Philosophy of Real Mathematics

February 7, 2006

Posted by Urs Schreiber

While I am struggling to understand elliptic cohomolohy, David Corfield, over on his weblog, has taken a look at the literature from the perspective of a philosopher of real mathematics.

“Real” here is meant in the sense of “what active mathematicians are really concerned with”, as opposed to an eternal occupation with Russel’s paradox and Gödel’s incompleteness. Have a look at his book for more.

To me, the interesting point to be addressed here is how we actually go about identifying the structures that we feel should be out there. As in: “How should we really think about elliptic cohomology?”, or the outworn but still curiously elusive “What is string theory?”. And maybe this one: “Is there a relation between these two questions?”

Posted at February 7, 2006 4:56 PM UTC

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Re: Philosophy of Real Mathematics

What would interest me about your questions is not so much their ultimate answers, but what kinds of answer would satisfy you. E.g., would characterising elliptic cohomology in terms of some 2-categorical universal property be the right kind of thing. Similarly, would you be happy with ‘String theory is a 2-functor…’?

Posted by: David Corfield on February 8, 2006 9:30 AM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

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This is a hard question. As a first attempt, let me try to answer as follows.

Maybe one way to describe the sort of understanding of some concept X that I would hope for is of the kind

X is a Y-structure in the context Z.

So I guess I am thinking in terms of internalization here. I suspect that there is only a handful really fundamental structures Y, in some sense. Here Y is something expressed in terms of an abstract set of diagrams. Examples would be Y = group, Y = bundle, Y=functor with certain properties, etc.

The more examples for Y there are (i.e. the more X there are such that there is a Z such that X is a Y-structure in Z) in my field of interest, the happier I am to find yet more examples of Y structures.

In the context of elliptic cohomology, for instace, I find a statement like

X=Elliptic cohomology is a Y=K-theory structure in the Z=loop space context.

very satisfying. Alas, while this seems to capture a good deal of what is going on, it is not fully correct yet.

But since it is already pretty good, I’d suspect that if we refine our notion of Y, it becomes true after all. We know that we can think of K-theory as a classification of 1-D susy field theories. These again can be thought of as certain transport 1-functors to a category of $\mathbb{C}$ -modules.

Being extremely cavalier with technical details, it should be not too wrong to say that

X=K-theory is a Y=(classes of)transport functor(s) structure in $\mathrm{Cat}$ .

(I hope in the present context nobody is offended by the audacious vagueness of this statement. :-)

So I am hoping that in the end something like this is true:

X=Elliptic cohomology is a Y=(classes of)transport functor(s) structure in $2\mathrm{Cat}$ .

Anyway, no matter if the precise content here makes sense to anyone else right now, it are statements of this kind that I would find satisfying.

And I hope it is clear why I would find the satisfying. Because they allow me to understand the world in terms of a few archetypes and a zoo of realizations of these archetypes. Once I understand the archetype, it is essentially a straightforward (though possibly tedious) matter to work out any of its realizations in contexts Z.

So for instance right now I cannot say that I understand many of the details that various authors discuss in the context of elliptic cohomology. But with the archetype “K-theory” thought of in the context “loop space” in mind I can get pretty far with pretending that I have a detailed understanding.

Posted by: urs on February 8, 2006 2:19 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

The *aesthetic* you present here is something I’d like to write about. I too am very drawn to that idea of simple concepts manifesting themselves in different contexts. Like the Legendre transform being to the Laplace transform what the tropical rig R_max is to C.

I’m not sure how universal though is the appreciation of this phenomenon. I’m reminded of a discussion on sci.physics.research between Baez and Kuperberg on the meaning of deep. While they agree to differ, there does appear to be a difference between their respective aesthetics, in John’s case expressed in the belief that many apparently complicated constructions will be seen to be very simple when framed properly.

Posted by: David Corfield on February 9, 2006 2:16 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

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An nice example that I recently came across was Michael Müger’s new proof of the Doplicher-Roberts result. His claim was essentially that by merely using the right category theoretic concepts (the right archetypes) the originally $\gt 200$ pages of proof reduce to just a handful.

I think Feynman used to make fun of mathematicians by observing that the only thing they can prove are trivial results. Since whenever a mathetmatician would explain his proof to anyone else he would explain the situation and then point out how the result follows trivially.

But this is just the nature of understanding something. As long as it looks complicated you haven’t understood it. Understanding it means that you see the pattern, after which the phenomenon becomes “trivial”.

Posted by: urs on February 9, 2006 2:51 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

I have written more about this in a post today on my blog.

