218 A History ofMathematics
work needed to be done on the axioms. There were some disagreements about what should be done,
but Cantor’s set theory underpinning Frege and Dedekind’s arithmetic seemed to provide a good
programme.
Nonetheless, as Gray points out in a recent article (2004), there were signs of what could be called
‘anxiety’ about foundations. They could be traced back much earlier; in 1810 Bolzano had said:However the greatest experts of this science [i.e. mathematics] have in fact always answered, not only, that the edifice
of their science is still no completely finished and self contained building; but also, that the foundation wall of this
otherwise splendid building is itself not yet completely firm and regular; or, to speak without pictures, that even in the
elementary lessons of all mathematical disciplines many gaps and imperfections are to be found. (Bolzano, (1810))However, it was only at the end of the century that a general optimism about the power of mathem-
atics began to give way to doubt—and indeed it would be hard otherwise to account for the amount
of work that was by then under way in an attempt to shore up the building. Any familiarity with
critical work on the culture of the period will show that anxiety and modernism go together like
a horse and carriage. How far mathematics caught a general infection, and how far it contributed
to it are questions yet to be settled.
The crisis came naturally: it appeared that set theory as (following Cantor) it had been freely
used (a) was inconsistent and (b) demanded extra articles of belief which were hard to accept.
Inconsistency followed from the so-called ‘Russell paradox’ of 1903.^3 The problem—if you have
not seen it before—is the following. Dedekind and Cantor had introduced sets to deal with numbers.
However, if real numbers ‘were’ sets, one also wanted to deal with sets of sets, and so on (perhaps)
indefinitely. In Cantor’s general set theory, which he imagined worked, given any propertyPone
had a set of all things with propertyP(the ‘axiom of comprehension’).
The problem was that, once the language became this general, the subject became the province
of philosophers, who may choose to replace numbers by philosophical objects such as the golden
mountain or black swans. The way was wide open for Russell, standing between mathematics and
philosophy, to devise the simple example of the property:P: x is a set andxis not an element of itself(x∈/x).For example, the set of all sets is an element of itself (this already shows signs of an infinite regress).
So is the set of all things which are non-human; or the set of all things which can be described
in English sentences of less than 18 words. On the other hand, most sets (numbers, black swans,
students in the classroom) are not elements of themselves. IfSis the set determined by the property
P, then one can derive a contradiction both fromS∈Sand fromS∈/S. Our first quote shows
Russell himself, disturbed at the paradox, feeling both that grown men should not worry about
such things (but then who should?) and yet that they had to be settled.
Various attempts were made to be more restrictive about how sets were used; they had become
too much a part of how the leading mathematicians thought to be given up altogether. Zermelo
around the same time tried to produce a system of axioms for set theory which would both avoid
paradoxes and do what mathematicians needed. He came up with what was in a way as shocking
as Russell’s paradox: the ‘Axiom of Choice’ (1904). This could be thought of as a modern analogue- As so often, there is a priority question here; Zermelo had already described the paradox in a letter to Husserl.