Lecture #1:
There lives the dearest freshness deep down things
Welcome, everyone, to Math 190 / Literature 210: Mathematical Proof as Literature.
I realize that half of you are here only to fulfill the writing requirement, and the other half, the math requirement. Either way, I hope to cause you a great deal of intellectual discomfort.
As you know, I am a scholar of literature, with no more than a high school background in math. Yet together we shall reach up and touch the thinnest, most delicate branches in the canopy of modern mathematics. Most likely, we will snap them by mistake.
Anyway, we begin as moderns must: by venerating the ancients in a covertly self-serving manner.
In A Mathematician’s Apology, after a long preamble about mathematics as an Edenic garden of harmless beauty, G.H. Hardy finally turns to some actual math:
I will state and prove two of the famous theorems of Greek mathematics… They are ‘simple’ theorems, simple both in idea and in execution, but there is no doubt at all about their being theorems of the highest class. Each is as fresh and significant as when it was discovered—two thousand years have not written a wrinkle on either of them.
No surprise that Hardy calls the proofs “significant.” But why “fresh”?
Why advertise this proof, like a synthetic fabric, as “wrinkle-free”?
Perhaps he means that each proof still “sounds like new”? But if so, this is fatuous. Hardy’s presentation is new; he is not using Euclid’s precise logic, and certainly not his precise Greek. If he is praising the style of the proofs as “fresh,” then he is merely applauding his own wit and verve, his own ability to bring these dusty texts back to life.
Or, alternatively, is Hardy making a claim about the nature of mathematical thoughts–that, somehow, to think them is to refresh them? That seems a lovely idea: that a proof blooms anew in each mind that ponders it.
But what does it say about the tenuous ontology of mathematical objects, if they are freshened in the mere thinking?
If a proof unfurls in a forest, with no mind to perceive it, is it still logically sound?
Anyway, enough of this game, these deliberate misconstruals. I know (or think I know), what Hardy means. He is not talking about a fresh style, or the fresh ears of a new listener. He is talking about the proof itself, which boasts some intrinsic freshness.
But this, too, is troubling.
What sorts of things do we call “fresh”? Only those with the potential to wilt, fade, decay. Vegetables and breezes may be fresh. Stones and stars may not.
Does this not contradict the traditional image (which Hardy has painted mere pages earlier) of proofs as timeless works? We do not call the pyramid in Giza “fresh.” We do not call Stonehenge “timely.” How, then, can mathematics be both fresh and eternal?
Or, perhaps–and here, after all this meandering, I began to circle towards my own view of the matter–is freshness the very essence of math’s immortality? Does the permanence of mathematics lie not in some kind of artistic or practical relevance, but in its potential for perpetual surprise?
I leave this question to your discussion sections: What, exactly, is fresh in an ancient proof?
Lecture #2:
The self is a cage in search of a bird
Welcome back. I must confess that my first lecture was, in the strictest sense, a mathematical failure. I talked about a proof; I proved nothing.
Let us remedy that today, and consider the first of Hardy’s two specimens of freshness: the proof that there exist infinitely many prime numbers.
He opens with a definition:
The prime numbers or primes are the numbers
(A) 2, 3, 5, 7,11,13,17,19, 23, 29,…
which cannot be resolved into smaller factors. Thus 37 and 317 are prime. The primes are the material out of which all numbers are built up by multiplication: thus 666 = 2 ⋅ 3⋅ 3 ⋅ 37. Every number which is not prime itself is divisible by at least one prime (usually, of course, by several).
Next comes the proof:
We have to prove that there are infinitely many primes, i.e. that the series (A) never comes to an end.
Let us suppose that it does, and that
2, 3, 5,… , P
is the complete series (so that P is the largest prime); and let us, on this hypothesis, consider the number Q defined by the formula
Q = (2 ⋅3⋅5⋅ … ⋅ P) +1.
It is plain that Q is not divisible by any of 2, 3, 5,…, P ; for it leaves the remainder 1 when divided by any one of these numbers. But, if not itself prime, it is divisible by some prime, and therefore there is a prime (which may be Q itself) greater than any of them. This contradicts our hypothesis, that there is no prime greater than P; and therefore this hypothesis is false.
