Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-18T11:57:21.775Z Has data issue: false hasContentIssue false
  • English
  • Français

STRICT FINITISM, FEASIBILITY, AND THE SORITES

Published online by Cambridge University Press:  30 July 2018

WALTER DEAN
WALTER DEAN*
Affiliation:
Department of Philosophy, University of Warwick
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF WARWICK COVENTRY CV4 7AL, UK E-mail: W.H.Dean@warwick.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This article bears on four topics: observational predicates and phenomenal properties, vagueness, strict finitism as a philosophy of mathematics, and the analysis of feasible computability. It is argued that reactions to strict finitism point towards a semantics for vague predicates in the form of nonstandard models of weak arithmetical theories of the sort originally introduced to characterize the notion of feasibility as understood in computational complexity theory. The approach described eschews the use of nonclassical logic and related devices like degrees of truth or supervaluation. Like epistemic approaches to vagueness, it may thus be smoothly integrated with the use of classical model theory as widely employed in natural language semantics. But unlike epistemicism, the described approach fails to imply either the existence of sharp boundaries or the failure of tolerance for soritical predicates. Applications of measurement theory (in the sense of Krantz, Luce, Suppes, & Tversky (1971)) to vagueness in the nonstandard setting are also explored.

Keywords

03B5203H1503D15Strict finitismultrafinitismvaguenesssorites paradoxnonstandard modelsbounded arithmeticsemisetsYessenin-VolpinDummett
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 2 , June 2018 , pp. 295 - 346
Copyright
Copyright © Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Arora, S. & Barak, B. (2009). Computational Complexity: A Modern Approach. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Barker, C. (2002). The dynamics of vagueness. Linguistics and Philosophy, 25(1), 1–36.CrossRefGoogle Scholar
Barnes, J. (1982). Medicine, experience and logic. In Barnes, J., Brunschwig, J., Burnyeat, M. F., and Schofield, M., editors. Science and Speculation. Cambridge: Cambridge University Press, pp. 24–68.Google Scholar
Benacerraf, P. & Putnam, H. (editors) (1983). Philosophy of Mathematics: Selected Readings (second edition). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Bernays, P. (1935). Sur le platonisme dans les mathématiques. L’enseignement mathematique, 34, 52–69. Reprinted in Benacerraf and Putnam (1983), pp. 58–71.Google Scholar
Boolos, G. (1991). Zooming down the slippery slope. Nous, 25(5), 695–706.CrossRefGoogle Scholar
Borel, E. (1952). Les Nombres Inaccessibles. Paris: Gauthier-Villars.Google Scholar
Buss, S. (1986). Bounded Arithmetic. Naples: Bibliopolis.Google Scholar
Carbone, A. (1996). Cycling in proofs, feasibility and no speed-up for nonstandard arithmetic. Technical Report IHES/M/96/55, Institut des Hautes Études Scientifiques.Google Scholar
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.CrossRefGoogle Scholar
Cook, S. (1975). Feasibly constructive proofs and the propositional calculus (preliminary version). Proceedings of Seventh Annual ACM Symposium on Theory of Computing. New York: Association for Computing Machinery, pp. 83–97.CrossRefGoogle Scholar
Cook, S. & Nguyen, P. (2010). Logical foundations of Proof Complexity. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Dean, W. (2015). Computational complexity theory. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy, Fall 2015 Edition. Available at https://plato.stanford.edu/archives/win2016/entries/computational-complexity/.Google Scholar
Dean, W. & Walsh, S. (2017). The prehistory of the subsystems of second-order arithmetic. The Review of Symbolic Logic, 10(2), 357–396.CrossRefGoogle Scholar
Dietz, R. (2011). The paradox of vagueness. In Horsten, L., and Pettigrew, R., editors. Continuum Companion to Philosophical Logic. New York: Continuum, pp. 128–179.Google Scholar
