206 A History ofMathematics
I discovered [about 1890] that in addition to Euclidean geometry there were various non-Euclidean geometries, and
that no one knew which was right. If mathematics was doubtful, how much more doubtful ethics must be! (Russell,
cited in Richards 1988, pp. 204–5)The term ‘Whig history’, which is only used by historians to denigrate the work of other historians,
describes a way of constructing the narrative so that the process of history is seen as one of
development towards the present; which itself is seen as a good state of affairs, if not the best.
Mathematicians, who find it difficult enough to imagine that mathematics could be done in any
way which is better than our present one, are particularly given to Whig history, and the quotation
above (non-Euclidean geometries were good because they led to the construction of necessary
axiom-systems) is a mild example. In its defence, it does at least provide (a) a structure for the
bare record of events, and (b) a starting-off point for more sophisticated narratives, which can
criticize it. In this section, we shall examine how far the story of the discovery and assimilation of
non-Euclidean geometry, outlined above, can be made sense of as part of a wider development of
geometry.
The natural endpoint of that development, as implied by Greenberg, is not the mere acceptance
of non-Euclidean geometry, but the modernization of the subject as a whole. The problem is that
the latter—the development of axiom-systems, and the increasing insistence that geometry was
not about ‘space’ at all, but aboutanyentities which might satisfy the axioms—occurred later still,
during the years from 1890 to 1910. Peano (1889), was the first to produce an axiom-system
which he declared to be ‘free-standing’, that is, independent of any possible meanings which one
might give to terms like ‘point’, ‘line’, and so on.
We are given thus a category of objects called points. These objects are not defined. We consider a relation between
three given points. This relation, notedc∈ab, is likewise undefined. The reader may understand by the sign 1 any
category of objects whatsoever, and byc∈abany relation between three objects of that category[...] The axioms
will be satisfied or not, depending on the meaning assigned to the undefined signs 1 andc∈ab. If a particular
group of axioms is satisfied, all propositions deduced from them will be as well. (Peano 1889, quoted in Torelli
1978, p. 219)More picturesquely, Hilbert, whose axiom system became the most influential put it:If I conceive my points as any system of things, e.g. the systemlove, law, chimney-sweep,...and I just assume all my
axioms as relations between these things, my theorems, e.g. the theorem of Pythagoras, will also hold of these things.
(Hilbert, cited in Torelli 1978, p. 251)Thiswasa radical change in how geometry was thought about. That the views of Hilbert and his
followers were not generally accepted—and are not universally believed even today among research
mathematicians—is less important than that they were voiced, and carried weight. They outlined
a programme for a new view of geometry which (since Hilbert was not obscure, indeed he was the
most respected of mathematicians) had to be taken seriously.
Non-Euclidean geometry was ‘revolutionary’ in its successful construction of a geometry which
was not Euclidean, so much is obvious. How much in this subsequent development of geometry
can be attributed to it is an altogether more problematical question. From a broader point of
view, the axiomatization of geometry, while it clearly owed something to the problems which
had arisen, should be seen as a part of the wider tendency to axiomatization in mathematics
during the late nineteenth century. Peano, indeed, is more often remembered for his axioms
for the natural numbers than for his geometric ones, and Hilbert and Russell similarly had