## RESUMO / ABSTRACT

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.