It seems that I was somewhat unfair to number theorists (October 19) - they are making use of the Web. Also take a look at the incomparable number theory and physics archive, a website you may find hard to exit once you have entered.
In a Clay Mathematics Institute interview, Terence Tao speaks of the importance of "being exposed to other philosophies of research, of exposition, and so forth". Some mathematicians I have spoken to seem sheepish when they use the word 'philosophy' in this sense, as though they have no right to use it thus. I disagree. Tao also claims that "a subfield of mathematics has a better chance of staying dynamic, fruitful, and exciting if people in the area do make an effort to make good surveys and expository articles that try to reach out to other people in neighboring disciplines and invite them to lend their own insights and expertise to attack the problems in the area." This supports my argument that by not making this effort mathematicians are failing to be fully rational.
One talk I haven't mention from the 'Impact of Categories' conference was Bob Coecke's very enjoyable 'Kindergarten Quantum Mechanics'. Bob works in the Computing Laboratory in Oxford and is trying to strip down quantum information theory to its bare category theoretic bones. What results is an extraordinarily simple diagrammatic calculus. Proofs in this calculus can be seen as flips from one slide to the next. This is work very much in line with the kind of diagrammatic reasoning I discuss in chapter 10 of my book. In the October 4 entry I mentioned this paper. Lauda is showing here with glorious diagrams that the 2-category of open strings, planar open string worldsheets, and isotopy classes of worldsheets embedded in the cube, known as the 2-category of ‘3-dimensional thick tangles’, is the (semistrict monoidal) 2-category freely generated by a ‘categorified Frobenius algebra’.
Lauda, now in Cambridge working with Martin Hyland, is a student of John Baez, who has devoted much of his research effort to n-category theory, along with its role in reconciling quantum mechanics and general relativity. This is very evident through many weeks of his 'Quantum Gravity Seminar', but you can get a taste of his ideas in 'Quantum quandaries: a category-theoretic perspective', in S. French et al. (Eds.) Structural Foundations of Quantum Gravity, Oxford University Press, also here. The terminology may be different, symmetric monoidal with duals rather than strong compact closedness, but Baez and Coecke are talking about very similar things.
As a philosopher you can either study scientific movements from a distance, perhaps to see how they clash with each other, or else you commit yourself to a particular position. My interest in n-category theory began in the first vein, but it's very easy to slip over to the second. Aside from higher-dimensional category theory, I am also rooting for Bayesianism to succeed. I haven't studied these closely but there are responses, amongst others by Fuchs and Cave, to the charge against Bayesianism that quantum mechanics shows that at least some probabilities are out there in the world.
There have been some category theoretic attempts to capture probability theory, e.g., by Lawvere and Giry, but I don't think this has taken off yet. Fields medallist Vladimir Voevodsky
appears to be trying to do something similar. Perhaps then we could put together all the ideas in this post. For what it's worth, I have never seen it observed elsewhere that a Bayesian network can be seen as an arrow in a symmetric monoidal category with duals. The dual of an arrow between A and B is given by Bayes' rule.
Update: Voevodsky's lectures 'Categories, Population Genetics and a Little of Quantum Physics' have the following abstract:
Unfortunately, there appears to be no other trace of the contents of these lectures on the Web.
In these lectures I will tell about my work on two related but separate subjects. The first one is mathematical population genetics. I will describe a simple model which is useful for the study of the relationship between the history of a population and its genetic properties. While the positive results obtained in the framework of this model may have little use because of the model's simplicity the negative results are likely to remain valid for more complex real world populations. The second subject can be described as a categorical study of probability theory where "categorical" is understood in the sense of category theory. Originally, I developed this approach to probability to get a better understanding of the constructions which I had to deal with in population genetics. Later it evolved into something which seems to be also interesting from a purely mathematical point of view. On the elementary level it gives a category which is useful for the work with probabilistic constructions involving complicated combinations of stochastic processes of different types. On a more advanced level, applying in this context the old idea of a functor as a generalized object one gets a better view of the relationship between probability and the theory of (pre-)ordered topological vector spaces. This leads to the third topic mentioned in the title. But I am only beginning to understand this connection.
