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So I asked https://math.stackexchange.com/q/1524356/21820 and don't understand why people have voted to close it as unclear or opinion-based, and down-voted as well. I specifically ask:

I'm looking for explicit experimental confirmation of some physical phenomenon that can be explained using the Lebesgue measure but not say the Jordan measure.

This is a factual question. If anyone has an answer, it would be experimentally verifiable and hence not opinion-based. If no one has an answer, it is also not opinion-based but simply remains an open question.

Also, I don't see how it can be unclear unless one says that "explained using" is unclear, which to me is silly since we have been doing this in science from the beginning. For example motion (on a non-quantum scale) can be explained using Einstein's theory of relativity but not Newtonian mechanics, so far at least. Similarly, either there is a physical phenomenon that is explained by some theorem that requires using the Lebesgue integral in its proof, or there isn't, say because all sets that arise in modelling physical phenomena turn out to be Jordan measurable.

So what specifically is wrong with it? So far no one has commented on why they thought there is a problem with my question.

Edit: So now based on some of the feedback on the question as well as here, I've given my own definition of "physical interpretation", based on which the question should be more objective. (Though I still welcome answers that might disagree with my definition, so long as they provide their own and explain why their concept of physical interpretation is more meaningful than mine.) Is it reasonable now?

Edit 2: It seems that some people on Math SE detest such questions, as they have voted to delete it. Yet I don't see why the question is bad at all, not to say that it should be deleted. Even this meta question has more down-votes than up-votes, although it's not a stupid question. If this is the environment here on Math SE, that does not welcome any critical inquiry into questions about the choice of mathematical frameworks, then I'd rather leave.

Edit 3: Here are example that seem to provide evidence that the reasons given against my question are not consistently applied to questions on Math SE:

  1. What is the simplest mathematical concept that does not map to a physical phenomenon?
  2. What is a particle mathematically?
  3. Does every major genre of mathematics have applications?
  4. What are some examples of mathematics that had unintended useful applications much later?

So any answer to my meta-question should explain why my question is invalid despite being as clear and precise as I could make it, which is in stark contrast to the incredible vagueness of the above questions.

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    $\begingroup$ Physics is a thing $\endgroup$ – user147263 Nov 13 '15 at 0:12
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    $\begingroup$ @NormalHuman: Really sorry but I don't get you at all. Are you saying I should ask on Physics SE instead? I thought it is more on the math side because it's sensitive to the choice of set theoretic axioms like AC. $\endgroup$ – user21820 Nov 13 '15 at 0:17
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I voted to close it as "unclear what you're asking." After reading your question, it was not clear to me what would constitute an answer and what would not. The comment of:

Although this is what I gather so far, it isn't a convincing answer to my question (in particular to those who disagree). Did any mathematician ever make such a claim seriously?

that you left on an answer which seemed to satisfy what I thought you were asking furthered this impression of mine. Nothing on the question gives me a precise idea of what "physical interpretation" means. As it stands, I do not believe it is possible to answer your question.

If you wish to improve it, you need to clearly specify what constitutes physical interpretation. If it were me, I would think giving examples of other mathematical facts which do have such an interpretation would be a very good idea. It would also be good to speak more to why you don't consider applications in probability theory sufficient (especially when you are questioning something which is more or less implied by our desire for countable additivity and whatnot) - and why you consider the current answer unsatisfactory. You'd also do well to delete the second paragraph's tangent about ZFC, since it distracts from the primary question. It's also quite possible that the answer to this question is "No" and I wouldn't be so quick to reject that answer if it's well explained.

