Church-Turing Thesis, in Practice

  • Luca San MauroEmail author
Part of the Boston Studies in the Philosophy and History of Science book series (BSPS, volume 334)


We aim at providing a philosophical analysis of the notion of “proof by Church’s Thesis”, which is – in a nutshell – the conceptual device that permits to rely on informal methods when working in Computability Theory. This notion allows, in most cases, to not specify the background model of computation in which a given algorithm – or a construction – is framed. In pursuing such analysis, we carefully reconstruct the development of this notion (from Post to Rogers, to the present days), and we focus on some classical constructions of the field, such as the construction of a simple set. Then, we make use of this focus in order to support the following encompassing claim (which opposes to a somehow commonly received view): the informal side of Computability, consisting of the large class of methods typically employed in the proofs of the field, is not fully reducible to its formal counterpart.



A preliminary version of this paper appeared as a chapter of my PhD thesis. I would like to thank my supervisors, Gabriele Lolli and Andrea Sorbi, for their guidance and support. I have presented this work at several conferences. In particular, I am grateful to the participants of APMP 2014, in Paris, and of FilMat 2016, in Chieti, for their comments. Finally, Richard Epstein’s remarks were fundamental in rethinking the organization of the present material.


  1. Awodey, S. 2014. Structuralism, invariance, and univalence. Philosophia Mathematica 22(1): 1–11.CrossRefGoogle Scholar
  2. Black, R. 2000. Proving Church’s thesis. Philosophia Mathematica 8(3): 244–258.CrossRefGoogle Scholar
  3. Blass, A., N. Dershowitz, and Y. Gurevich. 2009. When are two algorithms the same? Bulletin of Symbolic Logic 15(02): 145–168.CrossRefGoogle Scholar
  4. Burgess, J.P. 2015. Rigor and structure. Oxford: Oxford University Press.CrossRefGoogle Scholar
  5. Buss, S.R., A.S. Kechris, A. Pillay, and R.A. Shore. 2001. The prospects for mathematical logic in the twenty-first century. Bulletin of Symbolic Logic 7(02): 169–196.CrossRefGoogle Scholar
  6. Carter, J. 2008. Structuralism as a philosophy of mathematical practice. Synthese 163(2): 119–131.CrossRefGoogle Scholar
  7. Church, A. 1936. An unsolvable problem of elementary number theory. American Journal of Mathematics 58(2): 345–363.CrossRefGoogle Scholar
  8. Davis, M. 2006. Why there is no such discipline as hypercomputation. Applied Mathematics and Computation 178(1): 4–7.CrossRefGoogle Scholar
  9. De Mol, L. 2006. Closing the circle: An analysis of Emil Post’s early work. Bulletin of Symbolic Logic 12(02): 267–289.CrossRefGoogle Scholar
  10. Dean, W.H. 2007. What algorithms could not be. PhD thesis, Rutgers University-New Brunswick.Google Scholar
  11. Descartes, R. 1628. Rules for the direction of the mind. In Selections. Trans. R.M. Eaton. New York: Charles Scribner’s Sons, 1927.Google Scholar
  12. Epstein, R.L., and W. Carnielli. 1989. Computability: Computable functions, logic, and the foundations of mathematics. Pacific Grove: Wadsworth & Brooks/Cole.Google Scholar
  13. Fallis, D. 2003. Intentional gaps in mathematical proofs. Synthese 134(1): 45–69.CrossRefGoogle Scholar
  14. Folina, J. (1998). Church’s thesis: Prelude to a proof. Philosophia Mathematica 6(3): 302–323.CrossRefGoogle Scholar
  15. Gandy, R. 1988. The confluence of ideas in 1936. In The Universal Turing machine: A half-century survey, ed. R. Herken, 55–111. Wien/New York: Springer.Google Scholar
