Math 696, by Bill Reinhardt
If we were to ask the question, "What is it that mathematicians do?", it would not be entirely trivial to answer, "Mathematics." The next question to ask would have to be, "What is mathematics?" On this matter we can say at least a couple of pretty good things. Whatever mathematics is, it is a textual activity. Mathematicians write papers, read them, make marks on paper and on blackboards, and perhaps most importantly, talk about mathe- matics. If we can take seriously the sentiments behind be- haviorism, we can say that mathematics is no more than this tex- tual activity. The sense of this is not that mathematicians do not undergo various cognitive processes, but merely that their doing mathematics does not depend on it. It is like, as Wit- tgenstein says, each of us having a black box into which no one else can look. This black box is the thinking, apprehension, or whatever each of us does before and while doing mathematics. It makes no difference to the practice of mathematics what the con- tent of this thinking, apprehension, etc. is (or indeed if our minds are empty), just so long as it is mathematics which we are doing. That is, as long as we are writing the things which get counted as mathematics, saying the things which get counted as mathematics, reading the things which get counted as mathematics, and so on. The things which get counted as mathematics do so, ultimately, for purley institutional reasons: because of the or- ganization of journals, faculties, and primary, secondary, ter- tiary education, and so on.
More than this may be said about mathematics. We have said that mathematics is entirely constituted by social institutions, which circumscribe its textual activity. However, it is still worth asking why the practice of mathematics is circumscribed the way it is, rather than in some other manner. To this we will tend to say things which sound much closer to what mathematicians them- selves say, than were the remarks of the above paragraph. We will tend to respond with facts about the history of mathematics, about known theorems and structures, about human cognition, and about the behavior of medium sized objects. We will also have quite a bit to say about the ideological and political organiza- tion of societies which do mathematics.
A distinction is often made in mathematics and philosophy about
mathematics between statements about which one is pretty sure,
and those about which one is less sure. Finitists believe that
statements about the natural numbers are the ones (in
mathematics) which we are most sure about. The rest have only
figurative or metaphorical meaning, and so we can not be quite
sure how to take them. Intuitionists believe, amongst other
things, that the only statements we are sure about are those for
which we have a proof. We cannot say,
statement P is so or
statement not(P) is so unless we have a proof of one or the
other. They are also concerned with statements involving the ex-
istential quantifier. A classical mathematician feels no qualms
about saying an object exists solely on the basis of an
hypothetical proof, which does not actually produce the object in
question. For an intuitionist this is problematic. What seems
to be common to all mathematicians is a belief that whatever else
we are sure about, we can be pretty sure about statements about
the natural numbers; that is, statements of arithmetic. This may
well have a great deal to do with the behavior of medium-sized
objects. The medium-sized objects with which we are well
aquainted obey the laws of arithmetic when considered only in
terms of their distinctness. In the case of arithmetic, much
more that in set-theory, or topology, or algebra, or even
geometries, it seems as if the statements we make must be born
out (or refuted) by the very way the world is. If we claim that
a number has a certain property, we can ultimately (according to
a certain abstraction) take a collection of objects of that num-
ber and actually try and see if it can be chopped up, rearranged,
combined with other things, or whatever as purported. Statements
about all the numbers or, what amounts to the same thing, about
the existence of certain kinds of numbers may be on somewhat less
steady ground; but these are not really statements about collec-
tions of things rather they are statements about the nature of
The area of textual activity circumscribed as mathematics has much to do with interests, in both senses of the word. The par- ticular areas and theorems which a mathematician pursues are of- ten matters of which areas are likely to gain institutional recognition and hence express the interests of an individual mathematician, in the sense of "serving her interests." On the other hand (and this cannot be considered independent), these particular areas are ones which are interesting to a given mathe- matician, for a variety of ill understood reasons. In my own case, for example, a great deal of what I find interesting in mathematics is that which I learned first, that which is most of- ten written about in philosophy, that which is behind the "new math" taught to me in primary school, and so on. Perhaps some deeper psychoanalytic or otherwise psychological reasons also ex- ist for why I fancy logic and set-theory over algebra and geometry.
