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Elective English–I

Notes weak, and the ramified theory for being too strong. For some, it was important that any
proposed solution be comprehensive enough to resolve all known paradoxes at once. For
others, it was important that any proposed solution not disallow those parts of classical mathematics
that remained consistent, even though they appeared to violate the vicious circle principle.
Russell himself had recognized many of these weaknesses, noting as early as 1903 that it was
unlikely that any single solution would resolve all of the known paradoxes. Together with
Whitehead, he was also able to introduce a new axiom, the axiom of reducibility, which
lessened the vicious circle principle’s scope of application and so resolved many of the most
worrisome aspects of type theory. Even so, some critics claimed that the axiom was too ad hoc
to be justified philosophically.

Of equal significance during this period was Russell’s defense of logicism, the theory that
mathematics is in some important sense reducible to logic. First defended in his 1901 article
“Recent Work on the Principles of Mathematics,” and then later in greater detail in his Principles
of Mathematics and in Principia Mathematica, Russell’s logicism consisted of two main theses.
The first was that all mathematical truths can be translated into logical truths or, in other
words, that the vocabulary of mathematics constitutes a proper subset of the vocabulary of
logic. The second was that all mathematical proofs can be recast as logical proofs or, in other
words, that the theorems of mathematics constitute a proper subset of the theorems of logic.

Like Gottlob Frege, Russell’s basic idea for defending logicism was that numbers may be
identified with classes of classes and that number-theoretic statements may be explained in
terms of quantifiers and identity. Thus the number 1 would be identified with the class of all
unit classes, the number 2 with the class of all two-membered classes, and so on. Statements
such as “There are at least two books” would be recast as statements such as “There is a book,
x, and there is a book, y, and x is not identical to y.” Statements such as “There are exactly
two books” would be recast as “There is a book, x, and there is a book, y, and x is not identical
to y, and if there is a book, z, then z is identical to either x or y.” It followed that number-
theoretic operations could be explained in terms of set-theoretic operations such as intersection,
union, and difference. In Principia Mathematica, Whitehead and Russell were able to provide
many detailed derivations of major theorems in set theory, finite and transfinite arithmetic,
and elementary measure theory. A fourth volume on geometry was planned but never completed.
Russell’s most important writings relating to these topics include not only Principles of Mathematics
(1903), “Mathematical Logic as Based on the Theory of Types” (1908), and Principia Mathematica
(1910, 1912, 1913), but also his earlier An Essay on the Foundations of Geometry (1897), and his
Introduction to Mathematical Philosophy (1919a), the last of which was largely written while
Russell was serving time in Brixton Prison as a result of his anti-war activities. Coincidentally,
it was at roughly this same time (1918–19) that Wittgenstein was completing his Tractatus
Logico-Philosophicus while being detained as a prisoner of war at Monte Cassino during World
War I.

2.3 Russell’s Work in Analytic Philosophy

In much the same way that Russell used logic in an attempt to clarify issues in the foundations
of mathematics, he also used logic in an attempt to clarify issues in philosophy. As one of the
founders of analytic philosophy, Russell made significant contributions to a wide variety of
areas, including metaphysics, epistemology, ethics and political theory. According to Russell,
it is the philosopher’s job to discover a logically ideal language—a language that will exhibit
the true nature of the world in such a way that we will not be misled by the accidental surface
structure of natural language. Just as atomic facts (the association of universals with an appropriate
number of individuals) may be combined into molecular facts in the world itself, such a

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