In other words, physicists were entirely correct to toss the infinite values, and mathematicians were later persuaded it was mathematically sound.

Nowadays, although physicists say they are "renormalizing", they still just toss the infinite values. They're justified by results of experiments. Mathematicians are limited because, typically, they have no handy universe to run experiments in, never mind budget for a supercollider.

Or to phrase this differently, physicists do not really understand what they are doing. They do some operation because it sometimes happens to produce results that match some physical experiments that they are trying to model. Their derivation rules are due to habit (in the sense of Hume), not due to inference or logic.

Almost certainly, a rigorous formulation will come about later and explain when these derivations should actually hold (rather than just assuming they hold "whenever necessary").

This is actually how a lot of mathematics works: researchers notice some relationship empirically, which gives some intuition and suggests some hypothesis, and then the serious mathematical work is trying to determine exactly what assumptions are necessary and proving it.

>Or to phrase this differently, physicists do not really understand what they are doing.

I think this is very unfair to say. I would like to remind who was it that discovered the renormalization group: Wilson, a physicist.

I think I agree with you, and I also think that, "physicist do not really understand what they are doing" is phrased a bit flippantly.

Physicists understand what they're doing perfectly well, and they're not, for the most part, producing rigorous math. They are identifying phenomena which can be described systematically, as well as the systematic representations of those systems. Rigor, and math as a whole, is meaningful for a physicist only insofar as it facilitates that outcome.

It's interesting that you say that, "almost certainly, a rigorous formulation will come about later..." You seem to agree that experiment is a useful test for the rigor of operations, at which point it comes down to simple division of labor. The physicist has an expectation that an operation is rigorous, and his job isn't to establish its rigor; it's to arrive at the expression. He does his job, and mentions to the mathematicians that this operation 'should' be rigorous, and it'll be interesting to find out exactly how.

I think it's exceedingly wrong to say, "Their derivation rules are due to habit (in the sense of Hume), not due to inference or logic." Derivation rules support only one goal: predict outcomes of systematic phenomena. Physicists are neither nor are they trying to be mathematicians. The property that guides derivation is physical intuition, which I would say mathematicians malign at their peril, since the universe has thus far proved to be a useful machine for checking rigor.

And that's the heart of it. The universe has, for whatever reason, proven to be a detector, or perhaps executor, of mathematical rigor, and in describing the universe with ever-increasing precision, we develop results that feed back in to be explored and made complete. We can say, "it must be mathematically rigorous that this infinity can be discarded, so long as the universe is still functioning as a system that exhibits mathematical rigor," and be satisfied because the universe has shown us it is so. This does amount mathematically to not understanding exactly what it is you're doing.

For interesting reading on this by someone smarter than myself: The Unreasonable Effectiveness of Mathematics in the Natural Sciences (https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...)

> In other words, physicists were entirely correct to toss the infinite values, and mathematicians were later persuaded it was mathematically sound.

Rather: Mathematicians were able to develop an axiomatic framework in which these transformations have a well-defined meaning.

Potato, potahto. If the experiment produced different results, mathematicians would not be asked for an opinion until the physicists came up with some other math. Because the experiment matched, mathematicians were obliged to invent a framework to justify it.

Physics has an aim: to provide a rigorous description of reality on such a way that predictions about future experiments can be made. Physicists found that by altering certain quantities they got rid of the divergences and obtained a theory of remarkable experimental accuracy. That's all that matters.

Fortunately, of course, for our understanding, more people came along soon after to elucidate this mechanism.

It seems like if shortcuts ‘just work’ in physics, that should be a good sign that there is probably a mathematical justification waiting to be found.

Other times, it is a good sign of edge-cases that are physically impossible or highly unlikely.

> if shortcuts ‘just work’ in physics, that should be a good sign that there is probably a mathematical justification

Or new theory.

Your comment seems to be: Who cares whether or not QED was developed when it was? They should have stopped for a decade or two while they figured out the theory.

Are you serious?

Heaviside's comment, famously, was "Shall I refuse my dinner because I do not fully understand the process of digestion?"

That's not what he's saying. He's just making an observation that physicists have a tendency to make mistakes with math if it's convenient for them, so it's generally not a good idea to use their proofs when you're wanting something mathematical.

I guess that's more charitable. But to my ears it sounded like a baldly pedantic critique of what is easily one of the greatest theoretical results of the past century.

Dirac?

My favorite example of bogus-but-effective mathematics in physics is the Dirac Delta "function".

That also has a formal mathematical interpetation as a distribution rather than a function now and the theory of distributions found many applications in PDEs.

An interesting thing is that before Dirac, Green was already informally using distributions in the 1830s to solve differential equations with what we now call Green's functions.

Closely related to Green's functions are the fundamental solutions to an elliptic operator, consider the Laplacian and its fundamental solution in Rⁿ (n≥3), what's the Laplacian of the fundamental solution? A Dirac's delta!

The big trouble with the delta function in quantum physics is that it's in some sense the natural basis for your continuous Hilbert space, but they aren't in the Hilbert space. Which necessitates a whole lot of mathematical explaining-away-formalism.