The interests which mathematicians have tend to link together many areas of mathematics. It has turned out that many areas of mathematics which started out as the products of seperate inter- ests have come to have more and more connections with one another. A theorem in one area of mathematics often has ramifications on altogether different portions of mathematics. This is, in fact, often behind our interest in a particular area of mathematics. For example, our interest in large cardinals in set-theory is motivated, at least in part, by the intuitions (and results) of mathematicians that properties of large cardinals will be used to prove properties of arithmetic. The inter- connection of different parts of mathematics started with Descartes' unification of algebra and geometry; it has progressed to the point where essentially all mathematics may be modelled in set-theory. Set-theory, in turn, is shown by the Lowenheim- Skolem results to be modelled in arithmetic (at least if we think of set-theory axiomatically). The interest exhibited by modern mathematics in unifying the various areas of mathematics is cer- tainly a matter worth examining. One point which we should notice is that the consequences of other areas on arithmetic seems to be treated as more inherently interesting than con- sequences in other directions. This is in some way in keeping with the finitists' sentiments.
Due to the formalization of modern logic by Frege, Peano and others; and due to the axiomization of arithmetic, set-theory, geometry, and so on by many mathematicians (notably, Zermelo, Frankel, Dedekind, Peano, Hilbert, and others), particularly around the start of the twentieth century, we are now generally able to cast the ramifications of areas of mathematics upon one another as purely syntactic facts. This increased technical rigor is seen by contemporary mathematicians as an advance. It becomes very easy at this point in time to see the activity of mathematics as the exploration of syntax. This view may be cast back in time without too great misreading; at least I believe as much. I believe it fairly plausible that the interest in mathe- matics has long been due to some aspect of human cognition which is fascinated with syntax. That earlier mathematicians were not so self-conscious of the syntax of their practice as we are of ours no more counts against syntax being the central motivation of mathematics than does the fact that some cultures are not self-conscious about the syntax of their language count against the language having a syntax. What I am proposing here is that the fact that many cultures do mathematics may show something universal about human cognition. This universal may be a fas- cination with syntax and this is also evidenced by the exist- ence of many formal games in many cultures. My proposal is analogous to that of Levi-Strauss, who claims that the nature of myth-making, kinship patterns, and other activities shows a fun- demental structural nature to human cognition.
Many of the remarks which I have made about the practice of math- ematics run counter to traditional views about mathematics, par- ticularly those which claim that the business of mathematics is to discover how things really are. Kurt Godel is a mathematician who is known, beyond his phenomenal contributions to logic, as a Platonist. It is this view that in a naive reading would be most opposed to the story I sketch about mathematical practice. However Godel's article, "What is Cantor's continuum problem?", seems in many ways to reflect a similar view of mathematical practice to that which I have. I should like to discuss this way of looking at Godel's article. Page references will be from, Philosophy of Mathematics, Benacarraf and Putnam - second edition.
My main claim might be stated in a somewhat Wittgensteinian way:
mathematics is simply the way in which mathematicians "go on".
This says that mathematics is not a system, but a practice in
the sense in which structural linguists after Saussure distin-
langue as system, and
parole as practice.
Godel does not argue for this, as to do so is not his concern
here and elsewhere, but this view seems to underlie his article.
In a way this picture is forced on us by the mathematics itself:
any system of axioms is shown by Godel's incompleteness proof to
leave questions undecidable (strictly speaking, any set of axioms
which suffice to create a model for arithmetic). However, more
than this underlies Godel's thinking. We could imagine conceiv-
ing of the practice of mathematics as a practice of deciding
questions only as they come up and are shown independent of what
we have already accepted. Our system of axioms would merely be
seen as one with elastic borders. For Godel the role of axioms
is less than this. The axioms are merely heuristic in trying to
partially schematize the practice of mathematics. The following
Regarding anew axiom: a probable decision about its truth is possible also in another way, namely, inductively by studying its "success." Success here means fruitfulness in consequences, in particular in "verifiable" consequences, i.e., consequences demonstrable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make possible to contract into one proof many dif- ferent proofs. (p.477)
Godel is also interested in particular areas of mathematics in- sofar as they have connections with and ramifications upon other parts. I have tried to characterize the practice in this manner, and Godel says as much explicitly. For example, Godel finds it significant that:
T]hese axioms [of large cardinalshave consequences far outside the domain of very great transfinite numbers, which is their im- mediate subject matter.
