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I study pure mathematics. In pure mathematics, we begin from some axioms and obtain theorems. I am also interested in studying physics. I have some questions about the relationship between physical theory and mathematics. I apologize before asking if the questions are not appropriate. And I will be grateful to anyone who responds.

  1. Special relativity (SR) was originally proposed by Albert Einstein in a paper published in 1905. But Minkowski spacetime as a Mathematical structure for formulating SR introduced in 1908 (three years later)! I am confused about this. Did not Einstein himself state the mathematical framework of his theory? What is the difference between SR as Einstein himself formulated and SR in Minkowski's point of view? In fact, I always thought that every physical theory is formed at the same time as its mathematical formulation. Can we have a physical theory without any mathematical formulation?
  2. I read in some textbooks or internet pages (I do not remember which book or page) that in Quantum theories in physics there are some calculations that led to correct answers but we have no mathematical explanation for why. How is such a thing possible? Aren't our calculations based on mathematics?
  3. Is there any book to explain the difference between SR in Einstein's mind and Minkowski's? I heard that Einstein was a terrific intuitionist. Ok, but is intuition enough to form a physical theory? Thanks a lot.
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    $\begingroup$ I think the problem is that by looking at things through a "lens of the present" you are seeing ahistorical views and axiomatic views that, while they might make sense now, would have been very mysterious back then. I recommend ignoring gee-whiz internet web pages and overly-enthusiastic published popular expositions, and carefully work through a standard introductory physics text (e.g. this book), or possibly better, The Feynman Lectures on Physics. $\endgroup$
    – Dave L Renfro
    Mar 4, 2022 at 19:03
  • $\begingroup$ Einstein was influenced by Lorenz, Poincare and some other people including Minkowski, $\endgroup$
    – markvs
    Mar 5, 2022 at 5:33
  • $\begingroup$ By "intuitionist" do you really mean "Intuitionist" or just that he was really smart? $\endgroup$
    – Spencer
    Mar 5, 2022 at 14:39

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I think it is not quite accurate to say that "mathematics is done from axioms", nor to say that "[physics] calculations are based on mathematics". Let me explain.

First, as @DavidLRenfro's comment suggests, the idea that mathematics "has a foundation" (at all!) is fairly new.

A keyword that, for me, clarifies many aspects of such situations, is "narrative": written mathematics is often a narrative of thinking about physics, or about mathematics, and figuring out how to write it down to communicate to other people.

That is, if you wanted to tell about an event, you'd probably not worry first about axioms or defining terminology.

I more-and-more have the impression that, especially historically, but also nowadays, physicists write in mathematical notation as an account of an event. Much of mathematics is the same: not deducing from axioms, but telling a story, about tangible events [sic!].

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  • $\begingroup$ Yeah, but this is just arguing formalism (or even intuitionism) over Platonism. $\endgroup$
    – Spencer
    Mar 5, 2022 at 14:36
  • $\begingroup$ @Spencer, I don't think "narrative about real events" is "formal", nor "intuitionist", nor in any serious way non-Platonistic, after all. I've grown disenchanted with those labels' relevance to the practice of mathematics, myself... $\endgroup$
    – paul garrett
    Mar 5, 2022 at 17:19

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