| Summary: | In his Critique of Pure Reason, Immanuel Kant claimed that, being grounded on the forms of sense intuition, arithmetic and geometric propositions are both synthetic (i.e. informative) and a priori true. Bernard Bolzano, followed by the logicist movement (from Gottlob Frege to Rudolf Carnap), answered that the generality and necessity of mathematical propositions and proofs can only be grounded on conceptual analysis.Even though, like Frege, Charles Sanders Peirce is one of the fathers of formal logic, he provides some semiotic reasons to think that Kant was right: diagrams do convey general meanings and provide some knowledge which is necessary and not 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 enables new knowledge to be acquired: “diagrams evolve what was involved”. This is why, according to Kant’s notion of intuitive construction, mathematical inferences are not merely deductive but are also inventive and ampliative.The paper 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 (symbols) and demonstrative (indices) components of structured information. In 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 object”. For this reason, even though they are singular, diagrams are “abstractions” in the sense that they represent relations rather than their terms.Only from this perspective can a second, and more studied, feature of diagrams be considered significant, namely their two-dimensionality, which serves to represent complex relations that cannot be seen in linear linguistic expressions.Finally, a third feature of diagrams lies in their imaginary rather than referential nature. Icons connote without denoting, and therefore they can be informational without this information being limited to singular individuals. Furthermore, this non-referential nature of icons is what makes them open to virtual exploratory manipulations that allow one to consider and investigate possibilities which, in turn, inform us on less obvious properties of the presently visible configuration.
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