You seem to conflate here the concepts of "Mathematics as a subject" and "Mathematics as the sum total knowledge of people who identify as Mathematicians." Possibly you would like to clarify this in your answer.

The Feynman integral is clearly well-founded mathematics. Just because it's different mathematics to classical calculus doesn't mean it's not mathematics.

Gosh, we're getting some interesting topics here, with affairs and love and people who identify as mathematicians... Never heard that one before.

@OrangeDog "Mathematics" is something specific: It means that we can prove things about the things we talk about. People in physics usually create ideas without too much concern for this. "Mathematics" is not "anything that has numbers in it" as you seem to be thinking.

@OrangeDog Also, why do you think that this guy doesn't know that "non classical mathematics can be mathematics"? He has an account on Math Overflow with some activity in there, MO is basically about recent (non classical) mathematics. I guess you're trying to (condescendingly) lecture very basic stuff to people who know way more than you do. Also, "classical calculus" was eventually found to be lacking by Cauchy, Weierstrass, Cantor, Dedekind and others. So it's kinda hilarious that you think that he is saying that "classical mathematics is better". Anyone in a math major learns this.

@OrangeDog only certain Feynman integrals are mathematically well defined, for most cases we only use them as a recipe for perturbation theory and don't know the correct mathematical formulation for the "full" object (or if it even exists)

@RedBanana we can prove things about Feynman integrals, so by your definition it is mathematical. I am addressing the content of the answer. The experience of the author is not relevant.

@OrangeDog We can prove some things about them. Just as AwkwardWhale pointed out. "The experience of the author is not relevant" - OF COURSE it is relevant, you yourself don't have any idea about what are the mathematical issues with Feynman integrals, you probaby just watched some documentaries on YT and think you are well informed about it.

@RedBanana that doesn't make them not mathematics. No, the author is not relevant. All StackExchange posts must stand on their own, regardless of who wrote them. If there's something I "clearly" have no clue about, then it should be in the answer.

@Him, you wrote: "You seem to conflate here the concepts of 'Mathematics as a subject' and 'Mathematics as the sum total knowledge of people who identify as Mathematicians'." I personally only have access to the latter. If you have access to the former, I would certainly be interested in hearing about that.

@RedBanana don't act like you know the level of my education because you like feeling smugly superior. Regardless, if the answer depends on the reader knowing this definition, then it should be in the answer, otherwise it is a bad answer.

@RedBanana, I added to my answer the clarification requested by that user.

@RedBanana what makes something "mathematics" is the definition I was referring to. That link indeed leads to a mathematical definition of Feynman integrals, which somewhat proves that they are mathematics, no?

@MikhailKatz "It is well known" is just maths for "citation needed". But thank you for actually addressing the content, rather than insulting me and my mathematics PhD. The key here I think is whether the undefined cases of the integral actually describe anything physical. If they do not, then your conclusion does not follow.

@Orange, Hall in his "Quantum Theory for Mathematicians" writes: "To have a chance to make rigorous sense of path integrals in quantum field theory, one has to employ a complicated regularization process known as renormalization. This process has, so far, been carried out in a rigorous fashion only for a very small number of field theories. One of the Clay Millennium Prize problems is to make rigorous sense out of the Yang-Mills field theory in four spacetime dimensions." (pages 451-452).

@MikhailKatz there is a whole subject called "metamathematics" that rigorously studies the limits and properties of mathematical systems. Godel's and Turing's results would fall under this umbrella. So, Godel's result is a statement about "what mathematicians can possibly ever know" as opposed to "what mathematicians currently do know". In this sense, one can know lots of things and ask lots of questions about "Mathematics as a subject" independent of what Mathematicians actually know.

@MikhailKatz if there is anything in this comment thread you feel is useful, you might want to add it to your answer. The whole thread is likely to be nuked by mods at any moment due to RedBanana's inflammatory rhetoric.

@Him, as far as I am aware, Goedel's result sheds no light at all on the problem of mathematical justification of the Feynman integral.

@Him, you wrote: "if there is anything in this comment thread you feel is useful, you might want to add it to your answer." : I did already an hour ago.

@MikhailKatz apologies, there's a lot going on in this thread. :D. I was referencing this comment. It turns out that we can ask and, amazingly, rigorously answer questions about Mathematics beyond simply what Mathematicians happen to know at any point in time. In this sense, the question "Can Mathematics describe...?" could be taken as a question about Mathematical systems generally, and not about what individuals do or ever will actually know.

@MikhailKatz I beleive Him's point is, in essence, there is a difference between a proof that something cannot be done, and not having done something yet. All of the open Millennium problems are currently in the latter category. However, this question can be answered using results from the former category.

@Orange, Goedel's results are certainly interesting (and even revolutionary), but I wonder to what extent it would answer the OP's query. Note that by Goedel, there are some statements about the natural numbers N that we will not be able to either prove or disprove. But does that shed any light on the physical universe? I am not sure.

The question is about the ability of mathematics to describe the universe, where "describe" is interpreted as capturing all physical phenomena. I really don't think that mathematics has any problems to capture all physical phenomena related to redness. This answer uses "describe" in a completely different way, more like "give the experience to somebody who has not experienced it", and thus answers a different question than was asked.

@user132647, I didn't say anything about "redness". Did you mean to place your comment under the answer by Speakpigeon?