Beyond Hilbert's reach?

Wilfried Sieg

pp. 449-483

in: Erik Palmgren, Krister Segerberg (eds), Logicism, intuitionism, and formalism, Berlin, Springer, 2009

Abstract

Work in the foundations of mathematics should provide systematic frameworks for important parts of the practice of mathematics, and the frameworks should be grounded in conceptual analyses that reflect central aspects of mathematical experience. The Hilbert School of the 1920s used suitable frameworks to formalize (parts of) mathematics and provided conceptual analyses. However, its analyses were mostly restricted to finitist mathematics, the programmatic basis for proving the consistency of frameworks and, thus, their instrumental usefulness. Is the broader foundational quest beyond Hilbert's reach? The answer to this question seems simple: "Yes & No". It is "Yes", if we focus exclusively on Hilbert's finitism; it is "No", if we take into account the more sweeping scope of Hilbert and Bernays's foundational thinking. The evident limitations of Hilbert's "formalism" have been pointed out all too frequently; in contrast, I will trace connections of Hilbert's work, beginning in the late 19th century, to contemporary work in mathematical logic. Bernays's reflective philosophical investigations play a significant role in reinforcing these connections. My paper pursues two complementary goals, namely, to describe a global, integrating perspective for foundational work and to formulate some more local, focused problems for mathematical work.

Cited authors

Hilbert David