Posted by: David Corfield on February 10, 2006 10:21 AM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

I certainly agree that many complicated constructions can be seen to be much simpler when framed properly. In fact, I quite appreciate arguments and presentations in mathematics that cut through the crap, so to speak.

However, I do not believe that category theory is a universal solvent that makes all things simpler. Ultimately most category theory is a form of bookkeeping, in my view. A solid investment in bookkeeping sometimes makes things much simpler. But sometimes bookkeeping that is promised as a simplification makes things much more complicated instead.

Posted by: Greg Kuperberg on February 12, 2006 5:54 AM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

It is clear that mathematicians vary in their attitude to the importance of categories and their higher-dimensional cousins. Much less is heard these days of an outright dismissal. But there’s a step up from Greg’s appraisal of it as a bookkeeping device to a view of it as an invaluable tool to aid in the construction of powerful new concepts, a view which it seems Urs shares with John Baez. This latter position expresses itself in a strong belief that the correct categorification of the mathematical ingredients of gauge field theory will lead to important new constructions for mathematics and physics.

Now, you might think that little hangs on where individuals position themselves on the scale. Truth will out. Everything will become clear in the wash. Either 2-gauge field theory will work or it won’t. But I’m not sure things are quite so simple. One would expect that those further committed to n-categories would be less inclined to appreciate results which don’t fit into a larger scheme. I imagine, Urs and Greg, you’d both recognise that the casting of Picard groups and Brauer groups of rings using weak 3-categories (TWF week 209) characterises them well. And both will presumably agree that just as there are totally unintersting physical facts, there are totally uniteresting mathematical *contingencies*. But I would also imagine that a result that might intrigue Greg, might seem less interesting to someone more committed to n-categories if it was perceived that the result did not fit its terms well. Accessing the extent to which one could describe one’s level of commitment to any framework, not just n-categories, as rational, rather than merely aesthetic, must surely be quite a subtle affair.

Posted by: David Corfield on February 12, 2006 1:08 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

I am just a lowly apprentice of mathematical physics. What do I know of the big questions that are being addressed here?! :-)

But for what it’s worth, this are some of my opinions - currently.

1) I guess it is trivially obvious that the advantage one finds in category theory depends on what questions one is concerned with.

2) In mathematical physics, proper bookkeeping goes a long way. Many concepts in mathematical physics result from some mathematician realizing the right bookkeeping tool that deflates physicist’s approach to a few central and powerful concepts. Take for instance the rephrasing of classical mechanics and its (geometric) quantization in terms of symplectic geometry.

3) Beyond mere bookkeeping there is category theory’s ability to guide you to ask the right questions and think about concepts from the right point of view.

A while ago I was trying to see the expected connection between 2-vector transport and CFT. Turns out that at the heart of it is the concept that a local trivialization of a transport functor is given by an adjunction. From this point of view everything easily falls into place. The s/t-channel duality which gave rise to the discovery of string theory appears as a consequence of general abstract nonsense from this point of view.

Posted by: urs on February 12, 2006 6:36 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

I guess I agree with all of Urs’ points. Beyond mere bookkeeping, category theory can sometimes guide you to the right questions.

All I would say is that category theory is an open-ended resource that is sometimes fundamental, that can sometimes be used in good or bad ways, and that is sometimes an outright distraction. It’s important, but I don’t think of it as a magic brain vitamin that makes hard things easy.

Consider, for example, how category theory might guide you to define quantum groups. As a first try, you might consider group objects in the contravariant category of non-commutative algebras. This is a natural proposal, but it isn’t the most fruitful one. A much better idea is to use Hopf algebras. Hopf algebras are a non-cocommutative generalization of group algebras, but they are not group objects. They have their own category-theoretic motivation (in terms of monoidal categories), but it takes hindsight to see the way in which category theory is important here.

I suppose that in the average case, mathematicians do pay more attention than physicists to abstract bookkeeping and definitions. There is more appreciation of the conceptual value of definitions in mathematics than in most areas of physics. But category theory is not the only such tradition in mathematics.

The famous historical example was Minkowski’s geometric picture of spacetime as a model of special relativity. To do this day, there are physicists who dismiss Minkowski’s insight as notational: “Oh, he invented 4-vectors”. Admittedly these physicists are generally not hep-th types; in high-energy theory this is an obsolete example.

A more germane example might be the clean model of quantum information theory that has developed among operator algebraists. It is better than what most quantum information theorists do, which is to keep disjoint notation for quantum states and classical probability distributions. Disjoint notation invites disjoint thinking, of course. The quantum information theorists are rediscovering a perspective that operator algebraists already have.

