Books that teach other subjects, written for a mathematician Say I am a mathematician who doesn't know any chemistry but would like to learn it.  What books should I read?
Or say I want to learn about Einstein's theory of relativity, but I don't even know much basic physics.  What sources should I read?
I am looking for texts that teach subjects that are not mathematics, but I do not want to read through standard high school, undergraduate (and beyond) material.  I am looking for recommendations of sources that teach a scientific theory from a basic level, but not from a basic mathematical level.  Strong preference would be to concise, terse texts that are foundational but totally rigorous.
Not sure if these exist, but I often wish they did.
 A: In response to the (personal) finance book for mathematicians, I would suggest looking towards investment and risk management books as they tend to be written by mathematicians, so any related financial market and economic material will be relatively concise, which is the issue I think mathematicians have with more regular finance texts.
I felt Investment Science by David Luenberger was a great find when I was starting out though it seems to be harder to get hold of now.
On the risk management front, PRMIA is a professional risk management organisation that offers digestible blocks of material that cover background finance material, a brief bridge of the mathematical foundations of risk, and more detail beyond:
https://www.prmia.org/Public/Public/Resources/PRM_Handbooks.aspx
A: The Geometry of Musical Rhythm by Godfried Toussaint:
The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good? is the first book to provide a systematic and accessible computational geometric analysis of the musical rhythms of the world. It explains how the study of the mathematical properties of musical rhythm generates common mathematical problems that arise in a variety of seemingly disparate fields.
A: Well, it's not exactly science, but
The Mathematics of Juggling by Burkhard Polster is written by mathematician and is for mathematicians. I may add that you can enjoy the book, even you can't juggle.
A: I can recommend Mathematical Linguistics by Andras Kornai.
From the preface:

The book is accessible to anyone with sufficient general mathematical maturity (graduate or advanced undergraduate). No prior knowledge of linguistics or languages is assumed on the part of the reader. The book offers a single entry point to the central methods and concepts of linguistics that are made largely inaccessible to the mathematician, computer scientist, or engineer by the surprisingly adversarial style of argumentation (see Section 1.2), the apparent lack of adequate definitions (see Section 1.3), and the proliferation of unmotivated notation and formalism (see Section 1.4) all too often encountered in research papers and monographs in the humanities. Those interested in linguistics can learn a great deal more about the subject here than what is covered in introductory courses just from reading through the book and consulting the references cited.

Edit. See also Ian Chiswell's A Course in Formal Languages, Automata and Groups.
A: A brand new book that might be of interest is Data Science for Mathematicians, edited by Nathan Carter (who also wrote Visual Group Theory). It assumes the audience is a mathematician (at, say, the graduate student level), then gives high level treatments of:

