Alien mathematics

Unlike what you might be expecting from the title, this post is not about Großendieck's fundamental reploughing of algebraic geometry, but about the following hypothetical question:

Imagine we discover another civilisation living in some corner of some galaxy, where we know that the physics is essentially the same than on earth. Would they have the same mathematics than us? And if yes, would they have the same mathematics history than us?

David Gross was proposing this thought exercise at the end of a public talk by Robbert Dijkgraaf at KITP, and I figured the best way to attack it was by taking a hot bath. In fact, the bath was so hot that I could feel my spirits evaporate and I felt that were I to think about this question in such circumstances, I would develop biased conceptions that would definitely screw all my chances to reach any interesting conclusions... Nevertheless I did it, and it's a mess.

(Note that Dijkgraaf wasn't too inspired by the question -- I guess this is the difference between a Nobelised and a non-Nobelised physicist: the former becomes philosophically oriented (remember Josephson))

Probably it is sensible to start by the second part of the question, assuming the answer to the first is positive. Then it is obvious that it would be quite challenging to defend the opinion that their math history is exactly the same as ours. They would have needed to have e.g. an Evariste Galois killed at 20 in an obscure duel, etc. But might they have had the same structure in the development of their mathematics?

What would Kant say about this question? I believe that it would be something along the line of "Their mathematics would resemble ours inasmuch as we are able to perceive them, and them us. (After all they could be made of "dark matter", in which case it would be strange if they had the same concepts as ours...)

Mehr Licht/Mehr Nicht

"Mehr Licht!" Tels sont les derniers mots traditionnellement attribués à Goethe sur son lit de mort. Mais les mauvaises langues ont prétendu qu'il fallait en réalité entendre "Mehr nicht!" -- je n'en puis plus. Cette anecdote m'est revenue en tête hier, et j'ai pensé qu'on pouvait y voir plus qu'une moquerie: et si les deux interprétations n'étaient pas inconciliables? Ma thèse est celle de la coincidence, dans l'esprit du créateur, de l'instant de l'illumination et de celui de l'épuisement. Après tout, comment pourrait-il en être autrement, étant donne que pour percer la coque opaque de l'inconnu par un éclair de lucidité, il faut avoir atteint la frontière lugubre du périmètre de sécurité de l'esprit? Prenons l'example de l'athlète qui établit un nouveau record: le mehr Licht autant que le  mehr nicht l'habitent, et se confondent en lui.

Dans The Crucible d'Arthur Miller, il y a un vieil homme honnête qui est exécuté par l'Inquisition par écrasement: on lui pose des pierres de plus en plus lourdes sur la poitrine jusqu'à ce que mort s'en suive... Ses dernières paroles furent : "More weight!" 

Ici aussi on peut le comprendre de plusieurs manières: la manière comique (qui n'est pas ma favorite, va sans dire) selon laquelle il se moquait de ses bourreaux, mais aussi la manière, disons, existentielle qui appelle à plus de poids dans les décisions et les attitudes de vie de ces concitoyens. Une vie plus lourde de sens.

Fun with the G-string

If you like parodies of academic articles (and I know you do), check out how the early fathers of string theory used to have fun: The Super G-String. I don't know who is behind it (it is hosted on Warren Siegel's webpage) but the fake authors are V. Gates ("wie geht's?", a reference to Jim Gates), Empty Kangaroo, M. Roachcock (cockroach), and W.C. Gall (let me know if you crack this one---my attempt is the French word calvitie...).

I came across this paper because it is mentioned in the introduction of Aspinwall's lectures on D-Branes on Calabi-Yau Manifolds, where he writes in a footnote about D-branes that "The first reference to such objects that the author is aware of is, oddly enough, section 4 of [1]."

And indeed, on section 4 of the parody, there is an allusion to what we would call nowadays braneworld model-building, with the open string ending on a four-dimensional submanifold. So humour can be serious sometimes! After all the researcher's work isn't that different in its essence from a ludic fantasy...

Failblog

There is a YouTube channel called failblog which is dedicated to the biggest possible failures. At first I laughed my ass off by watching some of those videos, but then I started to think that there might also be something very interesting there. After all this is a precious collection of what is know as "actes manqués" in psychology (parapraxis), which are the equivalent in action of what a Freudian slip is in conversation. Where it becomes interesting is that such failures are, according to standard psychanlaytic theory, expressions of subconscious pulsions, and as such it is possible to make sense out of them.

Let us watch a few examples and you'll see what I mean.

