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A historical introduction to the
philosophy of mathematics
2022 Fall semester
András MátéP 10:00-11:30, i/129First class: 16th September The
nature of mathematical objects and mathematical knowledge has been an
important question in European philosophy since Plato and Aristotle.
However, philosophy of mathematics as a substantive branch of
philosophy closely connected with foundational research in mathematics
originates with Frege's Foundations of Arithmetics (1884).
Frege's work - as well as the works of his contemporaries - answered a
problem situation formed by the developments of 19th century
mathematics, but it led to a new problem situation because
Frege's and Cantor's answer was encumbered by the same paradox. Their
followers tried to eliminate the possibility of occurrence of
paradoxes in mathematics in different ways.These endeavours
led to the formation of the schools that are called the classical
schools in philosophy of mathematics: logicism, formalism and
intuitionism. They are not just philosophical opinions about
mathematics, but research programs in the foundations of mathematics as
well. The course presents this historical process from the
problem situation in 19th century mathematics to the results of
foundational research in the nineteen-thirties.
For the mark,
the student should produce a presentation about some subject
connected with the topic of the course. It will be discussed at a
"house conference" in the exam period. (S)he should participate
in the discussion of the presentations of the other students, too.
Contents of the course:
- Developments and problems in 19th century mathematics
- Bolzano, Cantor and the infinite
- Frege’s logicism and his construction of natural numbers
- Dedekind’s construction of natural numbers
- New paradoxes of infinity – the first fall of logicism
- The logicism of Russell and Ramsey
- Hilbert’s program and the arithmetisation
- Brouwer’s intuitionism
- Gödel’s theorems and the second fall of logicism
- The paradox of the liar and the indefinability of truth
- Decision problem, Church-thesis, Church(-Turing)-theorem
Recommended readings:
Benacerraf,
P. – H. Putnam (eds.): Philosophy of mathematics, Cambridge U.P., 1983 van Heijenooort, J. (ed.): From Frege To Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard U. P.; reprinted with corrections, 1977.
Mancosu, P. (ed.): From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, 1998.
Presentations
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