V.I. Arnold: On teaching mathematics | Hacker News Reader

V.I. Arnold: On teaching mathematics

pauli.uni-muenster.de

32 pointsTriinT19 comments

I agree with virtually everything he says about teaching (this is ironic because I really hated his ODE lectures as a student). But the idea of "mathematics is a part of physics" isn’t well justified in the article and is, with all due respect to such a luminary, ridiculous. Instead of dozens of examples we all can think of here’s my own: I am working on applications of topology in computer vision and data analysis. No physics in sight! His attitude toward proofs (outside teaching) is also very odd.

i think what Arnold is trying to convey is mathematics needs to be "learned" intuitively, rather than in an abstract manner. Pure mathematics may, however, be practiced (though one can argue that intuition will continue to play a significant role )

The essential bit is about being able to visualize a mathematical concept, before being able to express it in mathematical language. For example, I think the visualizations of sorting algorithms give a intuitive understanding of how something should work, that you will be able to see quicksort in your mind, before you code it up.

Physics can easily be interchanged for computer vision as a tool for aiding in mathematical intuition. Learning is hard, practice is easy.

>Physics can easily be interchanged for computer vision as a tool for aiding in mathematical intuition.

I came from topology to computer vision, not vice versa. This experience showed to me (as if I needed another proof) the "unreasonable effectiveness" of abstract(!) mathematics.

So much of this article strikes me as utterly ridiculous. To start with, even if someone where to believe that mathematics was only of value in its application, it would still be ridiculous to say that it was a part of physics and worthless beyond physics.

From there it seems to focus on denegrating the value of pure and axiomatic mathematics focusing almost purely on the need to keep mathematics tied to physics.

VI Arnold has authored some of the best expositional texts on mathematical physics and differential equations on the planet, even if he's being hyperbolic, there is still insight to be gained from his work

Arnold's article is highly satirical. If I wrote such an article, it would be, indeed, utterly ridiculous. Arnold is such a great mathematician that he has earned the right to poke fun at his own field. His books on differential equations and classical mechanics are simply the best.

Moreover, Arnold is more of a geometer than an analyst. His article is also a not-so-subtle attack on the French fanatics (i.e., "bourbakists"), who tried to axiomatize all math and remove any geometrical and intuitive aspects from it. On the value of pure and axiomatic mathematics, I would like to quote Jacques Hadamard (who was french!):

"The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it."

Rigor should follow intuition, not vice versa.

> His books on differential equations and classical mechanics are simply the best.

I'm partial to his Topological Methods in Hydrodynamics myself.

In case you're a "school child" interested in mathematics, the lecture he mentioned, The Abel theorem in problems, was just recently translated into english by Sujit Nair.

http://www.cds.caltech.edu/~nair/abel.pdf

Thanks for the wonderful URL. Yes, I am a "school child" indeed, albeit a bit older (i.e., I am a grad student ;-)

Ah, perhaps it is because I am not familiar with him that I completely missed the satire. I had taken it to be quite literal.

This is wonderful reading; I especially enjoyed the recurring use of "torment" to describe the indignities inflicted upon modern math students.

I'm also amazed to discover that V.I. Arnold is still alive! His work has been important for so long that I simply assumed he (like Landau & Lifshitz) belonged to an earlier time. I'm delighted to discover I'm wrong.

You might enjoy reading this interview: http://www.ams.org/notices/199704/arnold.pdf

Thanks! But this is just too tantalizing:

Editor’s Note: As this article went to press, V. I. Arnol′d submitted an update on the interview, based on subsequent correspondence and events. It was received too late to be included in the article.

Will we ever know what the updated interview would have been? (Maybe I should email Arnol'd to find out. :-)

To say that mathematics only serves physics, seems wrong. Mathematics provides a basis and guarantees for every science. Physics, Biology, Computer Science... name it- I bet it uses tons of mathematics to justify its models.

In my opinion the beauty of the mathematics is that if you prove something- it is TRUE, or your assumptions are wrong.

I think, the way it was phrased, Physics "is" every science. That is, every other science can be expressed purely in terms of (lots and lots and lots of very, very tedious) physics, similar to the way that every computer program can be expressed in terms of the simplest possible Turing machine. To put it in that perspective, mathematics must serve empirical investigation into reality.

Unfortunately this is not the origin of the modern European mathematics. The math was tightly related to the field of mathematical physics until early 20 century. And the cause of split of physics and mathematics is Bourbaki's movement for pure math, which unfortunately leading the movement of pedagogical "new math" later that make all school children hating math.

The job of old mathematicians is not only to prove theorem, but also to design tools to solve physics problems.

V.I. Arnold was trying to bring this back together again. It is the same as we design algorithms for real world problems.

I keep thinking of how V.I. Arnold managed to teach group theory to Moscow schoolchildren in the 1960s. I also wonder why Landau & Lipschitz are such wonderful books. There's something special in the Russian way of teaching Math / Physics, I guess.

I don't know. Even Komolgorov's "Mathematics: Its Content, Methods and Meaning" is pretty easy to read.

My theory is that instead of teaching the abstract definition and proofs for group like general college math professors, V.I. Arnold might start lectures from some games/problems in daily life that we can apply group theory on to kids. (Of course those kids maybe gifted in math already) then from the special concrete case as basis, he then showed students the abstract stuffs behind it.

Of course Math serves many fields, but have you ever found a research paper that applied Math to, say, Biology and didn't use a bit too much hand-waving? Of all fields of application of mathematical ideas, Physics has been (by far) the most successful. Wigner wrote all about it.

For biology you need a particular branch of math: statistics. Of course the same is true of physics when you look at it on a small enough scale.

With the small datasets some biologists work with, you might not get a useful result, but at least you can quantify how unuseful it is!

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