From Consistency to Incompleteness: A Philosophical Study of Hilbert's Program and Goedel's Incompleteness Theorem

Dissertation, University of California, Berkeley (1997)
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Abstract

The main objective of this thesis is a philosophical study of Hilbert's Program and Godel's Incompleteness Theorem. For this purpose we pursue historical, metamathematical, and conceptual investigations of them. ;By tracing the historical origins and conceptual developments of Hilbert's Program and Godel's Incompleteness Theorem, we will argue that both have inherently philosophical motivations. Also, by considering the relevant metamathematical developments such as Reverse Mathematics, the Paris-Harrington Incompleteness Theorem and related materials, we will argue that Hilbert's Program and Godel's Incompleteness Theorem can be regarded as a philosophical program and a philosophical thesis, respectively. ;In the case of Hilbert's Program, as a combination of a consistency program and a conservation program, it is an epistemological program for the establishment of reliable mathematical knowledge; as a combination of a finitist program and a formalist program it is a philosophical program for Hilbert's positivistic philosophy of mathematics. ;Regarding Godel's Incompleteness Theorem, we will show that it can be conceived as an ontological thesis either indirectly as a refutation of another ontological thesis or directly as expressing an ontological thesis . As such it gets support from Godel's objectivistic philosophy of mathematics. ;Finally, using these philosophical aspects of Hilbert's Program and Godel's Incompleteness Theorem, we will show that Wittgenstein's puzzling remarks on Hilbert and Godel during his 'return' period are legitimate philosophical criticisms. They are not intended to be a 'philosophy of mathematics' in the sense of a 'theory', but just a 'philosophy for mathematicians', a 'therapy for mathematicians'

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