§ 1. Classical Differential Calculus
145
cesses, operational research, turbulent flows or the mathematics of quantum
mechanics, all of which are subjects that we are told today , half a century later, suppose a new conception of mathematics, at the opposite end
to ours. Incidentally, reading Dautray-Lions’s volumes or some recent talks
at the Bourbaki Seminar is far from confirming this point of view. Around
1948 together with Dieudonn´ e and Schwartz, I attended some superb lectures
by Jean Delsarte, a founding member of Bourbaki, on analytic number theory, following Hardy-Littlewood-Rademacher-Winogradov’s version ; though
at the opposite of the Bourbaki spirit, the subject did not exactly arouse
reactions of scorn from the audience. Dieudonn´ e’s superb article on analytic
number theory in the Encyclopaedia Universalis or, in a neighbouring field,
my talks at the Bourbaki Seminar (1952–1953) on Hecke’s work (zeta functions of number fields, modular functions), the first of their kind in France in
a field that had not yet been modernized, attest to this. One of the members
of the group during this period, Charles Pisot, was a transcendental number
specialist, a subject not very much in the Bourbaki line for the time.
As for the other sciences, we had far less prejudices against geology than
shown nowadays by M. Claude All` egre, a great specialist of the subject and a
recent education minister, for our version of mathematics: we merely ignored
him for obvious reasons. Generally speaking, we did not consider it our duty
to provide experimentalists with mathematics in its traditional form, which
they had most often learnt in their youth and were determined to preserve. In
fact, their physics has become in many aspects as abstract as our mathematics
and it is hard to see why we should have had to grant them the exclusivity
of “ modernism ”
The conclusion that follows from all this seems to me to be that it is above
all to Henri Cartan and to the enthusiasm of the members of the Bourbaki
group for “ modern ” mathematics that the French mathematical school owes
its post-war recovery of the position it had lost since Poincar´ e, Picard and
Lebesgue.
Another rarely acknowledged factor needs to be mentioned. The weakness and isolation of the French school before 1939 is often explained by
referring to the Great War, its massacres and to the hostility towards Germany that continued long after 1918 in particular because of ´
Emile Picard.
On the contrary, its renewal after 1945 is perhaps also due to the fact that,
during the war, the French – including Leray in his Oflag, including people
like Schwartz, Samuel and the young Grothendieck threatened by antisemite
policies, including Weil and Chevalley in the USA or in Brazil – had nothing
else to do apart from “ real ” mathematics at a time when their German,
English, American, Russian, etc. contemporaries were engaged in doing work
for war purposes of a far lower quality than they were capable of, like Polish
Jews, were ending their lives in Nazi concentration camps.
After 1945, mainly in the United States, but also in Japan, Germany,
the USSR where an excellent school of functional analysis had flourished for
a long time, many mathematicians who, at first, were unaware of even the
145
cesses, operational research, turbulent flows or the mathematics of quantum
mechanics, all of which are subjects that we are told today , half a century later, suppose a new conception of mathematics, at the opposite end
to ours. Incidentally, reading Dautray-Lions’s volumes or some recent talks
at the Bourbaki Seminar is far from confirming this point of view. Around
1948 together with Dieudonn´ e and Schwartz, I attended some superb lectures
by Jean Delsarte, a founding member of Bourbaki, on analytic number theory, following Hardy-Littlewood-Rademacher-Winogradov’s version ; though
at the opposite of the Bourbaki spirit, the subject did not exactly arouse
reactions of scorn from the audience. Dieudonn´ e’s superb article on analytic
number theory in the Encyclopaedia Universalis or, in a neighbouring field,
my talks at the Bourbaki Seminar (1952–1953) on Hecke’s work (zeta functions of number fields, modular functions), the first of their kind in France in
a field that had not yet been modernized, attest to this. One of the members
of the group during this period, Charles Pisot, was a transcendental number
specialist, a subject not very much in the Bourbaki line for the time.
As for the other sciences, we had far less prejudices against geology than
shown nowadays by M. Claude All` egre, a great specialist of the subject and a
recent education minister, for our version of mathematics: we merely ignored
him for obvious reasons. Generally speaking, we did not consider it our duty
to provide experimentalists with mathematics in its traditional form, which
they had most often learnt in their youth and were determined to preserve. In
fact, their physics has become in many aspects as abstract as our mathematics
and it is hard to see why we should have had to grant them the exclusivity
of “ modernism ”
The conclusion that follows from all this seems to me to be that it is above
all to Henri Cartan and to the enthusiasm of the members of the Bourbaki
group for “ modern ” mathematics that the French mathematical school owes
its post-war recovery of the position it had lost since Poincar´ e, Picard and
Lebesgue.
Another rarely acknowledged factor needs to be mentioned. The weakness and isolation of the French school before 1939 is often explained by
referring to the Great War, its massacres and to the hostility towards Germany that continued long after 1918 in particular because of ´
Emile Picard.
On the contrary, its renewal after 1945 is perhaps also due to the fact that,
during the war, the French – including Leray in his Oflag, including people
like Schwartz, Samuel and the young Grothendieck threatened by antisemite
policies, including Weil and Chevalley in the USA or in Brazil – had nothing
else to do apart from “ real ” mathematics at a time when their German,
English, American, Russian, etc. contemporaries were engaged in doing work
for war purposes of a far lower quality than they were capable of, like Polish
Jews, were ending their lives in Nazi concentration camps.
After 1945, mainly in the United States, but also in Japan, Germany,
the USSR where an excellent school of functional analysis had flourished for
a long time, many mathematicians who, at first, were unaware of even the