Changing the rig

Here for posterity is a conversation I had with John Baez about using non-standard rigs (rings without negatives). There's nothing I like more (well not much anyway) than this kind of chatting. I haven't put in all the initials to indicate who's speaking, but most the time it's pretty clear.

DC: Now for a problem for future students if you carry on ideas from your Fall 03 Quantum gravity seminar:
Litvinov in this paper points out that corresponding to the Fourier transform for C, there is the Legendre transform for the rig R_max (R union {-infinity}, max ,+, -infinity, 0).

JB: I think the Legendre transform more directly generalizes the Laplace transform. In fact, as Jim Dolan explained but I never got around to retelling in the Fall '03 seminar, the Legendre transform ("finding the minimum of energy") is the temperature -> 0 limit of the Laplace transform ("summing over states weighted by exp(-E/kT)").

In other words, classical statics, where we minimize energy, is the temperature -> 0 limit of statistical mechanics. (Litvinov is maximizing instead of minimizing, but that's no big deal.) And, the ultimate reason this works is that the rig (R union {+infinity}, min, +, +infinity, 0) has a one-parameter deformation, where the deforming parameter is temperature.

When we let this parameter become complex we get quantum mechanics and Fourier transforms....

DC: What is the corresponding construction for the rig of truth values?

JB: Probably something like "finding the possible outcomes". Finding what's possible to do is a simplified version of "finding the least energetic thing to do", which is in turn a simplified version of "doing everything, but doing something of energy E with probability proportional to exp(-E/kT)".

All this needs to be explained very clearly to the world, so everyone will realize how cool it is!

DC:
> When we let this parameter become complex we get quantum mechanics and
> Fourier transforms....

and it seems that the Mellin transform is not far away:" the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject." http://en.wikipedia.org/wiki/Laplace_transform

were you hinting this in week 217?

JB: I wasn't thinking about that; I was just reassuring people thatall the transforms I listed are related by simple changes of variable,so you shouldn't feel ignorant of one if you know about another.

>>> What is the corresponding construction for the rig of truth values?

>> Probably something like "finding the possible outcomes". Finding what's
>> possible to do is a simplified version of "finding the least energetic
>> thing to do", which is in turn a simplified version of "doing
>> everything, but doing something of energy E with probability
>> proportional to exp(-E/kT)".

> So maps X -->{T,F} are subsets of X, and you find out whether it's possible
> to get from x to y within X. Right: and this is a kind of "path integral".

We can compute the "least action for a path from x to y" as an integral in a rig where addition is minimization.

We can compute the "amplitude for a path from x to y" as an integral in a rig where addition is addition.

We can compute the "possibility of a path from x to y" as an integral in the rig of truth values, where addition is "or".

> Sounds a lot like homotopy theory to me.

Yes, I guess any topological space gives a boolean-valued 2-variable function "can you get from x to y along a path?"

> But then the equivalence relationship of path connectedness is reflecting that it's a
> groupoid enriched over {T,F}, or was that impoverished?

Right! The really interesting 2-variable function associated to a topological space X is "the space of paths from x to y". If X is a homotopy n-type, this function will take values in homotopy (n-1)-types. If we think of a homotopy n-type as an n-groupoid this function is just hom(x,y)! But, we can decategorify hom(x,y) down to a homotopy -1-type, aka a truth value, which is "true" if hom(x,y) is nonempty and "false" if it's empty.

David writes:
> So if there is a categorifying chain of values for paths between x and y
> which runs: truth values, set, ..., n-groupoid,...
> are there other chains like
> cost, cost of passing between paths,...
> or probability, probability of passing between paths,.....

or amplitude, amplitude of passing between paths....

Good point! You're shooting ahead of me here, and it's a bit embarrassing, because as you note:

> this just seems to be pointing to things like your higher-gauge theory.

Part of what's been bugging me a lot about higher gauge theory is that I don't understand how Lagrangians fit into it. Usually people write down a Lagrangian as a function of some fields, which lets you compute an action, and minimizing the action give you equations of motion. You can do this in higher gauge theory too. BUT, usually the action for an ordinary gauge theory is required to be INVARIANT UNDER THE GROUP OF GAUGE TRANSFORMATIONS, since then gauge transformations will map solutions of the equations of motionto solutions. In higher gauge theory we have a 2-GROUP of gauge transformations. What does it mean for an action to be "invariant" under a 2-group? 2-groups really want to act not on a mere set, but on a CATEGORY - and the proper notion of "invariance" is "weak invariance", i.e. invariance up to a specified isomorphism satisfying some (understood) coherence laws.This suggests that actions in higher gauge theory should really take values in a category. And so, presumably, should Lagrangians. But, what category or categories??? Some categorification of the real numbers, maybe.

The problem is, I don't see the physics pointing me towards any particular choice. Probably I'm just being dumb. It's especially galling because I already think I know what one*result* of path-integral quantizing a higher gauge theory mightbe: a 2-Hilbert space of states! I wrote a paper on 2-Hilbert spaces once....

Hmm, this suggests that the appropriate "categorified transition amplitudes" lie not in C but in Hilb!!!

> Maybe it would be good to think how the fundamental groupoid arises through Lagrangian reasoning. In the path connectedness case, we have a space X, and a Lagangian map from X to truth values, i.e. a subset of X. Then for any path in X, there is an action formed by integrating L along it. This tells you whether the path lies wholly in X. Now you form the integral over all paths with the same endpoints. This just sees whether there is any path in X between those two points.

> Up a dimension, we're looking for a set of homotopy classes of paths:

Let me postpone thinking about this, even though it sounds really cool.

> Hmm. I have a feeling this ought to be slicker. Also the first 'integrations'
> in each case were over truth values and yet were ANDs rather than ORs.

