The May Notices of the AMS are up. The lead article is Cells in Coxeter Groups. This month’s What is… is What is… Percolation?.

It also includes obituaries for Serge Lang and Raoul Bott.

The new Bulletin of the AMS is out. It has a review of Computational Homology, a book that I have not read, but was very tempted by at the bookstore. Sadly, my library doesn’t have it. Homology provides an interesting pedagogical challenge. If you just wanted to convey the idea of it, you would probably start with simplicial or cubical homology (I think this is the approach Rotman takes in his book), but if you wanted to train future researchers in the subject, you’d be tempted to skip that and go straight to singular or cellular homology. Most graduate courses probably opt for the latter, but perhaps we’ll begin to see applied courses that take the former route.

I’ve been doing some reading into alternatives to subjective probability, and one interesting alternative is to model an assignment of subjective probability by a convex set of probability distributions, rather than a single distribution. Convex sets encompass several natural situations where you have a vague sense of probabilities, but would be unwilling to specify an exact value. For example, a range of probabilities for an event can be expressed as a convex set, as well as the idea that one event is more likely than another (without expressing exact probabilities for each event). Convexity also has a natural probabilistic interpretation: if two distributions are in the set, then any mixture of the two is also in the set.

A nice introduction to the subject is Fabio Cozman’s online tutorial Introduction to the Theory of Sets of Probabilities. For some additional surveys on related approaches, see the homepage of the Imprecise Probabilities Project.

Have any of you ever heard of the two-envelope paradox? It’s a paradox so important that Wikipedia manages to have two articles on it: Two envelopes problem and Envelopes paradox. The only thing that puzzles me about it is that I’m having trouble seeing how it’s a paradox — unlike, say, the Monty Hall problem, the naive answer is the correct one.

An easy-to-read article on mathematics in biology.

Ronnie Brown, of the University of Wales at Bangor, UK, recently posted some very interesting remarks on mathematical speculation to the categories list. With his permission, I am reposting them here:

“The situation is more complicated in that what could be classed as speculation may get published as theorem and proof. For example, in algebraic topology, sometimes proofs of continuity are omitted as if this was an exercise for the reader, yet the formulation of why the maps are continuous (if they are necessarily so) may contain a key aspect of what should be a complete proof. This difficulty was pointed out to me years ago by Eldon Dyer in relation to results on local fibration implies global fibration (for paracompact spaces) where he and Eilenberg felt Dold’s paper on this contained the first complete proof. I have been unable to complete the proof in Spanier’s book, even the second edition. (I sent a correction to Spanier as the key function in the first edition was not well defined, after Spanier had replied `Isn’t it continuous?’) Eldon speculated (!) that perhaps 50% of published algebraic topology was seriously wrong!

van Kampen’s original 1935 `proof’ of what is called his theorem is incomprehensible today, and maybe was then also.

Efforts to give full details of a major result, i.e. to give a proof, are sometimes derided. Of course credit should be given to the originator of the major steps towards a proof.

Grothendieck’s efforts to develop structures and language which would reduce proofs to a sequence of tautologies are notable here. Colin McLarty’s excellent article on `The rising sea: Grothendieck on simplicity and generality ‘ is relevant.Some scientists snear at the mathematical notion of rigour and of proof. On the other hand many are attracted to math because it can give explanations of why something is true. But `explanations’ need a higher level of structural language than for what might be called proofs.

I can’t resist mentioning that one student questionaire on my first year analysis wrote `Professor Brown puts in too many proofs.’ So I determined to rectify the situation, and next year there were no theorems, and no proofs. However there were lots of statements labelled `FACT’ followed by several paragraphs labelled `EXPLANATION’. This did modify the course because something labelled `explanation’ ought really to explain something! I leave you all to puzzle this out!

In homotopy theory, many matters, such as the homotopy addition lemma, had clear proofs only years after they were well used.Surely much early algebraic topology is speculative, in that the language has not yet been developed to express concepts with rigour so that a clear proof can be written down. It would be a curious ahistorical assumption that there is not at this date another future level of concepts which require a similar speculative approach to reach towards them.”

(Ronnie Brown, posted 2006-03-14 to the categories list).

The April Notices of the AMS is up. The bulk of the issue is devoted to the centennial of Kurt Goedel’s birth. Sadly, the internet has managed the rather remarkable feat of making me sick of the Incompleteness Theorems, but the issue also has an article about one of my favorite mathematical topics, syzygies, in Roger Wiegand’s article, What is… a syzygy?.