[T]he axioms of the existence of inaccessible numbers .... implies new theorems about integers (the individual instances of which can be verified by computation). (p.483)
Or again, Godel says,
The simplest case of an application of the criterion under discussion arises when some set-theoretic axiom has number-theoretic consequences verifiable by computation up to any given integer. (p.485)
These quotes demonstrate not merely an interest in connecting the parts of mathematics, but an assymetric interest in the parts; with the greatest interest being in the natural numbers. This interest in the natural numbers seems, as I have said, to be con- nected with the possibility of considering the objects of the ac- tual world only in terms of their distinctness. Godel indicates this,
Such considerations, it is true, apply directly only to physical objects, but a definition of the concept "number" which would depend on the kind of objects that are numbered could hardly be considered to be satisfactory. (p.471)
Godel, in some sense, agrees with the finitists that we have the greatest surity about the natural numbers. Whether we accept a givien axiom hangs on its consequences, and the consequences with which we have the clearest ideas are about natural numbers.
Let us try again to characterize the way that mathematicians "go on". The axioms which "force themselves upon us a being true (p.484)" partially characterize our actions in mathematics. That they do so entails that syntax forces itself on us too. We do not merely accept axioms, but we accept their consequences also. That these consequences are determinate rests on the fact that our application of syntactic rules is determinate and uniform. Kripke, via Wittgenstein, questions our epistemic certainty about this determinacy; but somehow despite whatever arguments he might have, we know that we will continue to apply syntactic rules "in the same way." This certainty we have about syntactic uniformity is common to all mathematicians, specifically more so than is the acceptance of given axioms. Godel recognizes this.
[T]he axioms underlying the unrestricted use of the concept of set . . . have been formulated so precisely in axiomatic set- theory that the question of whether some given proposition fol- lows from them can be transformed, by means of mathematical logic, into a purely combinatorial problem concerning the manipulation of symbols which even the most radical intuitionist must acknowledge as meaningful. (p.475)
An intuitionistic mathematician, or some other sort, might doubt the "truth" (or usefulness) of a particular axiom (notably, the axiom of choice), but she could not doubt the syntactic con- sequences of the axiom (i.e. of these being actual consequences). In this sense, mathematicians are concerned with syntax before they are concerned with anything else.
The certainty of syntax is closely related to Hilbert's "equivalency thesis". Hilbert believes that we can not be more certain of anything than of the natural numbers, because the natural numbers only need the one basic notion of iteration. The equivalency thesis says that the metamathematical methods we have are also based on this same basis, merely with symbols rather than numbers as their object. In either case, iteration is basi- cally a syntactic notion. It is because of the certainty we have about the regularity and uniformity of syntax that we are certain about both the mathematics and metamathematics generated thus.
Mathematics, for Godel, essentially comes down to the intuitions which mathematicians have. We might think of our feeling about the regularity (uniformity) of syntax as an intuition, but it is an intuition on an even deeper level than is the rest of mathe- matics. This fact might in some way underlie the logicists' sen- timents. The other intuitions we have concern the truth or fal- sity of certain axioms or statements, and these may be disagreed upon to one extent or another. These intuitions are seen by Godel to fully determine the practice of mathematics.
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 hypothosis. What, however, perhaps more than anything else, justifies the acceptance of this criterion of truth in set-theory is the fact that continued appeals to mathe- matical intuition are necessary not only for obtaining unam- biguous answers to the questions of transfinite set-theory, but also for the solution of the problems of finitary number theory (of the type of Goldbach's conjecture), where the meaningfulness and unambiguity of the concepts entering into them can hardly be doubted. (p.485)
I must take slight point with Godel here, however. The existence of mathematical intuitions is not a "mere psychological fact", but is a negotiated fact. Our intuitions are not fixed; they may be swayed and changed. On a grand scale, the intuitins of many mathematicians were changed by the discovery of the paradoxes of set-theory. On a smaller scale, we may argue over, and perhaps convince one another, of the truth or falsity of particular axioms like choice or the existence of certain large cardinals (or for that matter, Euclid's fifth postulate).