Physicists wielding ignorance as a super power
A rant, brought to you by me being jet-lagged and awake at 3am.
You might have heard about how the sum of all positive integers is equal to -1/12. It is not. The Riemann zeta function, for complex numbers with a real part bigger than one can be written as a series, and that series converges and forms a complexly differentiable (aka holomorphic) function. Holomorphic functions, defined on an open set, can be extended further. It doesn't always work, but when it does (and in the case of the Riemann zeta function it does), that extension is unique, so you can discuss the value of the Riemann zeta function at places that have a real part smaller than 1, like for example -1. We can calculate what the value the function takes there is, it happens to be -1/12. But, and the math is very clear on this, that value has absolutely, definitely, positively nothing to with the series we used on the right half plane.
So what do quantum physicists do, when their little theory runs into the sum of all positive integers? They just ignore that fact and replace the sum by -1/12. So far, that would just be a cautionary tale of how you shouldn't ignore preconditions, but they then go ahead, go to their experimental physicist colleagues, and ask them to run an experiment. And nature, which, at this point has to be assumed to be in on the conspiracy to drive mathematicians mad, makes it so that the result of the experiment is -1/12 to some ridiculously low tolerance.
Physicists wielding ignorance as a super power
But it doesn't end here, oh no. Physicists aquire the super power of mathematical ignorance early on in their career, at the start of their undergrad.
You might have asked why, when calculating differentials we write d/dx as the operator. And maybe even have noticed that when calculating integrals, dx again shows up in the symbol for the operator. You might even have asked a mathematician about that, and they probably told you that it is merely a coincidence, just a funny symbol, with no deeper meaning, and you really should not ask more questions about that. It was technically a lie, but it was a lie meant to protect you. There is a deeper meaning behind the dx, but the theory you need for that, the theory of differential forms, is some seriously advanced stuff, using exterior products, derivations, exact sequences of modules, and cohomology. It's something that is better left for a graduate program.
Of course physicists are not deterred by that. Worse still, when you tell them, after thorough warning, about cohomology and exact sequences and all that, they yell BORING and go ahead and tell their freshmen to just use differential forms. And not only that, they even tell them to divide two differential forms and pretend the result is well defined, "it's perfectly safe" they say. Of course, if a mathematician ever dared to divide differential forms, it would immediately blow up in their face and they would get seriously hurt. The mathematics is very clear on that. You can fill many many blackboards with counterexamples (wearing proper protective equipment, of course).
But when the freshmen physicists do it, they somehow are just fine. Don't get me wrong, it could definitely explode. It just doesn't. Because nature is trying to drive mathematicians insane.
Physicists wielding ignorance as a super power
I've recently learned of jokes that people call "jokes in the sense of Littlewood" (I'm somewhat puzzled why a single example yielded the name though): https://mathoverflow.net/questions/38856/jokes-in-the-sense-of-littlewood-examples
