
On teaching mathematics, by V.I. Arnold (1997) - ssivark
http://pauli.uni-muenster.de/~munsteg/arnold.html
======
danharaj
> A smooth k-dimensional submanifold of the Euclidean space RN is its subset
> which in a neighbourhood of its every point is a graph of a smooth mapping
> of Rk into R(N - k) (where Rk and R(N - k) are coordinate subspaces). This
> is a straightforward generalization of most common smooth curves on the
> plane (say, of the circle x2 + y2 = 1) or curves and surfaces in the three-
> dimensional space.

I just wanted to provide a somewhat humorous convergece between super-abstract
mathematics and the concrete mathematics VI Arnold advocated: The category of
smooth manifolds is the idempotent completion of the category of open subsets
of Euclidean spaces:

[http://mathoverflow.net/questions/224569/smooth-manifolds-
as...](http://mathoverflow.net/questions/224569/smooth-manifolds-as-
idempotent-splitting-completion)

[https://ncatlab.org/nlab/show/smooth+manifold#patching_as_id...](https://ncatlab.org/nlab/show/smooth+manifold#patching_as_idempotent_splitting)

When you unravel the definition you get a very concrete idea of what a
manifold is without the bureaucracy of charts and atlases and whatnot. In
general, Lawvere's mathematical career is an ambitious program to cast
classical mechanics in categorical terms. I see this as the synthesis of the
two traditions.

Another way in which abstract mathematics is converging with concrete
mathematics is the Curry-Howard-Lambek-... correspondence between logics,
programming languages, and category theory. The most abstract branch of
mathematics happens to be very amenable to the very physical, concrete process
of computation!

I think abstract definitions that are far divorced from the concrete objects
they are trying to capture is a deficiency in the abstraction. The correct
abstraction somehow miraculously becomes far more concrete than any
intermediate definition. For example, homotopy theory has a very _ugly_
definition in set based mathematics, but by abstracting the essentials,
homotopy type theory looks very promising at reconciling the bureaucracy of
rigor required to do homotopy theory in sets and the very geometric intuition
one has for homotopical objects.

------
marcelluspye
I feel there needs to necessarily be a separation of the doing of mathematics
and the teaching of mathematics in these kinds of matters. In the teaching of
mathematics, especially in the more 'abstract' areas, there is not nearly
enough driving of intuition, and the 'problems' students are given often are
unrelated to the 'problems' the theory they are learning about was created to
solve. Pushing things in the concrete direction is probably the right
direction for pedagogy.

But in the _doing_ of mathematics is where I think Arnold is looking back at
the past too romantically. Prior to programs like Bourbaki, the lack of unity
in mathematical notation, definitions, ideas in methods was very present.
Often if you were not a student of one of the masters of the time, you would
have an incredibly difficult time understanding their work. The tradition that
has lead to an 'over-abstraction' of things, as Arnold would have seen it,
also lead to a much more widespread accessibility of contemporary mathematics.
when you can count Kolmogorov among your teachers, this is less relevant to
you. But for the 'unwashed masses' of mathematics this has made a world of
difference.

~~~
tnecniv
> I feel there needs to necessarily be a separation of the doing of
> mathematics and the teaching of mathematics in these kinds of matters. In
> the teaching of mathematics, especially in the more 'abstract' areas, there
> is not nearly enough driving of intuition, and the 'problems' students are
> given often are unrelated to the 'problems' the theory they are learning
> about was created to solve. Pushing things in the concrete direction is
> probably the right direction for pedagogy.

Indeed. By far the best teachers I've had for math courses spent a good deal
of time discussing the history of the topic and motivating its creation.

------
potbelly83
It's interesting comparing Arnold's very anti-abstract approach to Gromov's
(another great Russian mathematician from the same era and now a professor at
the IHES). Gromov in his later writings take a very categorical approach in
his work. For example in the following paper he attempts to define entropy
from a very abstract view point [http://www.ihes.fr/~gromov/PDF/structre-
serch-entropy-july5-...](http://www.ihes.fr/~gromov/PDF/structre-serch-
entropy-july5-2012.pdf)

~~~
4ad
Gromov's approach is not abstract, it's extremely concrete. It uses
abstractions for a very precise treatment, and it's probably not the best text
for an introduction to entropy, but it's essentially a very concrete
treatment.

It analyses very concrete observations using abstract methods in order to
construct an abstract definition. It doesn't simply spew out abstract
definition. Everything is motivated by concrete facts.

I would not say it's against Arnold's method at all, if anything, it uses the
same concrete to abstract treatment. It is, however, not the best way to learn
about entropy, but it's a good way to formalise it once you have become
familiar with it. It's poor for undergraduate level material, but it's good as
graduate level material.

------
MexicanReformis
Growing up I found Mathematics difficult to grasp for various reasons. One of
them required mental gymnastics of "hidden" rules and exceptions to the rules.
There would always be rules that were often forgotten. And one of the bigger
problems I had was forcing myself to practice these exercises. In order to
store this information one must actually practice these problems many times to
have it lodged into your brain for later use. Being a kid made it even more of
an obstacle since I always wanted to play outside or hangout with friends.

