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
Bellantoni S. and Hofmann M. (2002). A new “feasible” arithmetic. Journal of Symbolic Logic 67: 104–116
Carbone A. (2000). Cycling in proofs and feasibility. Transactions of the American Mathematical Society 352: 2049–2075
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
Davis M. (1982). Why Gödel didn’t have a Church Thesis. Information and Control 54: 3–24
Dubucs J. (2002). Feasibility in logic. Synthese 132: 213–217
Dubucs J. and Lapointe S. (2006). On Bolzano’s alleged explicativism. Synthese 150: 229–246
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
Dummett M.A.E. (1978). Truth and other enigmas. Duckworth, London
Dummett M.A.E. (1993). The seas of language. Clarendon Press, Oxford
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
Garey M.R. and Johnson D.S. (1979). Computers and intractability A guide to the theory of NP-completeness. Freeman & Co., New York
Girard J.-Y. (1995). Light linear logic. In: Leivant, D. (eds) Logic and computational complexity, pp 145–176. Springer, Berlin
Leivant D. (1994). A foundational delineation of poly-time. Information and Computation 110: 391–420
Mancosu P. (2000). On mathematical explanation. In: Grosholz, E. and Breger, H. (eds) The growth of mathematical knowledge, pp 103–119. Kluwer, Dordrecht
Marion M. (1998). Wittgenstein, finitism and the foundations of mathematics. Clarendon Press, Oxford
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
Mühlhölzer F. (2005). “A mathematical proof must be surveyable” what Wittgenstein meant by this and what it implies. Grazer Philosophische Studien 71: 57–86
Parikh R. (1971). Existence and feasibility in arithmetic. Journal of Symbolic Logic 36: 494–508
Prawitz D. (1965). Natural deduction. A proof-theoretical Study. Almqvist & Wicksell, Stockholm
Prawitz D. (1974). On the idea of a general proof theory. Synthese 27: 63–77
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
Prawitz D. (2006). Meaning approached via proofs. Synthese 148: 507–524
Pudlak P. (1998). The lengths of proofs. In: Buss, S.R. (eds) Handbook of proof theory, pp 547–638. Elsevier, Amsterdam
Schroeder-Heister P. (2006). Validity concepts in proof-theoretic semantics. Synthese 148: 525–571
Shieh S. (1998). Undecidability and anti-realism. Philosophia Mathematica 3rd Series 6: 324–333
Tatzel A. (2002). Bolzano’s theory of ground and consequence. Notre Dame Journal of Formal Logic 43: 1–25
Turing A.M. (1936). On computable numbers with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 2nd Series 41: 230–265
Wright C. (1993). Realism, meaning and truth (2nd ed.). Blackwell, Oxford
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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