Recently I was humbled; here it is:
Naturality vis-à-vis Categories
but not chagrined ;) in large part cuz dis ain’t new to me! Just to give one example, I took up the then Nature editor John Maddox’s challenge:
back-of-the-envelope calculations
(irritated by the implicit preaching-sans-practice; he, being a physicist/quant, could have squeezed one or two back-of-the-envelope calculations into his editorial, which is a shameless display by the scientific priesthood of their (i) wholesale misunderstanding of reductionism that prevails to this day, notwithstanding Newton emphasizing the indispensability of compounding/synthesis after resolution/analysis, and (ii) wholesome ignorance of the relation between quantitative and qualitative, notwithstanding mathematicians illustrating how qualities constitute a refinement of quantitative accounts, as they figure in the practice of science; F. William Lawvere, Perugia Notes, p. iv; see also Science of Knowing, pp. 3-4).
Back then I was a patch-clamp guy in the laboratory of Professor Lipton (Harvard Medical School, Boston), who was working on neuronal death. Drawing parallels to Chandrasekhar limit, which tells whether a dying star will end up a white dwarf or a black hole, I set out to calculate the threshold determining whether a dying neuron will undergo apoptosis or necrosis (setting aside all the well-meaning biophysicists elaborating how biology is not as easy as physics), which wouldn’t have been possible without the freedom I enjoyed in Lipton lab.
Fast-forward to Bengaluru: I’ll begin with something that everyone finds familiar:
a set N = {0, 1, 2 …} of numbers. Next, consider a binary operation:
a: N × N → N
defined as
a (n, n’) = n + n’
for every pair of elements of the factor set(s) N. With addition of numbers as the binary operation, we find that addition of numbers is associative since
(n + n’) + n’’ = n + (n’ + n’’) = n + n’ + n’’
for every triple of numbers.
Next, consider a point:
i: 1 → N
(where 1 = {•} is a single element set) defined as
i (•) = 0
which is both left- and right-identity of addition, i.e.
a (0, n) = 0 + n = n
and
a (n, 0) = n + 0 = n.
Putting it all together, we have a set N of numbers, a binary operation a: N × N → N of addition of numbers that is associative, and a number i: 1 → N that is both left- and right-identity of addition. This trinity <N, a, i> = <N, +, 0> is a monoid.
Beginning with monoid, we go via Cayley, Yoneda, et al., to geometry of figures (in terms of basic shapes and their incidence relations; see F. William Lawvere, Perugia Notes, pp. 90- ; Marmolejo/Lawvere and Schanuel, Matemáticas Conceptuales/Conceptual Mathematics, pp. 358/361, 363/365, 368-370/370-371).
Happy Deepavali :)