My question: Who is the protagonist of this proof?
(“Wait,” you say, “must a proof have a protagonist?” Well, that impertinent question is yours, and this lecture is mine, so let us proceed from my preferred assumption: that proof, like most forms of narrative, has a hero. Or at least an actor who’s first on the call sheet.)
In naming the protagonist, one might point to P, the ostensible largest prime. But read again. The true focal character is not P, but its antagonist Q, who appears only in the final act, and whose self-destructive nature is the narrative engine of the whole proof.
This proof does not have a hero. It has an antihero.
Q is an embodied contradiction. It is prime and not. Prime, because it is divisible by no prime; and not, because it is larger than any prime (under the assumptions of the proof) can be.
Hardy presents Q’s dilemma in layered and convoluted language, with a fog of ambivalence. All is couched in conditionals (“if not itself prime”) yet the conditionals reverse themselves (“therefore there is a prime… which may be Q itself…”). Like Gregor Samsa, Q awakes to find itself grotesque, transformed, negated. Q is a poor creature, conjured by unfeeling gods, for the sole purpose of refuting itself.
Hardy, of course, would disapprove of this reading. Why psychoanalyze the character Q? In Hardy’s mind, there is no Q, no character. That’s the whole point.
But in Hardy’s proof, there is such a character: a chimerical non-prime prime, as real as any figure in myth or character in fiction. Q is realer, or at least more enduring, than Hardy himself, or Euclid, or any of us slowly decaying organisms in this lecture hall whose brief lives by sheer historical happenstance catch the glimmer of the present moment.
Lecture 3:
You can have anything in life if you sacrifice everything else for it.
Last lecture, we explored Hardy’s proof (Euclid’s, really, but Hardy is exercising squatter’s rights over it) of the infinitude of the primes.
However, I omitted the passage’s most famous paragraph, a concluding comment from Hardy:
The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
The bravado is magnetic. But in what way, exactly, does a mathematician offer the game?
Hardy is referring to the pivotal moment — which he actually breezes past, with baffling nonchalance — when we posit the opposite of what we are trying to prove.
We have to prove that there are infinitely many primes, i.e. that the series (A) never comes to an end.
Let us suppose that it does, and…
Hear that? Not even a period, not even a full stop. For this existential risk, Hardy offers only a comma of punctuation, half a breath’s pause, before moving on.
But this moment deserves more. Let us linger here.
Hardy proposes to sacrifice precisely what he wishes to prove. The primes do not end; so let us suppose that they do.
Wise politicians know never to repeat an attack against them, not even to refute or negate it. To say something at all is to entertain it, to enliven it. Gossip does not spread because it is true; it spreads because it is spoken.
Hardy then, must be playing a different game than a gossip or a politician. He knows that, in his arena of logical proof, a false claim cannot long stand. It will trip over its own falsehood, get tangled in its own mendacious shoelaces.
What, then, does he risk? What does he sacrifice?
Nothing, really. In the literature of mathematics, all statements already exist, like distant stars. To author a proof is to guide our gaze along a constellation of these pre-existing statements, to reveal a meaningful shape in the otherwise meaningless scatter.
There is, of course, no risk of sacrificing the game.
Rather, what the mathematician sacrifices is herself. The mathematician surrenders everything: not to a human opponent, but to the game itself, to the fixed and merciless rules of logic. She throws her oars out of the boat, and lets the rapids carry her where they may, no matter what horrors await.
And make no mistake. Horrors await.
Endless primes. Rationals lost like grains of sand tossed in an irrational sea. Curves jagged at every point. Shapes we cannot measure. Logic even turns against itself, and proves its own limitations. To travel this landscape we must sacrifice, to varying degrees, everything human about us: intuition, vision, experience, personality, and in the end, even the very virtue that led us to begin the journey, the thirst for certain truth.
It is only a slight indulgence to say that to be a mathematician is to sacrifice oneself to math.
