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I was born in Warsaw but grew up in Sosnowiec, an industrial city
then in south-western Poland. A prewar European high school gave at
least some of the students a taste of matters intellectual. The
language of instruction in my Gymnasium was Polish, but I also
learned Hebrew, Latin and German. The literary component of Hebrew
was significant from the beginning--entering students were supposed
to know the basics. After a year and a half of Latin we read Caesar,
then Cicero, then Ovid and Vergil. It took somewhat
longer to get to reading substantial German materials, and as the
relations between Poland and Germany deteriorated English began to
look like an attractive alternative to German. I began its study on
my own. In those days I liked mathematics but I didn't get much of an
idea of its significance. I did get some vague notion of the
axiomatic basis of geometry.
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In 1947 I took the entrance examination to Brooklyn College. This
was a five-hour written exam, whose cutoff score was 80%; my score
was 83. What saved me was my ability to solve problems such as: "If
3/4 of a pound of bread costs 23 1/2 cents, how many pounds of bread
can you buy for $6.25?" My score on the English part of the exam was
10%, because I largely ignored the part that asked me to correct the
spelling of 50 words. At a midyear conference my advisor was far from
pleased with that 10%--and was positively speechless when I told him
that my midyear grade in English was A.
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I found physics, my first major, conceptually profound but also
extremely difficult, and I reluctantly but wisely switched to
mathematics. There I did well in the sense that I mastered the
techniques of the calculus and of differential equations, but I could
not defend these subjects on intellectual or utilitarian grounds. I
could have learned some of the intellectual rudiments of mathematics
in a one-term course on the "nature" of the subject, but this course
was widely viewed as a soft option beneath the dignity of serious
students.
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In 1950 I was admitted to the graduate school at NYU. I was
interviewed by the eminent mathematician Richard Courant--but
luckily I didn't know who he was. He looked at my file and said,
"It states here that you translate books on
mathematics. How well can you do this?" I groped for an answer and
finally blurted out, "As well as any student in the United States".
"That", said Courant, "is not good enough".
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Then I talked to a pleasant man who said his name was Fritz
John--another eminent mathematician. We
talked about differential geometry, and I told him that I found the
language of the older work in the subject vague. He laughed. "You can
learn differential geometry without rigor, or you can learn rigor without
differential geometry". What impressed me about both
of these people was that they treated me without any trace of
condescension.
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Of the many remarkable mathematicians I met and befriended at NYU I must
single out two: Lipman Bers and Wilhelm Magnus. Magnus was both a
distinguished scientist and an intellectual par excellence.
Conversations with him reinforced my tendency, always latent, to look
by preference at intellectual aspects of mathematics as opposed to
the purely technical. A kind of down-to-earth description of my
position appears in a note that prefaced the first instalment of "The
Evolution of...", which I have edited for the "American Mathematical
Monthly" since January 1994:
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"An English major may or may not be a novelist or a poet, but would
undoubtedly be expected to evaluate a novel or a poem. The term
'English major' implies some historical, philosophical and evaluative
training and competence. Integration of the narrowly mathematical
and historical, philosophical aspects of our discipline is bound to
to make it more meaningful not only to those who identify themselves
as mathematicians but also to those who have no more than a
tangential interest in the subject."
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