Devlin's Angle

August 1998

Mathematicians and Philosophers - Chalk and Cheese?

Confession time: I occasionally read philosophy books. Like many mathematicians, for most of my career I tried to stay away from the stuff. Then, when I turned forty, I started to be plagued by "Why?" questions. Again, I know of many colleagues for whom the Big-Four-O had a similar effect. Now, having passed the even bigger Five-O, I have learned to live at peace with my interest in those nagging questions about the nature of mathematics.

My latest foray into the world of philosophy was John Haugeland's recent tome, Having Thought: Essays in the Metaphysics of Mind. The subtitle says it all. Haugeland's book is a fairly heavy duty philosophy book. In working my way through it, I kept being struck by the differences between philosophers and mathematicians.

Any mathematician who has ever wandered, either by accident or design, into a philosophy seminar will have found the experience somewhat bizarre. In a mathematics seminar, the presenter will write some figures, equations, graphs, and cryptic notes on a blackboard -- or maybe project them onto a screen using an overhead projector -- and will spend three quarters of an hour describing what those figures, equations, graphs, and notes mean. The mathematician will generally move around a lot in an animated fashion, often waving her arms in the air and pointing to something displayed on the screen. Occasionally, she will glance at her notes, scribbled hurriedly on the train or plane on the way to the meeting. An outsider observing the proceedings might get the impression that the mathematician is struggling to explain the results to herself as much as to the audience. And often that impression is not far from the truth.

At a philosophy seminar, on the other hand, the presenter stands motionless behind a podium and reads, verbatim, a pre-written, neatly-typed manuscript which he has made available to everyone in the audience. That's right: he reads it word for word. From start to finish.

I am sure that even my philosopher friends (I had several before I wrote this article) will acknowledge that the presentation at the math seminar is much more lively than the philosophy seminar. Things switch around dramatically, however, when the presentation gives way to the question and answer session that follows. In the math seminar, chances are that no one followed much beyond the first ten minutes, so the questions -- if there are any at all -- are either trivial or else so far off the mark that the presenter can't provide a coherent answer. When the philosophy presenter stops talking, however, all hell breaks loose. A casual observer would wonder why on earth the organizers had invited the presenter in the first place, given that the entire audience seems not only to disagree wildly with the views just presented, but express grave doubts that the presenter has read all the appropriate literature.

Not only do mathematicians and philosophers go about their business in very different ways, on the whole they don't mix. Mathematicians who arguably would benefit from reading works in philosophy rarely do so -- books by Daniel Dennett and John Searle excepted -- and even philosophers of mathematics appear to read little contemporary work on mathematics -- books by Roger Penrose excepted. As Thomas Kuhn wrote in his book The Structure of Scientific Revolutions, ". . . normal science usually holds creative philosophy at arm's length, and probably for good reasons." Kuhn goes on to describe those "probable good reasons" -- in essence, to make progress in science it is generally better to ignore the potentially distracting meta-scientific questions that so interest the philosopher. Count mathematics in with science on that score.

I have already confessed to being a bit of a closet philosopher, in that I find many questions in philosophy deeply fascinating. My problem is always that, because of my training in mathematics, I find philosophy books so hard to read. Given the vast deluge of new research that any mathematician has to read about in order to stay abreast of his or her field, I suspect that we all develop the same approach: First read the introduction, then skip ahead to the conclusion, then glance at any figures or graphs, then rapidly guess at the main points in the author's argument, and finally skim the main body of the text quickly to confirm or correct that initial guess. An hour at the most.

Ninety-nine percent of the time, this method works fine in mathematics. But it's the kiss of death when you are faced with a paper or book in philosophy. There, the author has labored long and hard to craft an intricate and tightly woven argument. Every word counts, and every phrase is critical. You can't simply cut to the chase. This is the chase, and it's going to take time.

The main concern of the mathematics author is to get the results into print. If the results are sound, everyone will be able to figure out what is intended, even if the explanations are poor and incomplete. But the philosopher sets out to provide a defense against all anticipated attacks from all quarters. In all likelihood, the conclusion is obvious, and in any case unlikely to be surprising. The game is to supply a good supporting argument.

The mathematician establishes results by logical deduction. The philosopher constructs elaborate "thought experiments," often involving imaginary creatures from other planets who share some -- but not all -- human intellectual abilities.

The mathematician writes for posterity, for the next grant, and, in his dreams if young enough, for a Fields Medal. The philosopher writes for a dozen or so (other) leading philosophers who work in the same area, against whose future criticisms the author will go to great lengths to erect a solid defense.

What then did I make of Haugeland's Having Thought, a collection of essays written over many years, all having to do with the nature of human thought? (Incidentally, another distinction between mathematicians and philosophers is that the former usually provide their academic affiliation on the first page of any book they write, whereas philosophers -- probably unduly influenced by Saul Kripke's writings on names -- often prefer to appear all but anonymous. For the record, John Haugeland is a highly respected professor of philosophy at the University of Pittsburgh. You won't find that out from reading his book.)

Well, I found it heavy going, I'll say that. The reason why Dennett and Searle attract so many scientist readers is that they write in a simple, direct, everyday style. But plain, everyday prose is not at all the fashion in philosophy, and Haugeland makes no such concessions for the outsider. (A chapter in Haugeland titled "Understanding Dennett and Searle" struck me as harder going than either Dennett or Searle themselves, which leads me to suspect that Haugeland is not such a pushover for those two pros as I am.)

Despite the difficult prose, however, I found the book made me think, which is surely the author's main intention. The first chapter begins with what I found to be a persuasive and wonderfully uplifting argument explaining why cognitive psychology should be regarded as a genuine science. (The gist of the argument is to forget thinking of physics as the ideal science to which all others should aspire, and rethink what is meant by science. That clearly affects the role of mathematics in "science". Mind you, in typical scientist's fashion, I have just condensed 34 pages of finely crafted argument into a single sentence, so I dare say I have over- simplified Haugeland's account.)

As I argued myself in my book Goodbye Descartes, I think that it is inevitable that, as we head into the twenty-first century, the human sciences will increasingly come in from what has hitherto been the scientific cold, both in terms of status and funding. Many of the issues Haugeland addresses are undoubtedly going to occupy a central place in the science of the next millennium. That will not only change the relationship between mathematics and science, it will influence the kinds of mathematics that are done. That's another issue I address in Goodbye Descartes.

- Keith Devlin

Devlin's Angle is updated at the beginning of each month.
Keith Devlin ( is Dean of Science at Saint Mary's College of California, in Moraga, California, and a Senior Researcher at Stanford University. His book Goodbye Descartes: The End of Logic and the Search for a New Cosmology of the Mind (John Wiley, 1997) was released in paperback earlier this year.

John Haugeland's book Having Thought: Essays in the Metaphysics of Mind has just been published by Harvard University Press.