Foundations of Mathematics

I am Stephen G. Simpson, a mathematician at the Pennsylvania State University. My principal research interest is foundations of mathematics.

Click here to read my short essay explaining what is meant by the term foundations of mathematics.

My paper Logic and Mathematics is a survey of logic and foundations of mathematics, for the general reader. It was published in a volume distributed by the Book-of-the-Month Club. It is available in several formats: DVI (84K), PDF (362K), PS (1080K), HTML.

A key issue in foundations of mathematics is Hilbert's program of finitistic reductionism. My paper Partial Realizations of Hilbert's Program is available in several formats: LaTeX (55K), DVI (72K), PDF (264K), PS (772K), HTML.

One of my ongoing research projects is Reverse Mathematics, an attempt to answer the foundational question: ``Which set existence axioms are needed to prove specific mathematical theorems?'' I study this question in the context of subsystems of second order arithmetic. I have written a book on this subject. Many of my publications deal with Reverse Mathematics, and many of my Ph.D. students have written theses in this area.

I teach two courses on foundations of mathematics:

  1. MATH 558, Foundations of Mathematics I. This is an introductory graduate course on foundations of mathematics. A course announcement and lecture notes are available.

  2. MATH 565, Foundations of Mathematics II. This is an advanced graduate course on subsystems of second order arithmetic and Reverse Mathematics. A course announcement is available. I have published a book on this subject.

I founded and continue to be associated with FOM, an automated e-mail list for discussing foundations of mathematics.   /   7 April 2006