[FOM] Pure mathematics and humanity's collective curiosity

[Andre.Rodin@ens.fr]

Robert Black:

>  Actually, I'm a bit sceptical about Gauss and non-euclidean
> geometry in general. Nineteenth-century German historians of
> mathematics were just unable to accept that something so important
> could have come from a Hungarian or even worse a Slav and not from
> THE GREAT GERMAN MATHEMATICIAN, and were rather too willing to
> believe Gauss's claim (made *after* he had seen Bolyai's work) that
> he had come to the result years before and just not published it
> through fear of the 'Boetians'. But perhaps there's a proper
> historian of mathematics on this list who can refute my scepticism.
>
I'm inclined to take Gauss' claim seriously (even being myself a Slav) and not
reduce it to sociological reason, moreover one you mention. True, talking about
"Boetians" Gauss suggests such an interpretation himself. But a deeper reason,
in my view, could be that Gauss aimed at what latter was achieved by Riemann -
I mean, of course, his notion of manifold - rather than at Bolyai's or
Lobachevsky's kind of results belonging to the "theory of parallels". Gauss
might believe that the whole issue about parallels was not so central for the
new conception of space as it appeared in Bolyai's and Lobachevsky's works -
the point latter stressed by H. Weyl.  Riemannian manifold unlike Lobachevsky's
space was a fairly new general conception of space which still serves us as the
best generally accepted mathematical description of the physical space-time.
(The switch from space to space-time is, of course, essential but it doesn't
effect the argument as far as I can see.). One can hardly get the notion of
manifold out of Bolyai's work but one can get it out of Gauss's work as did
Riemann. So Gauss might have good reason indeed to see Boliay's results as non-
satisfactory. Perhaps  I read too much into Gauss here but in any event I
cannot see any reason why one shouldn't take Gauss' claim on its face value.

andrei