and formal aspects to mathematics, but the distinction seems less disturbing.
Brouwer led a philosophical movement, called intuitionism, that challenged some widespread assumptions about the foundations of mathematics, including Hilbert's claims. The word ``intuitionism'' seems to be chosen to contrast with ``formalism,'' claiming that mathematics is inherently an intuitive contentual enterprise. The name is a bit misleading as a tag for Brouwer's ideas. Many, perhaps most, mathematicians outside of Brouwer's camp agree that mathematics has inherent intuitive content. Brouwer was really arguing for a particular quality of intuition---one in which it makes no sense to assert the existence of something until we have a recipe for constructing it. In particular, Brouwer rejected from logic the law of the excluded middle, which says that every proposition is either true or false (even though we often don't know which). Brouwer was only willing to admit the disjunction ``P or not P'' when there is a procedure that in principle will eventually determine whether P is true or false.
In addition to rejecting the excluded middle (or did he exclude the rejected middle?), Brouwer did not consider formal presentations of intuitionistic mathematics to be significant. In spite of this, his student Heyting invented a formal calculus for deriving mathematical propositions according to intuitionistic ideas, and many people consider it to be a reasonable and useful presentation of the basic logical principles of intuitionism.
From the point of view where mathematics is the science of formal systems, Brouwer's rejection of all formalism appears silly, and probably illusory. That is, his intuitions appear to be intuitions about formal systems. On the other hand, his insistence that we may only assert something about mathematics when we have a procedure for verifying it looks very sensible, if not absolutely compelling. Ironically, to a nonvulgar formalist the need for a conservative system of logic seems to derive precisely from the formality of the objects under study, and constructions appear to be formal arrangements. Brouwer was certainly correct in noticing that there is no a priori way to describe with certainty all of the sorts of relations that might ever be used in the arrangements in formal systems. Perhaps he overemphasized that observation, and extrapolated it (incorrectly in th formalist view) to denying every sort of dependence on formalism.
Several nonintuitionists made even stronger restrictions on logic than Brouwer when working on the foundations of mathematics. For example, Hilbert only accepted a sort of hyperintuitionistic reasoning, called finitist for establishing the correctness of the foundations of mathematics.
 | |