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A historical introduction to the
philosophy of mathematics
2025 Spring semester
András MátéP 14:00-15:30, i/221First class: 14th February The
nature of mathematical objects and mathematical knowledge has been an
important issue in European philosophy since Plato and Aristotle.
However, philosophy of mathematics as a branch of
philosophy closely related to foundational research in mathematics began with Frege's Foundations of Arithmetics (1884).
Frege's work, like that of some of his contemporaries - was a response to a
problem situation created by the development of mathematics in the19th
century. But it led to a new problem situation because
Frege and Cantor's answer was burdened by the same paradox. The next
generation tried to eliminate the possibility of
paradoxes in mathematics in many ways.These efforts
led to the emergence of what are known as the classical
schools in philosophy of mathematics: logicism, formalism and
intuitionism. These are not just philosophical opinions on
mathematics, but also research programmes in the foundations of
mathematics The course traces this historical process from the
problem situation in 19th century mathematics to the results of
foundational research in the 1930s. Some insights will be given
into the trends in the contemporary philosophy of mathematics
(neo-vartiants of the classical schools, structuralism, philosophy of
mathematical practice).
To receive a grade,
the student must give a presentation on a topic related to the course theme. This will be discussed at a
"house conference" during the exam period. (S)he should also participate in the discussion of the other students' presentations
Contents of the course:
- Developments and problems in 19th century mathematics
- Bolzano, Cantor and infinity
- Frege’s logicism and his construction of natural numbers
- Dedekind’s construction of natural numbers
- The 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's 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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ANDRÁS MÁTÉ