Naming mathematical entities is an important business. Charles Peirce argued that a scientist had the right to name their discoveries, but that this right would be overturned if the naming turned out to be unwise. Mathematicians have tended to be rather conservative. Only occasionally does a term convey associated imagery well, such as 'sheaf' with its paper and wheat connotations matching the covering and fibre imagery. It is very common to use a mathematician's name as a token of gratitude, although Peirce might have seen this as a failure on the part of the discoverer/inventor to find a suitable term. Alan Weinstein and colleagues in a paper today, looking to name a generalisation of Hopf algebras, explain their choice:
We call our new objects hopfish algebras, the suffix "oid" and prefixes like "quasi" and "pseudo" having already been appropriated for other uses. Also, our term retains a hint of the Poisson geometry which inspired some of our work.We await the new children's book 'One fish, two fish, red fish, hopfish'. It's good to avoid "quasi" and "pseudo". These versions often become the ones you care about, leaving you with the lengthier names as the norm. "Weak" n-categories may be a case in point. Some just want to drop the "weak", and specify "strict" in the other case. As to whether names have effects on the careers of concepts, Alain Connes in 'Noncommutative Geometry' suggested that the "-oid" ending of "groupoid" had a detrimental effect, and led to the concept being "despised".
Returning to the Paris conference, it is noticeable how Anglophone philosophers interested in category theory are very well-informed about the history of mathematics. They seem to be united in their love of mathematics as a quest, so cannot rest happy with straightening out a timeless conception of what mathematics is about and what kind of entities it deals with. In Colin McLarty's excellent 'Mathematical Platonism’ Versus Gathering the Dead: What Socrates teaches Glaucon', Philosophia Mathematica 13, 115–134, he shows that this quest conception is to be found in Plato's thinking, especially in The Republic. According to McLarty, it is Glaucon whose views are closest to what we mean today by 'platonism'. His paper requires a subtle reading of the dialogue to avoid misattributing to Plato views expressed by some of the characters. Less excusably this kind of misattribution has occurred in commentary on Lakatos's dialogue 'Proofs and Refutations'.
To refute a simplistic conception of an opposition between Aristotle and Plato on the Forms, Alasdair MacIntyre's "The Form of the Good, Tradition and Enquiry," (Value and Understanding: Essays for Peter Winch, Raimond Gaita, ed. (London: Routledge, 1990), pp. 242-62) presents Aristotle's response to the tensions in Plato's work as part of a continuous tradition of enquiry. This sense of philosophy as quest partially explains my interest in MacIntyre.
Not all branches of mathematics use the Web to the same extent. For example, number theorists are known for their reluctance to use the ArXiv. Perhaps they have most closely identified with Gauss' sentiment pauca sed matura, on balance not a very helpful policy. Philosophers as might be expected tend to behave like the number theorists, one exception being philosophers of science (and mostly of physics) and their PhilSci Archive. Even here the deposition rate since it started in 2001 has wavered 149 (2001), 223 (2002), 136 (2003), 215 (2004), and this year so far 115.
I mentioned in the last entry my sense that MacIntyre's tradition-constituted enquiry needs some of form of supplementation in terms of the personal or individual. Jack Russell Weinstein has some thoughts along these lines here, where he looks to Adam Smith. As you might imagine, such a reconciliation could only occur if Smith had been seriously misunderstood. Weinstein believes he has.
For MacIntyre's account of a way through Mill and Kant's differences on lying, read his 1994 Tanner Lectures.
What might a merging of Alasdair MacIntyre and Michael Polanyi resemble? Seeking to locate MacIntyre's traditions within mathematics, they appear a little monolithic, like Lakatos's research programmes. Theoretical commitments are surely more flexible and varied. Perhaps we need something of Polanyi's focus on the personal. My latest version of How Mathematicians May Fail to be Fully Rational points to this need.
Once upon a time synthetic differential geometry threatened to show how category theoretic thinking could aim right for the heart of as central an area of mathematics as differential geometry. Then the feeling grew that it was never quite going to make it. Evidence, at last, of its potential is here.
Concerning yesterday's comment on 2-vector spaces, John Baez writes:
I designed my 2-vector spaces for the purposes of higher linear algebra - Lie 2-algebras, but also associative 2-algebras and so on. I couldn't resist telling Nils about them and suggesting that he try those as a replacement of the K-V 2-vector spaces in studying 2-K-theory. Aaron Lauda went to visit him and explained a bunch of stuff. But, apparently they don't do the job.
It's a pity, and a bit mysterious since Lie 2-algebras ARE related to the string group, which plays a key role in Stolz and Teichner's work on elliptic cohomology, which Baas was trying to simplify in his work on 2-K-theory.
So, I don't think we've gotten to the bottom of things. In particular, Baas' work on the K-theory of the ku spectrum does not capture all features of elliptic cohomology. So, he hasn't gotten to the bottom of it either.
