.... disavowing du = dq - dw is not going to help me here.
That was entirely true in the previous context ... but
I still think that eschewing du = dq - dw is a good idea
for other reasons. To wit:
The notion of infinitesimals was introduced in the late
1600s by Leibniz, over the objections of Newton and
generation upon generation of mathematicians since.
There are things you can "get away with" using this
approach, but it's sorta like playing football without
a helmet. You might get away with it for a while, but
I don't recommend it. More to the point, students are
particularly unlikely to know when they can get away
with it versus when they're gonna get clobbered.
Just to compound the problem, there are a lot of things
you can get away with in one dimension that you can't
get away with in higher dimensions. And thermodynamics
(except maybe in trivial cases) is conspicuously multi-
dimensional. This leads to tremendous amounts of negative
transference.
It is possible to do thermo in a way that makes sense,
mathematically, physically, and otherwise. The price
is small. The idea is to interpret d(...) not as an
infinitesimal but as a gradient vector in some abstract
space. This allows us to write
dE = T dS - P dV - voltage d(charge) ... et cetera
and to know quite precisely what that means. It just
doesn't allow us to rewrite T dS as dQ or indeed as
d(anything). This is a price I'm willing to pay. When
speaking about T dS, I am happy to call it T dS rather
than dQ. AFAICT calling it dQ has no upside but plenty
of downside.
But I'll leave it as a question to the group: Is
there anything you can do with "dQ" that you can't
do just as well (or better!) with T dS?