I just want to say something about *this*. This actually seems right to me. In physics, a path integral is an integral over paths of the EXPONENTIAL of the action, which in turn is obtained by integrating a Lagrangian along the path. The exponential turns addition into multiplication. In fact, it's often good to think of the "exponential of the action", as more fundamental than the action. It has a clearer meaning. In quantum physics, the exponentiated action exp(iS/hbar) tells you the RELATIVE AMPLITUDE for taking that path. In statistical mechanics there's a version where you get the RELATIVE PROBABILITY.

In the situation you're talking about, the exponentiated action is a truth value saying whether the path is continuous - i.e., the POSSIBILITY of following that path. Now about that "first 'integration'" that's bugging you. The exponentiated action is given by a "product integral" along the path. I don't know if you've thought about product integrals, but they're just like integrals with + replaced by times. I reinvented them when I was a kid so I have a certain fondness for them. Normally you get them by multiplying lots of numbers that are really close to 1, instead of adding lots of numbers that are really close to zero... but normally you can reduce them to ordinary integrals using "exp" and "ln".

You however are doing product integrals in the rig of truth values: you are computing the possibility of a certain path as a product ofpossibilities of lots of little paths! And in this case there's no "exp" and "ln" to save us - unless there's some logical operation nobody every told me about, that converts "or" to "and". It's also neat to think about product integrals in the rig of costs:the rig R^{min} = (R union +infinity, min, +infinity, +, 0). Here we compute the cost of a path as an ORDINARY integral along the path...but the ordinary integral uses +, which is really MULTIPLICATION inthe rig of costs. So, it's again a case of a product integral. And again there's no "exp" and "ln" to save us.

> Presumably homotopy theory is treating a space as though it's
>infinitely cheap to go through the space, and infinitely expensive
> to go outside. So is there a cheap way to get from x to y? Yes,
>so long as there's a path [OR] along which [AND] all points are cheap.

Right! I like to think of truth values as a funny version of the rigof costs where the only two prices are "free" and "you can't afford it". Anyway, now I should go back to your categorified version of the wholesetup:

> Up a dimension, we're looking for a set of homotopy classes of
>paths: We have a space X, paths and paths between paths.
>The Lagrangian takes paths to truth values. When we integrate
>the Lagrangian along a path between paths it tells us whether
>we can do this all within X. Then for a given path f, we integrate
>[form set union] over all paths with the same end points,
> collecting all those homotopic to f. Now we integrate [form
>set union] over all paths f, forming the union of homotopy classes.

and think of the final result as an ordinary integral of a product integral. Btw, the "state sum models" in TQFT are all done by multiplying anamplitude for each labelled simplex and then summing over labellings, so it's the same sort of deal.

> About the transformation between quantum and classical, the
> trouble I'm having is that according to your lectures Sets
> and Relations are already on the quantum side.

Yeah! But that's GOOD.

In today's talk I explained how for any rig R there's a PROP whose morphisms from x^n to x^m are n x m matrices with entries in R, with the "tensor product" of morphisms being direct sum of matrices. In other words,"finitely generated free R-modules, made into a symmetric monoidal category using direct sum"This lets us do "matrix mechanics" in the manner we've been discussing, and when R is the rig of truth values we get "finite sets and relations, made into symmetric monoidal category using disjoint union" But we can also use the rig of costs....

> Oh, I see that I'd already got the point. But isn't that
> all the same odd to call anything matrix-like 'quantum'?

Well, FinRel is a symmetric monoidal category where the product is not cartesian, and it's a *-category, so in many ways it more closely resembles Hilb than Set or FinSet. The superposition principle is a bit stunted given that the rig of truth values has just one nonzero element, but don't let that fool you.

> Putting it naively, the things in the sets of Rel are
> perfectly classical. If the mere fact that things are
> related is enough to make them quantum, isn't that a sign
> that my twins entanglement idea is right - that a chunk of
> the weirdness of entanglement is little more exciting than
> that a twin marrying 12000 miles away makes you instantly
> an in-law. Or is your worry here that this is just about
> information? But then Fuchs, Cave et al want to say this is
> really all EPR experiments are doing.

Well, the sexier features of QM probably require a more interesting rig.

I'm not sure this is the right analogy; in quantum mechanics entanglement is about tensor products of vector spaces, so in FinRel we should be looking at tensor products of modules over the rig of truth values....

Hmm, it might turn out that you're right, and that the analog of a "entangled state" boils down to a pair of sets with a relation between them. But this is something one just needs to calculate. For example, maybe a relation f: S -> T can be dualized to give a relation g: 1 -> S* tensor T just like a linear operator f: V -> W gives a linear operator g: C -> V* tensor W - which is the same as a state in V* tensor W. The identity operator f: V -> V gives a maximally entangled state g: C -> V* tensor V, so maybe we can do the same thing with relations.

But, first I'd need to figure out if there really is a tensor product of "modules over the rig of truth values" (probably), and what it is.

> Presumably a module over the rig of truth values has to
> look something like a vector of truth values. Imagine the
> vector answering the two questions Are you a parent? Are
> you an aunt/uncle? Any individual can be in one of 4
> states. For any two unrelated people as far as you know
> they could be in any one of 16 states. Finding out about
> one of them doesn't help you with the other. But for two
> siblings (with no other siblings) they can only be in 4
> states, and finding out about the state of one tells you
> about the other.

Okay, very sensible. Now I can translate what you said intomath lingo. We only need (for now) to think about FREE modules of the rig R, namely those of the form R^n. And, with any luck,the tensor product of R^n with R^m is always R^{nm}, where you tensor two vectors by multiplying them entrywise to get a rectangular array. And, "entangled states" are those that aren't expressible as a tensor product of two vectors. For R the rig of truth values, you've got entanglement wheneveryou've got a rectangular array A_{ij} of truth values that's notof the form (B_i and C_j). There are lots of these, but as usual, they're always expressible as a sum - an "or" - of unentangled states.

