Review of David Corfield's Toward a Philosophy of Real Mathematics

(Apology in advance. This may just be a hasty emotional outburst. I think I had a stronger negative reaction to the book then I should have. The author is clearly a very bright guy, and the book is better than I portray it. If you are interested in the philosophy of mathematics, read it.)

Imagine that you were Catholic, or at any rate well versed in Catholicism and Catholic teaching and practice. Now imagine you encountered a philosopher of religion. Would you have the following gripe?: Philosophers of religion don't seem to deal with what is really important to religious people. Philosophers deal with the existence of God, almost to the exclusion of lots of important philosophical problems.

So, because you don't get why the standard problems are oh-so-important you write a book. In it, you spend a chapter insisting that philosophers must look to the history of religion to understand the real problems of religion that bother its real practitioners. What problem faces practitioners? For example, you might say that looking at history, the Catholic mass started out in some language, I assume Greek (or not), and then turned to Latin, then to the vernacular, and now there is a proposal being floated to return to Latin. The history of why and how is long and complicated, and you bog chapter two down with scholarly minutiae of this history. This is a difficult and controversial question with many serious repercussions. Other religions had the same issues. Reform Judaism's split from orthodox Judaism involved the language of prayer, and Islam still is exclusively Arabic. So whatever western religion you are, chances are that the language issue is important. So why the heck do philosophers of religion act as if the only important questions are those about the nature of God, you ask repeatedly. There are serious arguments in favor of a Latin mass, you say. It is traditional, if was more fruitful in producing scholars, it forced congregants to focus on other parts of the religion, it retained an aura of mystery in the religion. . . Your book then picks on other problems that bother religious people, like food habits, concepts of purity, and the nature of fundamentalism. Each issue does contain some specks of philosophy, though few that would seriously interest most philosophers of religion. Naturally religious people like this approach, because sorting through the Ontological Argument is pretty foreign to their religion, but arguing over what language mass out to be said in is quite familiar to them. So you are hailed as a serious philosopher of religion who takes note of Real Religion, not the fake stuff studied by philosophers.

David Corfield performs a similar service for the philosophy of mathematics. He takes questions that are traditionally off the radar of philosophers of mathematics, and faults them for ignoring them. Then he elaborates on the history and nature of mathematics at great length and with serious depth, and attempts to show that there are real philosophical questions there. Are there? Yes. Am I just being conservative and scared of mathematics by thinking that those are not the important questions? I think not.

Corfield's book could be uncharitably described as warmed-over Lakatos. (I wouln't describe it as such, but I can see it.) I presume that Lakatos and his famous book Proofs and Refutations has more fans in mathematicians than in philosophers of mathematics, likely because the book is just one long mathematical problem, discussed from the perspective of the problem's history, and the history of the concepts contained in the objects of the problem. Lakatos makes some interesting philosophical points along the way: namely he tells us that our mathematical concepts change over time. But more on this in a moment.

But philosophers wonder where the important and interesting philosophy is. And that is the difference between him and traditional accounts of the philosophy of mathematics. His recent blog post about two cultures is thus odd. Besides for the fact that I do not know what philosophical problem is meant when he claims that his goal is to understand the rationality of mathematics through the history of its practice, and the fact that he caricatures traditional philosophy of mathematics by claiming that it is reducible to logicism, it misses the point. In reality it is just that there are different sets of questions that can be deemed philosophical. The ones that traditional philosophers think about are different from his, because they are not easily articulated as interesting philosophical problems, answering the traditional questions about epistemology, metaphysics, and methodology. Perhaps if they were articulated as such, they would appeal to more philosophers. As they are, they are phrased as long mathematics discussions, with some nod towards an ill defined philosophical problem.

Chapter one sets up the book and makes a case for looking at the history of mathematics as it has been practiced by real human mathematicians - with all the baggage that comes with it.

Chapter two gives us a lesson on automated theorem proves and suggests that they are so far only as good as the mathematicians that guide them, and that the real philosophical problems in mathematics ought to be what notation mathematicians need to use. You see, some notations are harder to work with and some are easier - hence the philosophical problem. Get it? Right.

Chapter three makes a philosophical problem out of thinking up conjectures. Can computers do it? Probably not so well.

Chapter four discusses the interesting problem of analogy in mathematics. Many branches in mathematics and many problems in mathematics display interesting analogies with other branches and parts of branches of mathematics. Sometimes it only gets interesting when the analogies fail to hold, sometimes a good analogy can make a whole branch of mathematics obsolete, as all of its questions get reduced to other questions.

Chapter five has an account of Bayesianism in mathematics. A Bayeseian approach as it is traditionally understood in the philosophy of science is there to try to solve the problem of induction. It does so by saying that the logic of science is not inductive in any straight forward way, but rather it is probabilistic. (This is a gross oversimplification.) This is clearly not a problem you will have in mathematics. So Corfield's Bayesianism is restricted to the nature of Bayesianism in mathematics given a subjectivist approach, that is measuring mathematicians' subjective assessments of the probabilities that a given theorem is true.

Chapter six gives a few case studies showing that theories of science often rely on their underlying theories of mathematics, or the outstanding mathematical problems, for their solutions. This is not overly newsworthy, though it is good to have a list of case studies and a nice analysis.