Dragalin, A. G. (1985). Correctness of inconsistent theories with notions of feasibility. In Skowron, A., editor. Computation Theory: Fifth Symposium, Zaborow, Poland, Lecture Notes in Computer Science, Vol. 208. Berlin: Springer, pp. 58–79.CrossRefGoogle Scholar
Dummett, M. (1959). Wittgenstein’s philosophy of mathematics. The Philosophical Review, 68(3), 324–348.CrossRefGoogle Scholar
Dummett, M. (1975). Wang’s Paradox. Synthese, 30(3/4), 301–324.CrossRefGoogle Scholar
Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., & Sieg, W. editors (2003). Kurt Gödel Collected Works. Vol. IV. Publications Correspondence A-G. Oxford Univeristy Press.Google Scholar
Fernando, T. & Kamp, H. (1996). Expecting many. In Galloway, T. and Spence, J., editors. Proceeding from SALT VI. Ithaca, NY: CLC Publications, pp. 53–68.Google Scholar
Field, H. (2008). Saving Truth from Paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
Fine, K. (1975). Vagueness, truth and logic. Synthese, 30(3), 265–300.CrossRefGoogle Scholar
Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Nebert.Google Scholar
Frege, G. (1884). Die Grundlagen der Arithmetik. Breslau: Koebner.Google Scholar
Frege, G. (1893, 1903). Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet. Jena: Pohle.Google Scholar
Gaifman, H. (2004). Non-standard models in a broader perspective. In Enayat, A. and Kossak, R., editors. Non-Standard Models of Arithmetic and Set Theory. Providence, RI: American Mathematical Society, pp. 1–22.Google Scholar
Gaifman, H. (2010). Vagueness, tolerance and contextual logic. Synthese, 174, 1–42.CrossRefGoogle Scholar
Gandy, R. (1982). Limitations to mathematical knowledge. In van Dalen, D., Lascar, D., and Smiley, J., editors. Logic Colloquium 80, Vol. 108. Amsterdam: Elsevier, pp. 129–146.Google Scholar
Ganea, M. (2010). Two (or three) notions of finitism. The Review of Symbolic Logic, 3(01), 119–144.CrossRefGoogle Scholar
Geiser, J. R. (1974). A formalization of essenin-volpin’s proof theoretical studies by means of nonstandard analysis. The Journal of Symbolic Logic, 39(1), 81–87.CrossRefGoogle Scholar
Gómez-Torrente, M. (2010). The sorites, linguistic preconceptions, and the dual picture of vagueness. In Dietz, R. and Moruzzi, S., editors. Cuts and Clouds: Vaguenesss, its Nature and its Logic. Oxford: Oxford University Press, pp. 228–253.CrossRefGoogle Scholar
Graff, D. (2001). Phenomenal continua and the sorites. Mind, 110(440), 905–936.CrossRefGoogle Scholar
Hájek, P. (1973). Why semisets? Commentationes Mathematicae Universitatis Carolinae, 14(3), 397–420.Google Scholar
Hájek, P. & Pudlák, P. (1998). Metamathematics of First-Order Arithmetic. Berlin: Springer.Google Scholar
Hallett, M. & Majer, U. (editors) (2004). David Hilbert’s Lectures on the Foundations of Geometry, 1891–1902. David Hilbert’s Foundational Lectures, Vol. 1. Berlin: Springer.CrossRefGoogle Scholar
Heck, R. G. (2014). Predicative frege arithmetic and ‘Everyday’ mathematics. Philosophia Mathematica, 22(3), 279–307.CrossRefGoogle Scholar
Hyde, D. (2011). The sorites paradox. In Ronzitti, G., editor. Vagueness: A Guide. Berlin: Springer, pp. 1–17.Google Scholar
Isles, D. (1981). Remarks on the notion of standard non-isomorphic natural number series. In Richman, F., editor. Constructive Mathematics. Berlin: Springer, pp. 111–134.CrossRefGoogle Scholar
Kamp, H. (1981). The paradox of the heap. In Monnich, U., editor. Aspects of Philosophical Logic. Berlin: Springer, pp. 225–277.CrossRefGoogle Scholar
Kanovei, V. & Reeken, M. (2004). Nonstandard Analysis, Axiomatically. Berlin: Springer.CrossRefGoogle Scholar
Kaye, R. (1991). Models of Peano Arithmetic. Oxford Logic Guides, Vol. 15. Oxford: Oxford University Press.Google Scholar
Keefe, R. (2000). Theories of Vagueness. Cambridge: Cambridge University Press.Google Scholar
Koellner, P. (2009). Truth in mathematics: The question of pluralism. In Linnebo, O. and Bueno, O., editors. New Waves in the Philosophy of Mathematics. London: Palgrave Macmillan UK, pp. 80–116.CrossRefGoogle Scholar
