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Radical anti-realism, Wittgenstein and the length of proofs

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Abstract

After sketching an argument for radical anti-realism that does not appeal to human limitations but polynomial-time computability in its definition of feasibility, I revisit an argument by Wittgenstein on the surveyability of proofs, and then examine the consequences of its application to the notion of canonical proof in contemporary proof-theoretical-semantics.

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References

  • Asperti A. and Roversi L. (2002). Intuitionistic light affine logic. ACM Transactions on Computational Logic 3: 1–39

    Article  Google Scholar 

  • Bellantoni S. and Hofmann M. (2002). A new “feasible” arithmetic. Journal of Symbolic Logic 67: 104–116

    Article  Google Scholar 

  • Carbone A. (2000). Cycling in proofs and feasibility. Transactions of the American Mathematical Society 352: 2049–2075

    Article  Google Scholar 

  • Carbone A. and Semmes S. (1997). Making proofs without modus ponens: Introduction to the combinatorics and complexity of cut Elimination. Bulletin of the American Mathematical Society 34: 131–159

    Article  Google Scholar 

  • Davis M. (1982). Why Gödel didn’t have a Church Thesis. Information and Control 54: 3–24

    Article  Google Scholar 

  • Dubucs J. (2002). Feasibility in logic. Synthese 132: 213–217

    Article  Google Scholar 

  • Dubucs J. and Lapointe S. (2006). On Bolzano’s alleged explicativism. Synthese 150: 229–246

    Article  Google Scholar 

  • Dubucs J. and Marion M. (2003). Radical anti-realism and substructural logics. In: Rojszczak, A., Cachro, J. and Hanuszewicz, S. (eds) Philosophical dimensions of logic and science. Selected Contributed Papers of the 11th International Congress of Logic, Methodology and Philosophy of Science, Krakòw, 1999, pp 235–249. Kluwer, Dordrecht

    Google Scholar 

  • Dummett M.A.E. (1978). Truth and other enigmas. Duckworth, London

    Google Scholar 

  • Dummett M.A.E. (1993). The seas of language. Clarendon Press, Oxford

    Google Scholar 

  • Gandy R.O. (1982). Limitations to mathematical knowledge. In: van Dalen, D., Lascar, D. and Smiley, T. (eds) Logic colloquium ’80, pp 129–146. North-Holland, Amsterdam

    Google Scholar 

  • Garey M.R. and Johnson D.S. (1979). Computers and intractability A guide to the theory of NP-completeness. Freeman & Co., New York

    Google Scholar 

  • Girard J.-Y. (1995). Light linear logic. In: Leivant, D. (eds) Logic and computational complexity, pp 145–176. Springer, Berlin

    Google Scholar 

  • Leivant D. (1994). A foundational delineation of poly-time. Information and Computation 110: 391–420

    Article  Google Scholar 

  • Mancosu P. (2000). On mathematical explanation. In: Grosholz, E. and Breger, H. (eds) The growth of mathematical knowledge, pp 103–119. Kluwer, Dordrecht

    Google Scholar 

  • Marion M. (1998). Wittgenstein, finitism and the foundations of mathematics. Clarendon Press, Oxford

    Google Scholar 

  • Marion, M. (2007). Interpreting arithmetic: Russell on applicability and Wittgenstein on surveyability. In P. Joray (Ed.), Contemporary perspectives on logicism and constructivism, Travaux de Logique (Vol. 18, pp. 167–184). Neuchâtel: CdRS.

  • Mathias A.R.D. (2002). A term of length 4 523 659 424 929. Synthese 133: 75–86

    Article  Google Scholar 

  • Mühlhölzer F. (2005). “A mathematical proof must be surveyable” what Wittgenstein meant by this and what it implies. Grazer Philosophische Studien 71: 57–86

    Google Scholar 

  • Parikh R. (1971). Existence and feasibility in arithmetic. Journal of Symbolic Logic 36: 494–508

    Article  Google Scholar 

  • Prawitz D. (1965). Natural deduction. A proof-theoretical Study. Almqvist & Wicksell, Stockholm

    Google Scholar 

  • Prawitz D. (1974). On the idea of a general proof theory. Synthese 27: 63–77

    Article  Google Scholar 

  • Prawitz D. (2005). Logical consequence from a constructivist point of view. In: Shapiro, S. (eds) The Oxford handbook of philosophy of mathematics and logic, pp 671–695. Oxford University Press, Oxford

    Google Scholar 

  • Prawitz D. (2006). Meaning approached via proofs. Synthese 148: 507–524

    Article  Google Scholar 

  • Pudlak P. (1998). The lengths of proofs. In: Buss, S.R. (eds) Handbook of proof theory, pp 547–638. Elsevier, Amsterdam

    Chapter  Google Scholar 

  • Schroeder-Heister P. (2006). Validity concepts in proof-theoretic semantics. Synthese 148: 525–571

    Article  Google Scholar 

  • Shieh S. (1998). Undecidability and anti-realism. Philosophia Mathematica 3rd Series 6: 324–333

    Google Scholar 

  • Tatzel A. (2002). Bolzano’s theory of ground and consequence. Notre Dame Journal of Formal Logic 43: 1–25

    Article  Google Scholar 

  • Turing A.M. (1936). On computable numbers with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 2nd Series 41: 230–265

    Google Scholar 

  • Wright C. (1993). Realism, meaning and truth (2nd ed.). Blackwell, Oxford

    Google Scholar 

Download references

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Correspondence to Mathieu Marion.

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Marion, M. Radical anti-realism, Wittgenstein and the length of proofs. Synthese 171, 419–432 (2009). https://doi.org/10.1007/s11229-008-9315-9

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  • DOI: https://doi.org/10.1007/s11229-008-9315-9

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