Posted by: Greg Kuperberg on February 12, 2006 11:03 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

Sure, there’s plenty of room for a range of programs of conceptual reformulation. To believe higher categories is the only show in town would smack of fanaticism. On the other hand, to realise the promise of a program requires some very dedicated people. Ross Street’s opening remarks from his talk at the IMA n-categories workshop were really moving:

“For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established.”

I’d like to see far more exposition of mathematicians’ views on their perception of the strengths and weakness of their and others’ programs.

In any event, the higher-category theory story is a good one to wean philosophers off their fascination with the set theoretic one.

Posted by: David Corfield on February 15, 2006 5:20 PM | Permalink | Reply to this

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Posted by: urs on February 15, 2006 5:34 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

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David Corfield quoted Ross Street as saying

For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians […]

Sometimes, when in a relaxed mood (say when chatting over a beer) I like to think:

Category theory is to math like string theory is to physics.

and chuckle to myself. It’s an analogy that, if you don’t like it (which may depend on the correlation between your esteem for both subjects), can easily be dismissed. But in some respects there is a truth to it, I think.

Interestingly, a while ago the blogosphere was playing the game “What is ‘the string theory’ of your field??” ( $\rightarrow$ ).

Maybe it would be interesting to furthermore ask

“What is ‘the category theory’ of your field??”

Posted by: urs on February 15, 2006 5:49 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

What I try to drive at in my publications is that there’s an importance to detailing one’s understanding of the direction of a body of research that shouldn’t be left to the kind of discussions one has when chatting over a beer, although that is fun. Imagine, say, a discussion between propoponents of noncommutative geometry (NCG) and higher-dimensional algebra (HDA). The can choose a range of topics, including common concerns such as QFT. The NCGer explains how they conceive of their progress in understanding QFT, in particular the Kreimer-Connes stuff on Feynman diagrams. They then ask the HDAist for an account of their own progress. The latter talks about stuff types etc., and their hopes for further achievements. They may even explain how they will integrate NCG into their own account (no doubt using your 2-NCG ideas). The NCGer may now point to what they see as the inherent weakness of this approach, and perhaps cast doubt on the value of stuff types. They may also suggest what they would need to see the HDAer do for them to be convinced of its value. So such a discussion needn’t be wholly offensive. Each side should point to what it is that troubles them most about their own program, and seek to find out whether the other side has something to say there.

The NCGer might then go on to discuss their progress towards the Riemann hypothesis, and wonder whether Z-sets (TWF 218) take you very far. The HDAist goes on to discuss their own finest moments.

It’s not that such dialogues should allow for a spot decision on who’s *winning*, but rather to engage in such activity is a component of what makes it rational to espouse a program.

Posted by: David Corfield on February 16, 2006 1:05 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

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in particular the Kreimer-Connes stuff on Feynman diagrams

This reminds me: I want to read

Kurusch Ebrahimi-Fard, Dirk Kreimer, Hopf algebra approach to Feynman diagram calculations, hep-th/0510202

but I still haven’t.

The NCGer may now point to what they see as the inherent weakness of this approach, and perhaps cast doubt on the value of stuff types.

I wonder if it is right to think of two alternative approaches. The HDA approach which you have in mind is at a more conceptual level. The goal here is (to my mind at least) to find a good conceptual understanding of (perturbative and/or non-perturbative) QFT. This in the sense of ‘good understanding’ that I sketched above. Namely, if successful, it would realize QFT as a ‘QM object’ in $\mathbf{Cat}$ , roughly. If true this would be ‘good’ since it should guide us along the right way to think about QFT.

Stuff types (one particular way to realize Hilbert spaces in $\mathbf{Cat}$ (or actually Hilbert spaces in $\mathbf{2Cat}$ , something which I still find confusing)) indicate how this point of view may really lead to an understanding of the nature of Feynman diagrams. (By, roughly, identifying Feynman diagrams as categorified Fock space operators.)

If all this actually works as one might hope for (which is a matter of further working out this approach, I think), and if it hence really has something to say about the sort of QFT we are interested in in practice, then structures like discovered by Connes-Kreimers should appear as a consequence.

In the ideal case, once all the dust has settled you would be able to lean back and say: “Ah, since we know that QFT is a $Y$ structure in $Z$ , Connes-Kreimers is just a consequence of this and that abstract nonsense.”

This would be, to my mind, the goal aimed at by a ‘HDA approach’ to QFT. The Very Big Picture (TM).

