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On Algorithm and Robustness in a Non-standard Sense

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New Challenges to Philosophy of Science

Part of the book series: The Philosophy of Science in a European Perspective ((PSEP,volume 4))

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Abstract

In this paper, we investigate the invariance properties, i.e. robustness, of phenomena related to the notions of algorithm, finite procedure and explicit construction. First of all, we provide two examples of objects for which small changes completely change their (non)computational behavior. We then isolate robust phenomena in two disciplines related to computability.

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Notes

  1. 1.

    See (Freer et al. 2011).

  2. 2.

    The words in italics have precise technical definitions to be found in e.g. (Soare 1987).

  3. 3.

    This explains why, in any real-world scenario invariably involving noise, the non-computability of P[Y | X] never manifests itself.

  4. 4.

    See (Bishop 1967) and (Bridges and Vîţă 2006).

  5. 5.

    See (Hacking1983, pp. 54-55), (Salmon 1984, pp. 214-220) and (Salmon 1998, pp. 87-88).

  6. 6.

    In light of Examples 1 and 2, we may rest assured that intermediate values and conditional probabilities will always be computable in practice, as actual computational practice suggests.

  7. 7.

    See (Church 1936) and (Turing 1937). Intuitively, a Turing machine is an idealized computer with no limits on storage and meomory.

  8. 8.

    See (Friedman 1975; 1976).

  9. 9.

    See (Simpson 2009) for an introduction to Reverse Mathematics and p. xiv for the quote.

  10. 10.

    See (Simpson 2009, Theorem I.10.3).

  11. 11.

    Thus, Reverse Mathematics is intimately tied to Recursion Theory and computability.

  12. 12.

    In particular, Weak König’s Lemma states the existence of an infinite path through an infinite binary tree. Even for computable infinite binary trees, the infinite path need not be computable. In other words, WKL is false for the recursive/computable sets. See (Simpson 2009).

  13. 13.

    See (Sanders 2011) for an introduction to ERNA and a proof of Theorem 3.

  14. 14.

    In particular, ERNA was introduced around 1995 by Sommer and Suppes to formalize mathematics in physics. See (Sommer and Suppes 1996; 1997).

  15. 15.

    For an introduction to Nonstandard Analysis, we refer to (Kanovei and Reeken 2004).

  16. 16.

    In ERNA, the Riemann integral is only defined up to infinitesimals.

  17. 17.

    A similar (and equally valid) principle is If RCA 0 ⊢ T(=), then ERNA ⊢ T(≈).

  18. 18.

    A similar principle is If RCA0 ⊢ T( = ), then ERNA + ∏2-TRANS ⊢ T(≋).

  19. 19.

    The first one is the removing of parameters in ∏ 1-TRANS and the second one is the assumption of a greatest relevant infinite element.

  20. 20.

    The title of this section is explained in Remark 16 below. The italicized concepts are introduced in Section 3.2.

  21. 21.

    See (Kreisler 2006).

  22. 22.

    See (Soare 1987, Lemma 3.2.)

  23. 23.

    The function (μk ≤ m)ψ(k) computes the least k ≤ m such that ψ(k), for ψ in Δ0. It is available in most logical systems.

  24. 24.

    See (van Heijenoort 1967) and (Bishop 1967).

  25. 25.

    See (Bridges 1999, p. 96) and (Richman 1990).

  26. 26.

    These results are taken from (Ishihara 2006).

  27. 27.

    See (Bishop 1967) or (Ishihara 2006).

  28. 28.

    Note that we use the word ‘translation’ informally: The definition of \(\mathbb{V}\) is inspired by the intuitionistic disjunction, but that is the only connection.

  29. 29.

    See (Sanders 2012) for a list of thirty translated theorems.

  30. 30.

    See e.g. (Bridges 1999, Axiom set R2).

  31. 31.

    See e.g. (Bishop 1977, p. 208) and (Bishop 1975, p. 513).

  32. 32.

    With the notable exception of Erik Palmgren in e.g. (Palmgren 2001).

  33. 33.

    See (Wattenberg 1988).

References

  • Bishop, E., 1967, Foundations of Constructive Analysis. New York: McGraw-Hill Book Co.

    Google Scholar 

  • Bishop, E., 1975, “The Crisis in Contemporary Mathematics”, in: Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics, pp. 507-517.

    Google Scholar 

  • Bishop, E., 1977, “Book Review: Elementary Calculus”, in: Bull. Amer. Math. Soc. 83, 2, pp. 205-208.

    Google Scholar 

  • Bridges, D. S., 1999, “Constructive Mathematics: A Foundation for Computable Analysis”, in: Theoret. Comput. Sci. 219, 1-2, pp. 95-109.

    Google Scholar 

  • Bridges, D. S. and Vîţă, L. S., 2006,Techniques of Constructive Analysis. Universitext, New York: Springer.

