There are good reasons for this. Young mathematicians, especially at
prestigious universities, are well aware of the adage “publish or perish.”
The final tenure decision on assistant professors generally occurs no later
than six years after the initial hiring, and no matter how good a teacher
you are, at a top-ranked university, you’d better have something to show,
publication wise, for those six years—or you’re going to be looking for
another job. As a result, the pressure to publish—even prematurely—is
enormous. Additionally, even for the tenured, getting something out
there is important because (1) it helps to make a contribution, and (2) by
doing so, you may supply the critical piece of the puzzle that can turn an
unproven result, or Someone Else’s theorem, into Yours and Someone
Else’s theorem. I know, because I read a paper by the esteemed Czecho-
slovokian mathematician Vlastimil Pták, had one of the few really good
ideas I’ve had,^6 and wrote up a short paper that appeared in the Proceed-
ings of the American Mathematical Society. The best result in this paper
was to become known as the Pták-Stein theorem (as far as I know, the
only thing that’s named after me)—and nine months later a paper ap-
peared elsewhere with the exact same result. As Tom Lehrer put it in his
hilarious song “Nikolai Ivanovich Lobachevsky,”
And then I write
By morning, night,
And afternoon,
And pretty soon
My name in Dnepropetrovsk is cursed,
When he finds out I published first!^7 Gauss had a long history of investigation of alternative geometries. At
age fifteen, he told his friend Heinrich Christian Schumacher that he
could develop logically consistent geometries besides the usual Euclidean
geometry. Initially, he set out along the road of trying to deduce the paral-
lel postulate from the other four, but eventually reached the same conclu-
sion he had at fifteen, that there were other consistent geometries. In
1 824, he wrote to Franz Taurinus, in part to correct an error in Taurinus’s
purported proof of the parallel postulate. After doing so, Gauss wrote,
“The assumption that the sum of the three angles of a triangle is less
than 180 leads to a curious geometry, quite different from [the Euclid-
ean], but thoroughly consistent, which I have developed to my satisfac-
tion.... The theorems of this geometry appear to be paradoxical and,
to the uninitiated, absurd; but calm, steady ref lection reveals that they
contain nothing impossible.”^8 It seems fairly clear that Gauss had not
Never the Twain Shall Meet 107