alternative algebras, which they had generalized from Hamilton’s quaternions.
Peirce was an American import of British mathematical and philosophical net-
works, who brought together their controversies into a full-fledged system.
- There is even a personal connection to the core of the Utilitarians: John Stuart Mill
was Russell’s godfather. This could hardly have produced any direct intellectual or
emotional influence, since Mill died when Russell was one year old. But it exem-
plifies how central Russell was in the late Victorian network structure, a fact which
must have shaped his intellectual trajectory. He was raised by his grandparents,
Lord Russell, the introducer of the Reform Bill in 1832 and subsequent prime
minister, and Lady Russell, an outspoken political liberal and promoter of Utili-
tarian causes.
- Kline (1972: 1031–33). At this time Whitehead was purely a mathematician; his
shift into philosophy did not occur until the 1920s.
- Kline (1972: 1197–1208; DSB, 1981: 14:613–616; Peckhaus, 1990). Zermelo’s
basic idea, showing the existence of relations by which every set in an infinite
system corresponds to one of its elements, was already formulated in 1900–1901
lectures. It burst into prominence in 1904 in the aftermath of Russell and in tandem
with Hilbert’s announcement of his formalist program. This was the formalist
network in action; Zermelo was a Dozent at Göttingen, a colleague of Hilbert and
Husserl, and a former pupil at Halle of Cantor, whose set theory Zermelo defended.
At the same moment, Hilbert proposed his drastic new program to cure mathe-
matical paradoxes, including these newest ones, by treating mathematical symbols
as nothing more than arbitrary but consistent sets of marks on paper. Dramatic
developments of this kind are not mysterious and sudden insights. Typically they
happen in the course of long and intense concentration on problems focused by
network conflicts. In this case Russell found his paradox, and Zermelo his axiom
of choice, by wrestling with dubious points in Cantor’s system. Russell describes
how he tried out every conceivable approach to his problems, just as W. R.
Hamilton discovered quaternions in a sudden insight after 15 years of working on
the problem (Coffa, 1991: 103, 115; Kline, 1972: 779).
- Wedberg (1984: 139–141). Because of his emphasis on the independence of logical
elements, Russell for a while endorsed Meinong’s 1904 doctrine of the types of
reality of all mental and extra-mental objects. Russell could vacillate among
various forms of realism—Platonist, sensory, and others—because his principle of
acquaintance covered very generally every kind of item that might be directly
known.
- Wittgenstein also rejects Neo-Kantianism, which he otherwise somewhat resem-
bles. The space-form is not merely a kind of spectacles through which we see the
world; it involves a “multiplicity of relations” which we have had to discover
through exploration of uses of symbol systems (Tractatus 4.0412). Wittgenstein
seems to have in mind that a fact such as space having three dimensions rather
than something else is not explained by Neo-Kantian a priority; logic permeates
the world, including its “empirical” aspects.
- Russell himself, who declared that his brain had been burned out by the struggle
Notes to Pages 709–717^ •^1015
