Experiments do not mechanically give rise to theory.
To: firstname.lastname@example.org, Subject: categories: Re: Sketches and Platonic Ideas, From: “F. William Lawvere”, Date: 05 Dec 2001
[…] In my 1972 Perugia Notes I had made an attempt to characterize the relation between these sorts of mathematical considerations and philosophy by saying that while Platonism is wrong on the relation between Thinking and Being, something analogous is correct WITHIN the realm of Thinking. The relevant dialectic there is between abstract general and concrete general.
Not concrete particular (“concrete” here does not mean “real”). There is another crucial dialectic making particulars (neither abstract nor concrete) give rise to an abstract general; since experiments do not mechanically give rise to theory, it is harder to give a purely mathematical outline of how that dialectic works, though it certainly does work. A mathematical model of it can be based on the hypothesis that a given set of particulars is somehow itself a category (or graph), i.e., that the appropriate ways of comparing the particulars are given but that their essence is not. Then their “natural structure” (analogous to cohomology operations) is an abstract general and the corresponding concrete general receives a Fourier-Gelfand-Dirac functor from the original particulars. That functor is usually not full because the real particulars are infinitely deep and the natural structure is computed with respect to some limited doctrine; the doctrine can be varied, or “screwed up or down” as James Clerk Maxwell put it, in order to see various phenomena.