(from a 1991 letter to a Rocket Scientist [no joke])

Bob,

I enjoyed talking with you the other night, and I'm sending some Xeroxed pages about deconstruction from a book by Jonathan Culler. Critical theory excites me because it ignores traditional disciplinary boundaries. In a way it's a lot like philosophy, except that lately, academic philosophy is one of the primary things it messes with. This drives philosophers crazy because they seem to take debunking and messing with people as kind of a sacred calling. As a former philosophy major, I have to admit that upsetting philosophers makes me happy.

It's no mistake that much recent thought comes out of France, because the French educational system, especially at its highest levels, doesn't have many of the disciplinary boundaries that ours does. One of a discipline's (and a profession's) chief claims to legitimacy is the ability regulate itself. When critical theorists start talking about art history or sociology, they challenge those disciplines' claim to authority. These theorists also view their willingness to depart from the established procedures of a discipline or profession as a good thing, a critical perspective.

Well, enough said about recent thought's colonizing tendencies. What I wanted to write about is mathematics. Actually I know just enough about math to clarify my thinking about it. I've never taken anything past first year calculus. I've never read any philosophical accounts of mathematics more recent than Kant, although I know a few things about John Stuart Mill, Russell, and Whitehead's slants on the field. If I knew more, I'd be aware of just enough subtleties, qualifications and problems to bog down my thinking. As it stands, I have the luxury of being sophomoric.

As I re-read the enclosed chapter on deconstruction, I began to speculate as to how it might operate in mathematics. I don't know enough about math to know where to go with these thoughts, or even if they'd fly, but I enjoyed myself thinking about it. I'm just going to throw them out in fragments and see if anything happens. If you find any of this thought-provoking or useful, let me know. If not, well I've had a good time thinking about math.

1. Math, as it's currently taught and codified, seeks to halt interpretation. Once something is proven, it's proven. Interpretation, from within a mathematical system, should theoretically be impossible.

Understandably then, math has the incredibly seductive appeal of that which is indisputably true. And transcendentally true, for all time. No wonder philosophers from Plato to Russell and Whitehead have looked to math and geometry as an ideal discourse.

2. To deconstruct mathematical systems, you would have to focus on elements which are taken to be givens (self-evidently true) and demonstrate that those elements are actually derived, not self-evident, and in fact constructed.

3. I don't think it's worth trying to show that math bears little relation to "reality" (i.e. that it's a form of representation). I think that most mathematicians know that (Gödel's theorem) and maybe even pride themselves on it.

3.A I also think that it might be more useful to steer clear of explicit givens in a mathematical system. Anything resembling an axiom in geometry acknowledges the fact that it can't be proven. It would be more interesting to interrogate the logical givens which make it possible to move from A to B in a particular proof.

4. It would be much more interesting (and more difficult) to demonstrate how math is derived from specific, contingent, historical circumstances. What would it be like, for instance, to relate Cartesian mathematics to specific historical and philosophical trends in France at the time? To examine all those weird, deistic beliefs that get written out of Newton's ideas? To do a psychoanalytic study of some more recent mathematicians, demonstrating how local context and childhood experience contributed to a particular proof?

These sorts of moves would seriously disrupt the narrative which mathematics tells about itself: that it is a logical discipline removed from history, contingency, and politics, answerable only to its own rules and self-governing principles. You could also undermine its claims to neutrality

Politics: there's a ripe field. Relate Cartesian mathematics to artillery shell projections. Which relates it to industrialism, imperialism, and the subjection of non-white peoples. Huh, I can imagine how that would sound in the textbooks: "Cartesian mathematics were developed in order to more precisely calculate the arcs needed to drop cannon balls on non-Europeans." (Okay, it's an oversimplification. Cartoon Marxism. But it gives me a sense of where to head. I wonder if "pure" math has its roots, like so many other purities, in Kant. )

Anyway, demonstrating how Cartesian mathematics were appropriated is not the same as showing how they were produced. You could, however, relate them to a specific moment in an emerging tradition of rationalist thought that arose in a dialectic with mercantile capitalism and European colonialism. Yes, that's closer to what I'm after.

