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Predicativity and Constructive 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

In this article I present a disagreement between classical and constructive approaches to predicativity regarding the predicative status of so-called generalised inductive definitions. I begin by offering some motivation for an enquiry in the predicative foundations of constructive mathematics, by looking at contemporary work at the intersection between mathematics and computer science. I then review the background notions and spell out the above-mentioned disagreement between classical and constructive approaches to predicativity. Finally, I look at possible ways of defending the constructive predicativity of inductive definitions.

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Notes

  1. 1.

    See Bridges and Richman (1987) for an introduction.

  2. 2.

    See e.g. Martin-Löf (1975), Myhill (1975), Aczel (1978), Martin-Löf (1984), Beeson (1985). Another approach to the foundations of constructive mathematics is Feferman’s Explicit Mathematics, which has been studied especially in proof theory (Feferman, 1975). A very recent development is Homotopy Type Theory (Univalent Foundations Program, 2013).

  3. 3.

    As further clarified in Sect. 11.4.1, the debate on the predicative status of inductive definitions has focused on generalised inductive definitions. In the following, unless otherwise stated, I omit the qualification “generalised”.

  4. 4.

    Arguably, from a different perspective, constructive systems are more flexible and less restrictive than traditional classical systems such as ZFC, as they allow for a variety of interpretations, including computational interpretations (see Sect. 11.2). See also Bridges and Richman (1987).

  5. 5.

    See Martin-Löf (1984), Aczel (1986), Palmgren (1992), Dybjer (2000), and Dybjer and Setzer (2003).

  6. 6.

    See, for example, Martin-Löf (1982), Coquand and Huet (1986), Constable and et al. (1986), Nordström et al. (1990), Gonthier (2008), AGDA (2020), and The Coq Development Team (2020).

  7. 7.

    The W type constructor is used to codify well-founded trees in type theory. It can therefore be used to codify Brouwer’s constructive ordinals (see Sect. 11.4.1). The Curry-Howard correspondence, also known as “formulas-as-types” correlates intuitionistic logic with type theories. See e.g. Troelstra (1999) for details and Crosilla (2019) for an informal discussion of its relation with predicativity.

  8. 8.

    This paradox affected an early variant of type theory, which included a type of all types. See Girard (1972). See also Coquand (1989) and Martin-Löf (2008) for analysis and Crosilla (2019) for philosophical reflections.

  9. 9.

    Universes in Martin-Löf type theory are powerful constructs which act as reflection principles. Roughly a universe is a type closed under certain type-forming operations.

  10. 10.

    In Coq there are two sorts (i.e. categories) of objects “Prop” and “Set”. Both had impredicative features in early versions of the system, so that, for example, one could quantify over all Sets to define a new set. Recent versions, however, retain an impredicative “Prop” but abandon the impredicativity of “Set”. These new restrictions are introduced to increase compatibility with classical mathematics (see e.g. Barbanera and Berardi (1996)).

  11. 11.

    See, for example, Poincaré (1905, 1906a,b), Russell (1906a,b, 1908) and Poincaré (1909, 1912).

  12. 12.

    Russell and Whitehead gave a number of renderings of the VCP. For example, “no totality can contain members defined in terms of itself” (Russell, 1908, p. 237) and ”[…] whatever in any way concerns all or any or some of a class must not be itself one of the members of a class” (Russell, 1973, p. 198). See also Gödel (1944) for an influential discussion, especially p. 454-5.

  13. 13.

    See Russell (1908) and Whitehead and Russell (1910–1913).

  14. 14.

    See Sect. 11.4.3.1 for more on Weyl (1918).

  15. 15.

    See e.g. Feferman (1988, 2004, 2013b) and Simpson (1988, 1999).

  16. 16.

    Some of the most significant steps in that development are recalled in Feferman (2005). See also Dean and Walsh (2016) and Crosilla (2017).

  17. 17.

    See Kreisel (1958), Feferman (1964), and Schütte (1965a,b). Note that this is not the only logical analysis of predicativity proposed in the 1950–1960s. Another approach (Kreisel, 1960) made essential use of work in recursion theory and definability theory, and identified the predicatively definable sets of natural numbers with the so-called hyperarithmetical sets. Here work by Kleene, among others, provided fundamental insights and the necessary tools for the analysis. See Moschovakis (1974) for the relevant notions, historical notes and references.

  18. 18.

