Finitism: An Essay on Hilbert's Programme

Dissertation, Massachusetts Institute of Technology (1991)

In this thesis, I discuss the philosophical foundations of Hilbert's Consistency Programme of the 1920's, in the light of the incompleteness theorems of Godel. ;I begin by locating the Consistency Programme within Hilbert's broader foundational project. I show that Hilbert's main aim was to establish that classical mathematics, and in particular classical analysis, is a conservative extension of finitary mathematics. Accepting the standard identification of finitary mathematics with primitive recursive arithmetic, and classical analysis with second order arithmetic, I report upon some recent work which shows that Hilbert's aim can almost be realized. ;I then discuss the philosophical significance of this startling fact. I describe Hilbert as seeking a middle way between two mathematically revisionary positions in the philosophy of mathematics--a kind of proto-intuitionism, and an extreme realism, associated with the views of Kronecker and Frege respectively. I outline a Hilbertian alternative to these positions. The result is a moderate realism that owes much to Quine. I defend it against certain objections, and display its virtues in a series of comparisons with alternatives currently influential in the literature. ;In Chapter Two, I discuss the special status the Hilbertian gives to finitary mathematics. I argue that two ways of justifying this special status--by claiming that finitary mathematics is ontologically special, since it is committed only to expressions, and by claiming that finitary mathematics is epistemologically special, since its results are especially evident--are in fact hopeless. I then defend an alternative justification, drawing in part on Godel's well known discussion of mathematical intuition. ;In Chapter Three, I discuss the implications of incompleteness for the Hilbertian philosophy of mathematics. I argue, against some recent work by Michael Detlefsen, that the incompleteness theorems show definitively that Hilbert's Programme cannot be carried out in full generality. Drawing on recent work by Warren Goldfarb, I show that this conclusion follows from the First Incompleteness Theorem, and can be established without any controversial appeal to the semantic value of undecidable sentences. However, I argue that the fact of incompleteness adds to, rather than detracts from, the attractiveness of the basic Hilbertian position on the nature of mathematics
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