Links

Over at Ars Mathematica there's a discussion about the merits of category theory. I mentioned in a comment there this site of preprints, of which my favourites are Lawvere's 1 and 8. I most enjoy the way category theory suggests that you transcribe pieces of reasoning into different 'keys', sometimes just to recover something you already knew, but hadn't viewed in this way, preferably to perform a new piece.

I'd like to use this kind of thinking in coming to understand information geometry. I should be able to learn from a series of papers by Chris Hillman, especially the ones on entropy and information. Another idea worth exploring is one which says that probability and optimization are in some sense dual. This is related to what I have posted about tropical or idempotent mathematics.

Dawid on probabilities

A few days ago our reading group ran through Phil Dawid's Probability, Causality and the Empirical World: A Bayes-de Finetti-Popper-Borel Synthesis and learned that not only does probability not exist for him, like for de Finetti, nor does causality. A common characterisation of positions concerning probabilities split them into four groups:

1) Frequentist: probabilities are limiting frequencies of outcomes in sequences of events.
2) Propensity theorist: an individual event has a propensity to display a certain outcome.
3) Subjective Bayesian: a subjective degree of belief in an outcome.
4) Objective Bayesian: relative to specific background knowledge, there is an objective value an agent should accord to their degree of belief in an outcome.

Dawid (pronounced 'David') holds a Bayesian position, made evident in his involvement in the Sally Clark case, in which a mother was unjustly jailed after her two children died. But for Dawid, while the position that probabilities are consistent assignments of degrees of belief is all well and good, and it avoids the problems of taking probabilities to exist out there in the world, at some point you should want to calibrate the probabilities that an individual is spewing out. I could lock myself in a dark room keeping my degrees of belief consistent, but if I have England as 99% likely to win the World Cup, and other oddities, you will want to have a framework in which you can criticise me. This idea of calibration takes place in weather forecasting. You might score the forecaster's rain predicitions by forming the sum of (x_i - y_i)², where x_i = 0 if it is dry on the i^th day, and 1 if it rains, and y_i is the forecasted percentage chance of rain - the Brier score.

Intuitively, if a forecaster believes in their forecasts for rain they ought to be happy to accept bets made either for or against it raining at the odds they give. You'd think then that we could winkle out the bad from the good forecaster by noting which ones would go broke within a short space of time. The trouble is in telling when a 'good' forecaster's being very unlucky, and when a 'bad' forecaster's being very lucky. You are forced to appeal to an infinitely long series of predictions. And this is Dawid's position. Probabilities don't exist, they're theoretical tools that mediate between our theories and the world. The only way they hook onto the world is by what they rule out as impossible (this is the Borel part of the synthesis). For example, among other things, a forecaster who says 30% chance of rain every day is ruling out the possibility that in the long run it will rain on 40% of the days. This has the paradoxical consequence that if, say, two weather forecasters make predictions for rain over the next century and agree on every day except tomorrow when one says 10% and the other say 90%, there is no way you can say whether one was right. They've both ruled out various sets of sequences of outcomes, like those for which the average differs from the limit of the average of their probabilities. But neither rules out anything that the other doesn't.

Leaving aside the problem that runs aren't infinite, this game theoretic interpretation of probability theory is certainly very interesting. Shafer and Vovk have written a book-length treatment of the idea in their Probability and Finance: It's Only a Game!. Game theoretic ideas are also used to understand maximum entropy distributions. They correspond to stable points in zero-sum games played between a decision maker and nature. Flemming Topsoe has many good papers on this, and there's also one by Dawid with Peter Grunwald. Different entropies match up with different loss functions, such as the Brier score above. Something to add to the pot is Dawid's The geometry of proper scoring rules, a longer version of a paper written with Steffen Lauritzen. Now we have game theory blending with the differential geometric approach of Amari known as Information Geometry, discussed in earlier posts. I wish I could understand all this properly.

Old correspondence

My old laptop returned to life today. It had refused to boot up a while ago, and so was left to gather dust on a shelf. While I was passing this morning I gave its ON button a small tap and with that it sprang into action. So I've had the chance to look into my correspondence of the period 1999-2001. It's intriguing to see nascent ideas, some since turned into papers, others since abandoned.

