cians as formal rules become treated as significant in their own right. In the
German universities, independence from applied work broke the link with
physical interpretations and eliminated a source of justification for intuitive
concepts. Rigor was not a sudden realization of old faults but a social shift in
relationships within the mathematical community.
In the previous century the lack of rigor had been criticized mainly from
outside the ranks of mathematicians, by enemies of the scientific worldview
such as Berkeley. Mathematicians could see the problem but did not consider
it important.^3 From their point of view they were right. The rigor of the
following generations did not invalidate any of the theorems of previous
arithmetic, algebra, or Euclidean geometry but only put them on a new basis;
even in analysis the old glaring expressions were remedied by more careful
statement, but the results continued to be what mathematicians had previously
known (Kline, 1972: 1026). It was not a practical matter of improving the
utility of mathematics which gave rise to rigor but an internal development in
the social game mathematicians were playing with one another. With Cauchy
in the 1820s came the recognition that rigor was a way of beating one’s
opponents and simultaneously opening up a new field on which mathemati-
cians could play. The increase in the numbers of mathematicians, their material
bases, and the means of rival publications all amplified the drive toward new
areas of innovation. The older belief that math was becoming stagnant and
exhausted gave way to a feeling of unlimited vistas.
Rigor and abstract mathematics fed each other. Rigor was promoted by
axiomatization, the return to systematic exposition and proof in the manner
of Euclid, which had constituted the pedantic mathematics of the schools two
centuries before.^4 Once the system of axioms was displayed, it became possible
to vary the set, negating or eliminating some axioms and exploring the mathe-
matical realm thereby opened up. In this way non-commutative algebras such
as quaternions were developed in the 1840s. The most famous of these devel-
opments were the non-Euclidean geometries which became widely known in
the 1860s (although formulated in the 1820s and adumbrated as far back as
the 1760s), because they forced the recognition that mathematics had taken
leave of physical interpretations.^5 In the new game of higher mathematics,
entities were deliberately constructed whose properties are paradoxical from
the point of view of common sense: continuous functions without derivatives
at any point over an interval (Bolzano 1834; Weierstrass 1861, 1872); curves
without length or curves which completely filled a space (Peano 1890); geome-
tries in n-dimensional space (Cayley 1843, Grassmann 1844; see Boyer, 1985:
565, 604, 645; Kline, 1972: 1025, 1029–30). Paradoxes gave even more
impetus to the movement for rigorization; distrusting spatial intuition and
becoming aware of the naiveté of accepting traditional assumptions as self-evi-
The Post-revolutionary Condition^ •^699