9. Modernity and its anxieties
1. Introduction
If in summing up a brief phrase is called for that characterizes the life center of mathematics, one might well say:
mathematics isthe science of the infinite. (Weyl 1949, p. 66)
Pure mathematics is the class of all propositions of the form ‘p implies q’, where p and q are propositions each
containing at least one or more variables, the same in the two propositions, and neither p nor q contains any constants
except logical constants. (Russell 1903, p. 3)The ‘long twentieth century’, which should end our narrative, has seen more mathematics, as well
as more changes to what mathematics means to those who do it or those who use it than the whole
of preceding history. Those who had tried to define what mathematics was in its long past had
certainly not come up with answers as extremist, or as ‘unmathematical’ in appearance as either
Hermann Weyl or Bertrand Russell; and yet both answers now seem to belong to a bygone era. Even
Alan Turing’s paper on computable numbers, which more than ever stands as a founding document
for ‘where we are now’ is hard for the modern reader to construe; not only because Turing was
writing in a difficult field, but because the problems he was addressing belong to a time which,
a mere 70 years later, has long disappeared. As a minimal strategy in managing the material, we
have had to divide it in two, taking Gödel’s 1931 paper as a useful cut-off point. This chapter, then,
will deal with the central concerns which led up to the crisis of the years from 1900 to 1930; who
was affected and how they dealt with it; and how, in some sense, it ended. At the same time, it
seems essential to remember that the crisis was the concern only of a few mathematicians, although
those were among the most important ones. Hence, in the interests of balance—and also because
foundational questions provide painfully few opportunities for pictures—we shall consider parallel
developments in algebra and topology, particularly knot theory. These are also part of the story, in
that if there is a ‘twentieth-century outlook’ characterized by increasing abstraction and formalism,
it can be seen spreading even to such apparently down-to-earth subjects as the classification of
knots. Naturally, a very large part of the field has still been omitted, most particularly all that has
to do with physics. There may be some compensation in the next chapter, but the reader must
remember the arbitrariness of our selection.
As we shall see, the natural beginning of the story precedes the twentieth century by some
30 years. The world of mathematicians by that time was substantially professionalized around
great institutions of teaching and research in Germany and France and lesser ones in many other
countries. While this state of affairs remained constant, it should be borne in mind at each stage
(a) that the number of people so employed was tiny in comparison with today and (b) that it was
more or less constantly growing in response to the demands of society—not for mathematicians
(who needs them?) but for engineers, accountants, statisticians, and the like. That this growing
community chose to concern themselves chiefly with the definition of the numbers, or with how to