Universal Mapping Properties and Artificial Intelligence
In the early 1960s there was a mathematical advance, an advance on par with Newtonian mechanism in physics and Darwinian evolution in biology. A mathematical theory, prior to F. William Lawvere’s Functorial Semantics of Algebraic Theories (Lawvere, 1963/2004/2013), was a list of statements, which together determined whether a given object is this or that. So, a theory of a universe of discourse, say, the category of graphs (consisting of dots and arrows), had no choice but to leave the given universe for one, with no readily discernible kinship with graphs, of arbitrary symbols, words, and sentences, i.e., language. Following Lawvere’s functorial semantics, a theory of a given category of objects is a [sub]category with their basic properties as objects and mutual determinations of properties as morphisms (Lawvere, 2003; see also Posina, Ghista, and Roy, 2017). Along with the functorial semantics of Lawvere, sketches of Bastiani and Ehresmann (1972), and Grothendieck’s descent (see Clementino and Picado, 2007/2008, p. 15) contributed to the monumental development of our mathematical understanding of mathematics, wherein the relationship between particulars, theory, models, presentations, and doctrine is spelled out in a spellbinding display of science: ever-proper alignment of reason with experience.