216 A History ofMathematics
In Dedekind’s definition, one takes√
2 to be the cut. This might be considered slightly vague too,
and later writers who subscribed to Cantor’s set theory (such as Russell) defined it to be the lower
set. Which you do is a detail. The two important points are:- Once the definition has been made, it is easy to do arithmetic (adding, multiplying, even taking
roots etc.) with such numbers. - The further operation of taking limits (e.g. of an increasing sequencex 1 ,x 2 ,...which is
bounded above) isequallyeasy. - On the other hand, however you look at it, you have ‘defined’ a number to be something which
isnota number. For thousands of years, mathematics has been about numbers and geometrical
figures. It now, suddenly, is about something else. Has it then changed?
Underlying all this were ideas which were to come much later: that the objects of mathematics
were not actual things-in-themselves (as one thinks of a triangle, say, or the number ‘7’), but the
rules which they obeyed. Any way of constructing objects which obeyed the rules endowed them
with existence, and two different ways of construction, if the results obeyed the same rules, could
be thought of as the same—we would now in suitable circumstances use the word ‘isomorphic’. We
have already seen this in Chapter 8, where the non-Euclidean plane was constructed as a surface
with new rules about what constituted ‘straight line’ and ‘angle’.
Even today, such metaphysics are considered beyond the scope of the high-school student or
(often) the first-year college student. At the time they were new, just beginning to be explored, and
only a strong feeling—backed up by examples—that intuitive ideas of number were not reliable
enough drove the process forwards. Nothing makes clearer the fundamental change underlying
the new outlook than the fact that it seemed immediately necessary to go back further and set the
natural numbers{0, 1, 2, 3,...}on a secure foundation, although they had previously troubled no
one. Gottlob Frege was in 1884 to define these as sets too. ‘3’, for example, meant the set of all sets
which could be put in a 1-1 correspondence with a (previously defined) set, sayS 3 , which had three
elements—so that ‘3’ meant ‘the set of all sets with three elements’. [It is Frege’s way of defining
S 3 which stops the definition from being circular.]
Bit by bit, among these mathematicians—mainly German, but to include Peano (Italian), Russell
(British),...—more and more things which had seemed obvious were to need proof; when part of
the edifice seemed sound, one started to worry about its underpinnings, so that by the 1920s we
find Hilbert, probably the ablest mathematician of the time, taking time out to show how one could
prove that 1+ 2 = 2 +1.^2 The drive for sound foundations was a strong one, and on the whole
fruitful; it is interesting that it is an episode which can be considered closed, in that mathematicians
have returned to a naive condition of assuming that what they do works (although the procedures
of physicists may still worry them). The process of investigation, however, brought the worlds of
mathematics and philosophy into a much closer relationship.
The relationship was by no means a new one; almost all philosophers since Plato had reflected on
mathematics, and many mathematicians (Descartes, Pascal, Leibniz, Bolzano) were philosophers as
well. But the dependence on set theory and logic introduced a new outlook into both mathematics- Hilbert, ‘The New Grounding of Mathematics’, (1922) in Mancosu (1998, p. 207). The implication of triviality is of course
unfair; Hilbert was showing how a formal minimal axiom system for arithmetic could be used to establish all necessary results. All
the same, the image is a striking one.