Klein 2-geometry

As a small experiment in collective, public thinking, I'm going to devote a post to the attempt to categorify Kleinian geometry, and update the date so it doesn't slip off the radar of 'Previous Posts'.

Update: I'm very happy for it to be more collective. Your comments are welcome.

So let's see what we have so far:

DC: Here's a thought: gaining new insights by categorifying the very simplest entities seems a good way to bring in the punters. You've treated various kinds of number, natural, rational, etc. [E.g., From finite sets to Feynman diagrams] Now, if you ask anyone what else they first learned at school, they'll say 2-dimensional Euclidean geometry (if they're old enough to have been taught properly). Where, they may ask, are categorified lines and circles? If all interesting equations are lies, what of Pythagoras' theorem? Or, when we say the intersection of any pair of altitudes of a triangle is the *same* point as that of any other, is there room for weakening?

I suppose you might give two responses:

1) Euclidean geometry although it came first is actually a complicated affair. First you need to categorify a stripped down 'geometry' such as differential topology.

2) OK. 2d-Euclidean space is a homogeneous space, the points corresponding to cosets of the quotient of the Lie group of Euclidean transformations by the stabilizer of a point. All we need is a Lie 2-group version.

Either way what prevents a Erlangen program for 2-groups?

JB (John Baez): Hi -

> Where, they may ask, are categorified lines and
> circles?

Interesting idea; one could take it in various directions, I suppose.

> 1) Euclidean geometry although it came first is actually a
> complicated affair. First you need to categorify a stripped
> down 'geometry' such as differential topology.

Mainly you need to see where the categories are, so you can see if there are interesting n-categories lurking beneath them.

> 2) OK. 2d-Euclidean space is a homogeneous space, the
> points corresponding to cosets of the quotient of the Lie
> group of Euclidean transformations by the stabilizer of a
> point. All we need is a Lie 2-group version.

> Either way what prevents a Erlangen program for 2-groups?

Nothing! Especially since the Erlangen program is just the flip side of Galois theory: (see especially the slide about the icosahedron), and Galois theory has already been n-categorified to powerful and still growing effect.

But you're right - nobody seems to have thought hard about Klein geometry with Lie 2-groups (or higher) replacing Lie groups. Somehow people have skipped straight to categorifying principal bundles, even though principal bundles are a stripped-down way of thinking about Cartan geometries, which generalize Klein geometries! Sometimes ontogeny fails to recapitulate phylogeny. So, maybe the "punters" should be handed a nice specific Lie 2-group, some 2-spaces on which it acts, and be asked to study the "incidence relations" between these figures. Incidence geometry could be given a whole new lease on life!

Best, jb

DC:
In a pleasanter world I'd be funded to think longer about such things. For those with the leisure time, you can read about incidence geometry at TWF 178, which treats incidence relations in projective geometry in terms of the Dynkin diagrams An. Perhaps we should start with projective rather than Euclidean geometry. Is there an obvious candiate for a Lie 2-group one step up from SL(n, C)? What then is projective 2-geometry? What are Dynkin 2-diagrams?

Or doing things axiomatically, perhaps we can categorify the axioms of projective geometry, such as those for the projective plane, taken from week 145:

A) Given two distinct points, there exists a unique line that both points lie on.
B) Given two distinct lines, there exists a unique point that lies on both lines.
C) There exist four points, no three of which lie on the same line.
D) There exist four lines, no three of which have the same point lying on them.

TL (Tom Leinster): David asks (or rather, has a hypothetical character ask) what categorified lines and circles are. John points out that this could be taken in various directions. Here's a possible beginning of an answer.

We have to decide which aspects of lines and circles we're interested in. Let's treat them as metric spaces. Then the "categorified circle" should be a categorified metric space. OK, so what's a categorified metric space?

Here we can follow Lawvere, who's done a lot to develop the thesis that everything is a category. (I'm exaggerating, but see the first page proper of his metric spaces paper.) If a thing can be regarded some kind of category, that increases our chances of being able to perform some useful sort of categorification. This is ironic, as Lawvere seems to disapprove of categorification...

Anyway, Lawvere interprets metric spaces as being categories enriched in R, the poset of non-negative reals ordered by >=, made monoidal by its additive structure. So it looks as if our task is to categorify R - for then a categorified metric space could be defined as a category (weakly) enriched in the categorified R.

As far as I know, there's no really compelling answer yet to "what is the categorification of the reals?"

DC: Could you also recapitulate Descartes' coordinate-based approach to geometry? With categorified reals you could carve out subcategories of R^2 in terms of isomorphisms such as X^2 + Y^2 is isomorphic to 1. Seems like you wouldn't be too far from Joyal's species.

JB:
> Is there an obvious candidate
>for a nice specific Lie 2-group?

I list a bunch of nice 2-groups in HDA5, and if I were going to categorify Klein geometry I would I would look at a bunch in parallel. The fun of course is seeing the effect and significance of the 2-morphisms (sorry, now I'm thinking of a 2-group as a 1-object 2-groupoid). We've got the ordinary groups, with no 2-morphisms to speak of; then we've got the 2-groups with only one morphism and an abelian group of 2-morphisms.In between these extremes, and more interesting, are groups built from a group G acting on an abelian group A; a great example is the "Euclidean 2-group" where G = SO(n), A = R^n. Or, it might be nice to let G be the whole Euclidean group and A some abelian group on which it acts: this would decategorify to the Euclidean group.

DC:
> I list a bunch of nice 2-groups in HDA5, and if I were
> going to categorify Klein geometry I would I would look
> at a bunch in parallel. The fun of course is seeing the effect and
> significance of the 2-morphisms (sorry, now I'm thinking
> of a 2-group as a 1-object 2-groupoid).

