Abstract
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.
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- 1.
The theorem is mentioned, among others, in [4].
- 2.
Barwise and Etchemendy quote this passage from an article by Tennant (see [28]). They believe it expresses the ‘dogma’ of logocentricity that they want to challenge.
- 3.
I am not denying here the possibility that there are three-dimensional diagrams. I only want to exclude this possibility for the moment, because I am inclined to think that it implies additional considerations.
- 4.
“It is not, however, the statical Diagram-icon that directly shows this; but the Diagram-icon having being constructed with an Intention […]. Now, let us see how the Diagram entrains its consequence. The Diagram sufficiently partakes of the percussivity of a Percept to determine, as its Dynamic, or Middle, Interpretant, a state [of] activity in the Interpreter, mingled with curiosity. As usual, this mixture leads to Experimentation.” In [21].
- 5.
For a detailed discussion of this case-study, see [27].
- 6.
Letter 103, Of Syllogism, and their different Forms, when the first Proposition is Universal. See [9].
- 7.
Still today, there are attempts to take this path, such as the pictionary ‘Point it: Traveler’s Language Kit’, by Dieter Graf.
References
Baldasso, R.: Illustrating the book of nature in the Renaissance: drawing, painting, and printing geometric diagrams and scientific figures. PhD thesis (2007)
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)
Bouligand, G.: Premières leçons sur la théorie générale des groupes. Vuibert, Paris (1932)
Brown, J.R.: Proofs and pictures. Br. J. Philos. Sci. 48, 161–180 (1997)
Brown, J.R.: Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge, London (1999)
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)
Corfield, D.: Towards a Philosophy of Real Mathematics. Cambridge University Press, Cambridge (2003)
Englebretsen, G.: Linear diagrams for syllogisms (with relationals). Notre Dame J. Form. Log. 33(1), 37–69 (1992)
Euler, L.: Letters of Euler to a German Princess: On Different Subjects in Physics and Philosophy (trans.: Hunter, H.). Thoemmes Continuum, London (1997)
Fallis, D.: Intentional gaps in mathematical proofs. Synthese 134(1–2), 45–69 (2003)
Folina, J.: Pictures, proofs, and ‘mathematical practice’: reply to James Robert Brown. Br. J. Philos. Sci. 50(3), 425–429 (1999)
Grosholz, E.: Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford University Press, London (2007)
Hartshorne, C., Weiss, P. (eds.): Collected Papers of Charles Sanders Peirce, vols. 1–6. Harvard University Press, Cambridge (1931–1935)
Lakoff, G., Nuñez, R.: Where Mathematics Comes from: How the Embodied Mind Brings Mathematics into Being. Basic Books, New York (2001)
Lambert, J.H.: Neues Organon. Akademie Verlag, Berlin (1990)
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)
Mancosu, P.: Mathematical explanation: problems and prospects. Topoi 20, 97–117 (2001)
Nelsen, R.: Proofs Without Words: Exercises in Visual Thinking (Classroom Resource Materials). Math. Assoc. of America, Washington (1997)
Nelsen, R.: Proofs Without Words: More Exercises in Visual Thinking (Classroom Resource Materials). Math. Assoc. of America, Washington (2001)
Neurath, O.: Visual education: a new language. Surv. Graph. 26(1), 25 (1937). http://newdeal.feri.org/survey/37025.htm
Peirce, C.S.: Prolegomena to an apology for pragmaticism. Monist 16(4), 492–546 (1906)
Polya, G.: Mathematics and Plausible Reasoning. Princeton University Press, Princeton (1968)
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)
Shimojima, A.: The graphic-linguistic distinction. Artif. Intell. Rev. 15, 5–27 (2001)
Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994)
Shin, S.-J.: Heterogeneous reasoning and its logic. Bull. Symb. Log. 10(1), 86–106 (2004)
Shin, S.-J., Lemon, O.: Diagrams. Entry in the Stanford Encyclopedia of Philosophy (2008). http://plato.stanford.edu/entries/diagrams/
Tennant, N.: The withering away of formal semantics? Mind Lang. 1(4), 302–318 (1986)
Acknowledgements
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.
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Giardino, V. (2013). A Practice-Based Approach to Diagrams. In: Moktefi, A., Shin, SJ. (eds) Visual Reasoning with Diagrams. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0600-8_8
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