At the time, I had contacted several mathematicians for the writing of my chapter on groupoids, including Saunders Mac Lane, who told me that they didn't feature at all in the early days of category theory. I also corresponded with the late George Mackey. During my time as a PhD student I had loved the story of maths he had told about in The scope and history of commutative and noncommutative harmonic analysis, so I asked him how he saw groupoids fitting into the picture. His reply included the remark:

At the moment I am occupied with developing some recent ideas I have had on a possible extensive development of my methods to apply to a much larger part of mathematics and produce more unification. I will explain more fully when I have made a bit more progress in seeking the proper formulation.

He spoke of some manuscripts he had written along these lines. Someone could do us a great service by digging these out.

I was also interested in diagrammatic reasoning at the time, so contacted Todd Trimble about his category theoretic reconstruction of the American philosopher Charles Peirce's existential graphs. He said:

One thing I would have emphasized, had I addressed your group, is Lawvere's revolutionary insight that the connectives and quantifiers in logic are controlled by *adjoint functors*. I think this is the key to further progress in geometrizing logic: higher-dimensional adjunctions are intimately connected with Morse theory, esp. the calculus of cancelling and rearranging critical points of Morse functions. (I don't think Gerry Brady and I fully connected the Beta graphs with this geometric aspect of adjunctions -- it ought to be done.) It is interesting to me that Peirce perceived, at a pre-formal level, the structure of connectives via adjunctions.

This connection of logic and singularity theory really needs exploring.

To end this short stroll down memory lane, I had forgotten that I had developed an interest in Dudley Shapere, a philosopher of science. He seemed to my mind to be asking the kinds of question about science that I wanted to ask about mathematics. I had included this in an e-mail:

Shapere suggests that one should be answering the following questions:

(1) What considerations (or, better, types of considerations, if such types can be found) lead scientists to regard a body of information as a body of information - that is, as constituting a unified subject matter or domain tobe examined or dealt with?

(2) How is description of the items of the domain achieved and modified at sophisticated stages of scientific development?

(3) What sort of inadequacies, leading to the need for further work, are found in the bodies of information, and what are the grounds for considering these to be inadequacies or problems requiring further research? (Included here are questions not only regarding the generation of scientific problems about domains, but also scientific priorities - the questions of importance of the problems and of the "readiness" of science to deal with them.)

(4) What considerations lead to the generation of specific lines of research, and what are the reasons (or types of reasons) for considering some lines of research to be more promising than others in the attempt to resolve problems about the domain?

(5) What are the reasons for expecting (sometimes to the extent of demanding) that answers of certain sorts, having certain characteristics, besought for those problems?

(6) What are the reasons (or types of reasons) for accepting a certain solution of a scientific problem regarding a domain as adequate. (Shapere1984, 277-8).

He reckons that only the last has been seriously examined by philosophers of science.

These still strike me as very good questions.

More on information geometry

Some blogs use different categories to sort out their posts. However, in my experience, things that interest me enough tend to show themselves to be related somewhere down the line. A while ago I was pondering the question -

How much of the mathematics used in physics is describing our knowledge and ways of observing and intervening, and how much the physical world itself?

- in the context of Caves and Fuchs' interpretation of quantum theory. Now I see Ariel Caticha has an article trying to understand general relativity in terms of information geometry:

The point of view that has been prevalent among scientists is that the laws of physics mirror the laws of nature. The reflection might be imperfect, a mere approximation to the real thing, but it is a reflection nonetheless. The connection between physics and nature could, however, be less direct. The laws of physics could be mere rules for processing information about nature. If this second point of view turns out to be correct one would expect many aspects of physics to mirror the structure of theories of inference. Indeed, it should be possible to derive the “laws of physics” appropriate to a certain problem by applying standard rules of inference to the information that happens to be relevant to the problem at hand.

Elsewhere, work is underway to generalise information geometry to the infinite dimensional spaces used in nonparametric statistics. There are important papers at Jun Zhang's site, including 'Nonparametric information geometry: Referential duality and representational duality on statistical manifolds.' Zhang is a psychologist who uses this mathematics to model psychometric testing.

Klein 2-geometry II

Update: I've floated this to the top as some new comments have been added.

This is June's continuation of the attempt to categorify the Erlanger Program. What I think would help enormously is a good candidate for a projective 2-group. The Euclidean 2-group and Poincare 2-group present themselves quite straightforwardly as they are semidirect products of a rotational part and an abelian translational part. So should one be looking to decompose projective transformations in a similar way. There are plenty of stabilizer subgroups to think about - stabilizers of: a point, a line, a point on a line, a point off a line, a triangle, etc.

