Does the Banach-Tarski paradox have any consequences to the applicability of real analysis to physics we'd expect to correspond to real systems/objects/phenomena?
Posted Sep 27, 2019, 1:48 PM UTC
121
Don't think so. That level of math is barely used by physicists.
3
Real analysis isn't used by physicists?
-1
I think most theoretical physicists wouldn't know the definition of a measure.
2
I'm not sure whether to laugh or cry. First of all, you're probably not correct. Secondly, if you are, do you listen to them?
The concept is not so complicated. Do you understand the consequences of the content of the article?
-5
The problem is that the real world is not composed of an uncountably infinite number of points that can be evenly divided and mapped into two identical spheres of the same size as the first. A rubber ball sitting on the table might look like it's smooth and continuous and containing an infinite number of points, but it in reality it is a finite number of discrete particles.
And, no, in my experience, real analysis doesn't see much use in theoretical physics beyond things like proving continuity for solutions of a differential equation. I'm sure that some string theorist or mathematical relativist has used measure theory and the like to write some oddball paper or prove a theorem with no practical application, but differential geometry, abstract algebra, and PDEs are about as crazy as most theoretical physicists get.
3
I think you are overestimating the consequences of this mathematical fact (or quirk) for physics and you're making a lot of drama about it ("I'm not sure whether to laugh or cry."). I say that as someone with degrees in both math and physics. If you really think this has a lot of consequences for physical modelling you need to make a better case of where you think it would matter that you disassemble a set and reassemble it into one of larger measure. We do have to be careful that mathematical models are well-defined (say, path integrals in qft) but you're just taking for granted that this poses a problem without doing anything to make the case other than mentioning it and berating people who are asking "what about it then?". This is why you are downvoted too.
edit: I really didn't know what you wanted to hear.. "yes we can't use real analysis for physics anymore, because of the banach tarski paradox (anno 1924) showing some 'pathologic' behaviour in measure theory. We have to rethink and start using integers and counting measures only." and all that because you happened to hear about it in a youtube video in 2019.
2
I know of nothing in physics that would require assigning measures to completely arbitrary sets.
Is there no assignment of measure to any sets at all?
I mean, physicists do integrals, so of course they assign measures to lots of sets, even if they don't think about it that way.
Considering physicists can integrate over sets of measure zero and get a finite result, measures mean nothing in physics. If you want to know more, read about the path integral, Wiener measure and Wick rotations. All that would be summarized in a couple of pages in a physics course when it is an ongoing mathetical open problem.
I don't want to assume your reasoning is faulty just because it appears to be so to me (just being transparent here because I am not someone who thinks I know very much), but you might be missing my question. It's: why do we think our ability to integrate over sets of measure zero is fundamentally applicable to arbitrary or even any physical phenomena or systems just because we can take a limit.
Not sure if you studied QFT, but I'll roughly describe the issue I mentioned. In quantum mechanics, a way to calculate the ocurrence of a process is summing over all the "possible" trajectories. This is done writing down the sum and taking the limit to an integral in a handwavy way. Turns out this naive trick mathematically leads to a integral of a set of measure zero, which is not what we get in with the handywavy way.
So it's the opposite than what you said. We take the limit even knowing that with our current math techniques we would get a wrong number.