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. However, it led to a new problem situation, since Frege's answer was burdened by a paradox (which was essentially the same as the paradox that arose in Cantor's naive set theory). 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). The presentations used in  class will be posted here.
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:

1. Developments and problems in 19th century mathematics
2. Bolzano, Cantor and  infinity
3. Frege’s logicism and his construction of natural numbers
4. Dedekind’s construction of natural numbers
5. The new paradoxes of infinity – the first fall of logicism
6. The logicism of Russell and Ramsey
7. Hilbert’s program and the arithmetisation
8. Brouwer’s intuitionism
9. Gödel’s theorems and the second fall of logicism
10. The paradox of the liar and the indefinability of truth
11. 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