Kölbel, M. (2010). Vagueness as semantic. In Dietz, R. and Moruzzi, S., editors, Cuts and Clouds: Vaguenesss, its Nature and its Logic. Oxford University Press.Google Scholar
Krantz, D., Luce, R., Suppes, P., & Tversky, A. (1971). Foundations of Measurement: Additive and Polynomial Representations, Vol. I. Academic Press.Google Scholar
Kreisel, G. (1958). Wittgenstein’s remarks on the foundations of mathematics. The British Journal for the Philosophy of Science, 9(34), 135–158.CrossRefGoogle Scholar
Kreisel, G. (1967a). Informal rigour and completeness proofs. In Lakatos, I., editor. Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 138–186.CrossRefGoogle Scholar
Kreisel, G. (1967b). Review of “Le programme ultra-intuitionniste des fondements des mathématiques.” (Volpin). Zentrablatt Math, 134, 9–10.Google Scholar
Kreisel, G. & Ehrenfeucht, A. (1967). Review of “Le programme ultra-intuitionniste des fondements des mathmatiques” (Volpin). The Journal of Symbolic Logic, 32(4), 517.CrossRefGoogle Scholar
Lassiter, D. (2011). Measurement and Modality: The Scalar Basis of Modal Semantics. New York: New York University dissertation.Google Scholar
Luce, D., Krantz, D., Suppes, P., & Tversky, A. (1990). Foundations of Measurement: Representation, Axiomatization, and Invariance, Vol. III. Mineola: Dover.Google Scholar
Luce, R. D. (1956). Semiorders and a theory of utility discrimination. Econometrica, Journal of the Econometric Society, 24(2), 178–191.CrossRefGoogle Scholar
Magidor, O. (2011). Strict finitism and the happy sorites. Journal of Philosophical Logic, 41(2), 1–21.Google Scholar
Mannoury, G. (1931). Woord en gedachte: een inleiding tot de signifika, inzonderheid met het oog op het onderwijs in de wiskunde. Groningen: Noordhoff.Google Scholar
Mostowski, A. (1950). Some impredicative definitions in the axiomatic set-theory. Fundamenta mathematicae, 38, 110–124.Google Scholar
Narens, L. (1985). Abstract Measurement Theory. Cambridge, MA: MIT Press.Google Scholar
Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.CrossRefGoogle Scholar
Nelson, E. (1986). Predicative Arithmetic. Mathematical Notes, Vol. 32. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Novák, V. (1992). The Alternative Mathematical Model of Linguistic Semantics and Pragmatics. Berlin: Springer.CrossRefGoogle Scholar
Parikh, R. (1971). Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36(3), 494–508.CrossRefGoogle Scholar
Parikh, R. (1983). The problem of vague predicates. In Cohen, R., and Wartofsky, M., editors. Language, Logic and Method, Berlin: Springer, pp. 241–261.CrossRefGoogle Scholar
Poincaré, H. (1906). Les mathématiques et la logique. Revue de métaphysique et de morale, 14(3), 294–317.Google Scholar
Putnam, H. (1980). Models and reality. The Journal of Symbolic Logic, 45(3), 464–482.CrossRefGoogle Scholar
Raffman, D. (1994). Vagueness without paradox. The Philosophical Review, 103(1), 41–74.CrossRefGoogle Scholar
Robinson, A. (1966). A Non-Standard Analysis. Amsterdam: North-Holland.Google Scholar
Rogers, H. (1987). Theory of Recursive Functions and Effective Computability (Second Edition). Cambridge, MA: MIT Press.Google Scholar
Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222–262.CrossRefGoogle Scholar
Sainsbury, M. (1991). Is there higher-order vagueness? The Philosophical Quarterly, 41(163), 167–182.CrossRefGoogle Scholar
Sainsbury, R. (1995). Paradoxes. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Sainsbury, R. M. (1990). Concepts without boundaries. In Keefe, R. and Smith, P., editors. Vagueness: A Reader. Cambridge, MA: MIT Press, pp. 251–264.Google Scholar
Sassoon, G. W. (2010). Measurement theory in linguistics. Synthese, 174(1), 151–180.CrossRefGoogle Scholar
Sazonov, V. (1995). On feasible numbers. In Leivant, D., editor. Logic and Computational Complexity. Springer Lecture Notes in Computer Science, Vol. 960. Berlin: Springer, pp. 30–51.CrossRefGoogle Scholar