    Schütte’s fundamental contribution to this analysis of predicativity is acknowledged by Feferman (2013a, p. 8–9) as follows: “[…] the determination by Schütte and me in the mid-1960s of Γ0 as the upper bound for the ordinal of predicativity simply fell out of his ordinal analysis of the systems of ramified analysis translated into infinitary rules of inference when one added the condition of autonomy.”

  19. 19.

    For discussion see Coquand (1989), Dybjer (2012), Palmgren (1998) and Rathjen (2005).

  20. 20.

    Note that while my focus in this note are intuitionistic theories, Lorenzen and Myhill have argued for a rather liberal notion of predicativity with respect to a quite general notion of constructivity (also in the context of theories with classical logic). See especially (Lorenzen, 1958; Lorenzen and Myhill, 1959). See also Wang (1959). For Martin-Löf type theory, see e.g. Palmgren (1998) and Rathjen (2005).

  21. 21.

    See the fundamental (Barwise, 1975; Moschovakis, 1974).

  22. 22.

    See Buchholz et al. (1981). The introduction gives an insight into the historical developments of ordinal analysis beyond predicativity. See Chapter 1 for background. See also Feferman (2013a); Martin-Löf (2008).

  23. 23.

    See e.g. Palmgren (1992, 1998), Rathjen et al. (1998), Dybjer (2000), Dybjer and Setzer (2003) and Rathjen (2005).

  24. 24.

    See Dybjer (2012) for discussion and references.

  25. 25.

    See the exposition in Buchholz et al. (1981), Chapter 1.

  26. 26.

    See Aczel (1977). Particularly appealing from a constructive point of view are deterministic rules. A rule is deterministic if for any conclusion a there exists exactly one set of premises X such that a is a consequence of X according to the rule.

  27. 27.

    Note that while the forms of predicativity considered in this article take the natural numbers as unproblematic, this assumption is not gone unchallenged. Dummett, Nelson and Parsons have (independently) argued for the impredicativity of the principle of mathematical induction (Dummett, 1963; Nelson, 1986; Parsons, 1992). Nelson (1986) develops a form of predicative arithmetic that substantially constrains mathematical induction, therefore giving rise to weak subsystems of Peano Arithmetic.

  28. 28.

    See Buchholz et al. (1981, p. 147).

  29. 29.

    See Buchholz et al. (1981, Chapter 1).

  30. 30.

    A “miniature” argument along these lines can be carried out already in the case of the natural numbers to argue for the impredicativity of the induction principle. This will be discussed in Sect. 11.4.3.2.

  31. 31.

    See Poincaré (1909, 1912). See also Kreisel (1960, p. 378). Note that I am here interested in the main ideas underlying this notion, rather than in an exegesis of Poincaré’s thought.

  32. 32.

    See also Cantini (2022) for discussion.

  33. 33.

    The axiom of exponentiation allows us to collect in a set all the functions from a set A to a set B. This is constructively weaker than the full powerset (Aczel, 1978; Myhill, 1975).

  34. 34.

    I would like to thank a referee for drawing my attention to this passage and to Lorenzen (1958).

  35. 35.

    A thorough discussion of this point would require careful consideration of Lorenzen’s work. See e.g. Lorenzen (1958). Note that one could argue that the term “predicativity” is now been used to refer to a different phenomenon altogether compared with that giving rise to the Γ0 limit. This seems to be Feferman’s point of view in Feferman (1964, p. 4–5), when discussing especially Lorenzen and Wang’s work on predicativity. I am persuaded this is a complex issue that would require careful consideration.

  36. 36.

    See Crosilla (2020).

  37. 37.

    See Dummett (1963), Nelson (1986) and Parsons (1992).

  38. 38.

    See e.g. Dummett (1963), Nelson (1986) and Parsons (1992).

  39. 39.

    See Nelson (1986) and Parsons (1992). See also Crosilla (2016, 2020) for a detailed analysis of the natural number case.

  40. 40.

    See also Linnebo (2018).

  41. 41.

    See also Dybjer (2012) and Dybjer and Setzer (2003).

References

  • Aczel, P. 1977. An introduction to inductive definitions. In Handbook of mathematical logic, volume 90 of Studies in logic and the foundations of mathematics, ed. J. Barwise, 739–782. Elsevier.

