214 A History ofMathematics
tell two knots apart, was their business; one can speculate on how this related to what was going on
in the economic sphere, and we shall try to raise some questions. However, their main function as
employees of the state (which they usually were) was to teach, and to uphold the prestige of their
institutions.2. Literature
The mathematics of the early twentieth century has been patchily studied. Because the crisis of
foundations provides such rich material for the historian, it is easily the best covered, with very
full sourcebooks edited by van Heijenoort^1 (1967) and Mancosu (1998), as well as chapters in
Grattan-Guinness (1980). Corry’s recent work on the origins of abstraction (2004), if rather dry,
is useful on ground which in the main will not be covered here. Added to these, and to a growing
volume of articles in journals, the period has naturally been an attraction to biographers who
have often had access to their subjects or to close friends; one could cite Cantor (Dauben 1990),
Russell (Russell 1967; Monk 1997, 2001), Hilbert (Reid 1970), Brouwer (van Dalen 1999), Weyl
(Wells 1988), Ramanujan (Kanigel 1991), Noether (Dick 1981), and so on. They are not properly
‘histories’, but they can be excellent sources. All this, as well as the texts themselves—which,
it must be said, are usually extremely difficult as should be expected of mathematics today—are
useful material. Two interesting early twentieth-century works of fiction have ‘mathematicians’ as
their heroes—Musil’s (1953) and Ford Madox Ford’s (2002); however, the fact is fairly marginal
to the lengthy unfolding of the two novels.
Because of their difficulty, the remarks made in Chapter 8 apply even more here. Almost no
twentieth-century mathematicaldiscoveriesfind their way into an undergraduate course, although
any course on linear algebra, or group theory, or analysis will be taught in a way that was only
settled around 1950. We shall face constant problems of presentation, and must hope for the
reader’s patience. Modern mathematics does not easily lend itself to being democratized; Hilbert
(1900), introducing his very difficult list of problems for the next century, attributed to an unnamed
‘old French mathematician’ the saying: ‘A mathematical theory is not to be considered complete
until you have made it so clear that you can explain it to the first man whom you meet on the
street’, but little of Hilbert’s own work, or of what has been done since, stands up to the test.
3. New objects in mathematics
Es steht schon bei Dedekind [That’s already in Dedekind]. (Emmy Noether (frequently), quoted in Dick 1981, p. 68)
As a clutch of Victorian professors, avuncular, ascetic and a little disheveled, they [Dedekind and Cantor] were
gathering unawares around the cradle of an infant Briar Rose that would one day be christened Modernism. (Everdell
1997, p. 31)
To arrive at real proofs of theorems (as e.g.
√
2√
3 =√
6), which to the best of my knowledge have never been
established before. (Dedekind 1948, p. 22)- It is usual to point out that Jean van Heijenoort was Trotsky’s secretary during the 1930s, only later becoming a distinguished
historian of mathematical logic. And it is indeed an interesting footnote.