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?.

I have to admit that I have a somewhat dismissive nature sometimes, and have been known to make critical remarks for non-public consumption; “Programmers cannot do math at ALL” (hi Dale!) Of course, this is more out of shock than a belief in any natural order. I firmly believe that anyone of everyday intelligence can learn math. I am of the opinion that I could teach calculus* to a dead twig if the twig where sufficiently motivated.

Which is why I was happy to see that someone had written a blog entry on learning math being what you make of it. The only thing I would criticize is that he is completely wrong.

Ok..just kidding, but I do have to say that I would not follow his councel on exercises. When I read a GTM on a new subject (papers don’t usually have many exercises :), I don’t really view the problem sets as seperate from the explanitory text – I do every single problem. This is because the author of the text did not view the problem sets as seperate either. It goes beyond “having been shown the idea, cement it in your mind with the excercises”. Most of the time, realizations that the author wants you to have are set up in the problem sets because they would be TOO padantic and verbose in the main text.

*This isn’t restricted to calculus of course.

The mathematicians Ron Brown and Tim Porter, at the University of Wales in Bangor, UK, have long been at the forefront of mathematics popularization in Britain. (This is in addition to their very valuable contributions to algebraic topology, category theory and theoretical computer science!) Their web-pages include a page of articles and links about popular mathematics, reasons for studying math, the teaching of math, etc.

Over at the BBC, Melvyn Bragg presents a weekly radio programme called In Our Time in which are discussed a wide variety of subjects ranging from art and philosophy to politics and drama. Surprisingly, considering the wide range of human culture covered, Bragg devotes a good proportion of episodes to mathematics. He’s just presented a programme on the Negative Numbers and recently presented one on Prime Numbers. If you dig into the science archives you’ll find recordings of programmes on Zero, Infinity, π, Chaos Theory and Renaissance Mathematics. Some of these are conveniently available as podcasts.

(Note that some of the comments about the mathematical topics on the In Our Time website are a little inaccurate, eg. “a team of researchers…calculated the highest prime number”, but the actual guests on these shows are generally mathematicians or historians of mathematics who know what they are talking about.)

Issue 38 of Plus is now out!

I’ll be on vacation for the next few days, so I leave you to the tender ministrations of my co-bloggers, and this anecdote (from Deirdre McCloskey’s Secret Sins of Economics):

I have a brilliant and learned friend who is an intellectual historian of note. He and I were walking to lunch in Iowa City one day and I said offhandedly, assuming he would of course know this, that mathematics was one of the great achievements of Western culture. He was so astonished by the claim that he stopped short and argued with me there on the sidewalk by the Old Capitol Mall: “Surely math is like plumbing: useful, but hardly in touch with deeper things; hardly a cultural achievement!” I tried to persuade him that he felt this way only because he had no acquaintance with mathematics, but I don’t think I succeeded.