Startseite Philosophie What Hilbert and Bernays Meant by “Finitism”
Kapitel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

What Hilbert and Bernays Meant by “Finitism”

  • William Tait
Veröffentlichen auch Sie bei De Gruyter Brill
Erkunden Sie dieses Fachgebiet So veröffentlichen Sie bei uns
Philosophy of Logic and Mathematics
Ein Kapitel aus dem Buch Philosophy of Logic and Mathematics

Abstract

“Finitism” (Tait 1981) presents an argument that finitist number theory is primitive recursive arithmetic (PRA). The argument is based on taking seriously the “finite” in “finitism”. But the question remained: what did Hilbert (and Bernays) mean in the early 1920’s through the early 1930’s by “finitism” and in particular, did they restrict finitist number theory to PRA. In his dissertation (Zach 2003), Richard Zach pointed out that Hilbert endorsed results as finitist that require more than PRA for their proofs. Tait 2002 and tait2005 argue that it is not clear that Hilbertwas aware that these results go beyond PRA. But that view is challenged in more recent times in Sieg/Ravaglia 2005 and by the editors of (the invaluable!) David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933 (Hilbert 2013). I will survey the old ground and then discuss the new challenge, which claims that, from the early 1920’s on, Hilbert accepted as finitist an enumeration function of the primitive recursive functions (which of course is not primitive recursive). The grounds for this are a reading of a passage in §7 of Grundlagen der Mathematik I and an argument for the consistency of PRA which goes back to 1922-1923 and is elaborated again in §7 of Grundlagen der Mathematik I. I will argue that their reading of the passage in question is a misreading and that the argument for the consistency of PRA uses, not an enumeration function for the primitive recursive functions, but rather mathematical induction on a Π02 predicate (i.e. of the form ∀x∃yϕ(x, y)), which was explicitly rejected by Hilbert as finitist - e.g. notably in Hilbert 1926.

Abstract

“Finitism” (Tait 1981) presents an argument that finitist number theory is primitive recursive arithmetic (PRA). The argument is based on taking seriously the “finite” in “finitism”. But the question remained: what did Hilbert (and Bernays) mean in the early 1920’s through the early 1930’s by “finitism” and in particular, did they restrict finitist number theory to PRA. In his dissertation (Zach 2003), Richard Zach pointed out that Hilbert endorsed results as finitist that require more than PRA for their proofs. Tait 2002 and tait2005 argue that it is not clear that Hilbertwas aware that these results go beyond PRA. But that view is challenged in more recent times in Sieg/Ravaglia 2005 and by the editors of (the invaluable!) David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933 (Hilbert 2013). I will survey the old ground and then discuss the new challenge, which claims that, from the early 1920’s on, Hilbert accepted as finitist an enumeration function of the primitive recursive functions (which of course is not primitive recursive). The grounds for this are a reading of a passage in §7 of Grundlagen der Mathematik I and an argument for the consistency of PRA which goes back to 1922-1923 and is elaborated again in §7 of Grundlagen der Mathematik I. I will argue that their reading of the passage in question is a misreading and that the argument for the consistency of PRA uses, not an enumeration function for the primitive recursive functions, but rather mathematical induction on a Π02 predicate (i.e. of the form ∀x∃yϕ(x, y)), which was explicitly rejected by Hilbert as finitist - e.g. notably in Hilbert 1926.