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  • $\begingroup$ But that's exactly my point. I believe the answer to my question is "No" but I cannot simply accept that answer you refer to because it agrees with my opinion, and I want some better explanation of why he/she says so. I do have an opinion of what constitutes a physical interpretation, but I felt that if I provide such a definition it would unnecessarily restrict the possible answers and actually even make it unfairly biased towards what I believe is the answer. $\endgroup$ – user21820 Nov 15 '15 at 4:07
  • $\begingroup$ Countable additivity is desirable from a mathematical point of view but not necessarily from a physical point of view. After all, can you show me any physical entity that is somehow interpretable (in any reasonable way you choose) as the rational numbers on the real line? $\endgroup$ – user21820 Nov 15 '15 at 4:09
  • $\begingroup$ Did you see my edited question? I've clearly specified my definition of "physical interpretation", and I've given examples, and I've never said I would reject a "No" answer if it is well-explained. I am looking for a well-explained answer, whether it is "Yes" or "No". I'm not looking for an answer that merely agrees with me. $\endgroup$ – user21820 Nov 15 '15 at 10:40
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    $\begingroup$ @user21820 In your edited question, you haven't defined what a reproducible statement is, what an empirical statement is, or what "the real world" is. Believe it or not, this is a website for mathematical questions, and these aren't words with a standard meaning in mathematics (and not even in physics or philosophy but that's besides the point). Mathematics is about precision, you can't just throw vague words around and expect an answer. $\endgroup$ – Najib Idrissi Nov 15 '15 at 10:58
  • $\begingroup$ @NajibIdrissi: In that case by your reasoning all of science (except mathematics) is badly defined, because the scientific method explicitly refers to "reproducibility" and "empirical measurements". How do you propose to define it? Mathematics is about precision, and I am a stickler for precision, but not when it cannot be done. I challenge you to define "if", that you certainly use everywhere in your mathematical work. $\endgroup$ – user21820 Nov 15 '15 at 11:04
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    $\begingroup$ @user21820 I don't have to propose to define "empirical" or "reproducible" in mathematical terms. You do. You're the one insisting on coming to this math website to get an answer to your question. It's true that other sciences are not as rigorous as math, but it's not a problem; however if you insist on getting a mathematical answer to your question, then I'm afraid that you will have to frame it in mathematical terms. If it "cannot be done", then maybe math.SE isn't the right place for your question. $\endgroup$ – Najib Idrissi Nov 15 '15 at 11:13
  • $\begingroup$ @NajibIdrissi: I'm fine if people just say that Math SE doesn't deal with such questions, but not if people say it is a bad question, since you yourself cannot define "if" which you use in mathematics, and then you hold me to a higher standard. $\endgroup$ – user21820 Nov 15 '15 at 11:18
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    $\begingroup$ @user21820 Oh, that will solve the problem and put an end to this charade? Fine then: "math.SE doesn't deal with such questions". Bye! (If you want a definition of "if", read textbooks about logic and foundations of mathematics) $\endgroup$ – Najib Idrissi Nov 15 '15 at 11:28
  • $\begingroup$ @NajibIdrissi: Sorry but I believe you know very little about logic and the foundation of mathematics. I do, and if you ever bother to learn a bit you will find that it is impossible to define "if". Hope you take your unfriendly attitude somewhere else. $\endgroup$ – user21820 Nov 15 '15 at 11:34
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    $\begingroup$ @user21820: I am perhaps more familiar with mathematical foundations than Najib, and I agree with him to some extent. Unlike experimental sciences, mathematics is not about empirical evidence. Therefore in the context of "empirically true mathematical theory" you have to be more precise. Induction is not empirically true, so I am reluctant to take anything mathematical as even remotely possible as empirically true. You can argue that $\frac12+\frac12=1$ is not even empirically true, since we can't quite measure $\frac12$ or when two things are equal (here we need two equal halves). $\endgroup$ – Asaf Karagila Nov 15 '15 at 12:22
  • $\begingroup$ @AsafKaragila: That's precisely why I mentioned about Kolmogorov complexity in my revised question, because it is possible to handle the 'issue' of finite precision and assign an information-theoretic value to a theory.. but why should I obscure the obvious meaning of my little question with a whole theory just to define the terms I use? Nay, the negative reaction is because I seem to be asking for experimental evidence of people's pet axioms or pet techniques, and when they cannot think of any they try to shoot the question down. Yet I'd be happy to be proven wrong! $\endgroup$ – user21820 Nov 15 '15 at 15:55
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    $\begingroup$ @user21820: Because that's the wrong question to ask to begin with. Because empirically we are almost entirely blind and deaf, and because empirically we are extremely stupid about what is going on in the world around us. But that is my opinion, and I think that math is, and should be, removed from "empirical reality" (because what is empirical reality? Some people say other things, and we call them psychotic schizophrenic paranoids, but maybe we're the crazy ones?). So I am not going to continue this discussion here. I just think that the question you're asking is the wrong one to begin with. $\endgroup$ – Asaf Karagila Nov 15 '15 at 15:57
  • $\begingroup$ @AsafKaragila: I know you don't want to continue the discussion here, but I find it very odd that you say that math is completely removed from empirical reality, since for example Fermat's little theorem is a theorem of elementary number theory and has been applied millions of times in thousands of applications including RSA and signature schemes, which you rely on whenever you use your email. I would agree that these only empirically verify the theorem for numbers up to $2^{4096}$ (this is the typical key-length so far), but it's still strong evidence of the theorem's physical meaningfulness. $\endgroup$ – user21820 Nov 16 '15 at 5:21
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One plausible reason for downvotes is that a significant fraction of your question is a complaint about the axiom of choice that has no bearing on the question your post is asking.

