- Cantor originated the method of one-to-one correspondence; Cantor’s concept of
the power or cardinal number of a set is similar to Frege’s Platonic object-numbers. - Here Frege converges rather surprisingly with Bradley, who also held in his very
different logic of 1883 that all true statements are about a single object, the
universe. The British Idealist and the German logical formalist pursue a similar
path insofar as they are rebounding off a common enemy, empiricist logic in the
manner of J. S. Mill. - When the full-blown logicist movement widened the search for predecessors,
Bolzano was also added retrospectively to the pantheon. Beginning in the 1810s
as a mathematician-philosopher in the pre-disciplinary mode of an old-fashioned
university (Prague) on the periphery of the German system, Bolzano set out to
refute Kant with a combination of Leibnizian metaphysics and infinitesimal calcu-
lus (Coffa, 1991: 26–32; Boyer, 1985: 564–566). The result included various
distinctions resembling those of Frege and his successors. Bolzano was ignored,
owing in part to his isolation in the networks of the time; his work lacked the
drive to omni-symbolism which became the mark of the later movement, and his
innovations were buried in a metaphysical system of archaic cast. As in other cases
where adumbrations were recognized only in retrospect, Bolzano points up the
importance of a full-scale intellectual movement to raise details from their sur-
roundings and focus attention on them as landmarks of a new worldview. - Almost simultaneously, in 1844 Grassmann produced an even wider generalization
of complex numbers, connected to n-dimensional geometry, and dropping not only
the commutative but also the associative law. Grassmann’s work was little recog-
nized until the 1860s. The difference in fame was due to the fact that Hamilton’s
work came as the climax of controversy around the fundamentals of British
mathematics; whereas Grassmann was working in a side area far from the topics
on which German and French mathematicians were focusing attention. Grassmann
originated not in the network of leading German mathematicians, but as a theology
pupil of Schleiermacher. Under prevailing conceptions of mathematics, such non-
naturalistic innovations were likely seen in a theological rather than a strictly
mathematical context. Hamilton too had German Idealist connections as an inti-
mate of Coleridge; he claimed quaternions had cosmic rather than merely mathe-
matical significance (Hankins, 1980). - Sir William Hamilton and Bolzano (see note 12) are structural parallels: both were
philosophical conservatives at provincial universities who found their stimulus in
opposing the influx of fashionable Kantianism. In both cases fundamental innova-
tions in logic were set in motion in the 1830s which were not recognized until
later. - Again in 1863 Mill was to promote his systematic philosophy by using Sir William
Hamilton as a foil. Hamilton was especially well known in America, where Scottish
faculty psychology dominated the colleges; there he became the subject of discus-
sions from which emerged yet another innovation in logic, that of Peirce in the
1870s. As we saw in Chapter 12, Peirce’s starting point in mathematics, via his
father’s work, was an extension of Cayley and Sylvester’s method for inventing
1014 •^ Notes to Pages 703–707