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Truth and the Philosophy of Mathematics

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Objects, Structures, and Logics

Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 339))

Abstract

Is truth – qua a primitive notion – fit to play an independent role in the philosophy of mathematics and in the foundational investigations? The problem is handled by surveying axiomatic theories of truth and their implications, with a main concern for ontological and epistemological issues.

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Notes

  1. 1.

    Research partially supported by PRIN 2017.

  2. 2.

    Cantini (2017) being dedicated mainly to Feferman’s work.

  3. 3.

    See Feferman (1998a) pp.123–24.

  4. 4.

    The model we have in mind derives from practice, it is simply either the set of informal elucidations that precede the informal presentation or development of a theory, axiomatic or not.

  5. 5.

    Let me cite Tarski himself, Tarski (1944), p.352: When a language is unable to define truth, we then have to include the term “true” or some other semantic term, in the list of undefined terms of the meta-language, and to express fundamental properties of the notion of truth in a series of axioms. There is nothing essentially wrong in such an axiomatic procedure, and it may prove useful for various purposes.

    In Tarski (1956), p.266, Tarski similarly writes that for some of the languages for which truth cannot be defined, we can nevertheless make “consistent and correct use” of the concept of truth by way of taking truth as a primitive notion, and giving it content by introducing the relevant sorts of axioms.

  6. 6.

    The link between semantical investigations and foundations of mathematics is certainly not new, see the work following Kripke (1975) by Feferman and others (as documented in the references of Feferman (2008)) and all the recent investigations of axiomatic theories of truth (Cantini, 2017).

  7. 7.

    We have in mind constructive set theory as developed by Aczel and Rathjen (2001), explicit mathematics à la (Feferman, 1998c), metapredicativity in the sense of the Bern school (see Jäger 2005).

  8. 8.

    But consider the approach in Parsons (1971, 1974), Cantini (1996), and recently Schindler (2018).

  9. 9.

    See Weyl (1910).

  10. 10.

    We quote Hilbert’s text as translated in Parsons (2002), p.378; see also p.74, Mancosu (1998), and see Hilbert (2013).

  11. 11.

    E.g. permutations, duplications, identification, expansion of variables.

  12. 12.

    For all these theories, we refer to the monograph (Halbach, 2011).

  13. 13.

    Formulas generated from atomic formulas via Boolean operations and bounded quantifiers and ∃x < t… with <  representing the standard ordering on natural numbers.

  14. 14.

    See Theorem 3.2 in Fischer (2014).

  15. 15.

    Its most recent version being the so-called explicit mathematics which was introduced in the Mid Seventies by Feferman (1979) and is still being developed by the Bern school and others. See also Feferman (1998b) and the comments in Parsons (1971) and Cantini (2016).

  16. 16.

    These are neatly analyzed probably for the first time by Weyl (1910).

  17. 17.

    Note that this excludes the principles about very large cardinals, determinacy, etc.

  18. 18.

    Typically this happens for the Liar sentence L.

References

  • Aczel, P. 1980. Frege structures and the notions of proposition, truth and set. In The Kleene Symposium, ed. J. Barwise, H. Keisler, and K. Kunen, 31–59. Amsterdam: North Holland.

    Chapter  Google Scholar 

  • Aczel, P., and M. Rathjen. 2001. Notes on constructive set theory. Technical Report 40, Institut Mittag-Leffler, Djursholm.

    Google Scholar 

  • Burgess, J. 2014. Friedman and the axiomatization of Kripke’s theory of truth. In Foundational adventures: Essays in honor of Harvey M. Friedman, ed. N. Tennant, 125–128. London: College Publications.

    Google Scholar 

  • Caldon, P., and A. Ignjatovic. 2005. On mathematical instrumentalism. Journal of Symbolic Logic 50: 778–794.

    Article  Google Scholar 

  • Cantini, A. 1985a. A note on a predicatively reducible theory of iterated elementary induction. Bollettino dell’Unione Matematica Italiana 6(2): 1–17.

