
On teaching mathematics by V.I. Arnold (1997) - adamnemecek
https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
======
joker3
"Mathematics is a part of physics."

Try telling that to a statistician, or a computer scientist, or an economist,
or....

I think it's worth noting that category theory, which was initially derided as
abstract nonsense even by other mathematicians, is now a central field of
study in math and is also becoming a major part of theoretical physics. If we
only studied things that are immediately applicable to some particular
science, we would not be able to advance the sciences as quickly as we can
now.

I'm not that wild about the Bourbaki style of exposition, which does emphasize
rigor over intuition, but I think that there is a lot of value in having
students go through that phase. Terry Tao has a good explanation of why at
[https://terrytao.wordpress.com/career-advice/theres-more-
to-...](https://terrytao.wordpress.com/career-advice/theres-more-to-
mathematics-than-rigour-and-proofs/), and I'd encourage anyone interested in
the topic to read it.

~~~
gowld
What Arnold is saying is that mathematics is part of the natural world, with
immutable truths that can be discovered.

He is saying that the science of mathematics is part of the natural sciences.
There's no reason for a statistician, or a computer scientist, or an economist
to agree with that -- these fields depend on the natural truths discovered by
mathematical science just as surely a biologists depends on the natural truths
discovered by chemists.

~~~
perl4ever
Hasn't it been proven that there must be areas of math that can never be
connected to other areas of math? Saying it's part of science, or the natural
world, seems like saying all math is part of one great whole. That's something
that's been passé since 1931 or so, I thought.

~~~
harry8
I don't know the answer to your question but want to state the a differing
point of view in the same vein which is how I understood it as the contrast
might be something I learn from.

There can never be practically useless mathematics. Someone will always find a
practical use in the physical world for any new mathematics that is developed.
The story I heard illustrating this was that in the shadow of WW1 various
mathematicians who lived through that horror became (understandably) pacifists
(Hardy?) and decided they wanted to make sure their work would never be useful
in weaponry and so devoted themselves to the utterly useless number theory.
Sadly for them in a world of crypto number theory has a major use and that is
military.

~~~
perl4ever
This is a layman's view, but I thought Gödel's incompleteness theorem means
there must be in principle mathematics that can never be connected to other
parts of mathematics. And if something is part of the universe, it must be
connected to something we can perceive, so I conclude that not all math can be
connected to that which concerns something real.

In other words, even though math that was thought to be useless has repeatedly
been applied to physics, I think the claim that it is "always" applicable is
too strong and in fact proven false in principle.

~~~
joker3
That's not what GIT says. See [https://plato.stanford.edu/entries/goedel-
incompleteness/](https://plato.stanford.edu/entries/goedel-incompleteness/)
for a description of what it does say.

------
ahartmetz
I can ecpecially relate to the part where intuitively obvious things were
probably invented for intuitively obvious reasons, then the derivation was
thrown away and the thing was described axiomatically like the blind man
describes an elephant.

The non-explanation of the determinant in my linear algebra textbook was
especially infuriating: "There is this thing, right? We will later see its
uses. It happens to have the following properties: ..., If you swap two
columns its sign changes, ..."

~~~
gizmo686
The important thing is that a _reasonable_ derivation is presented. It need
not be historically accurate.

For instance, if I was teaching category theory to CS students, I would try to
motivate it with topology.

Similarly, if I were teaching group theory, I would not try motivating it with
solving polynomial equations.

~~~
jacobolus
> _Similarly, if I were teaching group theory, I would not try motivating it
> with solving polynomial equations._

Why not? That’s a fine (and historically important) reason to study group
theory, and everyone who goes through high school is fairly deeply familiar
with polynomial equations.

While we are discussing Arnold, you might like this book created out of
Arnold's lectures to some bright high school students in the 60s,
[https://www.mathcamp.org/2015/abel/abel.pdf](https://www.mathcamp.org/2015/abel/abel.pdf)

~~~
gizmo686
>Everyone who goes through high school is fairly deeply familiar with
polynomial equations.

A high schooler is not nearly familiar enough with polynomials to understand
the group theory component of Galois Theory. The central observation of Galois
theory is that, in certain circumstances, it is possible to permute values
without affecting the correctness of any polynomial (with coefficients in a
certain field).

For example, suppose you have two unknown values, A and B, and knew that the
following polynomial equations held:

    
    
        A + B = 0
        AB = 1
        A^2 = -1
        B^2 = -1
    

There are two possibilities: A=i and B=-i or A=-i and B=i.

Further, no matter what polynomials (with rational coefficients) I give you,
it is impossible to distinguish these two possibilities. The group in this
case is all the possible ways of permuting A and B which do not effect any
polynomials (with rational coefficients). There is a lot of work involved in
actually developing the theory I summarized above.

EDIT:

Perhaps more damning for using this approach to teaching group theory is that
it is, in general, difficult to actually compute what the Galois group is for
a given polynomial.

~~~
kazinator
> _There is a lot of work involved in actually developing the theory I
> summarized above._

Still, Galois had it licked by age 20, and was able to write it down the night
before engaging in a deadly duel that took his life.

------
aphextron
This has hit the front page at least 3 times in the last year. Why?

------
projectileboy
He lost me at the beginning. Paraphrasing Wolfgang Pauli, “That is not only
not right; it is not even wrong.” Certainly much of math was born from
science. But the notion of pure mathematics goes back to antiquity.

~~~
romwell
I would advise you to go and read his statement in full. His point is quite
valid, even if some of the phrasing is not well-perceived.

~~~
projectileboy
Thanks for the response. I took your advice and was glad I did. After reading
the entire piece, I agree with your statement.

~~~
romwell
Thank you for following through!

