Startseite Philosophie 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
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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

  • Susan Edwards-McKie
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Philosophy of Logic and Mathematics
Ein Kapitel aus dem Buch Philosophy of Logic and Mathematics

Abstract

I shall build on my paper “Following a Rule without the Platonic Equivalent: Wittgenstein’s Intentionality and Generality” (in The Philosophy of Perception and Observation: Contributions to the 40th International Wittgenstein Symposium, 2017) which explored the relation of the iterative operation to the potential infinite. Firstly, focussing on the principle of contextuality, I look at similarities and differences between Wittgenstein and Frege, which harmonize in interesting ways with the Dedekind cut and the actual infinite when viewed from the Fregean standpoint, but form a distinctly non-Dedekind paradigm when viewed from Wittgenstein’s standpoint. I shall consider the principle of composition through Frege’s critical question to Wittgenstein: “What cements things together?” with questions of range, part and whole. Wittgenstein’s idea that it is the Eigenschaft of “5” to be the Gegenstand of the rule “3 + 2 = 5” is contrasted with Frege’s Platonic work in “Der Gedanke”. Questions of the role of the Tractarian Gegenstand in developing rules of iteration, compositionality and use, and McGuinness’ and Pears’ retranslation of Sachverhalte from “atomic fact” to that which is in-potentia (state of affairs) is briefly highlighted. Lastly, I provide a Nachlass discovery which suggests Wittgenstein continued to work on the highly mathematical TS 222, which later becomes Remarks on the Foundations of Mathematics, later than hitherto thought by scholars, precisely in the areas we have considered in the previous sections.

Abstract

I shall build on my paper “Following a Rule without the Platonic Equivalent: Wittgenstein’s Intentionality and Generality” (in The Philosophy of Perception and Observation: Contributions to the 40th International Wittgenstein Symposium, 2017) which explored the relation of the iterative operation to the potential infinite. Firstly, focussing on the principle of contextuality, I look at similarities and differences between Wittgenstein and Frege, which harmonize in interesting ways with the Dedekind cut and the actual infinite when viewed from the Fregean standpoint, but form a distinctly non-Dedekind paradigm when viewed from Wittgenstein’s standpoint. I shall consider the principle of composition through Frege’s critical question to Wittgenstein: “What cements things together?” with questions of range, part and whole. Wittgenstein’s idea that it is the Eigenschaft of “5” to be the Gegenstand of the rule “3 + 2 = 5” is contrasted with Frege’s Platonic work in “Der Gedanke”. Questions of the role of the Tractarian Gegenstand in developing rules of iteration, compositionality and use, and McGuinness’ and Pears’ retranslation of Sachverhalte from “atomic fact” to that which is in-potentia (state of affairs) is briefly highlighted. Lastly, I provide a Nachlass discovery which suggests Wittgenstein continued to work on the highly mathematical TS 222, which later becomes Remarks on the Foundations of Mathematics, later than hitherto thought by scholars, precisely in the areas we have considered in the previous sections.

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/10.1515/9783110657883-027

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
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Heruntergeladen am 17.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/9783110657883-027/html
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