    Google Scholar 

  • Cantini, A. 1985b. Una nota sulla concezione semi–intuizionistica della matematica. Rivista di Filosofia 69: 465–486.

    Google Scholar 

  • Cantini, A. 1989. Notes on formal theories of truth. Zeitschrift f. Math. Logik u. Grundlagen der Mathematik 35: 97–130.

    Article  Google Scholar 

  • Cantini, A. 1990. A theory of formal truth arithmetically equivalent to I D1. Journal of Symbolic Logic 55: 244–259.

    Article  Google Scholar 

  • Cantini, A. 1996. Logical Frameworks for Truth and Abstraction. Amsterdam: North Holland.

    Google Scholar 

  • Cantini, A. 2016. About truth and types. In Advances in proof theory, volume 28 of Progress in computer science and applied logic, ed. R. Kahle, T. Strahm, and T. Studer, 287–314. Cham: Birkhäuser–Springer.

    Google Scholar 

  • Cantini, A. 2017. Feferman and the truth. In Feferman on foundations, volume 17 of Outstanding contributions to logic, ed. G. Jäger and W. Sieg, 31–64. Cham: Springer.

    Google Scholar 

  • Cantini, A. 2020. A fixed point theory over stratified truth. Mathematical Logic Quarterly. to appear, 17 pages.

    Google Scholar 

  • Cantini, A., and L. Crosilla. 2010. Elementary constructive operational set theory. In Ways of proof theory, ed. R. Schindler, 199–240. Frankfurt: Ontos Verlag.

    Chapter  Google Scholar 

  • Cellucci, C. 2017. Rethinking knowledge: The heuristic view. Cham: Springer.

    Book  Google Scholar 

  • Cieśliński, C., M. Łełyk, and B. Wcisło. 2017. Models of PT− with internal induction for total formulae. Review of Symbolic Logic 10: 187–202.

    Article  Google Scholar 

  • Crosilla, L. 2017. Predicativity and Feferman. In Feferman on foundations, volume 17 of Outstanding contributions to logic, ed. G. Jäger, and W. Sieg, 423–447. Cham: Springer.

    Google Scholar 

  • Dummett, M. 2004. Truth and the past. New York: Columbia U.P.

    Google Scholar 

  • Ehrenfeucht, A., and J. Mycielski. 2020. Abbreviating proofs by adding new axioms. Bulletin of the American Mathematical Society 77: 366–367.

    Article  Google Scholar 

  • Enayat, A., M. Lełyk, and B. Wcisło. 2020. Truth and feasible reducibility. Journal of Symbolic Logic 85: 367–421.

    Article  Google Scholar 

  • Enayat, A., and A. Visser. 2015. New constructions of satisfaction classes. In Unifying the philosophy of truth, Number 36 in Log. Epistemol. Unity Sci., ed. T. Achourioti, H. Galinon, J.M. Martinez, and K. Fujimoto, 321–35. Cham: Springer.

    Chapter  Google Scholar 

  • Feferman, S. 1979. Constructive theories of functions and classes. In Logic colloquium ’78, ed. M. Boffa and D. van Dalen, 159–224. Amsterdam: North Holland.

    Google Scholar 

  • Feferman, S. 1982. Iterated inductive fixed-point theories: Application to Hancock’s conjecture. In Patras Logic Symposion ’80, ed. G.E.A. Metakides, 171–196. Amsterdam: North Holland.

    Chapter  Google Scholar 

  • Feferman, S. 1998a. In the light of logic. Oxford: Oxford University Press.

    Google Scholar 

  • Feferman, S. 1998b. Weyl vindicated: Das Kontinuum 70 years later. In In the light of logic, 249–283. Oxford: Oxford University Press.

    Google Scholar 

  • Feferman, S. 1998c. Working foundations ’91. See Feferman (1998a ), 105–124.

  • Feferman, S. 2008. Axioms for determinateness and truth. Review of Symbolic Logic 1: 204–217.

    Article  Google Scholar 

  • Fischer, M. 2014. Truth and speed-up. Review of Symbolic Logic 7: 319–340.

