I enjoyed the post a lot (at least the parts that didn’t pass right over my head). But I never met a physicist with lingering unease about dimensional analysis. We get that beaten into us until it’s as natural as breathing.
There is very little that physicists have lingering unease about when it comes to (ab)using maths to get their way with the physical world.
Examples abound, and have actually led to many a new avenue in math.
Distributions (as in the Dirac distribution) are a very good example of this (IIRC convolution didn't have an identity and they needed one, and they just made up "functions" that were allowed to have infinite values as long as they had finite sums or something along those lines).
If your math objects breaks other properties of math that you don't use then it doesn't matter, you don't use those so your math is still correct.
A programmer analogy would be that you implement a special kind of list for your project, but don't implement the whole list interface because you don't need the rest. That makes the mathematicians fume, "it isn't a list!!!", and they invent a new interface name for your implementation, like "distribution" instead of function.
and you can't derive a contradiction
> you can do whatever you want with math.
Fixed.
However, regarding the Dirac delta function an other generalized functions, even though they were introduced by physicists (I think Heaviside was an early proponent) in a hand wavy fashion, they were later put in rigorous mathematical footing in the theory of distributions. Distributions are nowadays used by mathematicians without any hesitation.
This also not the first instance of something like this, and it won't be the last. Physicists have come up with ad hoc methods that work, but they can't justify why with rigor. Some time later mathematicians formalize it and it becomes part of their tool set.
Yeah some French dude called Laurent Schwarz that got the Fields medal for it.
IIRC he built the set of distributions as the dual vector space of the tiniest vector space he could think of: extremely well behaved functions (C-infinite functions with finite support - weird mathematical beasts that go to zero at the extremities of their support with all derivatives also going to zero there as well).
I never really managed to grok the intuition behind the formalization of Schwarz, whereas the hand-wavy physicist way is pretty straighforward to understand.
It counts discrete things that combine as integers, but are not convertible (commensurate) between different instances of the unit, like apples and oranges.
A hen lays 3 eggs a week (1egg/T), and a car factory makes 4,000 cars a week (1car/T). Dimensional analysis says they are both 1/T, but we know they are really not commensurable.
P.S. Things that do not combine as integers: water drops (1+1=1); rabbits (1+1=2^t for some time constant t).