During the end of the XIXth century and the beginning of the XXth , the relations between intuitionism and formalism were much discussed; and a great part of these discussions were concerned with the role played by figures and other graphic tools in geometrical demonstrations. There were, no doubt, good reasons behind the success of a purely linguistic conception of proof, such as the one defended in Hilbert's Foundations of Geometry. However, recently, the legitimacy of heterogeneous proofs is being defended again; that is, of proofs that include legitimately graphic tools. Thus, the distinction between graphic and linguistic is nowadays fundamental. Nevertheless, and somehow paradoxically, the following consensus has been reached: the distinction among graphic and linguistic signs, or among graphic and linguistic representation, has a gradual nature. It would be almost possible to speak of a sort of continuum from "depictions" of pictures or photos to the descriptions of natural language, going through intermediate instances such as maps, geometric diagrams, logical diagrams (like those of Venn), etc. Within this context, linked to the general problem of visualization, I would like to address the issue of demonstration.

Henceforth, after introducing the distinction between heterogeneous demonstrations à la Euclid and homogeneous demonstrations à la Aristotle, I shall introduce Leibniz's notions of Ecthetic and Catholic demonstrations. I advance that the distinction between ecthetic and catholic points out to the way in which natural languages and artificial languages represent concepts or conceptual relations, designating on the first case, exposing or exhibiting on the second. Moreover, I advance that at least on some cases the use of graphic tools would also involve a way of Ecthetic representation. I shall conclude with a brief analysis of Hilbert's conception of demonstration as a figure.