(A more appropriate title for this post would probably be “I hate Bourbaki,” but I like it as is.)
I spend a lot of my free time reading research papers, usually in combinatorics; those tend to require the least background. Today I decided to read everything I could find written by one of the great champions of combinatorics, Gian-Carlo Rota, and in his philosophical writings I found the explicit declaration of an opinion I’ve held for some time now.
Consider the following passage from Syntax, Semantics, and the Problem of the Identity of Mathematical Objects:
The real line has been axiomatized in at least six different ways. Mathematicians are still looking for further axiomatizations of the real line, too many to support the justification of axiomatization by the claim that we axiomatize only in order to secure the validity of the theory.
Whatever the reasons, the variety of axiomatizations confirms beyond a doubt that the mathematician thinks of one real line, that is, the identity of the object is presupposed and in fact undoubted.
The mathematician’s search for further axiomatizations presupposes the certainty of the identity of the object, but recognizes that the properties of the object can never be completely revealed. The mathematician wants to find out what else the real line can be. He wants ever more perspectives on one and the same object, and the perspectives of mathematics are precisely the various axiomatizations, which lead to a variety of syntactic systems always interpreted as presenting the same object, that is, as having the same models.
Or the following passage, from Combinatorics, representation theory, and invariant theory: The story of a ménage à trois:
A lot of mathematical research is spent in finding suitable definitions to justify statements that we already know to be true. The most famous instance of such a situation is the Euler-Schlafi-Poincare formula for polyhedra, which was believed to be true in great generality long before a suitably general notion of polyhedron could be defined.
At least one hundred years of research were spent on singling out a definition to match the Euler-Schlafi-Poincare formula. Meanwhile, no one ever entertained any doubt of the formula’s truth. The philosopher Imre Lakatos has documented the story of such a search for a definition in thorough historical detail. Curiously, his findings, which were published in the book ‘Proofs and Refutations‘, were met with a great deal of anger on the part of a section of the mathematical public, who held the axiomatic method to be sacred and inviolable. Lakatos’s book became for a while anathema among philosophers of mathematics of the positivistic school. The truth hurts.
Or the following two passages, from Mathematics and Philosophy: the Story of a Misunderstanding:
The axiomatic method of mathematics is one of the great achievements of our culture. However, it is only a method. Whereas the facts of mathematics, once discovered, will never change, the method by which these facts are verified has changed many times in the past, and it would be foolhardy to expect that it will not change again at some future date [emphasis mine].
The facts of mathematics are verified and presented by the axiomatic method. One must guard, however, against confusing the presentation of mathematics with the content of mathematics. An axiomatic presentation of a mathematical fact differs from the fact that is being presented as medicine differs from food. It is true that this particular medicine is necessary to keep the mathematician at a safe distance from the self-delusions of the mind. Nonetheless, understanding mathematics means being able to forget the medicine and enjoy the food. Confusing mathematics with the axiomatic method for its presentation is as preposterous as confusing the music of Johann Sebastian Bach with the techniques for counterpoint in the Baroque age.
I think modern undergraduate math classes have a tendency to fetishize the axiomatic method. I know people who, when prompted, would tell you that mathematics consists entirely of deducing theorems from axioms. Certainly this is a step up from people who would tell you that mathematics consists entirely of computation; nevertheless, it’s an excessively stifling way to look at mathematics. As Terence Tao points out in his career advice, there’s more to mathematics than rigour and proofs:
It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that “fuzzier” or “intuitive” thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as “non-rigorous”. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education.
When you think of axioms as the focal point of mathematics you miss out on the fact that, as Rota makes clear, a particular set of axioms only offers a particular perspective on an underlying phenomenon that we recognize as worth of study. Two examples that come to mind are matroids, where there is no preferred set of axioms at all, and cluster algebras, where the axioms alone are gibberish without a thorough discussion of the motivating examples. More tantalizingly, lots of interesting work has been done suggesting that, morally, certain branches of topology are essentially algebraic in nature, which is good news for people like me; I never found the topology axioms particularly natural.
Perhaps that’s what Rota meant when he said the following:
Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.
In combinatorics, we have the assurance that we aren’t working with the “wrong” axioms; it’s hard to get the definition of a finite set wrong. Who knows if pathological Banach spaces are really worth studying (inasmuch as they might appear in other branches of mathematics)? Who knows if the definition of a measure will suffice for future applications? Even in group theory the situation isn’t so clear-cut. Some people believe that the sporadic groups are just the tip of an iceberg involving a suitable generalization of the notion of a group; this is in some sense the philosophy underlying this thesis.
At least part of the problem is pedagogical; fetishizing axioms might just come naturally if your textbooks all present axioms before examples. On this point I agree wholeheartedly with Tim Gowers when he says to put examples first, as I think it places the emphasis on the empirical side of mathematics, as well as on the correct historical order of mathematical achievement: first we have interesting phenomena we wish to understand, and only then do we decide on an appropriate set of axioms by which we can understand it.
Perhaps the most unfortunate consequence of an emphasis on the axiomatic method is that it denies modern-day Eulers, Ramanujans, and Feynmans their right to intuit first and rigorize later.