The Arithmetic Mentality arises in the Ice Age and reaches its climactic formation in Sumerian civilization; the Geometric, which arises in Egypt and Babylon, reaches its climactic formation in the classical civilizations of Greece, Rome, Persia, India, China, and Mesoamerica; the Algebraic, which arises in Islamic Baghdad, reaches its climactic medieval formulation in the Mediterranean cultural ecology; the Galilean Dynamical, which arises in Renaissance Italy, reaches its climactic formation in the late nineteenth century; and, finally, the Complex Dynamical Mentality arises in Paris in 1889 with Poincaré, and has now become a widespread mentality among a scientific and artistic avant-garde. For each of these mentalities there is a cultural ecology in which it was embedded at its emergence, and for each there is an archetypal object that embodies the unique character of its performative life in the new culture.

If we go back to the beginnings of art and signs in the Chauvet Cave of 35,000 BCE – this is double the age of Lascaux – we can observe that the beautifully painted image of the European bison is superimposed upon the spot that had already been selected by the cave bear as the place of its clawed markings. Here we can see that Derrida was right, that the engraving, the gramé, is coeval with ritual or chant, and there is no archaic privileging of the oral as prior to the engraved.