Foundations of mathematics: an optimistic viewpoint

Stephen G. Simpson Department of Mathematics Pennsylvania State University http://www.math.psu.edu/simpson simpson@math.psu.edu

November 13, 2015

This document consists of an abstract and references for my Public Lecture scheduled for January 6, 2016 in Lecture Theatre 31 at the National University of Singapore.

Abstract

Historically, mathematics has been regarded as a “role model” for all of science – a paragon of abstraction, logical precision, and objectivity. The 19th and early 20th centuries saw tremendous progress. The great mathematician David Hilbert proposed a sweeping program whereby the entire panorama of higher mathematical abstractions would be justified objectively and logically, in terms of finite processes. But then in 1931 the great logician Kurt Gödel published his famous incompleteness theorems, leading to an era of confusion and skepticism. In this talk I show how modern foundational research has opened a new path toward objectivity and optimism in mathematics.

References

[1] Hippocrates G. Apostle. Aristotle’s Philosophy of Mathematics. University of Chicago Press, 1952. X + 228 pages.

[2] C.-T. Chong, Q. Feng, T. A. Slaman, and W. H. Woodin, editors. Infinity and Truth. Number 25 in IMS Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore. World Scientific, 2014. IX + 234 pages.

[3] S. Feferman, C. Parsons, and S. G. Simpson, editors. Kurt Gödel: Essays for his Centennial. Number 33 in Lecture Notes in Logic. Association for Symbolic Logic, Cambridge University Press, 2010. X + 373 pages.

[4] Stephen G. Simpson. Partial realizations of Hilbert’s program. Journal of Symbolic Logic, 53(2):349–363, 1988.

[5] Stephen G. Simpson. Subsystems of Second Order Arithmetic. Perspectives in Logic. Association for Symbolic Logic and Cambridge University Press, 2nd edition, 2009. XVI + 444 pages.

[6] Stephen G. Simpson. The Gödel hierarchy and reverse mathematics. In [3], pages 109–127, 2010.

[7] Stephen G. Simpson. An objective justification for actual infinity? In [2], pages 225–228, 2014.

[8] Stephen G. Simpson. Toward objectivity in mathematics. In [2], pages 157–169, 2014.

[9] Richard Zach. Hilbert’s Program. Stanford Encyclopedia of Philosophy, January 2015. http://plato.stanford.edu/entries/hilbert-program.