Instructor: Dr. Kálmán CZISZTER
Text:
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S. Shapiro: Thinking about Mathematics, The Philosophy of Mathematics, Oxford University Press 2000\
P. Benacerraf and H. Putnam (eds.): Philosophy of mathematics, selected readings, Cambridge University Press 1983\
A. George and D.J. Velleman: Philosophies of Mathematics, Blackwell 2002, Oxford\
handouts will be distributed with all the relevant readings
Prerequisite: none
Course description:
During history philosophers were always fascinated by mathematics partly because they regarded it as the model of absolute certainty and rigorous science but also partly because the nature of the mathematical objects and their connection to the empirical world seemed highly obscure and elusive. Many mathematicians have also participated in these philosophical debates which sometimes even gave them motivation and inspiration for their mathematical work. The course aims at reviewing some of the major interaction points between philosophy and mathematics and to give a vantage point for working mathematicians from which they can regard their field as part of a broader perspective.
Topics:
The following list of subjects is only tentative and it can be largely adapted to the actual interests of the participants.
- Mathematics in the antiquity. The pythagoreans and Zeno.s paradoxes. Platonism vs. Aristotle.
- The Newton-Leibniz controversy about the foundations of Analysis. Debates on the notion of infinity.
- Kant's rationalist account of mathematics and Mill.s empiricist criticism of it.
- Attempts to clarify the concept of number. Frege.s Grundgesetze der Arithmetic.
- Russell.s paradox and the development of modern set theory.
- Formalism, deductivism and Hilbert.s Programme.
- Gödel.s incompleteness theorems and their interpretation as epistemological limitations.
- The intuitionism of Brouwer and Heyting: what is wrong with the Principle of Excluded Middle or the Axiom of Choice? .