228 A History ofMathematics
124 3Fig. 9Two knots, each with four crossings. The crossings on knot 1 are labelled 1, 2, 3, 4 (and obviously those on knot 2 could be
labelled similarly).7 Outsiders
I have not trodden through a conventional university course, but I am striking out a new path for myself. I have made
a special investigation of divergent series in general and the results I get are termed by the local mathematicians as
‘startling’. (Ramanujan to Hardy, January 1913)
If one proves the equality of two numbersaandbby showing first thata≤band then thata≥bit is unfair; one
should instead show that they are really equal by disclosing the inner ground of their equality. (Noether, quoted in
Weyl 1935)The community of mathematicians in 1900 was restricted almost entirely to white males, as one
might expect. Today, the restriction is less complete, although one would not want to make any very
strong claims about the progress which has been achieved. At the beginning of the century stand
two very different figures, Srinivasa Ramanujan and Emmy Noether, whose stories are often told
as exemplary, and who certainly show the different ways in which unusual individuals could break
into the closed world of mathematicians; what they could achieve, and what were the necessary
limits of that achievement. They were both quite exceptional—and would be today—for completely
different reasons.
When Ramanujan came from Madras to study with the already famous number theorist
G. H. Hardy in Cambridge in 1914, he came from a situation where mathematical research was
not far advanced, and Indians had little chance of making headway in it. (Despite this, he had
had articles published in theJournal of the Indian Mathematical Society.) In the well-known story, he
was forced to leave his wife, to make an unwelcome adjustment in Cambridge to a hostile climate
and uneatable food, and finally to ruin his health, in order to study with the only mathematician
who had taken the trouble to respond to his extraordinary letters. Rather than looking back on the
outsider status which he had to endure as something belonging to a distant past, one might wonder
how likely it is today that a clerk without a university degree, writing to a professor at Princeton
(say) in such terms would be fortunate enough to get a similar response.
It is a cliché in writing about Ramanujan to describe the difficulty of assessing what he con-
tributed to modern mathematics—quite aside from the difficulty of the mathematics themselves.