    Google Scholar 

  • Aczel, P. 1978. The type theoretic interpretation of constructive set theory. In Logic colloquium ’77, ed. A. MacIntyre, L. Pacholski, and J. Paris, 55–66. New York: Amsterdam.

    Chapter  Google Scholar 

  • Aczel, P. 1986. The type theoretic interpretation of constructive set theory: Inductive definitions. In Logic, methodology, and philosophy of science VII, ed. R.B. Marcus, G.J. Dorn, and G.J.W. Dorn, 17–49. New York: Amsterdam.

    Google Scholar 

  • AGDA. 2020. Agda wiki. Available at http://wiki.portal.chalmers.se/agda/pmwiki.php.

    Google Scholar 

  • Barbanera, F., and S. Berardi. 1996. Proof-irrelevance out of excluded-middle and choice in the calculus of constructions. Journal of Functional Programming 6(3): 519–525.

    Article  Google Scholar 

  • Barwise, J. 1975. Admissible sets and structures. An approach to definability theory. Berlin: Springer.

    Google Scholar 

  • Beeson, M. 1985. Foundations of constructive mathematics. Berlin: Springer.

    Book  Google Scholar 

  • Benacerraf, P., and H. Putnam. 1983. Philosophy of mathematics: Selected readings. Cambridge University Press.

    Google Scholar 

  • Bishop, E. 1967. Foundations of constructive analysis. New York: McGraw-Hill.

    Google Scholar 

  • Bridges, D.S., and F. Richman. 1987. Varieties of constructive mathematics. Cambridge University Press.

    Book  Google Scholar 

  • Buchholz, W., S. Feferman, W. Pohlers, and W. Sieg. 1981. Iterated inductive definitions and subsystems of analysis. Berlin: Springer.

    Google Scholar 

  • Cantini, A. 2022. Truth and the philosophy of mathematics. This volume.

    Google Scholar 

  • Constable, R.L., and et al. 1986. Implementing mathematics with the nuprl proof development system. Englewood Cliffs: Prentice–Hall.

    Google Scholar 

  • Coquand, T. 1989. Metamathematical investigations of a calculus of constructions. Technical report, INRIA.

    Google Scholar 

  • Coquand, T., and G. Huet. 1986. The calculus of constructions. Technical Report RR-0530, INRIA.

    Google Scholar 

  • Coquand, T., G. Sambin, J. Smith, and S. Valentini. 2003. Inductively generated formal topologies. Annals of Pure and Applied Logic 124(1): 71–106.

    Article  Google Scholar 

  • Crosilla, L. 2016. Constructivity and Predicativity: Philosophical foundations. Ph. D. thesis, School of Philosophy, Religion and the History of Science, University of Leeds.

    Google Scholar 

  • Crosilla, L. 2017. Predicativity and Feferman. In Feferman on foundations: Logic, mathematics, philosophy, Outstanding contributions to logic, ed. G. Jäger and W. Sieg. Springer. Forthcoming.

    Google Scholar 

  • Crosilla, L. 2019. The entanglement of logic and set theory, constructively. Inquiry 0(0), 1–22.

    Google Scholar 

  • Crosilla, L. 2020. From predicativity to intuitionistic mathematics, via Dummett. Unpublished Manuscript.

    Google Scholar 

  • Dean, W., and S. Walsh 2016. The prehistory of the subsystems of second-order arithmetic. Review of Symbolic Logic 10: 357–396.

    Article  Google Scholar 

  • Dummett, M. 1963. The Philosophical Significance of Gödel’s Theorem. Ratio 5: 140–155.

    Google Scholar 

  • Dybjer, P. 2000. A general formulation of simultaneous inductive-recursive definitions in type theory. The Journal of Symbolic Logic 65(2): 525–549.

    Article  Google Scholar 

  • Dybjer, P. 2012. Program testing and the meaning explanations of Martin-Löf type theory. In Epistemology versus Ontology, Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf, ed. P. Dybjer, S. Lindström, E. Palmgren, and B. Sundholm.

    Google Scholar 

  • Dybjer, P., and A. Setzer 2003. Induction–recursion and initial algebras. Annals of Pure and Applied Logic 124(1): 1–47.

    Article  Google Scholar 

  • Feferman, S. 1964. Systems of predicative analysis. Journal of Symbolic Logic 29: 1–30.

