The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited
Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development.
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analysis argument arithmetic theory axiomatic axioms Bernays bounded arithmetic claim complete induction consistency proofs consistency statement construction contentual David Hilbert’s deﬁne deﬁnition demonstration Detlefsen difﬁculty Eloise ematics encoding epistemological evaluation expression extensional ﬁnd ﬁnitary ﬁnite ﬁnitism ﬁrst ﬁrst-order formal formula Frege functions Godel’s second theorem grounds Herbrand Herbrand’s theorem Hilbert Hilbert’s philosophical Hilbert’s program Hilbert’s thought idea induction intensionally correct interpretation intuition investigations justiﬁed Kreisel logic mathe mathematical activity mathematical autonomy mathematical induction mathematical methods mathematical techniques mathematicians matics meta-mathematics metatheoretical metic naturalistic nature no-counterexample notion numbers ofits ofmathematics ofthe ofthis one’s ordinary mathematics paradoxes philosophical philosophy of mathematics predicate principles proof theory provability prove Pudlak’s purely mathematical quantiﬁer question reason recursive reﬂection refutation relativized scientiﬁc second-order secure semantic sentence set theory signiﬁcance skepticism speciﬁc standard sufﬁce sufﬁciently syntactic cut T-provability T’s consistency Tarski tency theory’s tion unprovability veriﬁcation Weyl Wittgenstein