First a very awkward frisk by a policeman who is probably repressing his homosexual tendencies. It's called "Frisk Fail" :

What is very striking is that the gesture of the policeman does not look at all like an inadvertent slip but rather like an intentional pass. The policeman is overwhelmed by a subconscious desire and loses touch with admissible behaviour. When he realises what he did and his conscious self takes control again, he certainly cannot believe it and apologizes repeatedly.

This interpretation is corroborated by the discussion that they are having, which resembles some kind of seduction game :

Cop -- Spread your legs for me please... I see you're shaking, I make you nervous/That makes me nervous.

Guy -- It's cold outside.

Cop (laughing, at ease) -- Yeah? It's cold? ...

The policeman is trying to relieve the tension in this delicate situation, full of sexual implications, but when the guy acknowledges his sensitivity to the cold like a shy girl adorably shaking in her first evening dress he's getting so loose that repressed emotions take over and push him to make the fatal move.

Now let's look at this other failure, also of sexual origin I'm afraid (most of them probably are---but not all !---which misled Freud to think that sexual repression alone could account for the entirety of human behaviour).

The failure here is unbelievable : how could she fail to see the pole just in front of her ? There is apparently something big going on here ! But there is not far to look for to find it (some YouTubers also suggested it in the comments of this video in fact). She is obsessed by her sexual frustration in her little village in Alaska. When she's faced to a symbol of her object of desire (a phallus), she fails to recognise it. (Notice the similarity with the cop video...)

Now the "Best Man Fail" :

The obvious thing to say here is that the best man is probably secretly in love with the bride, to the point that he does not even admit it to himself. His subconscious then forces him to do what he should have had the guts to do in full possession of his senses : throw himself at her feet to tell her his love and interrupt the wedding...

But what I find more interesting here is that there is a controversy in the comments of this video (like for many other videos on failblog) about its authenticity [I always find that very disappointing, not being able to trust something as a piece of reality, but I have to live with it you know; after all everything is fiction, etc.]. Someone is writing that the best man is acting like a bad actor in a Bollywood movie when he shouts "Nooooo!", and another that everyone is behaving totally unnaturally. And it is indeed quite weird how they all repeat "Oh my god!" one after the other... However, I do not think that this video is a fake, but that it is an illustration of how we Americans are preconditioned to behave in a stereotypical manner in our everyday life.

Here comes a video that amazed me deeply, "Proposal Fail" (you might enjoy it more without the sound first) :

This girl, unless her great emotion and the pressure of the crowd, is still able to swim against the tide, and to do it with the grace of an angel. I was moved to the soul by her lovely body language : the sparkle in her eyes and the little lateral movement of her mouth just before she starts speaking, her hand gently laid on her unfortunate lover's raised arm while she speaks, the other protecting her genitalia, her quiet precipitation and awe once she's spoken...

Alas, alas ! This girl was not for real : the proposal was a prank, as guessed by the reporters, and confirmed by a commentator who remembers reading it in the paper the next day.

(See here for a genuine reaction -- note the larsen effect at the crucial moment...)

I thougt I'd also show an example that is not related to any kind of Freudian bullshit :

Actually I failed. Notice his reaction : "Oh ffffffff.. Shit. Oh my god." This guy manages to repress his impulse to say "fuck", but only to replace it by another similarly vulgar word, namely "shit". The "oh my god" comes only after, once the first emotion has passed. No doubt Freud would have concluded that his sexuality is repressed, and expresses itself in a scatological way...

Sorry for this tasteless post.

Compactification and singularity

On the train from Lausanne to Geneva Airport I recalled the way Polchinski introduces his chapter on toroidal compactification :

"In general relativity, the geometry of spacetime is dynamical. The three spatial dimensions we see are expanding and were once highly curved. It is a logical possibility that there are additional dimensions that remain small."

J. Polchinski, String Theory, volume I, CUP 1998 (p. 231)

My personal version of this argument is to say that in GR space can be curved and hence it is a "logical possibility" that there are dimensions so curved that they actually close on themselves. I find that it is a very seducing way to introduce the notion of extra dimensions that are hidden because of their compactness, since it follows from the well-established curvature of space-time.

Polchinski then goes on to explain Kaluza-Klein theory with a periodic dimension without further explanations. This leaves the following question open :

What kind of massive object curves some dimensions so as to make them compact while leaving other dimensions non-compact ?

I've been drawing a lot since then and realized a few things that amazed me for five seconds before revealing their complete triviality... Nevertheless, here are some of my reflections.

At first the idea that an object curves only some dimensions and not all of them seems bizarre. Imagine a toy universe that has only two dimensions so that one can envision curvature as the bending of an elastic membrane with a ball on it. This video which is part of a documentary about Hawking has a nice illustration of this after 3:00.