Scott, D. & Suppes, P. (1958). Foundational aspects of theories of measurement. Journal of Symbolic Logic, 23(2), 113–128.CrossRefGoogle Scholar
Shapiro, S. (2011). Vagueness and logic. In Ronzitti, G., editor. Vagueness: A Guide. Berlin: Springer, pp. 55–81.CrossRefGoogle Scholar
Sheard, M. (1998). Induction the hard way. The American Mathematical Monthly, 105(4), 348–353.CrossRefGoogle Scholar
Simpson, S. (2009). Subsystems of Second Order Arithmetic (second edition). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Skala, H. J. (1975). Non-Archimedean Utility Theory, Vol. 9. Dordrecht: D. Reidel.CrossRefGoogle Scholar
Solovay, R. M. (1976). Interpretability in set theories. Letter to Petr Hájek.Google Scholar
Sorensen, R. (2001). Vagueness and Contradiction. Oxford: Oxford University Press.Google Scholar
Sorensen, R. A. (1988). Blindspots. Oxford: Clarendon.Google Scholar
Steiner, M. (2011). Kripke on logicism, Wittgenstein, and de re beliefs about numbers. In Berger, A., editor. Saul Kripke. Cambridge: Cambridge University Press, pp. 160–176.CrossRefGoogle Scholar
Suppes, P., Krantz, D., Luce, D., & Tversky, A. (1989). Foundations of Measurement: Geometrical, Threshold, and Probabilistic Representations, Vol. II. San Diego, CA: Academic Press.Google Scholar
Tait, W. W. (1981). Finitism. Journal of Philosophy, 78(9), 524–546.CrossRefGoogle Scholar
Tye, M. (1994). Sorites paradoxes and the semantics of vagueness. Philosophical Perspectives, 8, 189–206.CrossRefGoogle Scholar
Tzouvaras, A. (1998). Modeling vagueness by nonstandardness. Fuzzy Sets and Systems, 94(3), 385–396.CrossRefGoogle Scholar
Unger, P. (1979). There are no ordinary things. Synthese, 41(2), 117–154.CrossRefGoogle Scholar
van Dantzig, D. (1955). Is 0000${10^{{{10}^{10}}}}$ a finite number? Dialectica, 9(3–4), 273–277.CrossRefGoogle Scholar
van Rooij, R. (2011). Vagueness and linguistics. In Ronzitti, G., editor. Vagueness: A Guide. Berlin: Springer, 123–170.CrossRefGoogle Scholar
Vopěnka, P. (1979). Mathematics in the Alternative Set Theory. Leipzig: Teubner.Google Scholar
Wang, H. (1958). Eighty years of foundational studies. Dialectica, 12(3–4), 466–497.CrossRefGoogle Scholar
Wang, H. (1990). Computation, Logic, and Philosophy. Boston: Kluwer.CrossRefGoogle Scholar
Weber, Z. & Colyvan, M. (2010). A topological sorites. The Journal of Philosophy, 107(6), 311–325.CrossRefGoogle Scholar
Williamson, T. (1994). Vaugeness. London and New York: Routledge.Google Scholar
Wright, C. (2010). The illusion of higher-order vagueness. In Dietz, R. and Moruzzi, S., editors. Cuts and Clouds: Vagueness, its Nature, and its Logic. Oxford: Oxford Univeristy Press, pp. 523–549.CrossRefGoogle Scholar
Wright, W. & Pitt, F. (1934). Hue-discrimination in normal colour-vision. Proceedings of the Physical Society, 46(3), 459.CrossRefGoogle Scholar
Yessenin-Volpin, A. (1961). Le programme ultra-intuitionniste des fondements des mathématiques. Infinitistic Methods, Proceedings of the Symposium on the Foundations of Mathematics. Oxford: Pergamon Press, pp. 201–223.Google Scholar
Yessenin-Volpin, A. (1970). The ultra-intuitionistic criticism and the antitraditional program for the foundations of mathematics. In Kino, A., Myhill, J., and Vesley, R., editors. Intuitionism and Proof Theory. Amsterdam: North Holland, pp. 3–45.Google Scholar
Yessenin-Volpin, A. (1981). About infinity, finiteness and finitization (in connection with the foundations of mathematics). In Richman, F., editor. Constructive Mathematics. Berlin: Springer, pp. 274–313.CrossRefGoogle Scholar
Yessenin-Volpin, A. (2011). Letter to Brouwer (dated 23-7-1958). In van Dalen, D., editor. The Selected Correspondence of Luitzen Egbertus Jan Brouwer. Berlin: Springer, pp. 457–459.Google Scholar
Zambella, D. (1996). Notes on polynomially bounded arithmetic. The Journal of Symbolic Logic, 61(3), 942–966.CrossRefGoogle Scholar
Zermelo, E. (1929). Über den Begriff der Definitheit in der Axiomatik. Fundamenta Mathematicae, 14, 339–344.CrossRefGoogle Scholar