It might also help to get a feel for smallish 2-groups. One would think that just as there is an adjunction between sets and groups which sends groups to their underlying sets, and sets to the free group with elements as generators, there is a 2-adjunction between categories and 2-groups. I.e., is there such a thing as the 2-group freely generated by the category C? Then we could look to impose relations.

Strict 2-groups are also known as crossed modules and as Cat¹-groups. In his doctoral thesis, Urs Schreiber gives a nice introduction to them in section 3.1. So, a strict 2-group is a pair of groups, G and H, with an action a of G on H, and a homomorphism t from H to G, satisfying the conditions:

a(t(h))(h') = hh'h^-1
t(a(g)(h)) = gt(h)g^-1

In the thesis you see why these conditions are natural.

In his doctoral thesis, Murat Alp uses computer algebra to calculate the smallest such entities. On pages I-2 and 3, he gives some constructions for forming 2-groups out of two ordinary groups, including these four:

1) H is abelian, the image of t is contained in the centre of G, and G acts trivially on H.

2) H is a normal subgroup of G, t is the inclusion, G acts on H by conjugation.

3) t is a surjection whose kernel lies within the centre of H, and g in G acts on H by conjugation with t ^-1g.

4) G is a subgroup of Aut(H) which contains the inner automorphisms of H, and t maps h to conjugation by h.

Yet more good exposition on strict 2-groups, and other interesting stuff on Matt Noonan's site.

Toby Bartels has finished his doctoral thesis - Higher Gauge Theory: 2-bundles. Section 2.4 which discusses 2-groups acting on 2-spaces should be useful for the project.

Information geometry and entropy

Information geometry (here and here) is a program which aims to apply the techniques of differential geometry to statistics. So we find statistical manifolds, such as the 2-dimensional manifold of normal distributions (mu, sigma), for which there are notions of Riemannian metrics, connections, curvature, etc. (see this for an analysis of standard cases: Bernoulli, Poisson, Gaussian). What I have yet to find is a very clear overview of the program as a whole. There are plenty of highly sophisticated presentations, and there are also some very gentle introductions, such as these videoed talks which towards the end of the second lecture touch on general Bregman divergences. But what I'd love is for someone to give a Baez-style sketch of the big picture.

Overfitting of data occurs when a set of model is so rich that members can be found which easily accommodate the data. These papers (here, here and here) by Vijay Balasubramanian (University of Pennsylvania) discuss in geometric terms what it is about a statistical manifold of models that corresponds to its capacity to represent data distributions.

Entropy is in the air, since the different 'alpha-connections' on the statistical manifolds correspond to different divergences. Now, entropy has a slippery character. Just as you think you're coming to terms with it, there's something new to take into consideration. An important component of the big picture comes from dynamical systems theory, where one finds the notion of the entropy of map. There's even an entropy for braids. (See the conjecture that braids of maximum entropy have either 3 or 4 strands.)

No doubt we should imagine how as data comes in we move about our statistical manifold. My guess is that the entropy of the mapping corresponding to this updating has something to do with the notion of entropy/divergence at play in the first paragraph. And perhaps, in view of his expertise in dynamical systems, I ought to try to understand Smale's position.

This post marks my great state of confusion. If I could penetrate the fog, I'd have a good grip on what statistics has to tell us about learning.

The Scope of Categorification II

I wrote last month about an apparent difference of opinion in the scope of categorification. Without wishing to give the impression of there being some great ideological divide, we might designate its two wings Frenkelian and Baezian. (How unfair it is that some surnames lend themselves so well to being turned into adjectives. What chance has a Higginbotham to establish a school of thought?). On the ArXiv today we have a paper - Open-closed TQFTs extend Khovanov homology from links to tangles - by former Baez student Aaron Lauda and Hendryk Pfeiffer which penetrates right into the Frenkelian territory of Khovanov homology. The abstract begins

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even.

Whenever you see that 'extended' in front of TQFT, you know you're heading up the ladder to 2-categories. Extended TQFTs are looking to attach algebraic objects to cobordisms between manifolds with corners, and there seems to be no better way to treat these than with higher-category theoretic tools.

In his recent post, Peter Woit wonders why "The math blogosphere seems to my mind somewhat weirdly dominated by those with an interest in category theory", and Walt from Ars Mathematica comments:

What’s even odder is that it’s not just a fascination with categories, but with n-categories. I think part of it is that John Baez has always been such an effective advocate of his n-categorical point of view that he’s both attracted people to the subject and inspired them to follow his example and post about it on-line.

