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I am self-studying physics, and I just finished Linear Algebra by Howard Anton, Vector Calculus by Susan Jane Colley, and Differential Equations by Boyce DiPrima.

In order the learn the mathematical formulation of quantum mechanics and general relativity, I need to take a second course in linear algebra and learn real analysis, then topology and metric spaces, which will be followed by differential geometry. This is the route mathematicians take when studying differential geometry and advanced linear algebra.

But I don't want to study proofs. I want to learn the mathematics behind physics in a rigorous way and understand all the concepts thoroughly, which is why I avoid any "mathematical methods" book, but I just simply can't stand proofs, especially the trivial ones. For example, a lot of the content in Sheldon's Axler's linear algebra is relevant to quantum mechanics, but the entire book is proof-based and that's not really helpful. Similarly, proofs are at the heart of Real analysis but ain't gonna help me in physics. I am really confused because I can never study mathematics from the "mathematical methods" books because they just take away all the beauty of mathematics, but don't want to study proof-based books either. I am looking for books that are mathematically rigorous, and explain all the concepts thoroughly, but aren't proof-based.

Arham Zahid
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    In my humble opinion, you don't learn mathematics without proofs, you just read it. If what bothers you is the "trivial proofs" in some elementary text-books, then just open less-elementary text-books – Didier Feb 26 '23 at 22:36
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    "Math without proofs is dead." James 2:17. (Though in all seriousness, do you have an example of a book you did like?) – eyeballfrog Feb 26 '23 at 22:37
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    Welcome to StackExchange. I believe there is a strong argument to be made that your perspective is based on a lack of experience, but at there is also a kernel of truth to it on the "highly experienced" end of the spectrum. I'll let others mark that debate, and simply answer your question with one of the best textbooks at the "intersection" of mathematics and physics: Arnold's legendary Mathematical Methods of Classical Mechanics. (Read some of Arnold's quotes here, e.g. here: https://www.quora.com/What-was-Vladimir-Arnolds-opinion-of-Grothendieck-and-his-work) – snar Feb 26 '23 at 22:38
  • I sometimes say in mathematics classes: "Proof is, for mathematicians, their scalpel, their microscope, their measuring stick, their paintbrush, and the product of their efforts. Thus, it is of some importance in the discipline. Proof combines the art of insight with the science of technique. " I am personally not convinced that you can learn mathematics in a rigorous way while avoiding proofs. – Jamie Radcliffe Feb 26 '23 at 22:39
  • Arnold's book is meant for those who are mathematically erudite. It requires prerequisite mathematical knowledge that I don't possess yet. I am trying to find routes for learning advanced linear algebra and differential geometry through books that are not heavily proof-based. – Arham Zahid Feb 26 '23 at 22:51
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    I am confused. Can you explain what does "rigorous" mean to you? – Arctic Char Feb 26 '23 at 22:54
  • Rigorous as in, doesn't gloss over details and definitions just for the sake of covering more content. Explains all concepts thoroughly, and emphasizes the computational part with a good deal of exercises. – Arham Zahid Feb 26 '23 at 22:57
  • I don't think it's unreasonable to ask for texts in these areas that take a less abstract and more computational / numerical problem-focused approach instead of emphasizing theorem-proving. – Jair Taylor Feb 26 '23 at 23:08
  • Thank you, Jair. Do you have any suggestions in mind? – Arham Zahid Feb 26 '23 at 23:11
  • You could try Strang's Linear Algebra. "Real analysis" to my mind is proof-based almost by definition (that's what separates it from the "calculus" sequence in typical US universities.) Topology is also abstract by its nature so I don't know if there are more computational approaches. As for differential geometry, I do not know. – Jair Taylor Feb 26 '23 at 23:18
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    The level of rigor is essentially defined by how proof based it is. You can't have a rigorous text without proofs and vice versa. Personally I don't think you can even understand what physics is without first understanding how Noether's theorem works. It's too important. – CyclotomicField Feb 26 '23 at 23:26
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    Maybe worth looking at is Some Modern Mathematics for Physicists and Other Outsiders by Paul Roman (2 volumes, 1975). Information about his book is given in this answer. Copies for purchase might be hard to find, but I've seen it in several U.S. university libraries, so if you're in the U.S. and it's not at a library near you, then interlibrary loan might be an option. – Dave L. Renfro Feb 26 '23 at 23:38
  • Incidentally, undergraduates in physics and engineering typically take (at least in the several universities I'm familiar with) a 2-semester "Advanced Mathematics for Engineers/Physicists/Scientists" course during their last two years, and given your background, one of the standard texts for such a course is probably best for you at this time. For example, Boas or Kaplan or Kreyszig or Zill or many others like these. Then go from there – Dave L. Renfro Feb 26 '23 at 23:53
  • It is possible to learn classical quantum mechanics without rigorous proofs. There are many textbooks that present this well. On the other hand, general relativity is requires a lot of mathematical background. Still, you could first try to read a GR book written by a physicist and see how far you can get. Often, after an effort like this, you appreciate better the need for a deeper and more rigorous understanding of the math. – Deane Feb 27 '23 at 03:15
  • The math books you have studied from are all pretty much plug-and-chug, formula-oriented, with relatively little theory. Colley's vector calculus book is only slightly more "mature" than the typical Calc I - Calc III textbook. You may "hate" proofs, but you have yet to read/learn from a serious mathematics text. – Ted Shifrin Feb 27 '23 at 03:46
  • I mean, personally, I doubt you can rigourously learn math without going near proofs. – c0m3tBl4d3 Apr 13 '23 at 08:56

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Personally, as others have stated, I disagree with this idea that you can learn mathematics without understanding proofs. Every book that is mathematically rigorous, and every treatment of some field which is mathematically rigorous will rely on proofs in some form or another, as that is what makes something mathematically rigorous, the act of proving things.

That being said, if you wish to learn physics from a mathematical perspective, there are a number of good books you may enjoy:

  • The Geometry of Physics by Theodore Frankel
  • Gauge Fields, Knots, and Gravity by John Baez
  • Quantum Theory for Mathematicians by Brian C. Hall
  • Quantum Theory, Groups and Representation by Peter Woit

It is important to note that these books are not devoid of proofs, and that the last two books have a heavy mathematical flavor to them, but I believe they are all aimed at both mathematicians and physicists (maybe less so the third one) and thus may help bridge this gap you are experiencing. In particular, since the books are focused on physics, the proofs may feel less like proofs, and the text surrounding the proofs has a great deal of exposition.

At the end of the day though, if you're not good at understanding physics without a rigorous foundation in the math the physics is using, you really ought to become comfortable with proofs.

Chris
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