Research Areas
Philosophy and History of Mathematics
Since ancient times, there has been an intimate connection between philosophical and mathematical thought; Plato, Descartes, Leibniz, and Kant are notable examples whose mathematical and philosophical reflections illuminate this deep relation. The second half of the 19th century witnessed a revolutionary transformation of mathematics through the work of Dirichlet, Riemann, Dedekind and others. It was accompanied by similarly dramatic changes in logic, brought about by Boole, Frege, Peano, Peirce, and Schröder, among others. Complemented by the continuing evolution of thesciences in the work of e.g. Mach and Einstein, this forms the background for the development of early analytic philosophy and modern mathematical logic.
Wilfried Sieg, Steve Awodey, and Jeremy Avigad have undertaken historical work in these areas. Sieg's and Awodey's work is further reflected in, and supported by, their extensive editorial work. Drawing on this basis, all three attempt to gain sound mathematical and historical perspectives for the philosophy of mathematics.
More specifically, Sieg and Awodey are directly involved in researching, editing, translating, and interpreting unpublished materials by Frege, Hilbert, Bernays, Carnap, and Gödel. In this connection, extensive use is also made of the University of Pittsburgh's valuable Archive of Scientific Philosophy, for instance in the publication of the Full Circle series. In addition, Avigad has worked on the historical background of contemporary proof theory.
This research on the 19th century transformation of mathematics and the roots of mathematical logic and analytic philosophy has resulted in conceptual analyses of the notions of "effective calculability", "completeness", and "categoricity". Both Sieg and Awodey have also contributed to the development of a historically and mathematically informed conception of structuralism.
Avigad has studied the development of nineteenth century number theory. In particular, he has used Dedekind's work in algebraic number theory to clarify the role of set-theoretic and nonconstructive methods in modern mathematics.