None of us at arsmath are big category theory fans, so we’ll just have to single-handedly restore the balance.

From my perspective, it can't harm mathematics to have a few great visions. Once you've seen that ladder heading up to omega-categorical heaven, it's hard to stop on the first rung.

So, I think the issue is why you would want to leave the ground in the first place. A large part of the attraction is a love of discovering common constructions going on behind the scenes. Here's Robin Houston having Fun with Rel:

One of the great joys of category theory is the way you can so often watch familiar structures emerge unexpectedly from general constructions.

NB. As they say, Lauda and Pfeiffer are using a 'special kind of 2-dimensional extended Topological Quantum Field Theories' which means they are only working explicitly at the 1-categorical level.

We have chosen to work with open-closed TQFTs mainly because the extension of a 2-dimensional TQFT to an open-closed TQFT is much better understood than the corresponding question for the Temperley–Lieb 2-category. (p. 9)

This 2-category is treated by Khovanov in A functor-valued invariant of tangles. They note, however, related work in their earlier paper:

Various extensions of open-closed topological field theories have also been studied. Baas, Cohen, and Ramırez have extended the symmetric monoidal category of open-closed cobordisms to a symmetric monoidal 2-category whose 2-morphisms are certain diffeomorphisms of the open-closed cobordisms. This work extends the work of Tillmann who defined a symmetric monoidal 2-category extending the closed cobordism category. (p. 4)

And observe that:

All of the technology outlined above is defined for manifolds with faces of arbitrary dimension. Thus, our work suggests a natural framework for studying extended topological quantum field theories in dimensions three and four. Using 3-manifolds or 4-manifolds with faces, one can imagine defining a category (most likely higher-category) of extended three or four dimensional cobordisms. (p. 45)

So, probably, even special extended 3d-TQFTs will require another rung of the ladder.

Brandom's Analytic Pragmatism

Robert Brandom (Pittsburgh) has been giving the John Locke Lectures in Oxford over the past few weeks, and has made available the texts of all six lectures. His position is neither one of those rare philosophies with which I find myself instantly admiring every part, nor is it one of those all too common ones which feel quite foreign. As such it should be good for me to think hard about what he's trying to do.

Here he is characterising the pragmatism of the later Wittgenstein:

At every stage, what practical extensions of a given practice are possible for the practitioners can turn on features of their embodiment, lives, environment, and history that are contingent and wholly particular to them. And which of those developments actually took place, and in what order, can turn on any obscure fact. The reason vocabulary-kinds resist specification by rules, principles, definitions, or meanings expressed in other vocabularies is that they are the current time-slices of processes of development of practices that have this character-and that is why the collection of uses that is the cumulative and collective result of such developments-by-practical-projection is a motley. If that is right, then any codification or theoretical systematization of the uses of those vocabulary-kinds by associating with them meanings, specifiable in other vocabularies, which determine which uses are correct will, if at all successful, be successful only contingently, locally, and temporarily. Semantics on this view is an inherently Procrustean enterprise, which can proceed only by theoretically privileging some aspects of the use of a vocabulary that are not at all practically privileged, and spawning philosophical puzzlement about the intelligibility of the rest. On this conception, the classical project of analysis is a disease that rests on a fundamental and perennial kind of misunderstanding-one that can be removed or ameliorated only by heeding the advice to replace concern with meaning by concern with use. The recommended philosophical attitude to discursive practice is accordingly descriptive particularism, theoretical quietism, and semantic pessimism. (lecture 2)

Phrased like this, you can see why Wittgenstein has been taken to heart by a certain strand of social epistemology, such as that maintained by David Bloor or Martin Kusch.

Brandom's way is to cut back towards analytic philosophy:

I want to show how pragmatism can be turned from a pessimistic, even nihilistic, counsel of theoretical despair into a definite, substantive, progressive and promising program in the philosophy of language: indeed, how it can be understood as simply the latest phase of the analytic project. (lecture 2)

But to do so it must take on board the pragmatist concern of what one is doing when one uses a vocabulary. However, had we look closely enough, we should have already known that analytic philosophy knew this:

supplementing the traditional philosophical analytical concern with relations between the meanings expressed by different kinds of vocabulary by worrying also about the relations between those meanings and the use of those vocabularies in virtue of which they express those meanings-as I recommended in my first lecture-is not so much extending the classical project of analysis as it is unpacking it, to reveal a pragmatic structure that turns out already to have been implicit in the semantic project all along.

