Four years ago I was invited to contribute an article for the Princeton Companion to Mathematics on the general theme of "mathematics and postmodern thought." Halfway through the article I realized I didn't have that much to add to the articles I had already written on the topic (e.g. this one or this one), and strayed from my initial assignment to consider the broader question of how mathematicians justify the practice of mathematics - to philosophers and to society at large, as well as to one's colleagues and oneself. The comment posted on Terence Tao's blog by the colleague who hides behind the modest pseudonym "Le fant^ome de Descartes" has unwittingly revealed why my initial assignment remains timely. Many readers are satisfied with caricatures of what is understood by the term "postmodern thought," such as those presented in polemical books on the subject. There is no doubt that some of the authors designated (often abusively) postmodernist are particularly easy (and fun) to caricature. But the trend did not arise in a vacuum.

Some readers seem altogether lacking in curiosity as to why authors write in a way they find objectionable — and more to the point, why anyone would be willing to read such authors. Such readers should therefore probably not bother reading articles that attempt to place this question in context. I don't claim to have got to the bottom of the mystery in my own articles, of limited scope and subject to fairly stringent space limitations. The ultimate disinterested history of the "postmodern turn" in French thought, and its odd reception in North American campuses, has yet to be written. Without presuming to anticipate its conclusions, I suspect that ridicule will not turn out to be an adequate analytical tool.

My personal beliefs are largely irrelevant to this discussion, but let me state for the record that I do not consider myself a postmodernist and would not even if I had a clear idea what the word meant. (I am fond of some fiction that is sometimes called postmodern, less enthusiastic about postmodern architecture, and find a great deal of interest in the writings of Foucault and the rest; all that doesn't add up to any "ism.") None of my articles can be remotely construed as a defense of what some understand to be the postmodern approach to mathematics. One point I try to make in my PCM article is that there is in fact no such thing, except in the imaginations of the anti-postmodern polemicists. It's a matter of record that neither Foucault nor Derrida had much to say about mathematics (though Derrida did write a long introduction to Husserl's Origin of Geometry at the beginning of his career), but because the so-called Science Wars have created some confusion on this point I spent a few pages of my PCM article trying to sort it out. On the other hand, the Science Wars, together with the obscure dynamics of North American university campuses, have had the effect of lumping French thinkers typically labelled postmodern with social constructivist tendencies in the sociology of science, presumably because of a perception that the two share a proclivity for skepticism. So, although this has little to do with "postmodern thought, " I provided my own (largely skeptical) reading of a few texts in the sociology of mathematics with a constructivist slant.

Of course skepticism has been around since antiquity, which makes the "postmodern" label look particularly silly. But there's no doubt that some people get worked up defending mathematics from the dangers of skepticism, either to maintain their own metaphysical equilibrium or, more mundanely, to protect the public perception of the discipline, especially insofar as the latter influences research budgets. The bulk of my PCM article is devoted to an attempt to look past both metaphysical and budgetary bottom lines to perceive what genuinely motivates (pure) mathematicians, as indicated by some of their (mostly informal) writings on the topic. Whether or not I was successful is for the reader to judge. I would hope to be judged on the basis of my actual intentions.

(Regarding the excerpt from my article posted by "Descartes's ghost," here it is in context:

a metaphysical abstraction like "essence", like a mathematical abstraction like "set", designates nothing in itself, but in each case refers to a canonical body of specialized texts in which the term in question plays a central role. I would like to argue that the nothing designated by "set" is somehow different, and more fruitful, than the nothing designated by "essence."

And here is the living Descartes, on the "essence" of a triangle:

"...when I imagine a triangle, even though there may perhaps be no such figure anywhere in the world outside of my thought, nor ever have been, nevertheless the figure cannot help having a certain determinate nature, or form, or essence, which is immutable and eternal, which I have not invented and which does not in any way depend upon my mind." (Fifth Meditation, my emphasis).

My point was that the notion of set provides a more fruitful understanding of triangles than Descartes' talk of essences, even though one can only make sense of sets in connection with a certain "canonical body of specialized texts" (the axioms of set theory), just as the term "essence" as used by Descartes refers implicitly to the corpus of Western philosophy as a whole.)