February 27, 2014 at 4:30 pm

Philosophy of Science: Hilbert’s Axiomatic Ways: Structural and Formal, March 14

Lectures in the History of Philosophy of Science presents Dr. Wilfried Sieg on “Hilbert’s Axiomatic Ways: Structural and Formal,” on March 14 at 4 p.m. in Scripps 111.

Sieg is the Patrick Suppes Professor of Philosophy at Carnegie Mellon University and a Fellow of the American Academy of Arts and Sciences.

Abstract: “David Hilbert was among the great mathematicians who expounded the centrality of mathematics in human thought over a long, remarkable career. And yet, his programmatic “formalist” papers from the 1920s, often seriously misinterpreted, still shape almost exclusively the contemporary perspective on his views concerning (the foundations of) mathematics. Even his own earlier and quite different work on the foundations of geometry and arithmetic (of real numbers) is most often seen through that lens.

“In contrast, I will sharply distinguish between Hilbert’s existential or structural axiomatics from the 1890s and his formal axiomatics from the late 1910s. That distinction allows us, on the one hand, to connect structural axiomatics with Dedekind’s work and the related radical transformation of mathematics in the second half of the 19^th century and, on the other hand, to link formal axiomatics with Whitehead and Russell’s Principia Mathematica and the related dramatic changes in logic due, in part, to Frege. The strict formalizability of mathematics was basic for the finitist consistency program as articulated for the first time in 1922.

“The methodological innovations of the earlier mathematical developments are reflected in the well-known, but not very well understood correspondence between Frege and Hilbert; the articulation of structural axiomatics can be seen to correspond to Frege’s introduction of second-level concepts. Taking seriously the goal of formalizing mathematics in effective logical frameworks (and the limitations due to Gödel’s incompleteness theorems) leads to contemporary tasks, not just historical and systematic insights; those tasks I will briefly describe as “one direction” for fascinating work at the intersection of automated proof search, proof theory, and cognitive science.”