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I once heard someone (perhaps from someone famous -- anyone have a citation?) say that there ought to be a series of books called "__ for Mathematicians," each one of which would explain a different topic or discipline using the tools of mathematics. (The idea is that knowledge of higher mathematics helps to clarify the exposition or simplify issues which would otherwise be inaccessible to beginners.)

Even though this book series doesn't actually exist, there are certainly some books which fall into this category. (I've listed two below.) What are the best books of this type?

Some examples:

  • Economics with Calculus by Lovell (an introductory economics textbook which assumes calculus knowledge)
  • Mathematical Methods of Classic Mechanics by Arnold (adopts an axiomatic, mathematical approach to classical mechanics)

As you can see, the above examples vary pretty widely in what type of mathematical sophistication they expect from the reader; for me the key fact is that they both significantly alter the normal presentation of material to make it more suitable for readers comfortable with mathematical reasoning.

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Great question, I've always felt a disconnect with the underlying mathematics whenever I approach a new discipline. –  tskuzzy Jul 8 '12 at 20:25
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5 Answers

up vote 15 down vote accepted

Here are some suggestions:

  1. Quantum Field Theory for Mathematicians (Encyclopedia of Mathematics and its Applications) -- Robin Ticciati,
  2. Quantum Mechanics for Mathematicians (Graduate Studies in Mathematics) -- Leon A. Tahktajan,
  3. Lectures on Quantum Mechanics for Mathematics Students (Student Mathematical Library) - L.D. Faddeev and O.A. Yakubovskii,
  4. Physics for Mathematicians, Mechanics I -- Michael Spivak,
  5. Quantum Fields and String: A Course for Mathematicians -- Pierre Deligne,
  6. Economics for Mathematicians (London Mathematical Society Lecture Note Series) -- J.W.S. Cassels,
  7. A Course in Mathematical Logic for Mathematicians (Graduate Texts in Mathematics) -- Y. Manin, N. Koblitz and B. Zilber.

As for suggestion number 4, I know that the author has planned to write a whole series of books of the form Physics for Mathematics, [insert subject here]. He has also written series of books on differential geometry, so I guess he will probably continue with his project with at least one other book.

Furthermore, I would like to add a book that does not bear a title of the form [subject] for Mathematicians, but it seems to me that, with a little imagination, it can be interpreted as such. The book I am talking about is:

  • Pattern Theory -- The Stochastic Analysis of Real-World Signals (Applying Mathematics) -- David Mumford and Agnès Desolneux.

It is an interesting account of the applications of mathematics to the analysis patterns. I think it could prove very useful for those studying artificial intelligence. I therefore think it could be interpreted as Artificial Intelligence for Mathematicians.

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If you are interested in "Computer Science for Mathematicians" you may want to look at this post that is asking the same question and the plethora of mathematically biased books for understanding CS - http://mathoverflow.net/questions/51217/computer-science-for-mathematicians

A personal recommendation is "Structure and interpretation of Computer Programs" - it helps lay a math-oriented foundation for programming and is based/derived from lambda calculus

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"Structure and Interpretation of Computer Programs" is available online at mitpress.mit.edu/sicp. –  lhf Jun 25 '13 at 17:43
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And, of course, you can search for "Poetry for Mathematicians."

I am (as you can see from my work here) a very amateur mathematician who also likes to write poetry, with very little of it about math.

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This isn't exactly what you asked for, but I have a copy of "Mathematics and Sex" by Clio Cresswell (really! - look here: http://www.amazon.com/Mathematics-Sex-Clio-Cresswell/dp/1741141591/).

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