Now, engaging in any 'autonomous discursive practice' (i.e., using a vocabulary such that one need use no other), necessarily involves the activities of asserting and inferring. The task Brandom sets himself is to relate the logical, normative, modal, and intentional vocabularies to the practices and abilities necessary or sufficient to engage in such discursive activity. Perhaps the quickest way to point out the difference between his programme and what might be called 'orthodox' analytic philosophy is his treatment of modality. All talk of truth-makers is absent, so there's no recourse to possible worlds: I could have been unwell today. Some truth-maker must make this true. The truth-maker is some possible world in which I am unwell. Instead, modality is treated in terms of incompatibility of assertion. I am inconsistent if I assert "it is impossible for copper to melt at below 0 degrees centigrade" while also saying that my pipes will melt this winter when it freezes.

Now, rather than restrict ourselves to features of all discursive practices, I take it as important to look at specific privileged discursive practices, one of which many would agree is mathematics. So, I am interested in features of mathematics shared with the natural sciences, and other disciplined forms of enquiry, which may not be found in every discursive practice. But it seems to me that Brandom offers a way of accounting for my sense that analytic philosophy of mathematics is far too narrow by encouraging us to consider the full relevance of normative vocabulary to a practicing mathematician. It struck me reading these lectures that where most philosophers of mathematics want to stop is with the treatment of assertions of, and inferences between, mathematical propositions, things like 2 + 2 = 4. Where I have spent much of my philosophical time, however, has been on mathematical value judgements. In Brandom's language I could put it thus: one could not count as engaging in the autonomous discursive practice that is research mathematics, or such a practice that contains research mathematics, while not making value judgements about the organisation of concepts or predictions about how one's subfield will proceed.

Let's consider this by treating a piece of exposition selected for no special reason - Claus Michael Ringel's Some Remarks Concerning Tilting Modules and Tilted Algebras. Origin. Relevance. Future. Now, I can honestly say I know nothing about tilted modules or algebras. I know what a module is, and I know what an algebra is, but I have no clue what 'tilted' means here. Let us proceed then together in ignorance. Consider these assertions:

...at the time the Handbook was conceived, there was a common understanding that the tilted algebras (as the core of tilting theory) were understood well and that this part of the theory had reached a sort of final shape. But in the meantime this has turned out to be wrong: the tilted algebras have to be seen as factor algebras of the so called cluster tilted algebras, and it may well be, that in future the cluster tilted algebras and the cluster categories will topple the tilted algebras. The impetus for introducing and studying cluster tilted algebras came from outside, in a completely unexpected way. (p. 26, my emphasis)

What does that normative 'have to be seen as' mean? Something like: anyone sufficiently well-informed about the practice must see tilted algebras in this way, at pain otherwise of being charged with blindness, inconsistency, irresponsibility, or worse. It's a question of getting the concepts right. Here again is Brandom:

Where enough TOTE (test-operate-test-exit) cycles of this sort have been engaged in to produce a relatively stable discursive practice, objective facts about what actually follows from and is incompatible with what will have been incorporated in the material inferences and incompatibilities that articulate the concepts expressed by the vocabulary deployed according to the practical norms implicit in that practice. This essentially holistic process involves getting on to how things objectively are not just by making true claims, but also by acknowledging the right concepts. (lecture 6)

The idea of TOTE cycles is to provide a basic form of "practical involvement with objects exhibited by a sentient creature dealing skillfully with its world". With this pragmatist outlook we get away from the image of a language hoping to hook onto the world, and see instead world and word emerging as two poles of a practice.

To return to Ringel, we also find the language of modality, "...and it may well be, that in future...". Later in the article, he says:

It should be noted that some of the strange phenomena of tilted algebras disappear when passing to cluster tilted algebras. (p. 32)

So phenomena can be strange. They don't fall into lawlike patterns.

The cluster tilting theory has produced a lot of surprising results - it even answered some question which one did not dare to ask. (p. 32)

So, one would not have been counted irresponsible had one not asked these questions, until now.

