, Volume 186, Issue 1, pp 387–409 | Cite as

Diagrams as sketches



This article puts forward the notion of “evolving diagram” as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter “diagrams*”) in the context of “sketch theory,” a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, that it helps to overcome the rivalry between set theory and category theory as a general semantical framework, and that it suggests a more flexible understanding of the opposition between formal proofs and diagrammatic reasoning. Thus, the aim of the paper is twofold. First, it claims that diagrams* provide a clear example of evolving diagrams, and shed light on them as a general phenomenon. Second, in return, it uses sketches, understood as evolving diagrams, to show how diagrams* in general should be re-evaluated positively.


Mathematical diagrams Pictorialism Categorical diagrams Sketch theory Formal proof Semantics 


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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Département de Philosophie (IREPH)Université Paris OuestNanterreFrance
  2. 2.SPHERE (UMR 7219), Université Paris-DiderotParisFrance

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