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Philosophy of mathematics

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Abstract
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The paper critically examines interpretations of Wittgenstein's propositions on mathematics in the Tractatus, focusing on paragraphs 6.02-6.031 and 6.2-6.241. It highlights the need for accurate exegesis to reconsider the philosophy of mathematics effectively, addressing common misinterpretations by Black and Anscombe. By emphasizing the inductive definition of natural numbers and the operational scheme introduced by Wittgenstein, the paper seeks to clarify fundamental ideas about mathematical calculations and their philosophical implications.

Key takeaways
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  1. Wittgenstein's philosophy of mathematics highlights the importance of understanding mathematical concepts as rules rather than objects.
  2. The interpretation of numerical identities requires a clear reading of Wittgenstein's Tractatus to avoid misinterpretations.
  3. Proofs in mathematics serve as paradigms, establishing rules for correctness in sign manipulations.
  4. Wittgenstein critiques the extensional understanding of infinite sets, emphasizing the role of generative laws.
  5. The Law of Excluded Middle is not universally applicable in mathematics, particularly regarding unproven propositions.

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References (125)

  1. = ( 2(2+3) Ω' 1(2+3) Ω)'x [from definition [III] by substitution of "Ω" with " (2+3) Ω"]
  2. = (( (2+3) Ω'
  3. Ω)' (2+3) Ω)'x [from [B] by substitution of "Ω" with " (2+3) Ω"]
  4. = ((( 2 Ω' 3 Ω)'( 2 Ω' 3 Ω))'( 2 Ω' 3 Ω))'x [definition [IV]]
  5. = ((((Ω'Ω)'(Ω'(Ω'Ω
  6. '((Ω'Ω)'(Ω'(Ω'Ω
  7. = (((Ω'Ω)'(Ω'(Ω'Ω
  8. '((Ω'Ω)'(Ω'(Ω'Ω'
  9. = (((Ω'Ω)'(Ω'(Ω'Ω
  10. '((Ω'Ω)'(Ω'(Ω'Ω
  11. '(Ω'Ω)'(Ω'(Ω'Ω))'x [[C]]
  12. = (((Ω'Ω)'(Ω'(Ω'Ω
  13. '((Ω'Ω)'(Ω'(Ω'Ω
  14. '(Ω'Ω)'Ω'Ω'Ω'x [[C and A]]
  15. = (((Ω'Ω)'(Ω'(Ω'Ω
  16. '((Ω'Ω)'(Ω'(Ω'Ω
  17. 'Ω'Ω'Ω'Ω'Ω'x [[A]]
  18. = ((Ω'Ω)'(Ω'(Ω'Ω
  19. '((Ω'Ω)'(Ω'(Ω'Ω
  20. 'Ω'Ω'Ω'Ω'Ω'x [[C]]
  21. = ((Ω'Ω)'(Ω'(Ω'Ω
  22. '(Ω'Ω)'(Ω'(Ω'Ω))'Ω'Ω'Ω'Ω'Ω'x [[C]]
  23. = ((Ω'Ω)'(Ω'(Ω'Ω
  24. '(Ω'Ω)'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'x [[C and A]]
  25. = ((Ω'Ω)'(Ω'(Ω'
  26. 'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'x [[A]]
  27. = (Ω'Ω)'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'Ω'x [[C and A]]
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