    Google Scholar 

  • Church, A., 1936, “A Note on the Entscheidungsproblem”, in: Journal of Symbolic Logic 1, pp. 40-41.

    Google Scholar 

  • Freer, C. E., Ackerman, N. L., and Roy D. M, 2011, “Noncomputable Conditional Distributions”, in: Proceedings of the Twenty-Sixth Annual IEEE Symposium on Logic in Computer Science (Toronto, Canada, 2011): IEEE Press.

    Google Scholar 

  • Friedman, H., 1975, “Some Systems of Second Order Arithmetic and Their Use”, in: Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), vol. 1, Canad. Math. Congress, Montreal, Quebec, pp. 235-242.

    Google Scholar 

  • Friedman, H., 1976, “Systems of Second Order Arithmetic with Restricted Induction, I & II (Abstracts)”, in: Journal of Symbolic Logic 41, pp. 557-559.

    Google Scholar 

  • Hacking, I., 1983, Representing and Intervening: Introductory Topics in the Philosophy of Natural Science. Cambridge: Cambridge University Press.

    Google Scholar 

  • Ishihara, H., 2006, “Reverse Mathematics in Bishop’s Constructive Mathematics”, in: Philosophia Scientiae (Cahier Spécial) 6, pp. 43-59.

    Google Scholar 

  • Keisler, H. J., 2006, “Nonstandard arithmetic and Reverse Mathematics”, in: Bull. Symbolic Logic 12, 1, pp. 100-125.

    Google Scholar 

  • Palmgren, E., 2001, “Unifying Constructive and Nonstandard Analysis” (Venice, 1999), in: Synthese Lib., Vol. 306, Dordrecht: Kluwer, pp. 167-183.

    Google Scholar 

  • Kanovei, V., and Reeken, M., 2004, Nonstandard Analysis, Axiomatically. Springer.

    Google Scholar 

  • Richman, F., 1990, “Intuitionism as a Generalization”, in: Philosophia Math. 5, pp. 124-128.

    Google Scholar 

  • Sanders, S., 2011, “ERNA and Friedman’s Reverse Mathematics”, in: Journal of Symbolic Logic 76, pp. 637-664.

    Google Scholar 

  • Sanders, S., 2012, On the Notion of Algorithm in Nonstandard Analysis, Submitted.

    Google Scholar 

  • Salmon, W. C., 1984, Scientific Explanation and the Causal Structure of the World. Princeton: Princeton University Press.

    Google Scholar 

  • Salmon, W. C., 1998, Causality and Explanation. Oxford: Oxford University Press.

    Google Scholar 

  • Simpson, S. G., 2009, Subsystems of Second Order Arithmetic (Perspectives in Logic), 2nd ed., Cambridge: Cambridge University Press.

    Google Scholar 

  • Soare, R. I., 1987, Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Berlin: Springer-Verlag.

    Google Scholar 

  • Sommer, R. and Suppes P., 1996. “Finite Models of Elementary Recursive Nonstandard Analysis”, in: Notas de la Sociedad Mathematica de Chile 15, pp. 73-95.

    Google Scholar 

  • Sommer, R. and Suppes P., 1997, “Dispensing with the Continuum”, in: Journal of Mathematical Psychology 41, pp. 3-10.

    Google Scholar 

  • Turing, A., 1937, “On Computable Numbers, with an Application to the Entscheidungsproblem”, in: Proceedings of the London Mathematical Society 42, pp. 230-265.

    Google Scholar 

  • van Heijenoort, J., 1967, From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931. Cambridge (Mass.): Harvard University Press.

    Google Scholar 

  • Wattenberg, F., 1988, “Nonstandard Analysis and Constructivism?”, in: Studia Logica 47, 3, pp. 303-309.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics (S22), Ghent University, Krijgslaan 281, B-9000, Ghent, Belgium

    Sam Sanders

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  1. Sam Sanders
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Editors and Affiliations

  1. Aarhus University, Ny Munkegade 1520, Aarhus, 8000, Denmark

    Hanne Andersen

  2. Inst. History & Foundations of Science, Utrecht University, Budapestlaan 6, Utrecht, 3584 CD, Netherlands

    Dennis Dieks

  3. , Faculty of Humanities, University of A Coruña, Dr. Vazquez Cabrera street, w/n, Ferrol, 15.403, Spain

    Wenceslao J. Gonzalez

  4. , Philosophy School of Social Science, University of Manchester, Manchester, M13 9PL, United Kingdom

    Thomas Uebel

  5. Centre Artificial Intelligence (CENTRA), Department of Computer Science, New University of Lisbon, Monte de Caparica, Lisbon, 2829-516, Portugal

    Gregory Wheeler

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Sanders, S. (2013). On Algorithm and Robustness in a Non-standard Sense. In: Andersen, H., Dieks, D., Gonzalez, W., Uebel, T., Wheeler, G. (eds) New Challenges to Philosophy of Science. The Philosophy of Science in a European Perspective, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5845-2_9

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