David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Erkenntnis 53 (1-2):127-146 (2000)
Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has to prove a certain technical result. The other part of the program is a philosophical part. It is concerned with philosophical questions that are the real aim of the program. To carry out this part one, basically, has to show why the technical part answers the philosophical questions one wanted to have answered. Hilbert probably thought that he had completed the philosophical part of his program, maybe up to a few details. What was left to do was the technical part. To carry it out one, roughly, had to give a precise axiomatization of mathematics and show that it is consistent on purely finitistic grounds. This would come down to giving a relative consistency proof of mathematics in finitist mathematics, or to give a proof-theoretic reduction of mathematics on to finitist mathematics (we will look at these notions in more detail soon). It is widely believed that Gödel’s theorems showed that the technical part of Hilbert’s program could not be carried out. Gödel’s theorems show that the consistency of arithmetic can not even be proven in arithmetic, not to speak of by finitistic means alone. So, the technical part of Hilbert’s program is hopeless, and since Hilbert’s program essentially relied on both the technical and the philosophical part, Hilbert’s program as a whole is hopeless. Justified as this attitude is, it is a bit unfortunate. It is unfortunate because it takes away too much attention from the philosophical part of Hilbert’s program. And this is unfortunate for two reasons.
|Keywords||Philosophy Philosophy Epistemology Ethics Logic Ontology|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Graham E. Leigh & Carlo Nicolai (2013). Axiomatic Truth, Syntax and Metatheoretic Reasoning. Review of Symbolic Logic 6 (4):613-636.
Similar books and articles
N. Shankar (1994). Metamathematics, Machines, and Gödel's Proof. Cambridge University Press.
Paolo Mancosu (2003). The Russellian Influence on Hilbert and His School. Synthese 137 (1-2):59 - 101.
Yingrui Yang & Selmer Bringsjord (2003). Newell's Program, Like Hilbert's, is Dead; Let's Move On. Behavioral and Brain Sciences 26 (5):627-627.
Panu Raatikainen (2003). Hilbert's Program Revisited. Synthese 137 (1-2):157 - 177.
Jeremy Avigad, “Clarifying the Nature of the Infinite”: The Development of Metamathematics and Proof Theory.
Aleksandar Ignjatović (1994). Hilbert's Program and the Omega-Rule. Journal of Symbolic Logic 59 (1):322-343.
Richard Zach (2006). Hilbert's Program Then and Now. In Dale Jacquette (ed.), Philosophy of Logic. North Holland 5--411.
Richard Zach, Hilbert's Program. Stanford Encyclopedia of Philosophy.
Added to index2009-01-28
Total downloads48 ( #53,895 of 1,700,300 )
Recent downloads (6 months)14 ( #47,237 of 1,700,300 )
How can I increase my downloads?