12 Março 1997, 14:30
F. William Lawvere, State University of New York at Buffalo
Kinship and mathematical categories

Those concepts which are historically stable tend to be those which in some way reflect reality, and in turn tend to be those which are teachable. "Mathematical" should mean in particular "teachable", and the modern mathematical theory of flexible categories can indeed be used to sharpen the teachability of basic concepts. A concept which has enjoyed some historical stability for 40,000 years is the one involving that structure of a society which arises from the biological process of reproduction and its reflection in the collective consciousness as ideas of genealogy and kinship. Sophisticated methods for teaching this concept were devised long ago, making possible regulation of the process itself. A more accurate model than heretofore possible of kinship can be sharpened with help of the modern theory of flexible mathematical categories, and at the same time our established cultural acquaintance with the concept helps to illuminate several aspects of the general theory , which we will therefore endeavor to explain concurrently.

Abstracting the genealogical aspect of a given society yields a mathematical structure within which aunts, cousins, etc. can be precisely defined. Other objects, in the same category of such structures, which seem very different from actual societies, are nonetheless shown to be important tools in a societys conceptualizing about itself, so that for example gender and moiety become labelling morphisms within that category. Topological operations, such as contracting a connected subspace to a point, are shown to permit rationally neglecting the remote past. But such operations also lead to the qualitative transformation of a topos of pure particular Becoming into a topos of pure general Being; the latter two kinds of mathematical toposes are distinguished from each other by precise conditions. The mythology of a primal couple is thus shown to be a naturally-arising didactic tool. The logic of genealogy is not at all 2-valued nor Boolean, because the truth-value space naturally associated with the ancestor concept has a rich lattice structure. A finitary approximation to this theory is also considered and the corresponding category of structures is fully analyzed.