§ 1. Classical Differential Calculus
141
for all systems of curvilinear coordinates was involved. This only implied
compliance to Einstein’s conventions. The “ curvature ” and “ geodesics ” of
the space were calculated and finally, the tensor notation presented above
was systematically used and led almost automatic calculations. In barely a
month, I became an expert of absolute differential calculus without understanding anything. I gained nothing at the time, excepting gymnastics, but
it was not worse than spending two hours per day in front of the television,
Flight Simulator or Tomb Raider being as yet unknown.
There was also another exception in this collection: a far more modern
and difficult treatise on La g´ eom´ etrie des espaces de Riemann, by ´
Elie Cartan. It contained little calculation and indices, but mainly abstract and somewhat vague ides, for example the distinction between “ closed ” (i.e. compact)
spaces and “ open ” (i.e. non-compact) ones. Formulations invented after 1945
thanks to the clarification of the notion of a topological space, to the definitive
crystallization of the theory of abstract differential manifolds,
4 in particular
by Claude Chevalley,
5 were unknown at the time, and to the invention of fiber
spaces by algebraic topologists partly inspired by ´
Elie Cartan who, during the
war, invented a broad generalization of tensors connected to group theory. In
the 1950s, the theory acquire the perfect and abstract form that can be found
in all presentations of the subject, starting with N. Bourbaki’s Fascicule de
r´ esultats. Naturally all this only concerns its most basic aspects and did not
prevent its development in often unexpected directions too hard to present
in the Bourbaki style – a maximum of abstractions and generalities.
In the special issue on Bourbaki in the magazine Pour la Science, the
French version, but not a translation of Scientific American, Benoˆ ıt Mandelbrot is alleged to have said, p. 82, that he left the ´
Ecole Normale Sup´ erieure
for the ´
Ecole Polytechnique because thanks to my uncle,
6 I knew they were
a militant gang, that they were strongly prejudiced against geometry and science, and that they tended to despise and even humiliate those who did not
follow them; and the author of this declaration apparently left France for the
United States (and the IBM) in 1958 because of their stifling influence.
I am in a good position to appreciate the influence of Bourbaki on the
´
Ecole normale in 1944. Between 1940 and 1953, the one and only member of
4 i.e. that are no longer considered subsets of Cartesian spaces, as used to be the
case. The most popular example is the general relativity space; most people find
it hard to understand precisely because of this and the more so when it appeared
that it could be “ closed ” or bounded, i.e. compact. Mystics, a species that is
not endangered, wondered what there could be outside: dread is the feeling of
nothingness (Heidegger).
5 Theory of Lie Groups, Princeton UP, 1946.
6 Szolem Mandelbrojt, professor at the Coll` ege de France and a specialist of quasianalytic functions of one variable, a difficult subject that did not acquire the
importance of the great fields developed after the war; see chapter 19 of Rudin’s
book. Szolem Mandelbrojt belonged to the initial Bourbaki group, but quickly
left due to this very different conception of mathematics. This proves nothing
against Bourbaki nor against Mandelbrojt. Everyone is free.
141
for all systems of curvilinear coordinates was involved. This only implied
compliance to Einstein’s conventions. The “ curvature ” and “ geodesics ” of
the space were calculated and finally, the tensor notation presented above
was systematically used and led almost automatic calculations. In barely a
month, I became an expert of absolute differential calculus without understanding anything. I gained nothing at the time, excepting gymnastics, but
it was not worse than spending two hours per day in front of the television,
Flight Simulator or Tomb Raider being as yet unknown.
There was also another exception in this collection: a far more modern
and difficult treatise on La g´ eom´ etrie des espaces de Riemann, by ´
Elie Cartan. It contained little calculation and indices, but mainly abstract and somewhat vague ides, for example the distinction between “ closed ” (i.e. compact)
spaces and “ open ” (i.e. non-compact) ones. Formulations invented after 1945
thanks to the clarification of the notion of a topological space, to the definitive
crystallization of the theory of abstract differential manifolds,
4 in particular
by Claude Chevalley,
5 were unknown at the time, and to the invention of fiber
spaces by algebraic topologists partly inspired by ´
Elie Cartan who, during the
war, invented a broad generalization of tensors connected to group theory. In
the 1950s, the theory acquire the perfect and abstract form that can be found
in all presentations of the subject, starting with N. Bourbaki’s Fascicule de
r´ esultats. Naturally all this only concerns its most basic aspects and did not
prevent its development in often unexpected directions too hard to present
in the Bourbaki style – a maximum of abstractions and generalities.
In the special issue on Bourbaki in the magazine Pour la Science, the
French version, but not a translation of Scientific American, Benoˆ ıt Mandelbrot is alleged to have said, p. 82, that he left the ´
Ecole Normale Sup´ erieure
for the ´
Ecole Polytechnique because thanks to my uncle,
6 I knew they were
a militant gang, that they were strongly prejudiced against geometry and science, and that they tended to despise and even humiliate those who did not
follow them; and the author of this declaration apparently left France for the
United States (and the IBM) in 1958 because of their stifling influence.
I am in a good position to appreciate the influence of Bourbaki on the
´
Ecole normale in 1944. Between 1940 and 1953, the one and only member of
4 i.e. that are no longer considered subsets of Cartesian spaces, as used to be the
case. The most popular example is the general relativity space; most people find
it hard to understand precisely because of this and the more so when it appeared
that it could be “ closed ” or bounded, i.e. compact. Mystics, a species that is
not endangered, wondered what there could be outside: dread is the feeling of
nothingness (Heidegger).
5 Theory of Lie Groups, Princeton UP, 1946.
6 Szolem Mandelbrojt, professor at the Coll` ege de France and a specialist of quasianalytic functions of one variable, a difficult subject that did not acquire the
importance of the great fields developed after the war; see chapter 19 of Rudin’s
book. Szolem Mandelbrojt belonged to the initial Bourbaki group, but quickly
left due to this very different conception of mathematics. This proves nothing
against Bourbaki nor against Mandelbrojt. Everyone is free.