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On Formal Proofs

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Book cover Deduction, Computation, Experiment

Abstract

We address two questions:

  • what is the use of formal proofs?

  • how do we proceed from a formal proof to a computation?

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References

  1. P. Aczel and M. Rathjen: Notes on Constructive Set Theory, Draft available from the Internet, http://www.cs.man.ac.uk/~petera/

    Google Scholar 

  2. M. Aschbacher: Highly complex proofs and implications of such proofs. Philosophical Transactions of the Royal Society A 363 (2005) pp 2401–2406

    Article  Google Scholar 

  3. J. Avigad: Number theory and elementary arithmetic. Philosophia Mathematica 11 (2003) pp 257–284

    Google Scholar 

  4. J. Avigad: Mathematical Method and Proof. Synthèse 153 (2006) pp 105–159

    Article  Google Scholar 

  5. J. Avigad, K. Donnelly, D. Gray and P. Raff: A formally verified proof of the prime number theorem. ACM Transactions on Computational Logic, 9, 1:2 (2007)

    Article  Google Scholar 

  6. H. Barendregt: The Lambda Calculus. Its Syntax and Semantics (Elsevier, Amsterdam, 1984)

    Google Scholar 

  7. P. Bernays: Mathematische Existenz und Widerspruchfreiheit. In: Ètudes de Philosophie des Sciences en hommage à Ferdinand Gonseth, (Neuchatel, 1950) pp 11–25 (English Translation by W. Sieg, R. Zach and S. Goodman, available as Text No.19, Bernays Project)

    Google Scholar 

  8. P. Bernays: Die Schematische Korrespondenz und die idealieserte Strukturen. Dialectica 24 (1970) pp 53–66 (English Translation by E. Reck and C.Parsons available as Text No.27, Bernays Project)

    Article  Google Scholar 

  9. A. Cantini: The axiom of choice and combinatory logic. Journal of Symbolic Logic 68 (2003) pp 1091–1108

    Article  Google Scholar 

  10. A. Cantini: Remarks on applicative theories. Annals of Pure and Applied Logic 136 (2005) pp 91–115

    Article  Google Scholar 

  11. L. Carroll: What the Tortoise said to Achilles. Mind 4 (1895) pp 278–280

    Article  Google Scholar 

  12. E. Casari: Matematica e Verità. Rivista di Filosofia 78 (1987) pp 329–350

    Google Scholar 

  13. C. Cellucci: The nature of mathematical explanation, this volume (2008)

    Google Scholar 

  14. C. Cornaros and C. Dimitracopoulos: The prime number theorem and fragments of PA. Arch. Math. Logic 33 (1994) pp 265–281

    Article  Google Scholar 

  15. L. Crosilla, H. Ishihara, P. Schuster: On constructing completions. Journal of Symbolic Logic 70 (2005) pp 969–978

    Article  Google Scholar 

  16. S. Feferman: In the Light of Logic (Oxford University Press, Oxford 1998)

    Google Scholar 

  17. E. Friedgut (with an appendix by J. Bourgain): Sharp thresholds of graph prop-erties and the k-Sat problem. Journal of the American Mathematical Society 12 (1999) pp 1017–1054

    Article  Google Scholar 

  18. E. Giusti: Ipotesi alla base dell’esistenza matematica (Boringhieri, Torino 1999)

    Google Scholar 

  19. A. Goerdt: A Threshold for Unsatisfiability. Journal of Computer and System Sciences 53 (1996) pp 469–486

    Article  Google Scholar 

  20. J. Hafner and P. Mancosu: The Varieties of Mathematical Explanation. In: Visualization, Explanation, and Reasoning Styles in Mathematics, ed by K. Jorgensen et al. (Kluwer, Dordrecht 2005).

    Google Scholar 

  21. G. Hardy: A mathematician’s apology (Cambridge University Press, Cambridge, 1941)

    Google Scholar 

  22. 22. B. Intrigila and R. Statman: Some results on extensionality in lambda calculus. Annals of Pure and Applied Logic 132 (2005) pp 109–125

    Article  Google Scholar 

  23. R. Kahle: David Hilbert and functional self-application, preprint (2006)

    Google Scholar 

  24. U. Kohlenbach and P. Oliva: Proof mining: a systematic way of analysing proofs in mathematics. Proc. Steklov Inst. Math. 242 (2003) pp 136–164

    Google Scholar 

  25. U. Kohlenbach: Some Logical Metatheorems with Applications in Functional Analysis. Transactions of the American Mathematical Society 357 (2005) pp 89–128

    Article  Google Scholar 

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

    Google Scholar 

  27. G. Lee: Phase Transitions in Axiomatic Thought, Ph. Thesis, Westfalische Wilhelms-Universität Mönster (2005)

    Google Scholar 

  28. G. Lolli: Capire una dimostrazione (Il Mulino, Bologna 1988)

    Google Scholar 

  29. R. S. Lubarski and M. Rathjen: On the constructive Dedekind reals. To appear in Logic and Analysis, published online: 18 February (2008) pp 1–22

    Google Scholar 

  30. S. Mac Lane: Despite physicists, proof is essential in mathematics. Synthèse 111 (1997) pp 147–154

    Article  Google Scholar 

  31. P. Mancosu: Mathematical Explanation: Problems and Prospects. Topoi 20 (2001) pp 97–117

    Article  Google Scholar 

  32. 32. G. Pólya: Eine Erinnerung an Hermann Weyl. Mathematische Zeitschrift 126 (1972) pp 296–298

    Article  Google Scholar 

  33. D. Prawitz: Natural Deduction: a Proof—theoretical Study (Almqvist and Wiksell, Stockholm 1965)

    Google Scholar 

  34. D. Prawitz: Philosophical Aspects of proof theory. In: Contemporary philosophy. A new survey, vol.I (Nijhoff, The Hague 1981) pp 235–277

    Google Scholar 

  35. Y. Rav: Why do we prove theorems. Philosophia Mathematica 7 (1999) pp 7–41

    Google Scholar 

  36. G. Rota: The phenomenology of mathematical beauty. Synthèse 111 (1997) pp 171–182

    Article  Google Scholar 

  37. B. Russell: The regressive method of discovering the premises of mathematics. In: B. Russell: Essays on Analysis (London 1973) pp 272–283

    Google Scholar 

  38. S.G. Simpson: Subsystems of Second Order Arithmetic (Springer, Berlin Heidelberg New York 1999)

    Google Scholar 

  39. A. Weiermann: An application of graphical enumeration to PA. J. Symbolic Logic 68 (2003) pp 5–16

    Article  Google Scholar 

  40. A. Weiermann: Analytic combinatorics, proof-theoretic ordinals, and phase transition for independence results. Ann. Pure Appl. Logic 136 (2005) pp 189-218

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Filosofia, Università di Firenze, Via Bolognese 52, 50139, Firenze, Italy

    Andrea Cantini

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  1. Andrea Cantini
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Editor information

Editors and Affiliations

  1. Dipartimento di Filosofia, Università degli Studi di Bologna, Italia

    Rossella Lupacchini  & Giovanna Corsi  & 

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Cantini, A. (2008). On Formal Proofs. In: Lupacchini, R., Corsi, G. (eds) Deduction, Computation, Experiment. Springer, Milano. https://doi.org/10.1007/978-88-470-0784-0_2

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