1734 had attacked Newton’s infinitesimals, British mathematicians had been
on the defensive. Around 1830 there was a wave of controversy over the use
of imaginary numbers in algebraic practice, despite their lack of a physical
interpretation. Conservatives grew more aggressive during the escalating con-
flict against Continentally oriented reformers. Not only imaginary numbers but
negative numbers as well cannot be said to exist; how then could any valid
mathematics be built upon them? It was not the first time a conservative attack
has provoked fundamental innovation.
In 1830 Peacock attempted to put algebra on axiomatic foundations similar
to Euclidean geometry, which is to say on the model of the school classic and
favorite of academic conservatives. Peacock thereby brought into the open the
basic principles of arithmetic operations, the associative, commutative, and
distributive laws; by abstraction from real numbers, he attempted to justify
(but actually asserted only by fiat) that similar rules hold good for operations
with any magnitudes, including complex numbers. In the same year De Morgan
began to publish on double algebra (i.e., the algebra of complex numbers,
combining a real number with an imaginary). The intentions of Peacock and
De Morgan were traditional, in that they denied any other forms of algebra
were possible than those following the laws governing positive integers; their
model was empirical science, and they gave no truth value to an abstract
mathematics in its own right (Richards, 1980). This attempt to incorporate
higher mathematics into the conservative framework of non-specialist liberal
education still prevailing in the English universities led to revolutionary com-
binations.
Peacock’s and De Morgan’s work provoked efforts to extend the method
of representing complex numbers graphically as vectors on a plane. The Irish
physicist W. R. Hamilton in the 1830s gave a quasi-physical interpretation of
operations on complex numbers as rotations in a plane; for a number of years
he attempted to extend this method to rotations in three dimensions, and in
1843 realized that the method would work only if one dropped the commu-
tative law of multiplication. The resulting new form of algebra, quaternions,
became famous because of the shock it gave traditional belief that the laws of
arithmetic are natural. In fact the discovery came directly out of efforts to find
a physical justification for imaginary numbers. Hamilton’s methods attempted
to rescue them by a spatial analogy, but at the cost of giving up traditional
operations.^13
In logic the conservative situation of British education again set the direc-
tion of innovation. Just as Euclidean geometry dominated the mathematics cur-
riculum of the schools, logic—a field which had been abolished in the French
curriculum and transformed in Germany into metaphysics—still loomed large
in the philosophy course. The movement for modernization and scientific re-706 •^ Intellectual Communities: Western Paths