But as long as this VBP is not available yet, people have every right (they should, indeed) to be sceptical and try to find relevant structures inductively instead of deductively.

Posted by: urs on February 16, 2006 3:35 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

The nesting is getting too deep, but in reference to your Feb 16 comment, clearly with you I’m preaching to the converted. Part of any dialogue between two programs is the framing of their relationship. If one side sees opposition, the other may see subsumption. Where Cartier proposes merger (around p.402) between Connes and Grothendieck, Connes resists by giving the advantage to noncommutative geometry, “… the algebra associated to a topos does not in general allow one to recover the topos itself in the general noncommutative case.” (A view of mathematics: 21)). Frustratingly, this response is brief and not part of a more sustained dialogue. And, Cartier told me himself that he wouldn’t have written his piece had it not been commissioned as otherwise he would have had no confidence that it would have been published. This state of affairs is the kind of irrationality I’m aiming at.

If the VBP is right (it’s not a very snappy acronym like TOE or GUT, might it work better in German?) about “Ah, since we know that QFT is a Y structure in Z, Connes-Kreimers is just a consequence of this and that abstract nonsense.”, we’d expect strong links to be forged between the combinatorics of Feynman diagrams and other areas. I wonder if Bertfried Fauser’s ‘Renormalization: a number theoretic model’ math-ph/0601053 points in the right direction. His abstract speaks of functorality.

Posted by: David Corfield on February 17, 2006 12:55 PM | Permalink | Reply to this

Categories and Algebras of Graphs

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David,

that’s a very intersting discussion.

On the train I found the time to look at some literature. I came across something that looks relevant to the point you keep addressing.

In math.AG/0402015 Kevin Costello discusses “2D QFT” (Riemann surfaces, really) in terms of operads and categories of graphs.

He points out that a (modular) operad is the same as a functor from a category of graphs to some symmetric monoidal category.

Here the category of graphs has a pecuilar composition law. Morphisms are graphs, and composition is not by attaching free edges, but by inserting one graph into the other. Think of magnifying one vertex of a graph and seeing a small subgraph appearing in place.

(I’d have thought this category is well known, but Kevin Costello indicates that this way of describing operads is his idea.)

There is an issue which Kevin Costello seems to gloss over somewhat(?), namely that even if source and target of two graphs match (if one can be inserted into the other), this can generally be done in several different ways. Hence there is a choice of isomorphism involved when composing two composable graphs. (This choice is the very last line on p.4 of Costello’s paper). I guess one can deal cleanly with this extra freedom, and it is essential for the following.

Now, why is that interesting? Open Connes/Kreimers, e.g hep-th/0510202. In equation six of that paper they define the algebra of graphs (of Feynman diagrams, really) which underlies renormalization theory.

A little reflection shows that this algebra is essentially nothing but the category algebra of the category of graphs that Kevin Costello talks about!

(Where by category algebra I mean the algebra generated by the morphisms ofa category with product being their composition if defined and zero otherwise.)

So that’s cool. What Kevin Costello describes fits into the VBP that I mentioned above (no, unfortunately the german acronym here isn’t any better…). I may say more about this when I have more time. So it seems that there is a connection between Connes/Kreimers and VBP induced by passing from categories to category algebras and vice versa.

I have to run now. I hope to be able to write an entry on this topic soon. It’s fascinating.

Posted by: urs on February 18, 2006 11:19 AM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

Greg Kuperberg writes:

Ultimately most category theory is a form of bookkeeping, in my view. A solid investment in bookkeeping sometimes makes things much simpler.

Calling category theory “bookkeeping” is presumably intended as a way of “putting it in its place”. It creates the impression that category theory is useful, but in a really boring sort of way: nothing you’d want to make a career out of!

In fact, this is what most people think about all of mathematics. And, they’re not entirely wrong. Mathematics can indeed be considered as an elaborate form of bookkeeping - the “handmaiden of the sciences”. But, I don’t think this explains why people can get passionate
about mathematics: why it’s so beautiful.

The Mycenanean script Linear B occurs on tablets dating back to the 13th and 14th centuries BC. They were only deciphered in the 1950s. They turned out to be inventories and bureaucratic documents - lots of tables of numbers and sums. Basically just bookkeeping!

Later on, of course, writing was used for somewhat more interesting purposes.

Posted by: john baez on March 9, 2006 5:28 PM | Permalink | Reply to this

Re: Philosophy of Real Mathematics

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I’ll just provide a link to David Corfield’s latest entry on this issue: $\to$ .

Posted by: urs on March 9, 2006 6:00 PM | Permalink | Reply to this

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February 7, 2006