This is a paper, published in Logic and Logical Philosophy, on the concept of universals in philosophical logic–which includes the example of “Sophia Loren as “the” Italian women”. The always-self-predicative universals of category theory form the opposite bookend to the never-self-predicative universals of iterative set theory.
The development of the logic of partitions (dual to the usual Boolean logic of subsets) and logical information theory bring out a fundamental duality between existence (e.g., elements of a subset) and information (e.g., distinctions of a partition). This leads in a more meta-physical vein to two different conceptions of reality, one of which provides the realistic interpretation of quantum mechanics.
Classical physics and quantum physics suggest two different meta-physical conceptions of reality: the classical notion of a objectively definite reality “all the way down,” and the quantum conception of an objectively indefinite type of reality. Part of the problem of interpreting quantum mechanics (QM) is the problem of making sense out of an objectively indefinite reality. Our sense-making strategy is to follow the math by showing that the mathematical way to describe indefiniteness is by partitions (quotient sets or equivalence relations).
Saunders Mac Lane famously said that Bourbaki (in the person of Pierre Samuel) “just missed” developing the notion of an adjunction in 1948 a decade before Daniel Kan. When adjoints are defined using heteromorphisms or chimera morphisms (first done by Bodo Pareigis in 1970), then indeed Samuel was only a now-simple dualization away from defining adjoints. Since the heteromorphic treatment allows adjunctions to be factored into two, in general, independent universal mapping properties (UMPs), we also pose the question as to whether adjunctions or UMPs should be considered as category theory’s most important concept. It seems Mac Lane and most orthodox category theorists would say adjunctions but Grothendieck among others would say UMPs stated using representable functors.
We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. This is a preprint of the paper coming out in Axiomathes.
The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality–that is described mathematically by partitions. Our sense-making strategy is implemented by developing the mathematics of partitions at the connected conceptual levels of sets and vector spaces. Set concepts are transported to (complex) vector spaces to yield the mathematical machinery of full QM, and the complex vector space concepts of full QM are transported to the set-like vector spaces over ℤ₂ to yield the rather fulsome pedagogical model of quantum mechanics over sets or QM/sets.