I misread this when I first read it, but even the misreading raised a question. I thought you were talking about 2-morphisms BETWEEN 2-groups (should be 3-morphisms I suppose). Back on the level of Kleinian 1-geometry, what role could group homomorphisms play in the Erlangen program? I guess what's already treated is group inclusions, e.g., projective transformations within affine transformations within Euclidean transformations. But is there scope for more interesting homomorphisms?

JB: Good point. Sure! Each group determines, or we could say "is", a Klein geometry, but groups form a category - in fact a 2-category, since groups are categories - so we get a 2-category of Klein geometries.

Ignoring the 2-morphisms for a moment, though they're interesting and important, let's think about the morphisms: group homomorphisms, viewed as morphisms between Klein geometries.

The examples you mention are already very interesting, because they show how geometry is a unified subject, not just a bunch of isolated "geometries". They show that including a little group in a big one can be seen as making a geometry "more flexible", by adding new transformations.

But, there's another way inclusions of groups show up: in Klein geometry, a "figure" is more or less given by the subgroup that stabilizes it. But, a subgroup is an inclusion of groups! So, the examples you listed of group inclusions can also be seen as "figures"! The most famous example is getting affine geometry by taking projective geometry and restricting to the transformations that stabilize a "point at infinity", or "line at infinity", or...

I hadn't noticed this dual viewpoint, for some reason.

Anyway, some other examples come from outer automorphisms of groups: the outer automorphism of SL(n) is the "duality" that switches points and lines, and for Spin(8) one has 3! acting as outer automorphisms, called "triality". For simple Lie groups one can read these off from the Dynkin diagramn
symmetries.

There are also inner automorphisms, which are just "changes of reference frame". The difference between inner and outer automorphisms is nicely handled by the 2-categorical structure of Grp. (Ever figured out what a natural transformation between a functor betwen groups is?)

This leaves the epimorphisms of groups: quotient maps. When you mod out the Euclidean group by the translation subgroup, you get the rotation group. What does this short exact sequence mean for the corresponding 3 Klein geometries?

This is a lot of fun, and it will be even more fun to categorify.

JB:
If you want to keep talking about this, maybe you should pick a 2-group, and I'll tell you about it, and then we can start trying to develop the corresponding "Klein 2-geometry". Or, we could work in abstract generality. Or both: lots of generalities, but with a key example or two to light the way. That's usually what I do.

Of course the fun will start as soon as we try to generalize the concept of "figures" familiar from ordinary Klein geometry. Ordinarily, a "type of figure" is a subgroup H of our symmetry group G, and the space of figures of that type is G/H. So, the concepts of "sub-2-group" and "categorified quotient space" will soon need some clarification. And, we'll soon start wondering if in addition to "figures" there are some new "2-figures", or "morphisms between figures", or something....

For this it'll be good to describe the theory of ordinary Klein geometry quite clearly using category theory, so we can see how it categorifies.

DC: The "Euclidean 2-group" where G = SO(n), A = R^n looks like a good candidate. So, we need to find sub-2-groups and work out what their quotient 2-spaces are. A couple of obvious cases are G = SO(n), A = {0}, and G = {id}, A = R^n. And then there's G=SO(n-1), A = R^n. Hmm, have you worked out the theory of 2-cosets yet? Or is that co-2-sets, which I suppose ought to be cocategories, but is already taken.

The real entities of mathematics

It's easy to find yourself thinking that there's a natural distinction to be found between entities which are the proper subject matter of mathematics, and those used to give us access to the former. Perhaps the most famous expression of such thinking is Kronecker's "God created the integers, all else is the work of man."

From a discussion after this post over at The String Coffee Table back in February:

DC: In a comment to this post I raised the question of whether we might expect categorification of the special functions to appear with 2-representation theory. Something that might win over more people to higher-dimensional algebra would be the discovery of something as *concrete* as Bessel functions in their role as matrix elements of representations of important Lie groups. In the first few pages of this paper Cherednik exhibits this kind of attitude when he likens the difference between new concepts and new objects to that between the imaginary and the real.

Bruce Bartlett: I think David’s comment is quite relevant. I looked at Cherednik’s paper and found the introduction a most interesting, and shrewd, read. I will certainly keep this in mind from now on! Perhaps some further developments are necessary before the “real projection” of this stuff is nontrivial.

DC: I suppose an obvious question to raise about Cherednik’s real/imaginary dichotomy is whether to take it as timeless. It’s questionable that your average nineteenth century mathematician beamed to the future would take his word for it that characters of Kac-Moody algebras (Fig. 2) are “fundamental objects” because they “are not far from the products of classical one-dimensional theta-functions and can be introduced without representation theory”.

What is clear is that the expression of such a distinction is a part of a mathematician's philosophy. It would be quite consistent to expect such distinctions to alter through the centuries without allowing it to be a wholly subjective matter.

On another point, in the blog discussion, Bruce also made the prediction that:

Somehow, one feels that a lot of roads in this whole business of equivariant string theory, gerbes, elliptic cohomology, higher gauge theory, higher categories, etc. are going to merge at some point.

Something that would interest me in the event of such a merger is the just distribution of credit for any achievements. Can a research programme 'get lucky' by being in the right place at the right time to allow a breakthrough, when with the advantage of hindsight we can see that it would have been fairer had a rival got there first? Would this matter if there weren't the tendency overly to reward the first success?