Furthermore, your title sounds like you're seeking physical intuition about Lebesgue integration, but the opening paragraph speaks on a rather different topic, that of empirical testing the choice of technical mathematical details.

Your question would probably be better received if you picked a topic (ideally one you have a question about, rather than one you want to comment on) and stuck to it without drifting onto other topics.

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  • $\begingroup$ The problem is that they are all related... The subsets of the real line that are Lebesgue measurable depend heavily on the underlying axiomatic system. I'm simply asking why people repeatedly justify the Lebesgue integral by appealing to probability theory as applied to the real world, because it seems that the real world doesn't have such pathological objects that require the Lebesgue measure in the first place! "Physical interpretation" is different from "physical intuition", so I'd say my title accurately reflected my desire for empirical evidence for certain mathematical objects. $\endgroup$ – user21820 Nov 15 '15 at 4:15
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    $\begingroup$ @user21820: The real world is made up of real world stuff, and doesn't have such abstract things like measures, integrals, sets, or even numbers. $\endgroup$ – user14972 Nov 15 '15 at 4:24
  • $\begingroup$ That's why now I provide a reasonably precise definition of "physical interpretation/meaningfulness" in my edited question. Of course the real world is not made up of sets and does not even have natural numbers in the form of an inductive set or something like that, but the question is whether there is an interpretation that maps from the mathematical objects to real-world entities in the same way the symbols and sentences in a formal system are not objects in the domain of an interpretation of it. $\endgroup$ – user21820 Nov 15 '15 at 6:37
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Consider this: in the time the question was open, the best answer you got ended with the statement, "So there really isn't a physical motivation for one over the other."

Your response in a comment:

Although this is what I gather so far, it isn't a convincing answer to my question (in particular to those who disagree).

Fair enough, but the reason some people aren't convinced is because it's their opinion that one definition is better for physics applications than another. Quite possibly their opinions are based on reasons other than the ability to detect a difference experimentally. Are those reasons therefore worthless? Well, that's a matter of opinion too.

The people who closed this presumably felt it was unlikely that this conversation would advance beyond this "so what?" stage.

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  • $\begingroup$ Those reasons are not worthless, but nobody has given such reasons! The only answer I got so far didn't give a single reason, because just because something is "pathological" in mathematics does not imply anything about real-world meaning or lack thereof. $\endgroup$ – user21820 Nov 15 '15 at 4:17
  • $\begingroup$ "Those reasons" are reasons "other than the ability to detect a difference experimentally." You complained that someone had given such a reason--something about probability theory--although you did not provide a link to show who gave that reason or what they said about it. Meanwhile, the answer you got said "there isn't a physical motivation," so of course it didn't provide one. Personally, I think that's a completely satisfactory answer; but clearly your opinion differs from mine. $\endgroup$ – David K Nov 15 '15 at 6:32
  • $\begingroup$ Sorry if it wasn't clear. I wasn't expecting that answerer to provide a physical motivation, of course, but I was expecting at least a reason why he was so confident that there isn't a physical motivation. If not, then it's just opinion. In any case I've now given a rather specific definition of "physical interpretation/meaningfulness" so I think it's much less opinion-based than it appeared to be before. $\endgroup$ – user21820 Nov 15 '15 at 6:34
  • $\begingroup$ The linked questions give some hints that maybe it isn't completely satisfactory after all to just say there is no physical motivation; there are some specific examples of the use of Lebesgue integration with regard to Brownian motion and quantum mechanics. So there's a motivation in the sense that Lebesgue integration made some problems easy enough for some very smart people to have solved already. What's less clear is that L. integration is logically necessary to solve these problems. I just don't see how we can resolve this in this forum. $\endgroup$ – David K Nov 15 '15 at 7:00
  • $\begingroup$ But I do agree the modification to the question makes it slightly clearer what you're looking for. (And I don't think it was ever really unclear.) $\endgroup$ – David K Nov 15 '15 at 7:01
  • $\begingroup$ Yes you got my question; it's about logical necessity rather than convenience. Just like there are conservative extensions of formal systems, but that does not mean that the extensions mean anything.. even though it may drastically reduce the length of proof (for example cut-elimination) $\endgroup$ – user21820 Nov 15 '15 at 7:57

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