Dr. A., whose pale eyes and sharp nose and traces of gray hair are all very becoming–he is certainly the most attractive in the Math department, very likely the most attractive at Tech–is a little frightening. He speaks in a low voice with a comfortable Russian accent, the sort that is there to stay since its owner doesn’t mind it, and he speaks very slowly, in a way that makes everything he tells you in class seem like a matter of course, damned though you are if you can remember it. “Some people call me a killer,” he says, “and I’m not.” He says this with a genuine air of protest, as if he were really upset to be considered such by his students. He hands out the syllabus, and asks the class, in those low, Slavic-tinctured tones that hum about the sternum, if anyone has any questions. The pale blue eyes are restrained but impatient, and everyone can see it. So of course, no one does.

“This is your first real math class,” he told us, and it was exactly so. There was little calculation, almost none in the entire course. What I did in Basic Concepts, about a year ago now, is a long series of puzzles, thought experiments, glistening webs of tight-drawn axioms and definitions. It is over these rarefied nodes and members that practical maths flicker about like spiders. In fact, these spiders are adrift; Gödel saw that nearly a century ago, that even in the Platonic realms of absolutes and ideals, the outermost nodes link back to themselves: the whole sticky mess bent into a Klein bottle. Yet about the inner-outsides the spiders deftly creep, undaunted by paradox.

Were it not for the spiders, a web might be spun about a single point, looping back to the node with the blithe suggestion that 1 is no different from zero, everything is nothing and nothing is whatever you like. Or it might be that 1 is not zero, but 1 + 1 is. This one is called a finite field, and as such can claim qualities of interest that the whole, infinite (yet we call them countable) set of integers by themselves cannot.

Nearly four years of mathematics in college has taught me nothing so well as the understanding that truth does not, cannot exist. In the logic class I’m now grading for the third semester running, there is a simple problem that gives voice to what I think is a profound characteristic of mathematics itself. Given a few statements, students must turn up whatever conclusions they can using rules of logical inference. In the nearly one-hundred times I’ve graded this problem, only a handful of students accept that there are none to be made. We want our axioms to be useful.

For whatever reason, evolution has finished our minds to prefer certainty, even in realms where none exists. In mathematics, we suppose what seems to be the case. We examine the world and construct a rigid, Platonic web over which scientists and the more daring philosophers can scuttle and snatch what they call truths, until they discover some conclusion where the nodes of the web are tangled, where what they observe is impossible to calculate. Then at the tangle, with delicacy and a sensible reluctance to upset the web, a definition is revised, or, more daringly, an axiom withdrawn, and for the nonce a tangle is circumvented, to keep the spider’s legs from catching.

Trying to dodge the judgmental blue eyes of Dr. A. as you scuttle through the halls to some class or other, your gaze might fall on a poster outside his door. The details of it are opaque, mysterious, enticing but forbidden. For in addition to being the most handsome and intimidating professor in the Math department, he is probably the most impressively published. The face that looks out from the poster is a Dr. A of days gone by, perhaps twenty-five years ago or more. It lacks the flecks of gray in his hair, and the eyes (though the photograph is in black and white) bear a depth of hue that

for some reason impart an edge of rebellion: nothing like the fearsome font of wisdom he now seems.

But the picture is only there for show, for vanity even. Beside it, beneath the words and symbols that, to an undergraduate, are familiar but incomprehensible, a maddening conflation of everything he knows, there–you see that out on the very edges of the web was Dr. A. himself, reknotting tangle that caught at his legs.