Revealing reviews

Book reviews generally provide the most accessible accounts of practitioners' views of the workings of their field. The presence and lack of certain virtues are their constant themes. As an example, consider a review by Peter Johnstone of a mathematics book - Natural dualities for the working algebraist - reported here, which reveals something of his sense of the moral and intellectual virtues required of a mathematician. Here is an extract:

One might argue that the book's shortcomings are not of major importance in relation to its declared purpose. After all, it contains a wealth of detailed information on particular techniques for establishing duality theorems, and "working algebraists" (read: working universal-algebraists) will undoubtedly find it immensely useful to have all these techniques collected together in one place. I admit the force of that argument; but I also find it seriously worrying, for it carries the implication that universal-algebraists have given up the attempt to engage in dialogue with the rest of the mathematical community. As the authors remark at the end of their Preface, there is much that remains to be done in studying and classifying concrete dualities; but this book is not likely to inspire anyone outside the closed circle of universal-algebraists to take up the task. (my emphasis)

Any member of a tradition of enquiry should freely participate in such dialogue. Having noticed the absence from the bibliography of certain key references, Johnstone says he

... began studying the Bibliography more systematically, and soon realized its salient feature: all works written by category-theorists, or making serious use of categorical ideas, are excluded from it -- with the twin exceptions of Saunders Mac Lane's classic 'Categories for the working mathematician' (which, after all, the authors could hardly have left out, given their indebtedness to his title), and of Peter Freyd's 1966 paper on algebra-valued functors.

So, now an accusation of a lack of justice.

Maturity

Anyone looking to gain some insight into the mind of a leading mathematician, and quite possibly a future Fields' Medallist, should find plenty to interest them on Terence Tao's What's New? pages. His December 6, 2005 entry gives us access to a book he first wrote at the age of 15 on problem solving. Let me just contrast a couple of extracts from the two prefaces:

The 15 year old Tao writes,

Proclus , an ancient Greek philosopher, said: This therefore, is mathematics: she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings to light our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth . . .

But I just like mathematics because it's fun.

Proclus was a fifth century Neoplatonist, perhaps best known for his commentary on Euclid's Elements. Now, as a 30 year old, Tao writes:

Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience. As a primary school student, I was drawn to mathematics by the abstract beauty of formal manipulation, and the remarkable ability to repeatedly use simple rules to achieve non­trivial answers. As a high­ school student, competing in mathematics competitions, I enjoyed mathematics as a sport, taking cleverly designed mathematical puzzle problems (such as those in this book) and searching for the right "trick'' that would unlock each one. As an undergraduate, I was awed by my first glimpses of the rich, deep, and fascinating theories and structures which lie at the core of modern mathematics today. As a graduate student, I learnt the pride of having one's own research project, and the unique satisfaction that comes from creating an original argument that resolved a previously open question. Upon starting my career as a professional research mathematician, I began to see the intuition and motivation that lay behind the theories and problems of modern mathematics, and was delighted when realizing how even very complex and deep results are often at heart be [sic] guided by very simple, even common­-sensical, principles. The "Aha!'' experience of grasping one of these principles, and suddenly seeing how it illuminates and informs a large body of mathematics, is a truly remarkable one. And there are yet more aspects of mathematics to discover; it is only recently for me that I have grasped enough fields of mathematics to begin to get a sense of the endeavour of modern mathematics as a unified subject, and how it connects to the sciences and other disciplines.

"Very simple, even common­-sensical, principles" is precisely the topic of pages 206 and 207 of my book. They are essential components of mathematics operating at its highest level, and, as such, to overlook them as a philosopher writing on mathematics is to go astray. What would be very useful would be a generous sample of such principles. I gave this one from Timothy Gowers in my book:

if one is trying to maximize the size of some structure under certain constraints, and if the constraints seem to force the extremal examples to be spread about in a uniform sort of way, then choosing an example randomly is likely to give a good answer.

For a pithier example from higher-dimensional algebra: "All interesting equations are lies.", i.e., should be seen as projections of isomorphisms, or higher equivalences. A more technical one tells us that: "certain algebraic structures can be defined in any category equipped with a categorified version of the same structure." In my experience, there's always what I would call a degree of creative vagueness to these principles. While they can be given a formal dressing, this often leaves a residual capacity for future application.

Let's end with a couple of related principles from Tao himself:

If an object is not (pseudo-)random, then it (or some non-trivial component of it) correlates with a structured object.

If A is an arbitrary object, then A (or some non-trivial component of A) splits as the sum of a structured object, plus a pseudorandom error.

I'm left wondering what Tao's thoughts on Proclus are now.

Why and how to write history

My head's been filled the last few days with just how large a part of machine learning can be couched in terms of convex optimization - Fenchel duality, and all that - so it will be a relief just to spend a few minutes thinking about the history of mathematics. I have to do this as I 'm going to be speaking about it at a workshop in Berlin this September. I've promised to speak about:

Why and how to write a history of higher-dimensional algebra

In a recent paper 'How Mathematicians May Fail to be Fully Rational' [1], I advocated the adoption in the philosophy of mathematics of Alasdair MacIntyre's general notion of tradition-constituted enquiry. A central component of this notion requires of a rational tradition that it know the history of its successes and failures. This raises the question as to whether, were such a history to be written, it would fall foul of the criticism contemporary historians of mathematics have levelled at mathematicians' histories that they are largely 'Royal-road-to-me' accounts. I shall address this question in the context of a research programme known as 'higher-dimensional algebra', and consider the charge mathematicians may make in return that historians are unable to treat research programmes which run for decades, supported by tens or hundreds of mathematicians from many countries and institutions.

