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[5. Biographical Notes.]
Friedrich Waismann.
The philosopher Friedrich Waismann (Vienna
1896  Oxford, UK, 1959) studied mathematics and
philosophy at Vienna. In 1929, he became an
assistant to Schlick. He was one of the few members
of the Vienna Circle admitted to the meetings with
Wittgenstein. Waismann recorded several
conversations which were published posthumously in
F. Waismann, Wittgenstein und der Wiener
Kreis, 1967 (English translation
Wittgenstein and the Vienna Circle:
Conversations Recorded by Friedrich Waismann,
New York: Barnes & Noble Books, 1979). Waismann
proposed a logical interpretation of probability
inspired by Wittgenstein in his work "Logische
Analyse des Wahrscheinlichkeitsbegriffs"
(Erkenntnis, 1, 1930). In 1936 he published
his only book Einführung in das
mathematische Denken, about the philosophy of
mathematics. He immigrated to England in 1937,
where he taught philosophy of mathematics and
philosophy of science at Cambridge and, from 1940,
at Oxford. In England he contributed to the
development of analytic philosophy. His book,
The Principles of Linguistic Philosophy,
Oxford, 1965, an exposition to the philosophy of
the late Wittgenstein, was also posthumously
published.
The only book he published during his life dealt
with the interpretation of mathematics. Waismann
criticized both Logicism and Formalism. Logicism
argues that all mathematical truths are logical
truths, and it is based on Frege and
Russell’s definition of natural numbers,
namely a natural number is the class of all
equinumerable classes. According to Waismann, this
definition introduces an element of contingency in
mathematics, thus disturbing its a priori
character. Moreover, formal logic is by no means a
privileged calculus to which all mathematics is
reducible. Logic itself is a part of mathematics.
Waismann also rejected the formalistic
interpretation, because it is not interested with
the meaning of mathematical concepts. According to
Formalism, a natural number is whatever fulfils the
axioms of mathematics. But this approach neglects a
very important problem, namely the question whether
the axioms of mathematics identify the natural
numbers we really employ. The solution consists in
the study of the role that natural numbers play in
ordinary language (note the evident analogy with
Wittgenstein's assertion that meaning is use).