...we have exhibited the cluster tilted algebras without reference to cluster categories, in order to show the elementary nature of these concepts. But a genuine understanding of cluster tilted algebras...is not possible in this way. (p. 37)

Again we find this blending of intentional, modal and normative vocabularies. Perhaps Brandom's onto something with his "objective pragmatism" which

sees those features of discursive practice that are made explicit by modal vocabulary and those that are made explicit by normative vocabulary as complementary, as each in principle fully intelligible only in terms of its relation to the other. Its understanding is, as the slogan that forms the title of this lecture has it, that discursive intentionality is a pragmatically mediated semantic relation, that essentially involves both what one is doing in saying something, and what is said about how it is with what one is thereby talking about. (lecture 6)

With some space for Hegel (lectures 5&6) and a whiff of category theory (mentioned once) in his diagrammatic representations of vocabularies and practices, this is an extremely interesting programme.

Tsallis entropy

My research in the theory of machine learning has got me thinking hard about maximum entropy (MaxEnt). It transpires that a whole host of machine learning algorithms are applications of this idea - maximising the Shannon entropy of a distribution subject to various constraints. See, e.g., Y. Altun and A. Smola, Unifying Divergence Minimization and Statistical Inference via Convex Duality. Not only do we find a duality between maximum likelihood estimation and MaxEnt with point constraints, but also one between Maximum a posteriori probability (MAP) estimation and MaxEnt with approximate constraints on expectations.

This has led me into the terrifying vast literature on maximum entropy. But it doesn't stop there. There are many who hold that the Shannon entropy is just one entropy, and the corresponding Kullback-Leibler divergence just one divergence. An important new entropy on the scene is the so-called Tsallis entropy, see here for an introduction by Tsallis himself.

Now, readers of this blog will know I'm interested in so-called q-deformation of mathematics (here and here). So, I'm intrigued to find that this Tsallis entropy can be thought of a q-deformation of the Shannon entropy. Presumably, then, just as so many of the standard distributions (Gaussian, Poisson, etc.) can be expressed in MaxEnt terms , there are corresponding q-deformations. And presumably there are q-deformations of Gaussian processes, and, who knows, these may find their uses in machine learning.

Graphical logic

Linear syntax, it is often been felt, obscures the combinatorics of proof. The graphical representation of logic thus has a long history. The latest attempt to capture these combinatorics is Dominic Hughes' Proof without Syntax. Here is the abstract:

"[M]athematicians care no more for logic than logicians for mathematics." Augustus de Morgan, 1868.
Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a *combinatorial proof* of a proposition P as a graph homomorphism h : C -> G(P), where G(P) is a graph associated with P and C is a coloured graph. The main theorem is soundness and completeness: P is true iff there exists a combinatorial proof h : C -> G(P).

Subtler Symmetry

I'm surprised that when new tools are made available to capture more subtle aspects of symmetry, mathematicians don't move a little faster to exploit their potential. The diagram on p. 12 of Ronnie Brown's notes on Nonabelian Algebraic Topology shows some of these tools and how they relate to the workhorses of symmetry - groups. The great breakthrough that was the quantum group concept doesn't explicitly appear, although they have been found to relate to double groupoids. The 2-groups John Baez and I have been discussing, vis-a-vis Kleinian geometry, are again not mentioned explicitly in Brown's diagram, but are a special case of 2-groupoids.

In view of the extraordinary pervasiveness of groups throughout mathematics, see e.g., George Mackey's wonderful The scope and history of commutative and noncommutative harmonic analysis, there must be many opportunities to find applications for their relatives. It's interesting to read then, via Ars Mathematica, an article by M. Golubitsky and I. Stewart entitled Nonlinear Dynamics of Networks: The Groupoid Formalism, which looks to exploit the more subtle symmetry that groupoids can detect in networks. Section 3 on Animal Locomotion is especially interesting. After calculating the set of possible quadruped gaits, one was not recognised as occurring, until a video from the Houston Livestock Show and Rodeo was analysed and a bucking bronco found to be performing it.

Links

Ronnie Brown and Tim Porter have written a short paper whose title - Analogy, concepts and methodology, in mathematics - put me in mind of my PhD thesis title - Research programmes, logic, and analogy : three aspects of mathematics and its development. 'Research programme' was a term used by Lakatos in his account of the methodology of science. I adapted this methodology for mathematics.

A new blog about higher-dimensional algebra (aka, higher-dimensional category theory) - Bosker Blog: Categorical Maundering - has been born. It's run by Robin Houston, a PhD student in Manchester.

The construction of a categorified Kleinian geometry is not proceeding quite as fast as I'd imagined, but it will continue. Blogs aren't really designed for protacted discussions over weeks and months, but it shouldn't be too hard to link up a series of posts.