A few weeks ago I read a book by David Carr called Time, Narrative, and History 1986 Indiana University Press Bloomington, which I thoroughly recommend. The central thesis of the first part of the book can be summed up in a couple of quotations:

To be an agent or subject of experience is to make the constant attempt to surmount time in exactly the way the story-teller does. It is the attempt to dominate the flow of events by gathering them together in the forward-backward grasp of the narrative act. Mink and the other theorists are right to believe that narration constitutes something, creates meaning rather than just reflecting or imitating something that exists independently of it. But narration, intertwined as it is with action, does this in the course of life itself, not merely after the fact, at the hands of authors, in the pages of books. (61-62)

So, there's no pristine unnarrated experience which later undergoes narrative reformulation. Now, for the interpersonal dimension:

Thus concretely, when we recount to others what we are living through and what we are doing, such recounting, rather than the adventitious communication of an already prepared and clearly formulated message, is actually constitutive of the content of what is said, and through it constitutive of the temporal organization itself. Most people have had the experience that they do not quite know what they mean or intend until they try to communicate it to others. The content is all the more affected, of course, when the speaker is met by rejoinders, questions, and criticisms. Thus telling the story of my action or experience to others can reorganize it for me; telling the story of my life can serve to make a sense of it I have not been aware of before.

The social connection among persons, conceived in this way, is one of reciprocal communicative roles in the constitution of experiences, actions, and lives. Others are encountered by me, not only as audiences or sounding-boards for the sense-constitution of my own ongoing experiences and projects, but as engaged in their own narratives as agents and stroy-tellers, narratives to whose construction I may contribute in my role as audience and possibly critic. This at least one way to conceive of the social horizon of the individual's existence.

Now the concept of historicity, as put forward by Husserl and Heidegger, adds a crucial element to this picture. It affirms that my connection with the actions and experiences of others can take a special form, apart from the relation of reciprocal narration, a form we can describe as the relation of predecessors and successors. What is indicated is a priority, not only of time but also of accomplishemnt. In the case of the ongoing scientific project I take up the work already accomplished by others. The end of another's work becomes the beginning of my own. The other need not have finished his or her work, of course, but some conclusion has been reached which serves as my starting point. This is true whether the other's results can be used as a basis for my own, and built upon, or whether I must begin by undoing his or her work and starting over. In either case the work of others, rather than simply existing alongside my own, becomes its background and prior condition. (112-113)

And this leads us to the issue of a community's narrative. In a previous post, I suggested that a communal group of mathematicians can write a narrative of their research programme which will fall foul of historians' strictures, and yet which is nonetheless truthful. In fact, I rather think that to remain strictly at the level of historians' history would be to overlook some part of the truth. Let's continue with Carr:

...the narrative structure and narrational activity within communal existence is, as we have insisted, primarily practical in character; historical narrative, by contrast is cognitive and seeks an objective representation. The former is engaged in action and has an interest in its outcome; the latter is detached and disinterested, and aims only at truth. The second difference concerns the temporal standpoints of the narrators in each case. Our "practical" narrator is situated in medias res, whereas the historical narrator looks back at actions and events already completed. That gives the latter the well-known (and already discussed) advantage of hindsight over his subjects: he knows how things turned out, knows the difference between the intended consequences and the real consequences of their action, etc.

These differences between narrative agent or participant and narrative historian are operative and important: there is no denying the importance of temporal standpoint and of the difference in attitude (engaged or detached) in relation to a lived or performed sequence of human events. At the same time we should like to emphasize several respects in which these differences are mitigated. And we shall do this not by denying objectivity and hindsight to historical inquiry, but by attributing them to narrative-historical existence.

We have already pointed out ..., with respect to individual action and experience, that the narrativization that goes on there cannot be indifferent to truth where the past is concerned. Indeed, where the issue is not merely the shaping of an open future but the coherence of future, present, and past, it is important to be clear on what really happened; the past may be variously interpreted but it cannot be wished away or forcibly altered by an inventive narrative magination. So much of one's present capacities are in continuity with, and sometimes result from, past choices and experiences that getting straight one's past can be seem as a desideratum and even a necessary condition for a coherent life. This is, of course, one of the insights on which much psychotherapy is based, as we pointed out.

A concern for the truth of the past plays the same role in the case of the community. Members often debate the facts of the past, precisely because they are so important in the constitution of the present and the future. This is not to deny that the past is often manipulated, especially where social story-telling is political and persuasive in character. The personal past is often distorted too, deliberately or not. My point is merely that a genuine interest in the truth of the past is compatible with and indeed important for the practical narrative constitution of communal existence. Equally, objectively-oriented historical enquiry and research are not disqualified from playing a role in the ongoing political and social debnated of a community; on the contrary, they can and do contribute to them.

We are not commenting here, it should be noted, on the success with which truthfulness about the past is actually attained. Our point concerns the interest in or commitment to truth, and we are only saying that this is not restricted to history as a discipline. It is true that the discipline has among other things developed techniques for discovering and evaluating evidence in order to implement its commitment to truth. A justified suspicion that partisanship in the events of the day can distort our view of the past has led to the emphasis on detachment and objectivity. But these in turn, once achieved, can be put in the service of engagement in the present and the shaping of the future.

As for the hindsight which is characteristic of historical enquiry, this too is not exclusive to the latter, at least not formally. Socially constitutive narrative, like the narrative structure of individual life and action, has a prospective-retrospective form. In anticiapating the future, it aims at, and largely achieves, that quasi-hindsight that we characterized earlier, borrowing Schultz's term, as the future perfect. Far from waiting passively for things to happen, communities negotiate with the future and understand the present in the light of that future. 171-2

Carr's position would require historians and mathematicians to be brought into a much closer relationship, and rightly so:

Far from dealing with past events which are fixed and whose consequences are clear, historians here deal with events whose consequences are still being felt and are operative in the present. 173

The scope of 'categorification'

I mentioned here that there were rival conceptions of noncommutative geometry. This set me wondering whether there might be rival conceptions of categorification. The originator of the term appears to be Louis Crane, who was looking to form 4d topological quantum field theories by categorifying constructions used in 3d theories. Back at the beginning of 1993, John Baez refers (item 4) to several papers written or co-written by Crane, including a draft paper 'Categorification and the construction of topological quantum field theory', written with Igor Frenkel. But one can trace the motivation back earlier.