Chapter seven begins Corfield's discussion of Lakatos. He is generally critical, and addresses Fefferman's criticism, and also mentions Steiner's. In some sense he really does nitpick Lakatos to death. I say this as a compliment. Lakatos does make some real mistakes, and does not adequately or fairly represent the history of mathematics, even in the very restricted case he gives.

Chapter eight continues the critique of Lakatos, and this time focuses on the question of what the mathematical research program is. This looks a bit more familiar to philosophers of science. This follows the traditions of Kuhn, Popper, Lakatos, and Fayerabend in looking for the methodology behind scientific, and in this case mathematical research. Corfield claims, correctly, I think that Lakatos' view of the history of science cannot be simply carried over to the history of mathematics. The study of the methodology of mathematical research programs needs 1) some refinements, and 2) more nuance than has been given so far. This program is probably of some interest to philosophers of science and mathematics. The study started by Kuhn was very much in vogue for a bit, though philosophers of science have tended to loose interest in this in favor of the traditional problems in the philosophy of science. Kuhn and co were certainly more influential in the long run outside philosophy than they were in it.

Chapter nine is about the question of what lead to the acceptance of groupoids as an important field of study.

Chapter ten is perhaps the most disappointing. I say that because it shows the most promise and delivers the least. We are offered that tantalizing suggestion that higher-order algebra has something interesting to offer philosophers. What we are actually given is a description of roughly what higher-order algebra is, and thus suggestion that it is of a higher level of abstraction, and that some of the notation can be done in pictures. (Repeating some other philosophers who unconvincingly tried to show that pictures are philosophically interesting.) The idea of infinitesimals, sets, and modals held real promise as a field of philosophical research, and they panned out because people were able to show that there were questions that philosophers could ask and try to settle about them. I had hoped that we would at least see some real questions or suggestions about higher order category theory or algebra, but we did not. What is it about them that a philosopher might want to know about that is not there with all other branches of mathematics? That is the only thing I want to know. There is no satisfying answer given.

(As a criticism of the book, it is annoying that despite the existence of real criticism of some key ideas that he deals with, they do not appear in the book. For example, the notion of revolutions in mathematics is taken for granted, as is the legitimacy of Lakatos' idea in general. Joseph Dauben, a giant in the history of mathematics, whose essay must be well known to Corfield, famously disagreed with Crowe about the existence of revolutions in mathematics. There are also some very critical reviews of Lakatos' Proofs and Refutations in the literature that are well worth reading that Corfield ignores.)

To be honest, I do find many of the problems discussed in the book interesting, but I fail to see how many of them are philosophical. (I hope the author was not pulling one of those "let me show you a lot of math, therefore I must be right.) Many are really psychological or sociological. The problem with this is that there is a good reason why we do not see mathematics and science papers starting out "Well, first I thought X, then I realized that X was wrong, so I changed Y, and got a proof for Z. Then I had lunch and realized that I can get a proof for W if I only . . ." No one cares what got you to the theorem or the proof. Your personal history of conjectures, subjective probabilities, luncheons, conferences, and the clever insights you used to get you to the proof are not what makes up a philosophy of real mathematics. Real mathematics is what appears in mathematics journals. Everything else is part of a philosophy of real mathematicians. They are a fascinating bunch, but not necessarily a subject of philosophical reflection.

That being said, the structure of mathematical knowledge and discovery probably ought to be a part of the philosophy of mathematics. One wonders however how that could be done. Is there a logic that really describes the whole process for each case? I would doubt it, and perhaps that is what makes the problem so uninteresting - the fact that it is truly intractable.

Corfield may be doing mathematics from the inside, but he is doing philosophy from the outside. He misses the questions that are important to real philosophers. Chapter one approvingly quotes Diderot in an epigraph saying "To speak informatively about bakery you have got to have put your hands in the dough." By this token, Corfield hasn't said much about bakery, but rather he has given us a cookbook of recipes that people rarely use. Let me end with the following challenge to Corfield: Stephen Stich and Adam Morton write

Why is there so little philosophy of mathematics? . . .One explanation might be that mathematics is hard , and philosophy is hard, so the philosophy of mathematics might be doubly hard. . . . But it cannot be the whole story since very sharp and troubling points in the philosophy of mathematics can be made with elementary mathematical examples. One reason why a philosopher of mathematics needs to know some advanced mathematics is to call the bluff of others. Irrelevant by terrifyingly technical exposition is common in second rate work. . .

Corfield writes that

the prospective philosopher of mathematics quickly gathers that some arithmetic, logic, and a smattering of set theory is enough to allow her to ply her trade, and will take some convincing that investing the time in non-commutative geometry or higher-dimensional algebra is worthwhile. One of the main purposes of the book has been to argue against this.

I do not know who is right on this one. But if Corfield is, the burden of proof is still on him (or perhaps he would be magnanimous enough to show us even if it wasn't) to show that there are interesting enough philosophical problems that can only be solved if we are aware of the higher mathematics being done by groupoid theorists, and mathematicians who study higher-order algebras.
We see a few not-so-very-interesting problems, and a lot of fancy modern mathematics, but few clear convincing arguments that the two have met.