Crooked Timber of Mathematics
Straightening Mathematical Thought
I saw category theory when I looked at a line. Zooming out of the place-value notation of numbers, I saw functions, within which category theory appeared with a figural salience of the auditory modality, leaving me no choice but to perceive. Having had lines, place-value notations, and functions for god knows how long, why did we wait unit 1945 to get to category theory (we could have at least gotten to category sometime around 1917, as I recently learned from Professor Eklund, which would have spared us from the Confusionism—inconsistencies, impossibilities, and paradoxes—that we had to put up with)?
As is my wont, I asked: Why?
We all know, since our high school days, if two functions
f, g: A —> B
are not equal, then there is at least one element ‘a’ of the domain set A at which the values of the two functions are not equal, i.e.,
f(a) is not equal to g(a).
Related to this, we might/must have noticed that the number of elements of a set A
|A| = |A|^|1| = |A|
where |A|^|1| is the set of functions from a one-element set 1 = {*} to the given set A. This 1-1 correspondence between elements of a set and functions from the one-element set 1 to the set and the sufficiency of an element in testing the equality of a parallel pair of (possibly equal) functions are evocative of adequacy (Lawvere, Perugia Notes, p. 135). And yet it took light years, so to speak, to define adequate subcategory.
Along these lines, there is the popular 2^|A| calculation predating Lawvere element, and another (plausibly) familiar 1-1 correspondence between two-variable functions
A x B —> C
and one-variable functions with functions as values
A —> C^B and/or B —> C^A
which wasn’t recognized for aeons as an instance of adjointness.