Abstract

In his Critic of pure reason, Immanuel Kant notoriously claimed that, being grounded on the forms of sense intuition, arithmetical and geometrical propositions are both synthetic and a priori. Bernard Bolzano, followed in this by the logicist movement , answered that the generality and necessity of mathematical propositions and proofs can only be grounded on conceptual analysis. Even though, just like Frege, he is one of the fathers of formal logic, Charles Sanders Peirce provides some semiotic reasons to think that Kant was right: diagrams do convey general meanings and provide some knowledge which is necessary yet non-trivial. Unlike logical analysis, visual presentation of concepts in schemas or diagrams helps to explore concepts by stressing some of their “side” features in such a way that new knowledge is gained: « diagrams evolve what was involved » . This is why, according to Kant’s notion of intuitive construction, mathematical inferences are not merely deductive but are inventive and ampliative. My talk aims at identifying some iconic virtues of diagrams which, according to Peirce, explain their epistemic productivity. A first one lies in the “formal” nature of icons, which allows them to express syntactic relations between descriptive and demonstrative components of structured information. On this respect, even algebraic and ideographic expressions are icons exhibiting a general form – a “rheme” – in which places for indices are filled with variables “x” and “y” meaning “any individual”. For this reason, even though they are singular, diagrams are “abstractions” in the sense that they represent relations rather than their terms. Only with this in prospect can a second, and more studied, feature of diagrams matter, namely their two-dimensionality, which helps to exhibit complex relations that cannot be seen on linear linguistic expressions. Finally, a third feature of diagrams lies in their imaginary rather than referential character. Icons connote without denoting, and therefore they can be informational without this information being limited to singular individuals. Furthermore, this non referential character of icons is what makes them open to virtual exploratory manipulations that allow to consider and investigate possibilities which in turn inform us on not obvious properties of the presently visible configuration