Synthese

, Volume 186, Issue 1, pp 371–386 | Cite as

And so on . . . : reasoning with infinite diagrams

Article
First Online:
Received:
Accepted:

Abstract

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.

Keywords

Diagrammatic reasoning Infinite diagrams Formalizability thesis 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allwein, G., Barwise, J. (eds) (1996) Logical reasoning with diagrams. Oxford University Press, New YorkGoogle Scholar
  2. Avigad J., Dean E., Mumma J. (2009) A formal system for Euclid’s elements. Review of Symbolic Logic 2(4): 700–768CrossRefGoogle Scholar
  3. 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.Google Scholar
  4. Azzouni J. (2004) The derivation-indicator view of mathematical practice. Philosophia Mathematica 12(2): 81–105CrossRefGoogle Scholar
  5. 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 14–17Google Scholar
  6. Barwise J., Etchemendy J. (1996) Heterogeneous logic. In: Allwein G., Barwise J. (eds) Logical reasoning with diagrams. Oxford University Press, New York, NY, pp 179–200Google Scholar
  7. 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).Google Scholar
  8. Feferman S. (1998) In the Light of Logic. Oxford University Press, New York, NYGoogle Scholar
  9. Feferman S. (2000) Mathematical intuition versus mathematical monsters. Synthese 125(3): 317–332CrossRefGoogle Scholar
  10. 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 CompanyGoogle Scholar
  11. Hodges W. (1997) A shorter model theory. Cambridge University Press, CambridgeGoogle Scholar
  12. 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 YorkGoogle Scholar
  13. Jamnik M. (2001). Mathematical reasoning with diagrams, Vol. 127 of CSLI Lecture Notes. Stanford, CA: CSLI Publications. (With a foreword by J. A. Robinson).Google Scholar
  14. Jans J. P. (1964) Rings and homology. Holt, Rinehart and Winston, New YorkGoogle Scholar
  15. Kuratowski K., Mostowski A. (1968) Set theory. (Translated from the Polish by M. Maczyński). PWN-Polish Scientific Publishers, WarsawGoogle Scholar
  16. 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 263–299Google Scholar
  17. Mac Lane S. (1975) Homology. Springer-Verlag, BerlinGoogle Scholar
  18. 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 13–30CrossRefGoogle Scholar
  19. Manders K. (2008) Diagram-based geometric practice. In: Mancosu P. (eds) The philosophy of mathematical practice. Oxford University Press, New York, NY, pp 65–79CrossRefGoogle Scholar
  20. Manders K. (2008) The Euclidean Diagram (1995). In: Mancosu P. (eds) The philosophy of mathematical practice. Oxford University Press, Oxford, pp 80–133CrossRefGoogle Scholar
  21. Mumma, J. (2006). Intuition formalized: Ancient and modern methods of proof in elementary geometry. Ph.D. thesis, Carnegie-Mellon University.Google Scholar
  22. Nelsen R. B. (1993) Proofs without words: Exercises in visual thinking. Mathematical Association of America, Washington, DCGoogle Scholar
  23. 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.Google Scholar
  24. Pelc A. (2009) Why do we believe theorems?. Philosophia Mathematica 17(1): 84–94CrossRefGoogle Scholar
  25. Poincaré, H. (1952). Science and method (Translated by Francis Maitland. With a preface by Bertrand Russell). New York: Dover Publications Inc.Google Scholar
  26. Rav Y. (1999) Why do we prove theorems?. Philosophia Mathematica 7(1): 5–41CrossRefGoogle Scholar
  27. Rav Y. (2007) A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica 15(3): 291–320CrossRefGoogle Scholar
  28. 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/.

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations