COURSE DESCRIPTIONS

Code of course: BMI-LOTD-316E.02	
Title of course: A historical introduction into the philosophy of mathematics	
Lecturer: András Máté assoc. prof. (retired)	
General aim of the course: 
Intruduction to the problems of the philosophy of mathematics and its classical schools, with an outlook to the contemporary debates.
Content of the course:
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. 
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
Grading criteria, specific requirements:
For the grade, 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.

Required reading:
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.