*

*programming with data,

*linear algebra (and its applications to data analytics),

*basic statistics,

*clustering,

*operations research,

*dimensionality reduction,

*machine learning,

*deep learning, and

*topological data analysis

I should disclose that I wrote one of the chapters, but don't have any financial stake in the book. I recommend it because I think it's great, and will help mathematicians who want to embrace data science in their research, teaching, or as an alternative career.
A: I like Leonard Susskind's The Theoretical Minimum series a lot. I've read the Quantum Mechanics one and started the Classical Mechanics one. The series also covers Relativity, Statistical Mechanics, and Cosmology per Wikipedia, and there are also lectures available online.
I also like Feynman's QED: The Strange Theory of Light and Matter on Quantum Electrodynamics - it sounds very intimidating but it's actually very approachable and a very good introduction, knowing nothing going in.
Re: finance, Mandelbrot's The Misbehavior of Markets is quite interesting, and Flash Boys by Michael Lewis, while not specifically mathematical, gives a lot of insight into how modern digital markets work [or don't, depending on who's asking].
A: There are many sources on general relativity for mathematicians (see, for example, the lecture notes of Schoen and the textbook Geometric Relativity by Dan Lee).
It's been a while since I read any chemistry books, but I remember Atkins' textbook on physical chemistry being fairly readable for a mathematician.
A: This is something I'm trying to learn from, a (text)book on music, written by a mathematician, and for a mathematically literate reader, called Music: A Mathematical Offering.
A: There is an interesting book that teaches dance concepts from a mathematical viewpoint:
Dance: Mathematical inquiry in the liberal arts
Its online version is free. I took a glance at the book and well, I must say I got stuck for an hour or two!
A: Russian for the Mathematician helps you learn basic skills for that language. Not only do they use words like "number" instead of "house", say,  to teach declensions, they also use real-life mathematical sample texts and offer an overall more math-inclined approach to language learning.
A: There are three that I can think:
Brian Hall, Quantum Theory for Mathematicians.
and
Sachs & Wu, General Relativity for Mathematicians
Also
Saunders Mac Lane, Categories for the Working Mathematician
All three are excellent and are very readable.
A: Michael Spivak's book, Physics for Mathematicians, Mechanics I, would definitely fit the bill. The goal is to discuss the foundations in a way that lays clear the underlying physical principles but doesn't simultaneously have to teach the underlying ideas of calculus.
Quoting from the (as usual, entertaining) introduction, ..."Ah, so you're going to be writing about symplectic structures, or something of that sort. And I would have to say, No, I'm not
trying to write a book about mathematics for mathematicians, I'm trying to write a book about physics for mathematicians...."
Reading the first chapter, I felt like the book was written exactly for me!
A: There are a lot of good physics books for mathematicians. My personal favorite is Mathematical Foundations of Quantum Mechanics by Mackey.
Let me also plug Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics by Wald, especially for anyone who's had some exposure to C*-algebras. It's very readable. Since the quantum fields are free, the treatment is rigorous, but since the underlying space is curved the theory is not trivial. The book culminates in an account of Hawking radiation.
A: When I read Cormen, Leiserson, Rivest's book on Data Structures and Algorithms, it was eye-opening to me to see how (mathematically) rigorous computer science could be.
A: Robin Giles, Mathematical Foundations of Thermodynamics.  From the preface:

This monograph is an attempt to give an account of the foundations of thermodynamics which is more than usually rigorous, not only in its logical structure but also in the "rules of interpretation" in which physical meaning is assigned to the theoretical terms.

A: Like another answer says, there are lots of good physics books. My personal favorite is Quantum Field Theory: A Tourist Guide for Mathematicians by Gerald Folland
A: Programming for Mathematicians by Raymond Seroul.
I recommend reading the highly entertaining amazon review by Ian Jakovenko.  He refers to the book as "Euclid's Elements for Cybernauts."
A: Most of the good physics books have already been mentioned, so I'll add one about machine learning.

Understanding Machine Learning: From Theory to Algorithms, by Shai Shalev-Shwartz and Shai Ben-David.

Fully rigorous, and explains what the mathematical challenges are in machine learning.
A: The IMHO most useful such book hasn't been mentioned yet:
The 85 Ways to Tie a Tie is a book by Thomas Fink and Yong Mao about the history of the knotted neckcloth, the modern necktie, and how to tie both. It is based on two mathematics papers published by the authors in Nature and Physica
A: Here is one you may like:
Economics for Mathematicians by J.W.S. Cassels, London Mathematical Society. I am curious what economists have to say about it. Link to Mathscinet review here.
A: The answers show that there are many books on physics, especially on quantum mechanics written by mathematicians and for mathematicians. Let me add my favorite one:
L. Faddeev and and O. Yakubovskii, Lectures on quantum mechanics for mathematics students. (Russian original 1980, English translation: AMS, 2009).
Another classical book is
V. Arnold, Mathematical methods of classical mechanics.
A: I think in general what you are looking for doesn't tend to exist (with exceptions, such as those in other answers). As background, I am a theoretical physicist with what I suspect is a stronger mathematics background than most, but I think theoretical physics (and probably computer science) is the only field where you could really expect to find books that satisfy:

Strong preference would be to concise, terse texts that are foundational but totally rigorous.

