The Unapologetic Mathematician

Mathematics for the interested outsider

Chalk is a “Feelie”

Okay, so I’ll pile on with the interactive fiction chatter. I really should, since I’ve been playing IF games since I was a wee lad.

First someone pointed out that a long calculation is like a computer game where you have to save and keep backtracking to your saved states. Isabel at God Plays Dice then drew the more specific connection to interactive fiction. Then Mark at Inductio Ex Machina contributed this sample transcript of such a “game”.

I’d like to point out an unintended analogy here. It’s pretty well accepted within the IF community that any puzzle should be solvable at a first pass. That is, if you’ve done everything right you’ll have all you need to solve a given puzzle without guessing, failing, and backtracking. In fact, it’s the height of bad writing to include a puzzle that requires you to attempt it and fail to gain information needed to pass.

I think that the same holds true in mathematics. If you find that you must do a hard calculation with attendant backtracking, you’re asking the wrong question. When properly viewed, the solution to any problem should be inherent in the problem itself. Of course, it might be more convenient in context to bash your head against a wall than to look for the hidden doorway, but it’s really not the best way to go about things in the long run. I come back to my favorite passage from Grothendieck’s Récoltes et Semailles.

Prenons par exemple la tâche de démontrer un théorème qui reste hypothétique (à quoi, pour certains, semblerait se réduire le travail mathématique). Je vois deux approches extrêmes pour s’y prendre. L’une est celle du marteau et du burin, quand le problème posé est vu comme une grosse noix, dure et lisse, dont il s’agit d’atteindre l’intérieur, la chair nourricière protégée par la coque. Le principe est simple: on pose le tranchant du burin contre la coque, et on tape fort. Au besoin, on recommence en plusieurs endroits différents, jusqu’à ce que la coque se casse — et on est content. Cette approche est surtout tentante quand la coque présente des aspérités ou protubérances, par où “la prendre”. Dans certains cas, de tels “bouts” par où prendre la noix sautent aux yeux, dans d’autres cas, il faut la retourner attentivement dans tous les sens, la prospecter avec soin, avant de trouver un point d’attaque. Le cas le plus difficile est celui où la coque est d’une rotondité et d’une dureté parfaite et uniforme. On a beau taper fort, le tranchant du burin patine et égratigne à peine la surface — on finit par se lasser à la tâche. Parfois quand même on finit par y arriver, à force de muscle et d’endurance.

Je pourrais illustrer la deuxième approche, en gardant l’image de la noix qu’il s’agit d’ouvrir. La première parabole qui m’est venue à l’esprit tantôt, c’est qu’on plonge la noix dans un liquide émollient, de l’eau simplement pourquoi pas, de temps en temps on frotte pour qu’elle pénètre mieux, pour le reste on laisse faire le temps. La coque s’assouplit au fil des semaines et des mois — quand le temps est mûr, une pression de la main suffit, la coque s’ouvre comme celle d’un avocat mûr à point ! Ou encore, on laisse mûrir la noix sous le soleil et sous la pluie et peut-être aussi sous les gelées de l’hiver. Quand le temps est mûr c’est une pousse délicate sortie de la substantifique chair qui aura percé la coque, comme en se jouant — ou pour mieux dire, la coque se sera ouverte d’elle-même, pour lui laisser passage.

L’image qui m’était venue il y a quelques semaines était différente encore, la chose inconnue qu’il s’agit de connaître m’apparaissait comme quelque étendue de terre ou de marnes compactes, réticente à se laisser pénétrer. On peut s’y mettre avec des pioches ou des barres à mine ou même des marteaux-piqueurs: c’est la première approche, celle du “burin” (avec ou sans marteau). L’autre est celle de la mer. La mer s’avance insensiblement et sans bruit, rien ne semble se casser rien ne bouge l’eau est si loin on l’entend à peine… Pourtant elle finit par entourer la substance rétive, celle-ci peu à peu devient une presqu’île, puis une île, puis un îlot, qui finit par être submergé à son tour, comme s’il s’était finalement dissous dans l’océan s’étendant à perte de vue…

Le lecteur qui serait tant soit peu familier avec certains de mes travaux n’aura aucune difficulté à reconnaître lequel de ces deux modes d’approche est “le mien” — et j’ai eu occasion déjà dans la première partie de Récoltes et Semailles de m’expliquer à ce sujet, dans un contexte quelque peu différent. C’est “l’approche de la mer”, par submersion, absorption, dissolution — celle où, quand on n’est très attentif, rien ne semble se passer à aucun moment: chaque chose à chaque moment est si évidente, et surtout, si naturelle, qu’on se ferait presque scrupule souvent de la noter noir sur blanc, de peur d’avoir l’air de combiner, au lieu de taper sur un burin comme tout le monde… C’est pourtant là l’approche que je pratique d’instinct depuis mon jeune âge, sans avoir vraiment eu à l’apprendre jamais.

In case you haven’t yet passed your French language qualifier, I’ll give a rough translation.

Take, for example, the task of proving a theorem. I see two extreme approaches one could take. The first is that of hammer and chisel, wherein the problem posed is seen as a large nut, hard and smooth, which contains a nourishing meat protected by the shell. The principle is simple: one puts the edge of the chisel against the shell and hits it hard. If necessary, one tries again in many different places, until the shell cracks — and one is happy. This approach is especially appealing when the shell shows a rough or bumpy patch where it can be grasped. In some cases, such places to grab the nut jump to the eye. In other cases, one must use all one’s senses and search carefully before finding a point of attack. The most difficult case is that where the shell is perfectly round and evenly firm. When hit strongly, the edge of the chisel just scratches the surface — one ends up merely tired. Sometimes the nut will finally crack through mere strength and stamina.

