Beyond Hilbert's Reach?
… historical reflection serves, in the end, to shape the present and
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 and No.” It is “Yes” if we focus exclusively on Hilbert's fintism; 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 nineteenth century, to contemporary work in mathematical logic. Bernays's reflective philosophical investigations play a significant role in reinforcing these connections.
It is a fact of intellectual history, perhaps a curious one, but nonetheless a fact, that the Grundlagenstreit of the 1920s colors even now our perspectives on the foundations of mathematics and beyond. In those early