Now if some dimensions are to be so strongly curved that they become compact, this must be the effect of a very massive object. So let us give the ball a tremendous mass; the way I picture what happens is that the weight of the ball stretches the membrane in the form of an extremely long tube. So neglecting the extremities, the membrane has now the topology of a cylinder, which is the direct product of a non-compact manifold, a real line, with a compact one, a circle. What seems a bit weird is that the radial direction, which is curved by the ball, remains non-compact, whereas the angular direction, which is not curved, is compact ! But there is a fallacy here : to have an angular coordinate (with a finite range) does not mean there is a compact dimension. The plane is not any compacter in polar coordinates than in Cartesian coordinates...

I'd like to understand spaces that have a non-compact part and a compact part. For simplicity I'll take direct products of real lines and spheres. The simplest case is R x S^1. This is pretty clearly a cylinder if you think of it as an S^1 fibration over R, but you could also think of it as a R fibration over S^1, in which case one can see that is is also a cylinder by taking into account that the fibers must not intersect. In fact, only the topology matters here and the cylinder can be continuously deformed at will. The formal definition of this direct product is R x S^1 = {(x, theta) | x \in (-\infty, \infty), theta \in [0, 2\pi]}. So you just need to specify a couple of coordinates, one for the real line (x), and one for the circle (theta).

An interesting deformation of the cylinder is achieved through a conformal mapping, where one defines a coordinate z = exp(x + i theta). This can be thought of as a radial coordinate r = exp(x), and an angular coordinate theta. The radial coordinate ranges from 0 to infinity but at r=0 the circle degenerates, which does not correspond to the cylinder bounded by circles at both its extremities at infinity. In consequence, the origin has to be removed. So one ends up with the manifold R^2 (in fact the complex plane...) with the origin removed : R x S^1 ~ R^2\{0}. R^2 is non-compact but by removing a point you make it isomorphic to a space that is the direct product of a compact space with a non-compact space. There is thus an equivalence between manifolds with a singularity and partially compact manifolds !

Coming back to our latex membrane, we understand now that in order to obtain a compact dimension the ball must be so heavy that it produces a black hole singularity.

Now consider R x S^2. It is hard to understand as a fibration : is it some kind of solid cylinder ? I guess that a particle can move arbitrarily inside the cylinder but as it approaches the surface it is forced to move tangentially to it (?) It is much easier to think about it after a conformal mapping, which leads to R^3\{0}. Again, the origin is removed for the angular coordinates to be well-defined everywhere. (Notice that the manifold is not complex anymore, given that it is odd-dimensional...)

Another simple case is R^2 x S^1, which can be reduced to the first case by rewriting it as R x R x S^1 ~ R x R^2\{0}. We get a manifold R^3 with a line singularity along the third axis. This is nice because it means we can get more than just one non-compact dimension (the radial direction), namely the dimensions of the singularity itself.

We are ready to generalize : after conformal mapping, the direct product R^n x S^d gives a (n-1)-dimensional singularity at the origin of a (d+1)-dimensional space.

String theory requiring 10 dimensions for its consistency while we only observe 4 non-compact dimensions, we might try the direct product R^4 x S^6. However, according to our reasoning, this is isomorphic to R^3 x R^7\{0}, so that the fourth non-compact dimension differs from the others in that its boundaries at infinity are six-spheres... (how bad is that ?)

If this has to be avoided, one must rather consider R^5 x S^5 ~ R^4 x R^6\{0}. This is pretty close to what appears in the context of the AdS/CFT correspondence ! There, the singularity would be black 3-branes and there would be a warping between R^4 and the radial coordinate to get AdS_5.

Remark :

The bending latex representation is misleading since it implies that the membrane is curved towards a third dimension. I don't think this is the proper way to view it, because otherwise the curvature of four-dimensional space-time would by itself imply the existence of extra dimensions... Rather, one should maybe picture a curved space by a density plot. The darker the color, the larger the curvature. The surface of the black hole would be a very dark circle. 

From the density plot the surface of the black hole is at a fixed radius, but from the curved membrane representation it forms a very long tube. It is folly to suppose the coordinate along the tube is the coordinate U = r/alpha' with r->0 defined by Maldacena.

Party String Theory

And of course you go to a party and of course someone asks you what you do and of course you say string theory and of course what is it ? and of course you don't know where to start. // Classic.

So it gets you thinking about how to present string theory and so you find one way and so you feel warm inside and so you post a post. // Classic.

A good way to introduce string theory is as a conceptual framework that unifies the two current pillars of theoretical physics : general relativity and quantum mechanics. To give a feel of what string theory is about, it suffices to focus the explanations on just one specific (but pretty central) aspect of each theory : black holes for GR, the uncertainty principle for QM.

