Modernity and itsAnxieties 223
the ruins:
Sitting next to the Nazis’ newly appointed minister of education at a banquet, he was asked, ‘And how is mathematics
in Göttingen now that it has been freed from the Jewish influence?’
‘Mathematics in Göttingen?’ Hilbert replied. ‘There is really none any more’. (Reid 1970, p. 205)
6. Topology
In an attempt to show some of what went on outside the world of foundational disputes, we consider
the rise of topology. This, it has to be said without any personalparti prisisthesuccess story of
twentieth-century mathematics, barely existing at the beginning of the century and intruding into
all other fields by the end. While there are obviously multiple reasons for this, we could give two:
first, that any problem which requires the passage from a simple local statement to a more difficult
global one (what can electromagnetic fields be like in the presence of currents? what shape can the
space-time of Einstein’s relativity have?) is a topological question; and second, that the machinery
was in place, or could be developed to solve such problems. A great many problems in topology
are hard, but not as hard as the continuum hypothesis, or Langlands’s conjecture on automorphic
forms; and a great many of the ablest mathematicians have devoted their time to them. It can
therefore be seen as a domain of Hilbertian optimism, in which questions are successively raised
and dealt with.
The time has now arrived when topologists consider their subject, however young, has a his-
tory; rightly so, and luckily there are two substantial contenders, Dieudonné (1989) and James
(1999). These provide the groundwork which more professional historians are now sifting over and
commenting. It is normal to consider the subject, properly considered, as just over a hundred years
old. Aside from two famous excursions by Euler, it was originated and given its name by Möbius and
Listing, classifying surfaces in the mid-nineteenth century; but serious methods of study became
available at the end of the century with Poincaré, who extended the field to higher dimensions and
gave it its first major theorem (‘Poincaré duality’) and its most enduring problem (the ‘Poincaré
conjecture’).
At this point, the methods of argument were not such as would have been recognized elsewhere,
for example, in algebra; and topology was perhaps fortunate in having as its originator Poincaré,
who was more interested in finding results than in defining exactly what he was talking about. His
object of study was:
(1) what Riemann would have called (and we now call) ‘manifolds’—curves, surfaces,...up to
any number of dimensions (think of the unit sphere inRn+^1 )^7
(2) under the relation of ‘homeomorphism’ (same shape); continuity is preserved, but not
distance.
To give standard examples: (a) manifolds of different dimensions are not homeomorphic (e.g. the
circle C and the torus T in Fig. 3); (b) nor are the sphere S and T; (c) but T and the knotted torus
T′are (Fig. 4). Poincaré’s method was to decompose a manifold into ‘cells’,rather like the faces of
a polyhedron; and to derive numbers from the cell decomposition.
- This field of acceptable objects was later to be substantially extended; no longer manifolds, no longer finite dimensional,...