    Article  Google Scholar 

  • Fischer, M., L. Horsten, and C. Nicolai. 2017. Iterated reflection over full disquotational truth. Journal of Logic and Computation 27: 2613–2651.

    Article  Google Scholar 

  • Fischer, M., L. Horsten, and C. Nicolai. 2019. Hypatia’s silence. Nous 20: 1–24.

    Google Scholar 

  • Fujimoto, K. 2011. Autonomous progressions and transfinite iteration of self-applicable truth. Journal of Symbolic Logic 76: 914–945.

    Article  Google Scholar 

  • Fujimoto, K. 2018. Truths, inductive definitions, and Kripke–Platek systems over set theory. Journal of Symbolic Logic 83: 868–898.

    Article  Google Scholar 

  • Fujimoto, K. 2019. Predicativism about classes. Journal of Philosophy 116: 206–229.

    Article  Google Scholar 

  • Girard, J. 1987. Proof theory and logical complexity. Naples: Bibliopolis.

    Google Scholar 

  • Glass, T. 1996. On power set in explicit mathematics. Journal of Symbolic Logic 61(2): 468–489.

    Article  Google Scholar 

  • Gödel, K. 1986. Über die länge von beweisen. In Collected works, Vol. I, ed. S. Feferman and et al., 123–153. Oxford: Oxford University Press.

    Google Scholar 

  • Halbach, V. 2011. Axiomatic theories of truth, 1st ed. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Halbach, V., and H. Horsten. 2006. Axiomatizing Kripke’s theory of truth. Journal of Symbolic Logic 71(2): 677–712.

    Article  Google Scholar 

  • Hilbert, D. 2013. David Hilbert’s lectures on the foundations of arithmetic and logic, 1917–1933. Cham: Springer.

    Google Scholar 

  • Horsten, L. 2011. The Tarskian turn. Deflationism and axiomatic truth. Cambridge: MIT Press.

    Book  Google Scholar 

  • Horsten, L., and G. Leigh. 2017. Truth is simple. Mind 126: 195–232.

    Google Scholar 

  • Jäger, G. 1997. Power types in explicit mathematics? Journal of Symbolic Logic 62: 1142–1146.

    Article  Google Scholar 

  • Jäger, G. 2005. Metapredicative and explicit Mahlo: A proof-theoretic perspective. In Logic colloquium 2000, Lecture notes in logic 19, ed. R. Cori, A. Razborov, S. Todorcevic, and C. Wood, 272–293. Urbana: Association for Symbolic Logic.

    Google Scholar 

  • Jäger, G., R. Kahle, A. Setzer, and T. Strahm. 1999. The proof-theoretic analysis of transfinitely iterated fixed point theories. Journal of Symbolic Logic 64: 53–67.

    Article  Google Scholar 

  • Kaye, R. 2005. Models of peano arithmetic. Lecture notes in logic 19. Urbana: Association for Symbolic Logic.

    Google Scholar 

  • Kotlarski, H., and A. Lachlan. 1981. Construction of satisfaction classes for nonstandard models. Canadian Mathematical Bulletin 24: 283–93.

    Article  Google Scholar 

  • Kreisel, G. 1970. Principles of proof and ordinals implicit in given concepts. In Intuitionism and Proof Theory, ed. A. Kino, J. Myhill, and R.E. Vesley, 489–516. Amsterdam: North Holland.

    Google Scholar 

  • Kripke, S. 1975. Outline of a theory of truth. Journal of Philosophy 72: 670–716.

    Article  Google Scholar 

  • Lachlan, A. 1981. Full satisfaction classes and recursive saturation. Canadian Mathematical Bulletin 24: 295–297.

    Article  Google Scholar 

  • Leigh, G.E. 2015. Conservativity for theories of compositional truth via cut elimination. The Journal of Symbolic Logic 80: 845–865.

    Article  Google Scholar 

  • Łełyk, M., and B. Wcisło. 2017a. Models of weak theories of truth. Archive for Mathematical Logic 56: 453–474.