Baez and Dolan's paper Categorification lays out the scope of their sense of the term:

It is clear, therefore, that the set-based mathematics we know and love is just the tip of an immense iceberg of n-categorical, and ultimately ω-categorical, mathematics. The prospect of exploring this huge body of new mathematics is both exhilariting and daunting. (p. 46)

One can try to categorify anything and everything. The continuation of the discussion about categorying Klein's Erlangen Program, I recorded here (in post and comment), went as follows:

DC:
> I list a bunch of nice 2-groups in HDA5, and if I were
> going to categorify Klein geometry I would I would look
> at a bunch in parallel. The fun of course is seeing the effect and
> significance of the 2-morphisms (sorry, now I'm thinking
> of a 2-group as a 1-object 2-groupoid).

I misread this when I first read it, but even the misreading raised a question. I thought you were talking about 2-morphisms BETWEEN 2-groups (should be 3-morphisms I suppose). Back on the level of Kleinian 1-geometry, what role could group homomorphisms play in the Erlangen program? I guess what's already treated is group inclusions, e.g., projective transformations within affine transformations within Euclidean transformations. But is there scope for more interesting homomorphisms?

JB: Good point. Sure! Each group determines, or we could say "is", a Klein geometry, but groups form a category - in fact a 2-category, since groups are categories - so we get a 2-category of Klein geometries.

Ignoring the 2-morphisms for a moment, though they're interesting and important, let's think about the morphisms: group homomorphisms, viewed as morphisms between Klein geometries.

The examples you mention are already very interesting, because they show how geometry is a unified subject, not just a bunch of isolated "geometries". They show that including a little group in a big one can be seen as making a geometry "more flexible", by adding new transformations.

But, there's another way inclusions of groups show up: in Klein geometry, a "figure" is more or less given by the subgroup that stabilizes it. But, a subgroup is an inclusion of groups! So, the examples you listed of group inclusions can also be seen as "figures"! The most famous example is getting affine geometry by taking projective geometry and restricting to the transformations that stabilize a "point at infinity", or "line at infinity", or...

I hadn't noticed this dual viewpoint, for some reason.

Anyway, some other examples come from outer automorphisms of groups: the outer automorphism of SL(n) is the "duality" that switches points and lines, and for Spin(8) one has 3! acting as outer automorphisms, called "triality". For simple Lie groups one can read these off from the Dynkin diagramn
symmetries.

There are also inner automorphisms, which are just "changes of reference frame". The difference between inner and outer automorphisms is nicely handled by the 2-categorical structure of Grp. (Ever figured out what a natural transformation between a functor betwen groups is?)

This leaves the epimorphisms of groups: quotient maps. When you mod out the Euclidean group by the translation subgroup, you get the rotation group. What does this short exact sequence mean for the corresponding 3 Klein geometries?

This is a lot of fun, and it will be even more fun to categorify.

END

In this sense of the word 'categorify' you can try it on just about anything. Now, someone else who is very much involved with categorification is Dror Bar-Natan, a former doctoral student of Edward Witten. From here and here, we glean the following:

categorification (a bold suggestion of I. Frenkel, that much of math is the Euler characteristic of some "higher math", much like much of algebra is q-algebra at q=1)

Conjecture: (I. Frenkel, though he may disown this version)
1. Every object in mathematics is the Euler characteristic of a complex.
2. Every operation in mathematics lifts to an operation between complexes.
3. Every identity in mathematics is true up to homotopy at complex−level.

Now, it seems a shame to have this position confined to a couple of hand-outs, expressed in a few lines, and attributed to someone else who might not agree with it. I would like to find out whether there are any differences of conception either in terms of scope or emphasis. No doubt this would become clear if I could attend this conference on categorification.

A lot of work going by the name categorification has centred on categorifying the Jones polynomial, in the understanding of which Witten played such an important part. Bar-Natan's Khovanov's Homology for Tangles and Cobordisms describes this work. Here is an extract from the paper which contains an enormous amount for a philosopher to ponder:

1.2. The plan. A traditional math paper sets out many formal definitions, states theorems and moves on to proving them, hoping that a “picture” will emerge in the reader’s mind as (s)he struggles to interpret the formal definitions. In our case the “picture” can be summarized by a rather fine picture that can be uploaded to one’s mind even without the formalities, and, in fact, the formalities won’t necessarily make the upload any smoother. Hence we start our article with the picture, Figure 1 on page 5, and follow it in Section 2 by a narrative description thereof, without yet assigning any meaning to it and without describing the “frame” in which it lives — the category in which it is an object. We fix that in Sections 3 and 4: in the former we describe a certain category of complexes where our picture resides, and in the latter we show that within that category our picture is a homotopy invariant. The nearly tautological Section 5 discusses the good behaviour of our invariant under arbitrary tangle compositions. In Section 6 we refine the picture a bit by introducing gradings, and in Section 7 we explain that by applying an appropriate functor F (a 1+1-dimensional TQFT) we can get a computable homology theory which yields honest knot/link invariants.

While not the technical heart of this paper, Sections 8–10 are its raison d’être. In Section 8 we explain how our machinery allows for a simple and conceptual explanation of the functoriality of the Khovanov homology under tangle cobordisms. In Section 9 we further discuss the “frame” of Section 3 finding that in the case of closed tangles (i.e., knots and links) and over rings that contain 1/2 it frames very little beyond the original Khovanov homology while if 2 is not invertible our frame appears richer than the original. In Section 10 we introduce a generalized notion of Euler characteristic which allows us to “localize” the assertion “The Euler characteristic of Khovanov Homology is the Jones polynomial”.

The final Section 11 contains some further “odds and ends”.

If you would like to see the picture in glorious technicolor, then look here.

