Written: Aug. 25, 2010

The ICM has just promoted four mathematicians from the rank of "excellent mathematician"
to that of "truly great mathematician". Being a good mathematical citizen, I tried to understand,
at least superficially, what they did. Except for Stas Smirnov, with whose work
I was already familiar (and that I find the most exciting, being kind-of-combinatorics), I had no clue.
So I turned to
Terry Tao's blog entry and got *some* very general idea.

"To be honest, I don't really understand Elon's mathematics. There are probably only twenty mathematicians in the whole world who canreallyunderstand his work."

This made me feel much better. Maybe I am not so dumb after all. If one of the greatest functional analysts in the world, whose research area is much closer to Elon's area of Dynamical Systems than to mine, has no clue, what would you except from a simple-minded discretian like myself?

And that is the **problem**. "Mainstream" Mathematics has gotten so *fragmented*, so **specialized**,
so out-of-reach-with reality and so *boring*. While I have (hardly) any clue what
the new Fields medalists achieved, I know very well what they did **not** achieve:

- Proof of the Goldbach conjecture
- Proof of Collatz's 3x+1 conjecture
- Proof that e+π is irrational
- Proof of the Riemann Hypothesis
- Proof (or disproof) of the Jacobian conjecture
- etc. etc.

(Speaking of "rigor", this is a soon-to-be-obsolete hang-up of 19th and 20th-century mathematics that did some good, but much more harm by hindering its progress. We often brag about how "useful" mathematics is in science and engineering, true, but usually the scientists and engineers discover it all by themselves (e.g. physicists Seiberg and Witten in topology and engineers Hamming and Golay in coding theory), and Einstein would have easily developed Riemannian Geometry, his way if Riemann didn't do it before).

I was particularly unimpressed by the "almost" solution of the Littlewood conjecture.
**"Almost" does not count**!. The whole concept of "Lebesgue measure" gives me the creeps. It is so
artificial and in fact an artifact of mathematicians' superstitious belief in the infinity
and their fanatic insistence on (the appearance) of "rigor".
"Almost" proving something is often a piece of cake (for example that almost all x wind up at 1
after iterating the 3x+1 map). I am sure that in the case of the Littlewood conjecture
it was a major technical feat, so I am not saying that what Elon et. al.
did is *trivial*.
Quite the contrary, it is *extremely* deep, in fact *too deep for my taste*.
It is so deep that I (and 5 billion people
take away 20) couldn't care less about.

But, the point of this opinion is not to put you down or discourage you. All the four of you are so brilliant. Such great minds are terrible things to waste on current "main-stream", over-specialized, mathematics, that you have been doing so far.

First and foremost, learn how to program! well!, by yourself, and not just let students
do it for you. Once you will learn how to think *algorithmically* you would
be much better off.
If you would have taken my Experimental Math class
either in 2010
or 2009,
or 2008,
or 2007,
or 2006,
and learned how to program Maple (or Mathematica, but not with me), you would have
been able do do so much more, than what you accomplished by mere paper-and-pencil.

You would also realize that not all mathematical results
could be proven with full rigor, and sometimes one has to settle with
semi-rigorous, and even non-rigorous proofs. Look at the great mathematics
done by physicists Ken Wilson, Leo Kadanoff, and others when they developed
the "renormalization group", and the attempts by some mathematicians to make
it rigorous is "who cares?". I admit that *sometimes* attempt
to make things "rigorous" leads to beautiful new insights, and sometimes
"non-constructive" approaches (like Furstenberg's ergodic approach to Ramsey theory
that lead to insights into purely combinatorial proofs) are worthwhile.
But other times attempts to "rigorize" a piece of mathematical physics while
technically very challenging, is a futile and pointless exercise.

Please *diversify!*, **expand!**,
and try to prove (in whatever level of rigor you can master) results that
are (i) interesting (ii) that I and my fellow mortals can understand and **appreciate**.

We have to thank Professor Fields for the 40-year-old upper bound. You still
have (at least!) forty years of productive mathematical life.
I agree with G.H. Hardy that *mathematics is a young man's game*
(except for the "man" part), but "young" does not mean "under thirty", and
not even "under forty". "Young" means, "young at heart", and willing
to learn new things and new methodologies and master new technology.
If you will follow my advice, I am sure that you would achieve much more than
your already very impressive (albeit mostly boring) feats, and your future
achievements will not only be technically challending, but also **exciting**,
not just to you and to your thirty cronies, but to all of us common folks.

Opinions of Doron Zeilberger