A Practice-Based Approach to Diagrams

  • Valeria Giardino
Part of the Studies in Universal Logic book series (SUL)


In this article, I propose an operational framework for diagrams. According to this framework, diagrams do not work like sentences, because we do not apply a set of explicit and linguistic rules in order to use them. Rather, we become able to manipulate diagrams in meaningful ways once we are familiar with some specific practice, and therefore we engage ourselves in a form of reasoning that is stable because it is shared. This reasoning constitutes at the same time discovery and justification for this discovery. I will make three claims, based on the consideration of diagrams in the practice of logic and mathematics. First, I will claim that diagrams are tools, following some of Peirce’s suggestions. Secondly, I will give reasons to drop a sharp distinction between vision and language and consider by contrast how the two are integrated in a specific manipulation practice, by means of a kind of manipulative imagination. Thirdly, I will defend the idea that an inherent feature of diagrams, given by their nature as images, is their ambiguity: when diagrams are ‘tamed’ by the reference to some system of explicit rules that fix their meaning and make their message univocal, they end up in being less powerful.


Diagrammatic reasoning Practice-based philosophy of mathematics Peirce’s diagrams Manipulative imagination Productive ambiguity 

Mathematics Subject Classification (2010)

00A66 03A05 97C30 



I want to thank Mario Piazza, Achille Varzi, Roberto Casati, and two anonymous referees who gave me very useful suggestions in order to improve this article. Many thanks to Christopher Whalin for having proof-read the final version. Special thanks to the editors, Amirouche Moktefi and Sun-Joo Shin, for their careful work.


  1. 1.
    Baldasso, R.: Illustrating the book of nature in the Renaissance: drawing, painting, and printing geometric diagrams and scientific figures. PhD thesis (2007) Google Scholar
  2. 2.
    Barwise, J., Etchemendy, J.: Visual information and valid reasoning. In: Allwein, G., Barwise, J. (eds.) Logical Reasoning with Diagrams, pp. 3–25. Oxford University Press, London (1996) Google Scholar
  3. 3.
    Bouligand, G.: Premières leçons sur la théorie générale des groupes. Vuibert, Paris (1932) Google Scholar
  4. 4.
    Brown, J.R.: Proofs and pictures. Br. J. Philos. Sci. 48, 161–180 (1997) zbMATHCrossRefGoogle Scholar
  5. 5.
    Brown, J.R.: Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge, London (1999) zbMATHGoogle Scholar
  6. 6.
    Byrne, O.: First Six Books of the Elements of Euclid, in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners. William Pickering, London (1847) Google Scholar
  7. 7.
    Corfield, D.: Towards a Philosophy of Real Mathematics. Cambridge University Press, Cambridge (2003) zbMATHCrossRefGoogle Scholar
  8. 8.
    Englebretsen, G.: Linear diagrams for syllogisms (with relationals). Notre Dame J. Form. Log. 33(1), 37–69 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Euler, L.: Letters of Euler to a German Princess: On Different Subjects in Physics and Philosophy (trans.: Hunter, H.). Thoemmes Continuum, London (1997) Google Scholar
  10. 10.
    Fallis, D.: Intentional gaps in mathematical proofs. Synthese 134(1–2), 45–69 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Folina, J.: Pictures, proofs, and ‘mathematical practice’: reply to James Robert Brown. Br. J. Philos. Sci. 50(3), 425–429 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Grosholz, E.: Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford University Press, London (2007) zbMATHGoogle Scholar
  13. 13.
    Hartshorne, C., Weiss, P. (eds.): Collected Papers of Charles Sanders Peirce, vols. 1–6. Harvard University Press, Cambridge (1931–1935) Google Scholar
  14. 14.
    Lakoff, G., Nuñez, R.: Where Mathematics Comes from: How the Embodied Mind Brings Mathematics into Being. Basic Books, New York (2001) Google Scholar
  15. 15.
    Lambert, J.H.: Neues Organon. Akademie Verlag, Berlin (1990) zbMATHGoogle Scholar
  16. 16.
    Macbeth, D.: Diagrammatic reasoning in Euclid’s Elements. In: Van Kerkhove, B., De Vuyst, J., Van Bendegem, J.P. (eds.) Philosophical Perspectives on Mathematical Practice, pp. 235–267. College Publications, London (2010) Google Scholar
  17. 17.
    Mancosu, P.: Mathematical explanation: problems and prospects. Topoi 20, 97–117 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Nelsen, R.: Proofs Without Words: Exercises in Visual Thinking (Classroom Resource Materials). Math. Assoc. of America, Washington (1997) Google Scholar
  19. 19.
    Nelsen, R.: Proofs Without Words: More Exercises in Visual Thinking (Classroom Resource Materials). Math. Assoc. of America, Washington (2001) Google Scholar
  20. 20.
    Neurath, O.: Visual education: a new language. Surv. Graph. 26(1), 25 (1937). Google Scholar
  21. 21.
    Peirce, C.S.: Prolegomena to an apology for pragmaticism. Monist 16(4), 492–546 (1906) CrossRefGoogle Scholar
  22. 22.
    Polya, G.: Mathematics and Plausible Reasoning. Princeton University Press, Princeton (1968) zbMATHGoogle Scholar
  23. 23.
    Rota, G.C.: The phenomenology of mathematical proof. Synthese 111(2), 183–196 (1997) (Special Issue on Proof and Progress in Mathematics edited by A. Kamamori) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Shimojima, A.: The graphic-linguistic distinction. Artif. Intell. Rev. 15, 5–27 (2001) zbMATHCrossRefGoogle Scholar
  25. 25.
    Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994) zbMATHGoogle Scholar
  26. 26.
    Shin, S.-J.: Heterogeneous reasoning and its logic. Bull. Symb. Log. 10(1), 86–106 (2004) zbMATHCrossRefGoogle Scholar
  27. 27.
    Shin, S.-J., Lemon, O.: Diagrams. Entry in the Stanford Encyclopedia of Philosophy (2008).
  28. 28.
    Tennant, N.: The withering away of formal semantics? Mind Lang. 1(4), 302–318 (1986) CrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Institut Jean Nicod (CNRS-EHESS-ENS), Pavillon JardinEcole Normale SupérieureParisFrance

Personalised recommendations