Add Jacob Lurie's version of elliptic cohomology by doing algebraic geometry in a monoidal omega-category into the mix, and the 'bottom' should be quite impressive when located.
I'm just back from Paris and 'The Impact of Categories' conference held at the ENS. While mathematics may approximate an international activity, philosophy is far from being so, but we still managed to learn a little from each other. Many of the Anglophone talks were aimed implicitly or explicitly at the issue of how to think about foundations. There are some subtle issues here. I agree with Jean-Pierre Marquis (Montreal) that merely saying set theorists/logicians' sense of the term 'foundations' is different from category theorists' and so they need not be seen as rivals is not the right way to look at things. Rather the different senses of the term must be viewed as interconnected. Where people differ is over the best way to organise this interconnection. It would be worth comparing Marquis' 1995 paper "Category Theory and the Foundations of Mathematics: Philosophical Excavations", Synthese, 103, 421-447, with Alasdair MacIntyre's 'First principles, final ends and contemporary philosophical issues' to appear in The Tasks of Philosophy (it already has appeared in Kevin Knight's 'MacIntyre Reader').
Steve Awodey (Carnegie-Mellon) told us that Saunders Mac Lane was 12 years earlier than Quine in finding that Carnap had failed to find a principled way to distinguish the logical from the non-logical in a formal theory. This is very important when you recall the role this observation plays in 'Two Dogmas of Empiricism'. We were left to draw our own conclusions from the fact that Mac Lane and Quine were well enough acquainted to go sailing together.
Nils Baas (mathematician from Oslo) stood in at the last moment for Alain Badiou. He reported interesting results in trying to form a 2-K theory using 2-vector bundles, as a way of approaching elliptic cohomology and homotopy data at chromatic filtration level 2 (see Baez on this). Baas claims his 2-vector spaces work here where Baez's don't. The next step is to sort
out the 2-functor from the 2-category of surface elements of a space to 2-Vect, before pushing on to 3-vector spaces. Once we have this, and a 3-category of cobordisms with corners, edges and boundaries, more will be convinced that mathematics beyond 2-categories exists "in nature".
On his Algebraic Topology Problem List, Mark Hovey remarks "…even if the problems we work on are internal to algebraic topology, we must strive to express ourselves better. If we expect our papers to be accepted in mathematical journals with a wide audience, such as the Annals, JAMS, or the Inventiones, then we must make sure our introductions are readable by generic good mathematicians. I always think of the French, myself--I want Serre to be able to understand what my paper is about. Another idea is to think of your advisor's advisor, who was probably trained 40 or 50 years ago. Make sure your advisor's advisor can understand your introduction. Another point of view comes from Mike Hopkins, who told me that we must tell a story in the introduction. Don't jump right into the middle of it with "Let E be an E-infinity ring spectrum". That does not help our field." And naturally Mike Hopkins abides by his own advice.
He gave an interesting introduction to homotopy theory at the ICM 2002. The following passage is very relevant to chapter 3 of my book, where I discuss Ronnie Brown and Tim Porter's schema:
geometry-->underlying process-->algebra-->algorithms-->computer implementation
For some questions the homotopy theoretic methods have proved more powerful, and for others the geometric methods have. The resolutions that lend themselves to computation tend to use spaces having convenient homotopy theoretic properties, but with no particularly accessible geometric content. On the other hand, the geometric methods have produced important homotopy theoretic moduli spaces and relationships between them that are difficult, if not impossible, to see from the point of view of homotopy theory. This metaphor is fundamental to topology, and there is a lot of power in spaces, like the classifying spaces for cobordism, that directly relate to both geometry and homotopy theory. It has consistently proved important to understand the computational aspects of the geometric devices, and the geometric aspects of the computational tools.
My fascination with 'Higher dimensional algebra', the study of n- and omega-categories continues. Something it's especially good at is providing a framework to understand theories which use algebra to extract topological information. Not just the 'old-fashioned' algebraic topology, but also 'quantum topology', where the algebra has to tailored to the dimension of the topological object. This framework is being applied in physics and computer science. As John Baez puts it, "we're starting to see a unification of logic, singularity theory/topology, and physics."
An interesting recent paper in this programme is Aaron Lauda's 'Frobenius algebras and planar open string topological field theories'. The diagrammatic proofs such as the one on p. 61 will become increasingly common over the coming years. Those 2-arrows could be represented by two-dimensional surfaces. Can we expect papers of the future to contain java applets showing surfaces and higher-dimensional entities transforming as parts of calculations and proofs?