    Article  Google Scholar 

  • Feferman, S. 1975. A language and axioms for explicit mathematics. In Algebra and logic, volume 450 of Lecture notes in mathematics, J. Crossley, 87–139. Berlin: Springer.

    Google Scholar 

  • Feferman, S. 1988. Weyl vindicated: Das Kontinuum seventy years later. In Temi e prospettive della logica e della scienza contemporanee, ed. C. Cellucci and G. Sambin, 59–93.

    Google Scholar 

  • Feferman, S. 2004. Comments on ‘Predicativity as a philosophical position’ by G. Hellman. Review Internationale de Philosophie 229(3).

    Google Scholar 

  • Feferman, S. 2005. Predicativity. In Handbook of the philosophy of mathematics and logic, ed. S. Shapiro. Oxford: Oxford University Press.

    Google Scholar 

  • Feferman, S. 2013a. The proof theory of classical and constructive inductive definitions. A forty year saga, 1968–2008. In Ways of proof theory, ed. R. Schindler, 7–30. De Gruyter.

    Google Scholar 

  • Feferman, S. 2013b. Why a little bit goes a long way: Predicative foundations of analysis. Unpublished notes dating from 1977–1981, with a new introduction. Retrieved from the address: https://math.stanford.edu/~feferman/papers.html.

  • Friedman, H. 1973. The consistency of classical set theory relative to a set theory with intuitionistic logic. Journal of Symbolic Logic 38: 315–319.

    Article  Google Scholar 

  • Girard, J. 1972. Interprétation fonctionnelle et élimination des coupures de l’arithmétique d’ordre supérieur. Ph. D. thesis, These d’Etat, Paris VII.

    Google Scholar 

  • Gödel, K. 1944. Russell’s mathematical logic. In The philosophy of Bertrand Russell, ed. P.A. Schlipp, 123–153. Northwestern University, Evanston and Chicago. Reprinted in Benacerraf and Putnam (1983). (Page references are to the reprinting).

  • Gonthier, G. 2008. Formal proof–the four-color theorem. Notices of the American Mathematical Society 11(55): 1382–1393.

    Google Scholar 

  • Kreisel, G. 1958. Ordinal logics and the characterization of informal concepts of proof. In Proceedings of the International Congress of Mathematicians (August 1958), 289–299. Paris: Gauthier–Villars.

    Google Scholar 

  • Kreisel, G. 1960. La prédicativité. Bulletin de la Societé Mathématique de France 88: 371–391.

    Article  Google Scholar 

  • Linnebo, O. 2018. Generality explained. Unpublished manuscript.

    Google Scholar 

  • Lorenzen, P. 1958. Logical reflection and formalism. The Journal of Symbolic Logic 23(3): 241–249.

    Article  Google Scholar 

  • Lorenzen, P., and J. Myhill. 1959. Constructive definition of certain analytic sets of numbers. Journal of Symbolic Logic 24: 37–49.

    Article  Google Scholar 

  • Martin-Löf, P. 1975. An intuitionistic theory of types: Predicative part. In Logic Colloquium 1973, ed. H.E. Rose and J.C. Shepherdson. Amsterdam: North–Holland.

    Google Scholar 

  • Martin-Löf, P. 1982. Constructive mathematics and computer programming. In Logic, methodology, and philosophy of science VI, ed. L.J. Choen. Amsterdam: North–Holland.

    Google Scholar 

  • Martin-Löf, P. 1984. Intuitionistic type theory. Naples: Bibliopolis.

    Google Scholar 

  • Martin-Löf, P. 2008. The Hilbert–Brouwer controversy resolved? In One hundred years of intuitionism (1907 – 2007), ed. E.A. van Atten, 243–256. Publications des Archives Henri Poincaré .

    Chapter  Google Scholar 

  • Moschovakis, Y. 1974. Elementary induction on abstract structures (Studies in logic and the foundations of mathematics). American Elsevier Pub. Co.

    Google Scholar 

  • Myhill, J. 1975. Constructive set theory. Journal of Symbolic Logic 40: 347–382.

    Article  Google Scholar 

  • Nelson, E. 1986. Predicative arithmetic. Princeton: Princeton University Press.

    Book  Google Scholar 

  • Nordström, B., K. Petersson, and J.M. Smith. 1990. Programming in Martin-Löf’s type theory: An introduction. Clarendon Press.

    Google Scholar 

  • Palmgren, E. 1992. Type-theoretic interpretation of iterated, strictly positive inductive definitions. Arch Math Logic 32: 75–99.

