
V.I. Arnold: On teaching mathematics - TriinT
http://pauli.uni-muenster.de/~munsteg/arnold.html
======
pixcavator
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.

~~~
sandGorgon
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.

~~~
pixcavator
>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.

------
timwiseman
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.

~~~
TriinT
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_.

~~~
onedognight
> 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>

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

------
mhartl
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.

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

~~~
mhartl
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. :-)

------
mitko
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.

~~~
eugenejen
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.

~~~
TriinT
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.

~~~
eugenejen
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.