    Article  Google Scholar 

  • Łełyk, M., and B. Wcisło. 2017b. Notes on bounded induction for the compositional truth predicate. Review of Symbolic Logic 10: 455–480.

    Article  Google Scholar 

  • Mancosu, P. 1998. From Brouwer to Hilbert the debate on the foundations of mathematics in the 1920s. Oxford/New York: Oxford University Press.

    Google Scholar 

  • Parsons, C. 1971. Ontology and mathematics. The Philosophical Review 80: 151–176.

    Article  Google Scholar 

  • Parsons, C. 1974. Sets and classes. Noûs 8: 1–12.

    Article  Google Scholar 

  • Parsons, C. 2002. Realism and the debate on impredicativity. 1917–1944. In Reflections on the foundations of mathematics, Lecture notes in logic 15, ed. W. Sieg, R. Sommer, and C. Talcott, 372–389. Natick: ASL and A.K.Peters.

    Google Scholar 

  • Rathjen, M. 2016. Indefinitess in semi-intuitionistic set theories: on a conjecture of Feferman. Journal of Symbolic Logic 81: 742–754.

    Article  Google Scholar 

  • Russell, B. 1906. On somes difficulties in the theory of transfinite numbers and order type. The Proceedings of the London Mathematical Society 4: 29–53.

    Google Scholar 

  • Russell, B. 1919. Introduction to mathematical philosophy. London: Allen and Unwin.

    Google Scholar 

  • Schindler, T. 2018. Some notes and comprehension. The Journal of Philosophical Logic 47: 449–479.

    Article  Google Scholar 

  • Simpson, S. 1999. Subsystem of second order arithmetic. Berlin/Heidelberg: Allen and Unwin.

    Book  Google Scholar 

  • Strahm, T. 2000. Autonomous fixed point progressions and fixed point transfinite recursion. In Logic colloquium ’98, ed. S. Buss, P. Hájek, and P. Pudlák, 449–464. Association for Symbolic Logic.

    Google Scholar 

  • Tait, W. 1983. Against intuitionism: constructive mathematics is part of classical mathematics. The Journal of Philosophical Logic 12: 449–479.

    Article  Google Scholar 

  • Tait, W. 1986. Truth and proof: The platonism of mathematics. Synthese 69: 341–370.

    Article  Google Scholar 

  • Tarski, A. 1944. The semantic conception of truth and the foundations of semantics. Philosophy and Phenomenological Research 4: 341–376.

    Article  Google Scholar 

  • Tarski, A. 1956. The concept of truth in formalized language. In Logic, Semantics, Metamathematics: Papers From 1923 to 1938, ed. J. Woodger, 152–278. Oxford: Clarendon Press.

    Google Scholar 

  • Weyl, H. 1910. Über die Definitionen der mathematischen Grundbegriffe. In Gesammelte Abhandlungen vol.1, Mathematisch-naturwissenschaftliche Blätter 7:93–95/109–113, 298–304.

    Google Scholar 

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Acknowledgements

I would like to thank the anonymous referees for helpful comments and the editors of this volume, Stefano Boscolo, Gianluigi Oliveri and Claudio Ternullo for their determination in bringing the project of this volume to completion.

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  1. DILEF, Università di Firenze, Firenze, Italy

    Andrea Cantini

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  1. Accademia Nazionale delle Scienze, Lettere ed Arti di Palermo, Università di Palermo, Palermo, Italy

    Gianluigi Oliveri

  2. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain

    Claudio Ternullo

  3. Dipartimento di Filosofia e Beni Culturali, Università di Venezia Ca' Foscari, Trevio, Italy

    Stefano Boscolo

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Cantini, A. (2022). Truth and the Philosophy of Mathematics. In: Oliveri, G., Ternullo, C., Boscolo, S. (eds) Objects, Structures, and Logics. Boston Studies in the Philosophy and History of Science, vol 339. Springer, Cham. https://doi.org/10.1007/978-3-030-84706-7_12

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