    Article  Google Scholar 

  • Palmgren, E. 1998. On universes in type theory. In Twenty–five years of type theory, ed. G. Sambin and J. Smith. Oxford: Oxford University Press.

    Google Scholar 

  • Parsons, C. 1992. The impredicativity of induction. In Proof, logic, and formalization, ed. M. Detlefsen, 139–161. London: Routledge.

    Google Scholar 

  • Poincaré, H. 1905. Les mathématiques et la logique. Revue de Métaphysique et Morale 1: 815–835.

    Google Scholar 

  • Poincaré, H. 1906a. Les mathématiques et la logique. Revue de Métaphysique et de Morale 2: 17–34.

    Google Scholar 

  • Poincaré, H. 1906b. Les mathématiques et la logique. Revue de Métaphysique et de Morale 14: 294–317.

    Google Scholar 

  • Poincaré, H. 1909. La logique de l’infini. Revue de Métaphysique et Morale 17: 461–482.

    Google Scholar 

  • Poincaré, H. 1912. La logique de l’infini. Scientia 12: 1–11.

    Google Scholar 

  • Rathjen, M. 2005. The constructive Hilbert program and the limits of Martin–Löf type theory. Synthese 147: 81–120.

    Article  Google Scholar 

  • Rathjen, M., E. Griffor, and E. Palmgren. 1998. Inaccessibility in constructive set theory and type theory. Annals of Pure and Applied Logic 94: 181–200.

    Article  Google Scholar 

  • Russell, B. 1906a. Les paradoxes de la logique. Revue de métaphysique et de morale 14: 627–650.

    Google Scholar 

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

    Google Scholar 

  • Russell, B. 1908. Mathematical logic as based on the theory of types. American Journal of Mathematics 30: 222–262.

    Article  Google Scholar 

  • Russell, B. 1973. Essays in analysis, ed. D. Lackey. New York: George Braziller.

    Google Scholar 

  • Sambin, G. 1987. Intuitionistic formal spaces – a first communication. In Mathematical logic and its applications, ed. D. Skordev, 187–204. Plenum.

    Chapter  Google Scholar 

  • Schütte, K. 1965a. Eine Grenze für die Beweisbarkeit der Transfiniten Induktion in der verzweigten Typenlogik. Archiv für mathematische Logik und Grundlagenforschung 7: 45–60.

    Article  Google Scholar 

  • Schütte, K. 1965b. Predicative well–orderings. In Formal systems and recursive functions, ed. J. Crossley and M. Dummett. North–Holland, Amsterdam.

    Google Scholar 

  • Simpson, S.G. 1988. Partial realizations of Hilbert’s program. Journal of Symbolic Logic 53(2): 349–363.

    Article  Google Scholar 

  • Simpson, S.G. 1999. Subsystems of second order arithmetic. Perspectives in Mathematical Logic. Springer.

    Google Scholar 

  • The Coq Development Team. 2020. Coq. https://coq.inria.fr.

  • Troelstra, A.S. 1999. From constructivism to computer science. Theoretical Computer Science 211: 233–252.

    Article  Google Scholar 

  • Univalent Foundations Program, T. 2013. Homotopy type theory: Univalent foundations of mathematics. Institute of Advanced Studies.

    Google Scholar 

  • Wang, H. 1959. Ordinal numbers and predicative set theory. Zeitschr. f. math.Logik und Grundlagen d. Math. 5: 216–239.

    Article  Google Scholar 

  • Weyl, H. 1918. Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Veit, Leipzig.

    Book  Google Scholar 

  • Whitehead, A.N., and B. Russell. (1910, 1912, 1913). Principia mathematica, 3 Vols., Vol. 1. Cambridge: Cambridge University Press. Second edition, 1925 (Vol 1), 1927 (Vols 2, 3); abridged as Principia Mathematica to *56, Cambridge: Cambridge University Press, 1962.

    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 Olivieri and Claudio Ternullo for their determination in bringing the project of this volume to completion. I am grateful to Andrea Cantini and Øystein Linnebo for comments on an earlier version of this article. The research leading to this article has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 838445.

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  1. Department of Philosophy, IFIKK, University of Oslo, Blindern, Norway

    Laura Crosilla

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    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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Crosilla, L. (2022). Predicativity and Constructive 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_11

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