Kapitel in diesem Buch

  1. Frontmatter I
  2. Contents V
  3. Preface IX
  4. Part I: Philosophy of Logic
  5. Link’s Revenge: A Case Study in Natural Language Mereology 3
  6. Universal Translatability: An Optimality- Based Justification of (Classical) Logic 37
  7. Invariance and Necessity 55
  8. Translations Between Logics: A Survey 71
  9. On the Relation of Logic to Metalogic 91
  10. Free Logic and the Quantified Argument Calculus 105
  11. Dependencies Between Quantifiers Vs. Dependencies Between Variables 117
  12. Three Types and Traditions of Logic: Syllogistic, Calculus and Predicate Logic 133
  13. Truth, Paradox, and the Procedural Conception of Fregean Sense 153
  14. Wittgenstein and Frege on Assertion 169
  15. Assertions and Their Justification: Demonstration and Self-Evidence 183
  16. Surprises in Logic: When Dynamic Formality Meets Interactive Compositionality 197
  17. Part II: Philosophy of Mathematics
  18. Neologicist Foundations: Inconsistent Abstraction Principles and Part-Whole 215
  19. What Hilbert and Bernays Meant by “Finitism” 249
  20. Wittgenstein and Turing 263
  21. Remarks on Two Papers of Paul Bernays 297
  22. The Significance of the Curry-Howard Isomorphism 313
  23. Reductions of Mathematics: Foundation or Horizon? 327
  24. What Are the Axioms for Numbers and Who Invented Them? 343
  25. Part III: Wittgenstein
  26. Following a Rule: Waismann’s Variation 359
  27. Propositions in Wittgenstein and Ramsey 375
  28. An Unexpected Feature of Classical Propositional Logic in the Tractatus 385
  29. Ontology in Tractatus Logico-Philosophicus: A Topological Approach 397
  30. Adding 4.0241 to TLP 415
  31. Understanding Wittgenstein’s Wood Sellers 429
  32. On the Infinite, In-Potentia: Discovery of the Hidden Revision of Philosophical Investigations and Its Relation to TS 209 Through the Eyes of Wittgensteinian Mathematics 441
  33. Incomplete Pictures and Specific Forms: Wittgenstein Around 1930 457
  34. „Man kann die Menschen nicht zum Guten führen“ – Zur Logik des moralischen Urteils bei Wittgenstein und Hegel 473
  35. Der Status mathematischer und religiöser Sätze bei Wittgenstein 485
  36. Gutes Sehen 499
  37. Wittgenstein’s Conjecture 515
  38. Index of Names 535
  39. Index of Subjects 539
https://doi.org/10.1515/9783110657883-015

Kapitel in diesem Buch

  1. Frontmatter I
  2. Contents V
  3. Preface IX
  4. Part I: Philosophy of Logic
  5. Link’s Revenge: A Case Study in Natural Language Mereology 3
  6. Universal Translatability: An Optimality- Based Justification of (Classical) Logic 37
  7. Invariance and Necessity 55
  8. Translations Between Logics: A Survey 71
  9. On the Relation of Logic to Metalogic 91
  10. Free Logic and the Quantified Argument Calculus 105
  11. Dependencies Between Quantifiers Vs. Dependencies Between Variables 117
  12. Three Types and Traditions of Logic: Syllogistic, Calculus and Predicate Logic 133
  13. Truth, Paradox, and the Procedural Conception of Fregean Sense 153
  14. Wittgenstein and Frege on Assertion 169
  15. Assertions and Their Justification: Demonstration and Self-Evidence 183
  16. Surprises in Logic: When Dynamic Formality Meets Interactive Compositionality 197
  17. Part II: Philosophy of Mathematics
  18. Neologicist Foundations: Inconsistent Abstraction Principles and Part-Whole 215
  19. What Hilbert and Bernays Meant by “Finitism” 249
  20. Wittgenstein and Turing 263
  21. Remarks on Two Papers of Paul Bernays 297
  22. The Significance of the Curry-Howard Isomorphism 313
  23. Reductions of Mathematics: Foundation or Horizon? 327
  24. What Are the Axioms for Numbers and Who Invented Them? 343
  25. Part III: Wittgenstein
  26. Following a Rule: Waismann’s Variation 359
  27. Propositions in Wittgenstein and Ramsey 375
  28. An Unexpected Feature of Classical Propositional Logic in the Tractatus 385
  29. Ontology in Tractatus Logico-Philosophicus: A Topological Approach 397
  30. Adding 4.0241 to TLP 415
  31. Understanding Wittgenstein’s Wood Sellers 429
  32. On the Infinite, In-Potentia: Discovery of the Hidden Revision of Philosophical Investigations and Its Relation to TS 209 Through the Eyes of Wittgensteinian Mathematics 441
  33. Incomplete Pictures and Specific Forms: Wittgenstein Around 1930 457
  34. „Man kann die Menschen nicht zum Guten führen“ – Zur Logik des moralischen Urteils bei Wittgenstein und Hegel 473
  35. Der Status mathematischer und religiöser Sätze bei Wittgenstein 485
  36. Gutes Sehen 499
  37. Wittgenstein’s Conjecture 515
  38. Index of Names 535
  39. Index of Subjects 539
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 17.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/9783110657883-015/html
Button zum nach oben scrollen