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Remarks on Two Papers of Paul Bernays

  • Charles Parsons
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Philosophy of Logic and Mathematics
This chapter is in the book Philosophy of Logic and Mathematics

Abstract

The paper comments on two papers in French that Paul Bernays derived from lectures at a conference on mathematical logic held in Geneva in June 1934. The first, the well-known “On Platonism in mathematics,” sets forth a methodological version of Platonism and observes that it can be implemented in some branches of mathematics and not others. He notes that by his definition Brouwer’s intuitionism rejects all Platonism,while HermannWeyl’s reconstruction of analysis retains it for generalizations about natural numbers but not for generalizations about real numbers or higher-type objects. He rejects the then widespread idea of a crisis of foundations and argues that the questions raised are philosophical. Bernays’ second paper, “Some observations on metamathematics,” is a technical sequel to “On Platonism.” It describes some basic points in the Hilbert school’s proof theory and sketches some results, such as a simple application of Herbrand’s theorem, Gödel’s proof that if intuitionistic first-order arithmetic is consistent, then so is classical, and Gödel’s second incompleteness theorem. A full proof of the latter and a correct proof of Herbrand’s theorem only appeared in 1939, in volume II of Hilbert and Bernays, Grundlagen der Mathematik.

Abstract

The paper comments on two papers in French that Paul Bernays derived from lectures at a conference on mathematical logic held in Geneva in June 1934. The first, the well-known “On Platonism in mathematics,” sets forth a methodological version of Platonism and observes that it can be implemented in some branches of mathematics and not others. He notes that by his definition Brouwer’s intuitionism rejects all Platonism,while HermannWeyl’s reconstruction of analysis retains it for generalizations about natural numbers but not for generalizations about real numbers or higher-type objects. He rejects the then widespread idea of a crisis of foundations and argues that the questions raised are philosophical. Bernays’ second paper, “Some observations on metamathematics,” is a technical sequel to “On Platonism.” It describes some basic points in the Hilbert school’s proof theory and sketches some results, such as a simple application of Herbrand’s theorem, Gödel’s proof that if intuitionistic first-order arithmetic is consistent, then so is classical, and Gödel’s second incompleteness theorem. A full proof of the latter and a correct proof of Herbrand’s theorem only appeared in 1939, in volume II of Hilbert and Bernays, Grundlagen der Mathematik.

Chapters in this book

  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/10.1515/9783110657883-017

Chapters in this book

  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
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