  16. Gödel, K. 1946. Remarks before the Princeton bicentennial conference on problems in mathematics. In Kurt Gödel: Collected works, ed. S. Feferman, J. Dawson, and S. Kleene, vol. II, pp. 150–153. Oxford: Oxford University Press.Google Scholar
  17. Gurevich, Y. 2000. Sequential abstract-state machines capture sequential algorithms. ACM Transactions on Computational Logic (TOCL) 1(1): 77–111.CrossRefGoogle Scholar
  18. Hacking, I. 2014. Why is there philosophy of mathematics at all? Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  19. Hilbert, D. 1899. Grundlagen der geometrie. In Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, 1–92. Leipzig: Teubner.Google Scholar
  20. Kleene, S.C. 1952. Introduction to metamathematics. Amsterdam: North Holland.Google Scholar
  21. Lakatos, I. 1976. Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  22. Mancosu, P. 2008. The philosophy of mathematical practice. Oxford: Oxford University Press.CrossRefGoogle Scholar
  23. McLarty, C. 2008. What structuralism achieves. In Mancosu (2008).CrossRefGoogle Scholar
  24. Mendelson, E. 1990. Second thoughts about Church’s thesis and mathematical proofs. The Journal of Philosophy 87(5): 225–233.CrossRefGoogle Scholar
  25. Odifreddi, P. 1989. Classical recursion theory, vol. I. Amsterdam: North Holland.Google Scholar
  26. Olszewski, A., J. Wolenski, and R. Janusz. 2006. Church’s thesis after 70 years. Frankfurt/New Brunswick: Ontos Verlag.Google Scholar
  27. Post, E.L. 1936. Finite combinatory processes–formulation. The Journal of Symbolic Logic 1(03): 103–105.CrossRefGoogle Scholar
  28. Post, E.L. 1944. Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society 50(5): 284–316.CrossRefGoogle Scholar
  29. Rav, Y. 1999. Why do we prove theorems? Philosophia Mathematica 7(1): 5–41.CrossRefGoogle Scholar
  30. Resnik, M.D. 1997. Mathematics as a science of patterns. Oxford: Oxford University Press.Google Scholar
  31. Rogers, H., Jr. 1967. Theory of recursive functions and effective computability. New York: McGraw-Hill.Google Scholar
  32. Shapiro, S. 2006. Computability, proof, and open-texture. In Olszewski et al. (2006), 420–455.Google Scholar
  33. Shapiro, S. 2010. Mathematical structuralism. Internet Encyclopedia of Philosophy. 25 June 2018.
  34. Sieg, W. 1994. Mechanical procedures and mathematical experience. In Mathematics and mind, ed. A. George, 71–117. Oxford: Oxford University Press.Google Scholar
  35. Soare, R.I. 1987a. Recursively enumerable sets and degrees. Perspectives in mathematical logic, omega series. Heidelberg: Springer.Google Scholar
  36. Soare, R.I. 1987b. Interactive computing and relativized computability, In Computability: Turing, Gödel, Church, and beyond, ed. B.J. Copeland, C.J. Posy, and O. Shagrir, 203–260. Cambdrige: MIT Press.Google Scholar
  37. Turing, A.M. 1936. On computable numbers, with an application to the entscheidungsproblem. Proceedings of the London Mathematical Society 2(1): 230–265.CrossRefGoogle Scholar
  38. Turing, A.M. 1948. Intelligent machinery. In Collected works of A.M. Turing: Mechanical intelligence, ed. D.C. Ince, 107–127. Amsterdam: North-HollandGoogle Scholar
  39. Welch, P.D. 2007. Turing unbound: Transfinite computation. In Computation and logic in the real world, ed. B. Löwe, B. Cooper, and A. Sorbi, 768–780. Berlin: Springer.CrossRefGoogle Scholar
  40. Wittgenstein, L. 1980. Remarks on the philosophy of psychology. Oxford: Blackwell.Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryTechnische Universität WienViennaAustria

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