Iconicity and Visualization in Peirce's Philosophy of Mathematics

JAVIER LEGRIS

SALVADOR, BA

Congreso; Fourth Meeting of the Association for the Philosophy of Mathematical Practice (APMP); 2017

Association for the Philosophy of Mathematical Practice (APMP)

Within theframework of the Workshop ?Varieties of Visualization inmathematics?,the aim of this talk is to analyze the idea of visualizationarisingfrom the notion of iconicity, as it was devised by Charles S. Peirce inhis theoryof signs. The notion of iconicity is central in Peirce?s conceptionofmathematics. For example, he referred to ?mathematical, that is, [?]diagrammatical,or, iconic, thought.? (CP 3.429 ), and he wrote explicitly:?Mathematicalreasoning is diagrammatic. This is as true of algebra asofgeometry.? (CP 5.148). In his theory of signs, Peirce proposed manyclassificationsof signs. According to the way signs refer to the denotedentities,signs were classified into icons, indices and symbols. Diagrams fallin thecategory of icons, so that mathematical knowledge is the object of arichanalysis where the idea of visualization has a special role. Geometricalfigures,tables and formulas are all of iconic nature. Peirce conceivedmathematicalactivity as the construction, manipulation and observation oficons. Inthe case of algebra, Peirce wrote in a famous paper from 1885 ?thevery ideaof the art is that it presents formulae which can be manipulated,and that byobserving the effects of such manipulation we find propertiesnot to beotherwise discerned. [?] These are patterns which [?] are the iconsparexcellence of algebra.? (Peirce, CP 3.363). Two different aspects of iconicitywill behighlighted: the operational aspect and the purely topological one. Thefirstaspect focuses on icons as structural representations on the basis of visualproperties.In this respect a diagram is characterized as ?an Icon of a setofrationally related objects? (MS 293: 11) and referred to ?icon [or analyticpicture]?(Peirce CP 1.275). From the analysis and transformation of signs newknowledgeobtains. In the exposition the elucidation of operational iconicitydue toFrederik Stjernfelt will be also discussed, stressing its cognitiveimportance.The second aspect is related to the topological features of iconicity,which arestronger related to the idea of visualization. Some examples frommathematicallogic will be provided in order to show different diagrammaticsystemsthat are topological equivalent but have different cognitiveeffectsleading to differences in visualization. Finally, both aspects will beconnectedwith the tradition of symbolic knowledge stemming from Leibniz.Summing up,the presentation aims at contributing to the development ofconceptualframework for appreciating the various epistemic roles played bythe differentvarieties of visualizations in mathematical practices.Reference:Peirce,Charles Sanders. CP. Collected Papers. 8 vols, vols. 1- 6 ed. by CharlesHartshorne& Paul Weiss, vols. 7-8 ed. by Arthur W. Burks. Cambridge(Mass.), Harvard University Press, 1931-1958.