It seemed that mathematics required more of my time than any of the subjects.
So wouldn't spend the necessary time needed to actually store it in my head.
Later in life, I realized how fascinating math was, and fell in love with it.
Even though I am still not good at it, it's fun to solve puzzles.

~~~
JadeNB
> One of them required mental gymnastics of "hidden" rules and exceptions to
> the rules. There would always be rules that were often forgotten.

Could you elaborate on this? Mathematics, perhaps uniquely among human
intellectual endeavours, has _no_ hidden rules. Its rules may be complicated
and unintuitive at times, but they can be listed; and whatever follows from
these listed rules is true (to be precise: whatever follows from the axioms is
true in any model of the axioms), and whatever contradicts them is false,
regardless of how natural or un- it may seem.

In particular, I am curious what you mean by "There would always be rules that
were often forgotten." Do you mean that _you_ forgot them, or that your
_teachers_ forgot them (both of which were eminently believeable!), or that
practicing, professional mathematicians forget them? I am sceptical of the
last (although plenty of, indeed probably all, professional mathematicians
don't _need_ to keep all the rules explicitly in their heads at all times),
and, although I am eminently willing to believe the first two, I think that
that shows a fault in the teacher, not the subject.

------
valw
This article raises an important and interesting problem, but would benefit
from a lot more nuance. It seems to claim there is only one ubiquitous way
people teach, learn and do mathematics, but that is not the case.

In my view, using abstract notions for describing is in no way incompatible to
motivating them with concrete, visual notions, and that is what good teachers
do and good students seek. This is done imperfectly by many teachers, but
there is nothing new in stating that not everyone teaches perfectly.

It also really depends on the particular skills and desires of the student.
I'm a more abstraction-oriented person, and as such when studying physics I
often wished the teachers would rely more on abstract mathematics because _in
my view_ it made the phenomena clearer; I have no doubt that many students
feel otherwise.

------
nextos
I've read this article countless times when looking for his textbook
recommendations, cited at the end. All excellent.

While I also appreciate abstraction, very often mathematics is taught at the
wrong level of abstraction. A book I like a lot, in the spirit of his
recommendations, is Hubbard & Hubbard.

------
jl274
There are some reasonable points in here (problems with abstraction for
abstraction's sake) but on the whole this reads as someone who simply cares
about geometry and analysis, but not algebra or discrete mathematics.

Which is fine, he's entitled to his opinion. But the stronger argument that
"mathematics is only good when it's tied to physics" argument is wrong and
outdated -- belongs to the pre-computer, pre-information age.

Even on a purely mathematical level, some of his arguments don't hold. For
instance, for all his complaining about how abstraction doesn't contribute
anything -- Cayley's theorem on groups, Whitney's theorem on manifolds --
there ARE categories where the abstract definition is needed or is more
natural, like algebraic varieties (which can't always be embedded in complex
N-space for any N) and rings (I have no idea what better definition he'd be
proposing).

To me, it sounds like he doesn't like or know much about the algebraic side of
math, and resents how much it dominates in math today.

~~~
conistonwater
> _To me, it sounds like he doesn 't like or know much about the algebraic
> side of math, and resents how much it dominates in math today._

Given that Arnold was a student of Kolmogorov, was awarded the Wolf prize, and
was denied the Fields medal due to some Soviet-era politics
([https://en.wikipedia.org/wiki/Vladimir_Arnold#Honours_and_aw...](https://en.wikipedia.org/wiki/Vladimir_Arnold#Honours_and_awards)),
I think it's safe to say he understood algebra. You can also read his books,
and find out that he used abstract algebra quite well. That's not what he's
saying here, I think you misread.

> _Which is fine, he 's entitled to his opinion. But the stronger argument
> that "mathematics is only good when it's tied to physics" argument is wrong
> and outdated -- belongs to the pre-computer, pre-information age._

I think you're missing the point he's making and cherry-picking examples from
the article. Indeed, much of the article is directed at how very conventional
mathematics is taught at the school and undergraduate level, and one key point
is that the way it is taught is too far divorced from how it was discovered a
long time ago. He claims that teaching mathematics in the algebraic way makes
it unnecessarily more difficult to solve mathematical problems that used to be
essentially trivial, making people worse at mathematics for no good reason.

So you picking out the example of categories of algebraic varieties is
misleading because that's not at all the focus of the article, it seems like
an incidental example. I think the _determinant_ and _group_ examples are a
much clearer illustration of the difference he's talking about.

I also have no idea what you mean by your comment about _pre-computer, pre-
information age_. If anything, mathematics became much more of an experimental
science (thus closer in spirit to physics in Arnold's view) now that you can
run computational experiments so easily with this much computing power. Note
his example of the difference between _observation - model - investigation of
the model - conclusions - testing by observations_ and _definition - theorem -
proof_.