5. So, you look at a particular proof, school or system. You focus on the things that appear to be either self-evident or derived only from that proof/school/system's logical procedures. You take those givens and show how they are NOT derived from an ideal logic which is removed from history, politics, and contingency, but are instead derived from local, contingent circumstances.

To do this intelligently, you would have to focus on the obvious dialectic between current mathematicians and a history of mathematical thought. However, that history, like all histories, would be a highly selective one. The problems mathematicians focus on would vary historically and from culture to culture. What makes a mathematician would vary. How the culture (and political forces within it) support him or her would vary.

6. It would be interesting to study zero, which I believe Europeans cribbed from Islamic culture (Moors in Spain?). Absolute absence. An impossible, but productive, concept. What would modern math look like without zero, which was not derived from western mathematics' internal logic, but appropriated from Africans? Would any of the west's subsequent conquests, it's elaborate machinery of exploration and cartography, been possible without it? Without zero, this lacuna, this implausible gap that spans planes and anchors integers?

Did it come over with Aristotle? Was it an obscure monk, laboring over the [Arabic ?] texts that first grasped, dimly, it's significance? What would absolute absence mean to a monk? To Aquinas? Ex Nihil. God creating out of an absolute void, an idea which would have seemed nonsensical to the Greeks. Sure there was a void before Aquinas, but it wasn't the same void. Would these thoughts have occurred to Aquinas without zero?

The place to start might be conflicts about universals. Abelard castrated because of the appropriation of absence? Desire, hollow in its center, structured around hunger, absence, lack. Castration as being the final, logical embodiment of desire--the absence at it's inner core.

Having made this unexpected leap from Algebra to Abelard, I realize that I'm beginning to do what I previously claimed could not be done: interpret mathematics. Math becomes interpretable once you begin to focus on its production and its significance, instead of questions of validity and proof.

Once you stop asking "is it valid within its own system?" you can see another, more circumspect question: "what set up this system in the first place, certainly nothing inevitable about the universe, categories of thought, or unavoidable linguistic procedures."

Just as the constantive turns out to be a special case of the performative in semiotics, math's concern with logical proof comes to be a subset of larger questions. Questions like how math proofs/systems are produced, what significance they have in the culture, how they are appropriated, who determines what counts as valid, what narratives math tells about itsel(ves) and what those narratives ignore, erase and hide.

Well, Bob, I see that I've completely forgotten my audience and wandered into something between speculative thought and beat poetry. I don't know if this was interesting to you, but I'm getting some great ideas. I want to have a story where one character is writing a treatise on the history of zero in world culture. A mathematician scorned by his profession, but obsessed with a pattern, or perhaps a lack of pattern, he sees everywhere. A paranoiac, out of his gourd scholar --cross between Hesse and Pynchon, with a little Bertold Brecht tossed in. A term, delineated by a circle, the most economical of figures, which signifies absence. One stroke which signifies that which, by its very definition, cannot be signified. Yet which sits dead center in the Cartesian grid that makes systems, governments, missiles, fly. A term for the very gaps, the very absences, which make signification possible to begin with. How many modern mathematics do without it, and what are their anchoring principles?

Enough of this. Hope everything's okay up there in Ann Arbor and you're enjoying the PhD track after your stint at the JPL. I hope your students enjoy Calculus as much as I did. I liked math a lot, although I was always slow at it, a B student. The thing which I couldn't get over as I took calculus was that we kept solving things which I wouldn't have previously believed could be solved, by anybody. I remember this problem about the rate water was rising in a pool that I must have worked three hours on. Solving that was like learning to do a back flip or making out for the first time. It broadened my sense of the possible.

 Warren Hedges, English Dept., Southern Oregon University, 9/95