30 May 2010

Interview with Gromov

In the March 2010 issue of the Notices of the American Mathematical Society, there is a very good interview with Mikhail Gromov, prominent mathematician and the recipient of the 2009 Abel Prize of the Norwegian Academy of Science and Letters. I would like to quote just a couple of little excerpts from this interview, leaving aside the majority of it for future comments which will appear on this blog.
As you know, in the UK, in some
of the universities, there are faculties of homeopathy
that are supported by the government. They
are tremendously successful in terms of numbers
of students. And anybody can learn that nonsense.
It is very unfortunate.
Alas, this is true. However, it is easy to spot fake university programmes, like those offering courses on homeopathy. What is harder is to spot university programmes that hide themselves under the guise of science. Indeed, imagine, for instance, a university offering a course on the Mathematics of Homeopathy, say, luring students who aspire to learn applied mathematics. Such a hoax would be harder to spot.

A second excerpt, from the same interview, goes as follows. (It is a reply to the question on how education should change to get better adapted to very different minds.)
Look at the number of
people like Abel who were born two hundred
years ago. Now there are no more Abels. On the
other hand, the number of educated people has
grown tremendously. It means that they have not
been educated properly because where are those
people like Abel? It means that they have been
destroyed. The education destroys these potential
geniuses—we do not have them! This means that
education does not serve this particular function.
The crucial point is that you have to treat everybody
in a different way. That is not happening
today. We don’t have more great people now than
we had one hundred, two hundred, or five hundred
years ago, starting from the Renaissance, in spite
of a much larger population. This is probably due
to education. This is maybe not the most serious
problem with education.
What he says is absolutely true: Despite the fact that the population has grown by almost an order of magnitude in 200 years, despite the fact that the majority of people are able to (and do) enter a "higher education establishment", the number of thinkers like Abel has not grown in proportion to the world population. Education, says Gromov, is responsible for this. Indeed, many universities nowadays offer vocational training instead of university education. There is nothing wrong with vocational training: we need car mechanics, accountants, paramedics, fire-fighters, etc. What is wrong is to substitute university education with vocational training. And what is much worse is to claim that we are not doing so; that we are still offering "high quality" education, whereas, in reality, we are training people to do a specific job. This is were the problem lies. And this is why Gromov, in the first excerpt above, gave the homeopathy example as an example of what is happening in universities nowadays.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant