Giving credit

V. I. Arnold, On teaching mathematics:

What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? […]

What is a smooth manifold? In a recent American book I read that Poincaré was not acquainted with this (introduced by himself) notion and that the “modern” definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms.

For what sins must students try and find their way through all these twists and turns? Actually, in Poincaré’s Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the “abstract” one.

(Interesting talk, do read.)

Meanwhile…

Bill Poser at the Language Log:

Sir William Jones is incorrectly viewed as the discoverer of the Indo-European language family and founder of modern historical linguistics […]

The second and more important point is that Jones cannot be considered the founder of modern historical linguistics because he did not use the comparative method, the crucial innovation that distinguishes modern historical linguistics from its predecessors.

Sigh. Let’s not forget people who actually caused us to perceive the world differently, and leave it to pedantic types to define who invented what.

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Music and lyrics

I attended a talk today by Adriano Garsia, which was part of the MIT combinatorics seminar. It was called “A New Recursion in the Theory of Macdonald Polynomials”, and while I didn’t know what Macdonald polynomials were, I went to the talk anyway, because I like polynomials and I like recursion and I like combinatorics (but primarily because it was a way of procrastinating). :-)

Even though I understood almost nothing of the deep mathematics in the talk (and still don’t exactly know what Macdonald polynomials are), it was a very pleasant and refreshing talk, and I felt very good after hearing it. The reason is that it had, of all the talks I’ve attended in recent memory, probably the best “music”. What does that mean? As Prof. Doron Zeilberger invented the term:

Human beings have bodies and souls. Computers have hardware and software, and math talks have lyrics and music. Most math talks have very hard-to-follow lyrics, […]

But like a good song, and a good opera, you can still enjoy it if the music is good. The “music” in a math talk is the speaker’s enthusiasm, body-language, and off-the-cuff heuristic explanations.

Sometimes you can recognize a familiar word, and relate it to something of your own experience, whether or not the meaning that you attribute to it is what the speaker meant, and this can also enhance your enjoyment.

(read more at Zeilberger’s Opinion 78)

And so it was with this talk. Prof. Garsia clearly loved the subject, and even someone like me who had no idea what’s going on felt compelled to listen, fascinated. He told us how the problem came about (“long relationship with Jim Haglund: he makes brilliant conjectures and I prove them”), of false proofs they had had, of how their current proof was driven by heuristics and unproven conjectures, he even posed a problem and offered a $100 reward for an elementary/combinatorial proof. :-)
Far better than the talks with bad music and bad lyrics. (It also helped that although I couldn’t understand the lyrics, they sounded nice: permutations, Young tableaux, polynomials defined in terms of them…)

Edit: See also the recent research ‘showing’ that gestures help students learn mathematics.

Convex and concave

A mnemonic for ‘convex’ and ‘concave’:

A convex function

A conCAVE function

Two mnemonics are better than one.

Good Will Hunting: a mathematician’s review

Good Will Hunting, reviewed by Mark E. Saul in the Notices of the American Mathematical Society, April 1998. (The most interesting part is Daniel Kleitman’s box on “My Career in the Movies”; also, please read the book review of The Curious Incident of the Dog in the Night-time if you’ve read the book.)

It feels very good to read the Notices. All its issues since 1995 are available online.

“Every good theorem must have a good counterexample”

Abhyankar[1] attributes the quote to Severi.

[1]: Historical Ramblings in Algebraic Geometry and Related Algebra, Shreeram S. Abhyankar, The American Mathematical Monthly, Vol. 83, No. 6 (Jun. – Jul., 1976), pp. 409-448. Also available here, because it won a Lester R. Ford Award (“articles of expository excellence”) and also a Chauvenet Prize (“the highest award for mathematical expository writing”).

Abhyankar, after distinguishing between the flavours of “high-school algebra” (polynomials, power series), “college algebra” (rings, fields, ideals) and “university algebra” (functors, homological algebra) goes on to present his fundamental thesis (“obviously a partisan claim”):

The method of high-school algebra is powerful, beautiful and accessible. So let us not be overwhelmed by the groups-ring-fields or the functorial arrows of the other two algebras and thereby lose sight of the power of the explicit algorithmic processes given to us by Newton, Tschirnhausen, Kronecker, and Sylvester.

Perhaps for this reason, Dr. Z calls Abhyankar (“one of my great heroes”) “the modern prophet of high-school-algebra”.

Anyway, enough rambling. Back to “Every good theorem must have a good counterexample”. Discuss.

[Edited to add: The statement in its original context was referring to a phenomenon where a pleasing conjecture is found to have counterexamples, until it is resolved by realising that we must, say, count multiplicities the “right” way—the right way turning out to be whatever makes the conjecture true. Thus Bezout’s theorem, etc., and the quote just means, as he paraphrases, “don’t be deterred if your formula is presently invalid in some cases; it only means that you have not yet completely deciphered god’s mind”. On the other hand, what I (mis?)remembered was that one must know “counterexamples” to a theorem in the sense that one must know why the conclusion is not true if the hypotheses are weakened: thus one doesn’t really understand a theorem till one knows at least one “counterexample” (and at least two proofs).]