The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited
Format: PDF / Kindle (mobi) / ePub
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.
Certainty and the naturalistic epistemology depicted there – the view according to which “at the foundation of well-founded belief lies belief that is not founded” (§253). Wittgenstein suggests that, for anyone, some system of belief must be completely immune from doubt because it is the system from which the person weighs the truth or falsity of claims, the ground on which he or she stands in order even to express doubt: “I have a world picture. Is it true or false? Above all it is the
his mind about this matter during the course of the 1920s. But if the proof theory is properly mathematical in the sense demanded by Hilbert’s aims, should an unambiguous approach to this question not be evident? That one is not is evidence of a lack of rigor undermining the program’s principal aims. The failure of Hilbert’s program as it stood in the late 1920s is thus that by isolating a significantly strong form of induction for use at the meta-mathematical level, the proof theorist exempts an
Interpreting G2 for Q Q + ∀xψ(x) ∀xψ(x) 0 + Exp Proof. We need a proof-theoretical version of this theorem since it is more useful for our analysis. Accordingly we will prove actually the equivalence between 2 and the following condition: 3. There is a syntactic cut I (x) in Q such that Q ψ(x)). ∀x(I (x) → This condition is equivalent to 1. Moreover, we shall only use the direction 2 ⇒ 3, so we prove only this entailment. For our proof we will need to cite Wilkie’s result that if I 0 + E x p
and understand its methods and tasks without sharing in them. This is a radical idea. Sections 103–133 of the Investigations are not just descriptions of the tasks and methods of philosophy. They are pronouncements of what philosophy ought to be like, what it can hope to accomplish, and what its limits are. And, crucially, they are not pronouncements that Wittgenstein expects will sit comfortably with professional philosophers. So it can seem like an ironic moment in the Investigations when
from the practice of getting a clear picture of how words like “mean,” “know,” “understand,” and “obey” are used. Since Wittgenstein concedes our ability to engage in the former practice – since his misgivings 4 This is an oddity about philosophy: “It leaves everything as it is,” except for philosophy itself. Section 116 exemplifies this attitude: “When philosophers use a word – ‘knowledge,’ ‘being,’ ‘object,’ ‘I,’ ‘proposition,’ ‘name,’ – and try to grasp the essence of the thing, one must