1. The central result of Einstein's general relativity is that a very massive object -- or equivalently a very energetic object, remember E=mc^2 -- has the effect of curving the neighbouring space-time (which in fact implies gravity...). But there is a bound to this phenomenon, since if an object is too massive, it will curve space-time so much that it will end up tearing it apart and creating a black hole. (As an illustration, imagine space-time had only two dimensions, and think about it as a stretched piece of cloth ; putting heavy objects on it will curve it, and if an object is heavy enough it will make a hole in it.)

2. At the heart of quantum physics lies Heisenberg's uncertainty principle, according to which a particle cannot have both a precise position and a precise momentum (mass times speed). There is a tension between the precisions of those two properties : if the position of a particle is determined with great accuracy, unavoidably its momentum becomes very uncertain -- we know precisely where it is but not where it is going nor at which speed it is moving. And inversely, if the momentum is precise, it is the position which becomes uncertain, as if the particle were dissolved into some sort of a fuzzy cloud...

Now if you ask the philosophical question of knowing what is the ultimate building block of reality, and that you undertake to break objects down into little pieces, then in littler pieces, and then yet littler and so on, there will be a point where the piece you consider is so small that by the uncertainty principle its momentum (and so its energy) is so fuzzy that it can produce a black hole. From this point, the division process cannot be carried any further.

String theory aims at describing what is happening at this ultimate scale, where tiny black holes appear and disappear randomly by quantum fluctuations, and where the very notions of space and time lose their pertinence...

[Poetic suspension]

What we call "reality" is nothing but the surface of a chaos of black holes...

[Poetic suspension again]

And hence holography, AdS/CFT, and the like. Good.

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On a related note, people are usually amazed when I tell them that I switched from philosophy and art to physics. The most coherent explanation in my bag of tricks has to do with my interest in perception and the essence of reality, but I think I just found a new twist on it, as I was staring at the moon (beautifully speckled tonight). It goes something like this :

At some point in my early life it occurred to me that creation was the purest source of joy (this expression is in fact taken from my application letter to Oxford -- how naive I was...). So that might explain why I was interested in arts and why I even wanted to become a painter. But I was aware (I think) that the product of an artist's creation is not a work of art (although it is, don't get me wrong !), but rather a new perspective on existence.

[Which example could I find to illustrate that ??? I reckon that if you don't already know it, it will be quite impossible to convince you... For example when the Gothic cathedrals started to be built with thinner walls and with more light, it was not (even though maybe ingenuously it was) to make a "pretty" building, but to be in phase with the evolution of the societies from a feudal to a seigniorial organisation. (About this fascinating topic I recommend the brilliant book L'Art féodal et son enjeu social by André Scobeltzine.) But it is not a real good example, because the notion of artist was ill-defined at the time... Anyways !]

OK so the point is that what I am doing currently, namely theoretical physics, is also a creative venture. And what the theoretical physicist creates [notice my emphasis on the "theoretical" bit ? ; )] is not an article, not even a physical theory (although it is, don't get me wrong !), but -- reality itself !

Just like the philosopher in fact... And like the artist as well in fact... I'm afraid I would have utterly lost my party interlocutors by this time in fact... Perhaps that explanation ought to be substantiated more substantially in fact...


Addendum : Well my basic intuition arises roughly from the following type of considerations. Before Galileo the earth was the centre of the universe. It's not just that everybody wrongly thought it was. Go back to Timbuktu : if nobody sees the Dalmatian in the picture then there is no Dalmatian in the picture. Peace.

Time thickening

On Sunday I went to a Coffee Concert by John Myewrscough (cello) and Lara Dodds-Eden (piano). They played various compositions among which the mind-blowing Cello Sonata n° 1 by Alfred Schnittke (1934-1998). Here I just want to talk about a thought that came to me during the concert.

The enjoyment of music increases with the ability to "grasp" with the mind a piece of melody as a whole -- at once. I mean as a unique entity rather than as a succession of individual notes (Husserl tried to explain the possibility of such a thing in his Vorlesungen zur Phänomenologie des inneren Zeitbewusstseins). The broader the perception of a fragment of music, the deeper the pleasure (think about Mozart who was able to restitute an entire concerto after hearing it only once...). So listening to music encourages the development of the ability to grasp longer periods of time instantaneously, i.e. to have them "present" interiorly as sensations.

This is how music enriches our existences : by teaching us how to thicken our present time.

(The cellist's name contains the name "Myers", so I'll call that the timelike Myers effect ; )