The word "genius" is used in many senses. The American Heritage
Dictionary gives the following as its main definition: "Exceptional or
transcendent intellectual and creative power." Let's stick with that
definition for the purposes of this discussion. How do we know who
has such power? The proof of the pudding is in the eating. The only
test that means anything is lifetime achievement.
Transcendent: In other words, there are not millions of geniuses.
There are only a few. Beethoven, Bach, maybe a dozen others in
music. A similar number in other fields. We are talking about the
very summit of human achievement.
The question is: who reaches that summit?
In this post I'm just going to focus on one field, mathematics. There
were a few great architects who created mathematics: Archimedes,
Euclid, Fermat, Descartes, Newton, Leibniz, Euler, Lagrange, Gauss,
Cauchy, Galois, Abel, Riemann, Weierstrass, Poincare. That's the
whole list. They are the ones who reached the summit. They were all
I could extend this list to include other men who made contributions to
the structure of mathematics, less fundamental and far-reaching than
the work of the 15 listed above. There are approximately a hundred
names on this list: Pythagoras, Eudoxus, Apollonius, Diophantus,
Al-Khowarizmi, Levi ben Gershon, Nicolas Oresme, Tartaglia,
Cardano, Viete, Desargues, Blaise Pascal, Napier, John Wallis, Isaac
Barrow, James Gregory, de Moivre, Christiaan Huygens, N. Mercator,
James Bournouli, John Bournouli, Taylor, D'Alembert, Joseph Fourier,
Vandermonde, Legendre, Pierre de Laplace, Eisenstein, Moebus,
Pluecker, Dirichlet, Nikolai Lobachevsky, Janos Bolyai, Beltrami,
Jacobi, Hermite, Poisson, Hamilton, Cayley, Liouville, Dedekind,
Kronecker, Felix Klein, W. Dyck, Camille Jordan, Georg Cantor,
Sophus Lie, Borel, Henri Lebesgue, Elie Cartan, Maurice Frechet,
David Hilbert, Felix Hausdorff, Ramanujan, Georges de Rham,
Banach, G. D. Birkhoff, John von Neumann, Kurt Goedel, Hermann
Weyl, Andre Weil, Jean Dieudonne, Rene Thom, Liapunov, Pontrygin,
L. J. Mordell, Paul Cohen, ...
That's as far as I can get tonight. It's hard to decide which 20th
century mathematicians to include. The point is that there are only a
few mathematicians who have made fundamental contributions to the
subject, and they are all men. There are NO women, ZERO, whose
mathematical work comes anywhere near qualifying for this list.
Now, I could extend the list again, to include several hundred names of
individuals who have made seminal contributions at a less general
level. Note that I said "individuals" this time, instead of men. At this
level, there would be two women on the list: Sonja Kovalevsky and
Emmy Noether. Two. In the entire history of mathematics, those are
the only two women whose contributions are memorable at all, and
they are relatively minor. Two, out of hundreds.
Being a mathematician is not always easy. Let's look at one example.
Quoting from *Mathematics and its History* by Stillwell,
> Niels Henrik Abel was born in the small town of Finnoy, on the
southwestern coast of Norway, in 1802 and died in Oslo in 1829...
His father drank himself to death in 1820, leaving the family penniless.
Niels Henrik, now the oldest responsible member of the family, took
steps which were to save his sister Elisabeth and younger brother
Peder. He found another home for Elisabeth and took Peder with him
when he entered the University of Oslo in 1821.
> Before long, Abel had read most of the advanced mathematical
works in the university library, and his own research began in earnest.
By 1823 he had discovered the inversion that was the key to elliptic
functions, proved the unsolvability of the quintic, and discovered a
wonderful general theorem on integration, now known as Abel's
theorem, which implicitly introduces the concept of genus...
> By the end of 1826 Abel was running out of money and eating only
one meal a day. He feared he was losing touch with Crelly [his fiance],
as she had returned to Copenhagen and her letters were infrequent.
He left Paris for Berlin on December 29, while he still had money to
pay for the journey, and found a letter from Crelly waiting. Some good
news at last! Crelly stood by him as ever, and their plans for the future
were revived. ... Unfortunately, the university was still unwilling to
give him more than a temporary appointment, which paid barely
enough to meet his family's debts. ...
> In May 1828 Abel finally received a decent job offer from Berlin,
only to have it withdrawn two months later... Then a group of French
mathematicians petitioned the king of Norway - Sweden to use his
influence on Abel's behalf, but still the University of Oslo remained
unmoved. By now, time was running out. Abel's health worsened
and in January 1829 he began spitting blood. ... Abel died on April 6,
1829, just two days before the arrival of a letter informing him of his
appointment as professor in Berlin.
Thus he died in his 27th year. Hermite said "Abel left mathematicians
with enough to keep them busy for 500 years." That's genius.
They say women are discouraged from studying mathematics. Well,
I'm sorry. No doubt, the poor dears are discouraged. When I see them
eating one meal a day and spitting blood, then I'll know they are