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A historical introduction to the
philosophy of mathematics
2016 Fall semester
András Máté
P 14:00-15:30, i/129
First class:  23rd 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 mathematicsbut 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:

  1. Developments and problems in 19th century mathematics
  2. Bolzano, Cantor and the infinite
  3. Frege’s logicism and his construction of natural numbers
  4. Dedekind’s construction of natural numbers
  5. 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-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

16th September
23rd September
30th September
7th October
14th October
21st October
11th November
18th November
25th November
2nd December

9th December