Date: 03 Aug 2011
And so on . . . : reasoning with infinite diagrams
- Solomon Feferman
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This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a “pre” form of this thesis that every proof can be presented in everyday statements-only form.
This paper is based in part on a lecture delivered to the Workshop on Diagrams in Mathematics, Paris, October 9, 2008. Slides for that talk and another one on the same topic for the Logic Seminar, Stanford, February 24, 2009 are available at http://math.stanford.edu/~feferman/papers/Infinite_Diagrams.pdf. I wish to thank Jeremy Avigad, Michael Beeson, Hourya Benis, Philippe de Rouilhan, Wilfrid Hodges, and Natarajan Shankar as well as the two referees for their comments on a draft of this article. I wish also to thank Ulrik Buchholtz for preparing the LaTeX version of this article.
Allwein, G., Barwise, J. (eds) (1996) Logical reasoning with diagrams. Oxford University Press, New York
Avigad J., Dean E., Mumma J. (2009) A formal system for Euclid’s elements. Review of Symbolic Logic 2(4): 700–768CrossRef
Avigad, J., Donnelly, K., Gray, D., & Raff, P. (2007). A formally verified proof of the prime number theorem. ACM Transactions on Computational Logic 9(1), Art. 2.
Azzouni J. (2004) The derivation-indicator view of mathematical practice. Philosophia Mathematica 12(2): 81–105CrossRef
Barker-Plummer D., Bailin S. C., Ehrlichman S. M. (1996) Diagrams and mathematics. In: Selman B., Kautz H. (eds) Proceedings of the Fourth International Symposium on Artificial Intelligence and Mathematics. Fort Lauderdale, FL, pp 14–17
Barwise J., Etchemendy J. (1996) Heterogeneous logic. In: Allwein G., Barwise J. (eds) Logical reasoning with diagrams. Oxford University Press, New York, NY, pp 179–200
Feferman, S. (1979). What does logic have to tell us about mathematical proofs?. The Mathematical Intelligencer 2, 20–24. (Reprinted as Ch. 9 in Feferman 1998).
Feferman S. (1998) In the Light of Logic. Oxford University Press, New York, NY
Feferman S. (2000) Mathematical intuition versus mathematical monsters. Synthese 125(3): 317–332CrossRef
Hahn, H. (1933). The crisis in intuition. In Empiricism, logic, and mathematics: philosophical papers, Vol. 13 of Vienna circle collection (pp. 73–102). Dordrecht: D. Reidel Publishing Company
Hodges W. (1997) A shorter model theory. Cambridge University Press, Cambridge
Hrbacek K., Jech T. (1999) Introduction to set theory, Vol. 220 of monographs and textbooks in pure and applied mathematics. (3rd ed.). Marcel Dekker Inc., New York
Jamnik M. (2001). Mathematical reasoning with diagrams, Vol. 127 of CSLI Lecture Notes. Stanford, CA: CSLI Publications. (With a foreword by J. A. Robinson).
Jans J. P. (1964) Rings and homology. Holt, Rinehart and Winston, New York
Kuratowski K., Mostowski A. (1968) Set theory. (Translated from the Polish by M. Maczyński). PWN-Polish Scientific Publishers, Warsaw
Leitgeb H. (2009) On formal and informal provability. In: Bueno O., Linnebo Ø. (eds) New Ways in the Philosophy of Mathematics. Palgrave Macmillan, New York, NY, pp 263–299
Mac Lane S. (1975) Homology. Springer-Verlag, Berlin
Mancosu P. (2005) Visualization in logic and mathematics. In: Mancosu P., Jørgensen K., Pedersen S. (eds) Visualization, explanation and reasoning styles in mathematics, Vol. 327 of Synthese Library. Springer, Netherlands, pp 13–30CrossRef
Manders K. (2008) Diagram-based geometric practice. In: Mancosu P. (eds) The philosophy of mathematical practice. Oxford University Press, New York, NY, pp 65–79CrossRef
Manders K. (2008) The Euclidean Diagram (1995). In: Mancosu P. (eds) The philosophy of mathematical practice. Oxford University Press, Oxford, pp 80–133CrossRef
Mumma, J. (2006). Intuition formalized: Ancient and modern methods of proof in elementary geometry. Ph.D. thesis, Carnegie-Mellon University.
Nelsen R. B. (1993) Proofs without words: Exercises in visual thinking. Mathematical Association of America, Washington, DC
Nipkow, T., Paulson, L. C., & Wenzel, M. (2002) Isabelle/HOL—A proof assistant for higher-order logic, Vol. 2283 of LNCS. Berlin, New York: Springer.
Pelc A. (2009) Why do we believe theorems?. Philosophia Mathematica 17(1): 84–94CrossRef
Poincaré, H. (1952). Science and method (Translated by Francis Maitland. With a preface by Bertrand Russell). New York: Dover Publications Inc.
Rav Y. (1999) Why do we prove theorems?. Philosophia Mathematica 7(1): 5–41CrossRef
Rav Y. (2007) A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica 15(3): 291–320CrossRef
Shin, S.-J., & Lemon, O. (2008). Diagrams. In E.N. Zalta (Ed.) The Stanford encyclopedia of philosophy. CSLI, winter 2008 edition. URL: http://plato.stanford.edu/archives/win2008/entries/diagrams/.
- And so on . . . : reasoning with infinite diagrams
Volume 186, Issue 1 , pp 371-386
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- Diagrammatic reasoning
- Infinite diagrams
- Formalizability thesis
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- Solomon Feferman (1)
- Author Affiliations
- 1. Department of Mathematics, Stanford University, Stanford, CA, 94305–2125, USA