Terry Tao points to a beautiful article written by Michael Harris for the Princeton Companion to Mathematics, titled *Why Mathematics, You Might Ask*.

The titular question is the point of departure for a fascinating discussion on the foundations of mathematics, on the philosophy of mathematics, on post-modernism, on the “anthropology” approach to social science studies of mathematics, and on what mathematicians think they are doing, and why.

In general, I find articles on philosophical issues in mathematics to be more readable and enlightening when written by mathematicians. Perhaps it’s just that they lack the sophistication of the working philosopher, a sophistication which I mistake for unreadability. But I also find that mathematicians tend to bring up issues that matter more to me.

For example, the metaphysical discussions on the “reality” of mathematical objects and the “truth” of theorems are all well and good, but the really interesting questions seem to be different ones.

The *formalist* view of mathematics, for example, according to which mathematics is the derivation of theorems from axioms via formal proofs, or as Hilbert apparently put it, “a game played according to certain simple rules with meaningless marks on paper,” does not begin to capture what mathematics, just as “writing one sentence after another” does not capture what poetry is. (The analogy is due to Giancarlo Rota.) Indeed one of the main fallacies that follow by taking the extreme formalist position as anything more than a self-deprecating joke is to consider mathematical work as *tautological*. That is, to see a mathematical theorem as *implicit* in the axioms and so its proof as *not a discovery*. (Some of the comments in this thread relate to this point.) Plus, the view does not account for the difference between “recreational” mathematics and “real” mathematics, a difference that I don’t find it easy to explain in a few words, probably because I don’t have a coherent view of what mathematics really *is*.

It’s not quite related, but I am reminded of a conversation I had a long time ago with Professor X about faculty candidate Y.

[Not an actual transcript, but close enough]

X: so what do you think of theory candidate Y?Me: he is not a theory candidate.X: but his results have no conceivable application.Me: there is more to doing theory than proving useless theorems.X: that’s interesting! Tell me more

I enjoyed Harris’s suggestion that “ideas” are the basic units of mathematical work, and his semi-serious discussion of whether ideas “exist” and on their importance.

There are indeed a number of philosophical questions about mathematics that I think are extremely interesting and do not seem to figure prominently in the social studies of mathematics.

For example, and totally randomly:

- When are two proofs essentially the same, and when are they genuinely different?
- What makes a problem
*interesting*? What is the role of*connections*in this determination? - What makes a theorem
*deep*? - What does it mean when mathematicians say that a certain proof
*explains*something, or when they say that it does not?

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November 29, 2007 at 8:06 am

ScottI once mentioned to Umesh that I was reading Wittgenstein’s 1939 lecture notes, where he gets into repeated arguments with Alan Turing (who was auditing his class) about whether all theorems are tautologies and therefore devoid of meaning. Umesh remarked that Wittgenstein might, in a certain sense, have been correct:

all theorems he knewmight well have been tautologies.(Incidentally, Turing apparently had to drop Wittgenstein’s class halfway through, to work on meaningless tautologies at Bletchley.)

November 29, 2007 at 4:09 pm

tejasMe: there is more to doing theory than proving useless theorems.

X: that’s interesting! Tell me more

tell us more! :-)

i had an intense conversations with my officemate about what constitutes a proof: “convincing” another person that it solves the problem at hand, or “convincing” him/her that it can be written out as a formal proof. i felt that the first definition of proof makes it subjective since it depends on who you are telling it to.

November 29, 2007 at 4:46 pm

LucaScott, and Umesh, bring up another reason why I find philosophical arguments on mathematics to be more apt when coming from mathematicians, and I always take it to be a bad sign when examples to illustrate an argument come from elementary arithmetic or from Euclidean geometry.

Indeed, one would not expect an aesthetic of architecture to be based on the experience of putting up a drywall, and to focus too much on the “brickness” of bricks and on the fundamental act of laying one brick on top of each other.

November 30, 2007 at 6:31 am

AnonymousPerhaps it’s just that they lack the sophistication of the working philosopher, a sophistication which I mistake for unreadability.A classic.

December 2, 2007 at 3:18 am

AnonymousI actually didn’t like Harris’ article: too long-winded and hard to read (I took Harris for a philosopher), and it seems to me he never answered the title question. (What I got from the article is that, according to Harris, math is not a search for truth but instead an expression of beauty like art. So why art?)

I did like the idea, not exactly new but worth seeing again, that math is not about theorems but is instead about ideas.

December 2, 2007 at 7:24 am

andrejI just read Gowers’ post – many of his examples are too unfamiliar but here is one from CS that I find puzzling.

It is about the Goldreich-Levin theorem; there are two proofs and I have heard them described as “essentially the same proof”, but I can’t see why.

Say we want to list-decode f.

One proof (I think attributed to Rackoff) goes by guessing the value of the decoding at some carefully chosen points and then recovers the decoding by using the unique-decoding algorithm for the Hadamard code.

The other proof goes by splitting the Boolean cube into two subspaces, then estimating the correlation coming from each subspace. If the correlation is small it throws out the subspace. If not it splits again and again, zeroing in on the large Fourier coefficients.

Are these two proofs really the same?

December 2, 2007 at 6:01 pm

Fred RossYour last question about what it means for a theorem to “explain” something is broader. I recently stumbled across the boundary between physics and biology (alright, it was rather more ballistic, as I tunneled from mathematical physics to bench work with heavy duty pathogens), and the biggest issue I faced was reconciling what the various communities I was talking to — mathematicians, physicists, biologists — meant by “understood.”

So far as I can tell, everyone has a certain set of objects which, for their field, is considered fundamental (this is purely a question of consensus). Geneticists want loci in chromosomes; biochemists want some kind of protein. As soon as they have attained this level, the issue is “understood.” Now, it’s very important that the next person who wants to come along and reduce it to their version of understood come from a field sufficiently separated from the original one that the scientists in the original field can’t huff and puff and declare it a waste of time.

To say this is unsatisfying is an understatement. The most frustrating part is that the underlying concepts aren’t even very useful. Chromosomal loci, for instance, are dead ends. A useful piece of knowledge is one that makes other pieces of knowledge come into reach. Gaining knowledge by associating a chromosomal locus with a trait is idempotent.