2005/08/29 by Richard Zach · 71 citations
Arts and Humanities · Mathematics · Psychology · #Algebra over a field #Algorithm #Axiom #Calculus (dental) #Constructive proof #Discrete mathematics #Finitary #Foundations of mathematics #Hilbert space #Hilbert's fourteenth problem #History and Theory of Mathematics #Mathematical proof #Mathematics #Mathematics education #Philosophy and Theoretical Science #Philosophy of mathematics #Philosophy, Science, and History #Proof theory #Pure mathematics #Reproducing kernel Hilbert space #Rigged Hilbert space #Sketch #math.HO #math.LO #msc:00A30 #msc:01A60 #msc:03A05
paper · pdf · doi:10.1016/b978-044451541-4/50014-2
published in Elsevier eBooks, 411-447 (Elsevier BV) · 43 pages
arxiv created 2005/08/29 · openalex publication_date 2007/01/01 · arxiv updated 2015/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.