Ruskin: An early post-autistic economist

From Unto This Last:

The real science of political economy, which has yet to be distinguished from the bastard science, as medicine from witchcraft, and astronomy from astrology, is that which teaches nations to desire and labour for the things that lead to life: and which teaches them to scorn and destroy the things that lead to destruction. And if, in a state of infancy, they supposed indifferent things, such as excrescences of shell-fish, and pieces of blue and red stone, to be valuable, and spent large measures of the labour which ought to be employed for the extension and ennobling of life, in diving or digging for them, and cutting them into various shapes, or if, in the same state of infancy, they imagine precious and beneficent things, such as air, light, and cleanliness, to be valueless,-or if, finally, they imagine the conditions of their own existence, by which alone they can truly possess or use anything, such, for instance, as peace, trust, and love, to be prudently exchangeable, when the markets offer, for gold, iron, or excresrences of shells -- the great and only science of Political Economy teaches them, in all these cases, what is vanity, and what substance; and how the service of Death, the lord of Waste, and of eternal emptiness, differs from the service of Wisdom, the lady of Saving, and of eternal fulness; she who has said, "I will cause those that love me to inherit SUBSTANCE; and I will FILL their treasures."

Unhappily for the progress of the science of Political Economy, the plus quantities, or, -- if I may be allowed to coin an awkward plural -- the pluses, make a very positive and venerable appearance in the world, so that every one is eager to learn the science which produces results so magnificent; whereas the minuses have, on the other hand, a tendency to retire into back streets, and other places of shade, -- or even to get themselves wholly and finally put out of sight in graves: which renders the algebra of this science peculiar, and difficultly legible; a large number of its negative signs being written by the account-keeper in a kind of red ink, which starvation thins, and makes strangely pale, or even quite invisible ink, for the present.

In all buying, consider, first, what condition of existence you cause in the producers of what you buy; secondly, whether the sum you have paid is just to the producer, and in due proportion, lodged in his hands; thirdly, to how much clear use, for food, knowledge, or joy, this that you have bought can be put; and fourthly, to whom and in what way it can be most speedily and serviceably distributed: in all dealings whatsoever insisting on entire openness and stern fulfilment; and in all doings, on perfection and loveliness of accomplishment; especially on fineness and purity of all marketable commodity: watching at the same time for all ways of gaining, or teaching, powers of simple pleasure, and of showing "οσον εν ασφοδελω μεγ ονειαρ" -- the sum of enjoyment depending not on the quantity of things tasted, but on the vivacity and patience of taste.

(The Greek is from Hesiod, and means 'how great blessing lies in mallow and asphodel', i.e., in simple things which even the poor can enjoy.)

For contemporary post-autistic economics you can read this review.

Ruskin and the RAE

I'm just back from a short break to Coniston in the Lake District, where the art critic and social critic John Ruskin ended his days. There I began to read Unto This Last, the work he considered his most important. It opens,

Among the delusions which at different periods have possessed themselves of the minds of large masses of the human race, perhaps the most curious - certainly the least creditable - is the modern soi-disant science of political economy, based on the idea that an advantageous code of social action may be determined irrespectively of the influence of social affection.

Besides being beautifully written, it has been extremely influential. Proust and Tolstoy, Gandhi and Attlee all thought very highly of it. And yet while it was being published in instalments during 1860, it was lambasted by critics - 'the world simply refuses to be preached to by a mad governess' - to the extent that the magazine proprietor felt forced to curtail publication of the full series of essays. Fortunately, Ruskin had the wherewithal to publish it as a book in 1862, although even then not all of the thousand copies printed had been sold a decade later.

Back home, I find that my doctoral supervisor, Donald Gillies, has written a piece entitled Lessons from the History and Philosophy of Science regarding the Research Assessment Exercise. The Research Assessment Exercise (RAE), for those of you who don't know it, is an enormous piece of bureaucratic machinery whose function is to measure the quality and quantity of the research output of British University departments in order to calculate the level of research funding due in coming years. Gillies offers us a powerful critique of the RAE by means of some counterfactual scenarios of the kind: if the RAE had been in place in the time of Wittgenstein, Frege, Semmelweiss, Copernicus, their research would not have been funded. For Gillies the RAE has been devised to target the wrong kind of error. The primary aim of any reasonable funding system must be to ensure that every first rate researcher is funded. However, the RAE seems designed only to prevent bad research being done. In the effort to ensure that only good work is supported, it promotes a cautiousness whose effect is to obstruct the production of important and original ideas.

Gillies resorts to history as we cannot tell so close to the event what impact the RAE has had on the originality of research since it was introduced under Margaret Thatcher's government in 1986. This allows him diplomatically to omit telling us his views on the state of philosophy over the past couple of decades. Those wishing to take a brief glimpse of contemporary English-language philosophy could do worse than drop in on this month's Philosophical Carnival. If other disciplines face a difficult task reckoning how all of its branches are faring, you may imagine from the breadth of style and content what this is like for philosophy. It would seem a reasonable conjecture, then, that deviations from acknowledged philosophical orthodoxies would not be rewarded.

But, as the range of disciplines treated by Gillies suggests, this problem is not confined to philosophy. A piece written by the mathematician, Ronnie Brown, makes some very similar points. To take one line from The methodology of the RAE? as an example:

In mathematics, one can believe that Galois, Cantor, Grassman, ... , would have done badly in any supposed contemporary RAE.

Although Ruskin published prodigiously, in view of the reception of 'Unto This Last', a similar analysis might apply to him, especially as he made contributions to many fields. These works were strongly interconnected in his eyes, but the different RAE panels would have thought otherwise.

If, as Ruskin wrote, "All books are divisible into two classes, the books of the hour, and the books of all time", the RAE may be characterised as trying to ensure that the books of the hour are competently written. But in that it stands in the way of the writing of a single book of all time it is rightly condemned.