If it’s not too much trouble, would you mind posting this list?
Okay–his list has motivic cohomology; special cases of the Langlands functoriality conjecture; advanced number theory; quantum groups; infinite-dimensional Banach spaces; local and micro-local analysis of large finite groups; large and inaccessible cardinals; algebraic topology; superstring theory; and non-abelian reciprocity, automorphic respresentations, and modular varieties.
Out of curiosity what about Stephen Wolfram and his “change the world” self published book, “A New Kind Of Science”. Is he even on the radar screen of mathematical greats?
I’d have to cast my vote for a three way tie between Newton Gauss and Archimedes, with Euler coming in a close fourth. It doesn’t seem all that profitable to compare ancient and modern (post mediavel) mathematicians, since they had such vastly different tools to work with. However anybody who can calculate pi to two “decimal places” without a workable number system ( Archimedes) gets my vote. why he isn’t on the list Thudlow Boink refers to is beyond me.
And not even a runner-up for Cauchy? Also, Descartes was possibly the greatest philosopher of all time, but top 10 mathematician?
Not big in terms of impact, but for sheer genius, how about Ramanujam?
From this site:
http://freepages.genealogy.rootsweb.com/~jamesdow/Tech/mathmen.htm
“The name Srinivasa Ramanujan (1887-1920) is omitted for the opposite reason: he had great genius, but his work lacked historical importance.”
I have to agree with you, there are definitely good arguments to put Mr. Ramanujan in the top 10.
This is all very interesting to read. Thanks to all for their thoughts.
My opinion on Wolfram. He has played with some new stuff, and may eventually be credited as originating a new branch of mathematics. But he strikes me as the kind of guy best described by the phrase “If all you have is a hammer, the whole world looks like a nail.” I just don’t buy into all the applications he suggests.
Hijack here, or perhaps not.
For those who are interested, I can thoroughly recommend A History of Mathematics by Boyer and Merzbach. If you haven’t read it, it is fantastic. (And relatively straightforward for a non-expert like me.)
Let me qualify that by saying that I have only browsed briefly Wolfram’s A New Kind Of Science.
Larry Borgia writes:
> However anybody who can calculate pi to two “decimal places”
> without a workable number system ( Archimedes) gets my vote.
So did other people of the time. It’s easy to forget how much the people of late classical times knew. For instance, in astronomy, they knew that the Earth is a sphere, they knew the size of the Earth (within 10%), they knew that the moon is a sphere, they knew its size (within 10%), and they knew that the other planets, the sun, and the sun were at least much further away than the moon.
All the mathematicians on the 10 best list are post-Medieval and (were born) pre-20th-century (except for Euclid). It’s hard to know what a mathematician did in original work before modern times. People back then weren’t as big on precisely assigning the rights to mathematical (or any scientific) work. And any work done by someone born in the twentieth century is probably too recent to be certain of its importance.
If we’re going to give credit to anyone for cellular automata, it should go to John Conway, not Stephen Wolfram.
I’d leave him out of my top 10, but perhaps my reasons are good enough to convince others. No doubt he had the talent, but not only did his work lack historical importance, he was a specialist in just one area (Number theory) at a time when geniuses spanning multiple fields (like Poincare) were still thriving, and he required the help of other geniuses to prove his results (e.g., Hardy).
I’m not sure I’d include Poincare, but I’d probably have to draw up my own list to be sure. The problem I have with Poincare is his reaction to Special Relativity. He was one of two or three people who should have slapped themselves in the forehead and said “Damn! Why didn’t I think of that!” (You can Pepe Le Pew that if you wish.) Instead, there is no evidence he ever understood it. (Granted, Special Relativity is usually considered to be part of physics, but he did work along those lines, which is why something called Poincare Invariance bears his name.)
Let’s see, the SMT list
1, 2, 3: Archimedes, Newton, and Gauss
4 Riemann (yeah, I specialized in Relativity, and have some idea of how hard his stuff was).
5 Euler
6 Cauchy
7 Lagrange (probability, anyone?)
8 Hilbert
9,10 Abel and Galois
I guess that would Poincare in my 10-20 list.
This site puts de Fermat at #6.
http://freepages.genealogy.rootsweb.com/~jamesdow/Tech/mathmen.htm
I wasn’t quite sure about that choice until I read this:
Descartes attacked Fermat’s method of maxima, minima and tangents. Roberval and Étienne Pascal became involved in the argument and eventually so did Desargues who Descartes asked to act as a referee. Fermat proved correct and eventually Descartes admitted this writing:-
… seeing the last method that you use for finding tangents to curved lines, I can reply to it in no other way than to say that it is very good and that, if you had explained it in this manner at the outset, I would have not contradicted it at all.
To win a battle of wits with Descartes, he has to be top 10. 
Especially since Fermat was a judge, not a professional mathematician.