To find stuff like that you'd probably have to look at a very small selection of research papers (some of which I'll list at the end of this answer). However since these are research papers they tend to be focused on the subfield and probably assume some familiarity already (kind of defeating the purpose to learn from them).
I would say that the lack of these textbooks is not a bad thing, and that's because mathematics tends to be different to basically any other science. In mathematics you can start with your axioms and prove what you would find if you did experiments with that mathematical structure. While in basically every other field you start with knowing the results of experiments and have to try to deduce which axioms you should start with. There are probably exceptions to this but I think in general this argument holds.
To give a bit of a feel why this is different something I remember hearing about the theory of (I think) topologically ordered phases a bit over 5 years ago: when you read papers there are maybe 3 different ways that people define these phases, which aren't equivalent but kind of do the same thing in the end. While this might sound like there should be 3 different definitions used and introduced at the beginning of each paper, the end goal is to discuss and explain real experiments and so we only want and we only expect a single theory to exist. The different ways people have defined things the definition becomes a "I know it when I see it" situation and everyone working on the topic, even if using different definitions, should expect to see the same kinds of behaviours.
The other thing to keep in mind is that if you have a rigourous theory that uses sophisticated mathematics compared to a simple theory that only requires arithmetic, if the simple theory matches the experimental data better then the rigourous theory then that's the one that should be kept since the goal is to explain these experiments. Also if you can explain the same set of experiments with either simple or sophisticated mathematics with the only difference being rigour, unless your sophisticated mathematics approach is more powerful in explaining or making predictions for other experiments that the simple approach cannot, then the simple approach will remain dominate just because of the cognitive overhead associated with using more sophisticated mathematics.
The final thing is to think about is where would you start if you wanted to give a foundational description of genetics? we could start with central dogma of biology which is that DNA -> RNA -> proteins, explain what all these things are and build up how proteins effect the behaviour of cells and eventually an animal along with expressions of phenotypes (what it looks like). From this construction you could derive evolution. The problem with this approach is that we can't do it yet, there are heaps of things we don't understand about proteins and protein folding (which is important for how a protein changes cell behaviour), and when do you introduce the ability to inhibit/enhance different gene expressions (protein production) via the existence of certain proteins. This is a more complicated topic that if introduced too early distracts the reader from the core idea of DNA gives proteins which give looks/behaviour.
Instead it makes more sense to start with talking about the looks and behaviours you see and saying it is conjectured to have been caused by ... which is conjectured to be caused by ... and so on. This also means that new research, which is more likely to be at the molecular scale than animal scale, is only a correction to the end of your book, rather then the very first sentence. This would be different to a maths text where new research won't overturn the axioms but would be more likely to add new knowledge about the structure that arises from these axioms.
We could discuss the looks and behaviour in terms of other mathematical structures to give them a rigourous definition, but WHY? You add cognative overhead to those who don't know this mathematical structure while not adding any explanatory power. Furthermore for those who understand the structure you're possibly misleading them into thinking the choice of this structure is deep while preventing them from learning how to think about the field as everyone else does.
So longer then I initially expected (and missing points I was initially thinking of) but
tl dr: mathematics is unique in that in generally derives results from axioms rather than axioms from results like most other fields, this makes it very difficult to have concise terse texts that are foundational and totally rigourous (with exception of some theoretical physics and computer science).
Possible papers:
These might require too much knowledge about the field, however maybe the mathematics in them might make it easier to read and be able to pull out the terms and facts that these experts seems to care about for future papers.

*

*Knot theory in understanding proteins
J. Math. Biol. (2012) 65:1187–1213
DOI 10.1007/s00285-011-0488-3

Actually targeted at mathematicians trying to get them to work in the field and has some references to biology which might help to just gain a bit of familarity/knowledge of what's important before reading other texts.


*String-net condensation:
A physical mechanism for topological phases
PHYSICAL REVIEW B 71, 045110 2005

Probably not accessible without some knowledge of quantum physics, but argues about using the objects of fusion categories to describe phases of matter.


*CRICK, F., WATSON, J. Structure of Small Viruses. Nature 177, 473–475 (1956). https://doi.org/10.1038/177473a0
I think this was the paper that used group theory to explain the structure of a virus shell. There is a later paper that I think extended it to Kac-Moody algebras and quasi-crystals for an abnormal class of viruses but I can't quickly find it.