I can illustrate the second approach with the same metaphor of a nut to be opened. The first explanation that comes to mind is to immerse the nut in some softening liquid — water, for instance — and to rub it from time to time to allow the water to penetrate better, but otherwise to leave it alone. Over weeks and months, the shell softens — when the time is right, a flick of the wrist is sufficient, and the shell opens to it like a ripe avocado! Or again, one can leave the nut out in the sun and the rain and even through the icy winter. When the time is right, it is a delicate touch that breaks the shell — or to say it better, the shell will open itself to let one through.

The pictur that came to me recently was again different. The unknown thing one is trying to undertand seems to me like a stretch of land or a hard patch of earth, hard to dig into. One might go at it with picks or mining tools, or even with jackhammers: this is the first approach, that of “chisels” (with or without hammer). The other is that of the sea. The sea advances imperceptibly and noiselessly. Nothing seems to break, nothing moves… Yet eventually it surrounds the land. It slowly becomes a peninsula, then an island, then an islet, and finally it is submerged completely, dissolved into the ocean which stretches as far as the eye can see…

The reader who is familiar with some of my work will have no difficuly determining which of these two approaches is “mine” — and I have had occasion already in the first part of these “Reapings and Sowings” to explain myself on this subject, in a slightly different context. It is “the method of the sea”, by submersion, absorption, dissolution — that where, if one does not pay close attention, nothing seems to happen at any given moment: everything is at each moment so evident and so natural that one feels nervous to write it down in black and white for fear of being others’ disapproval, rather than banging away at a chisel like everyone else… Yet this is the approach that I instinctively took since I was young, never having really noticed it.

As for the title of this post, a feelie is a physical object — often some document — that was packaged with a game and containing information crucial to some puzzle you’d need to solve. That is, if you didn’t buy the game and get the feelie, you couldn’t get past a certain point. It provided a crude level of copy-protection back in the good old days, under the pretense of extending the game experience (more common in non-IF games was asking the user to type in some specified word from the documentation). Thus, a feelie was all too often a hack — a puzzle relying on them was awkward and inelegant, pulling you out of the experience of the game rather than immersing you in it as was hoped.

Blackboards full of equations serve the same obscuring purpose. True understanding never lies in a calculation. The chalk on the board should not be a map, but a lens, and the mathematics is not in the equations, but behind them.

October 19, 2007 Posted by | rants | 5 Comments

Chain Homotopies and Homology

Sorry about going AWOL yesterday, but I got bogged down in writing another exam.

Okay, so we’ve set out chain homotopies as our 2-morphisms in the category \mathbf{Kom}(\mathcal{C}) of chain complexes in an abelian category \mathcal{C}. We also know that each of these 2-morphisms is an isomorphism, so decategorifying amounts to saying that two chain maps are “the same” if they are chain-homotopic.

Many interesting properties of chain maps are invariant under chain homotopies, which means that they descend to properties of this decategorified version. Alternately, some properties are defined by 2-functors, which means that if we apply a chain homotopy we change our answer by a 2-morphism in the target 2-category, which must itself be an isomorphism. I like to call these “homotopy covariants”, rather than “invariants”. Anyway, then the decategorification of this property is an invariant, and what I said before applies.

The big one of these properties we’re going to be interested in is the induced map on homology. Let’s consider chain complexes A and B, chain maps f and g from A to B, and let’s say there’s a chain homotopy \alpha:f\Rightarrow g. The chain maps induce maps \widetilde{f}:H_\bullet(A)\rightarrow H_\bullet(B) and \widetilde{g}:H_\bullet(A)\rightarrow H_\bullet(B). I assert that \widetilde{f}=\widetilde{g}.

To see this, first notice that passing to the induced map is linear. That is, \widetilde{f}-\widetilde{g}=\widetilde{f-g}. So all we really need to show is that a null-homotopic map induces the zero map on homology. But if \alpha:f\Rightarrow0 makes f null-homotopic, then f_n=d^B_{n+1}\circ\alpha_n+\alpha_{n-1}\circ d^A_n. When we restrict f_n to the kernel of d^A_n, this just becomes d^B_{n+1}\circ\alpha_n, which clearly lands in the image of d^B_{n+1}, which is zero in H_n(B), as we wanted to show.

Now if we have chain maps f:A\rightarrow B and g:B\rightarrow A along with chain homotopies g\circ f\Rightarrow1_A and f\circ g\Rightarrow1_B, we say that A and B are “homotopy equivalent”. Then the induced maps on homology \widetilde{f}:H_\bullet(A)\rightarrow H_\bullet(B) and \widetilde{g}:H_\bullet(B)\rightarrow H_\bullet(A) are inverses of each other, and so the homologies of A and B are isomorphic.

This passage from covariance to invariance is the basis for why Khovanov homology works. We start with a 2-category \mathcal{T}ang of tangles (which I’ll eventually explain fore thoroughly). Then we pick a ring R and consider the 2-category \mathbf{Kom}(R\mathbf{-mod}) of chain complexes over the abelian category of R-modules. We construct a 2-functor K:\mathcal{T}ang\rightarrow\mathbf{Kom}(R\mathbf{-mod}) that picks a chain complex for each number of free ends, a chain map for each tangle, and a chain homotopy for each ambient isotopy of tangles. Then two isotopic tangles are assigned homotopic chain maps — the chain map is a “tangle covariant”. When we pass to homology, we get tangle invariants, which turn out to be related to well-known knot invariants.

October 19, 2007 Posted by | Category theory | 3 Comments