Science for Mathematicians?

Any recommendations of science books for mathematicians? I'm looking for books that address a mathematically-proficient audience and present the science accordingly. The focus would be on teaching the science, not teaching the mathematics. Books like Physics for Mathematicians, Mathematical Aspects of Quantum Field Theory, and Music: A Mathematical Offering are more what I'm looking for than books like Mathematics: A Simple Tool for Geologists. "Mathematically-proficient" may range anywhere from "knowing Calculus" to "knowing graduate mathematics".

In a broader sense, I'm looking for science books for the intelligent layman that aren't about the history of X or the political implications of X but about X itself. And The Complete Idiot's Guide to X, X for Dummies, and X Demystified aren't what I'm looking for. The campus bookstore here has a pretty big General Science section, but there don't seem to be many books there that just try to teach the science intelligently and without gimmicks.

Why not just get good textbooks directed at undergraduate majors? I don't think there will be much of a market for something like "Physics for Mathematicians" so your choices there will be limited, but mathematicians will surely be able to hack the math in "Physics for Physicists", right?

Physics is a pretty broad category. Any particular topic you're interested in?

>2 alco261:

Thanks for the list, but I think you've misunderstood me. I don't want books on the history of X.

>3 lorax:

The textbook route sounds promising, but I haven't had much success making it work. One problem is that the introductory textbooks (I'm a layman w.r.t. X, remember) tend to dumb down and/or gloss over the math, since underclassmen aren't assumed to have had the math that upperclassmen will.

I know that I'm looking for books targeted at a niche market, at best. That's why I'm asking here. Google informs me that someone else asked my same question on math.stackexchange.com and didn't get much response, so that may be a sign that I should just give up!

>5 alco261:

It's only an odd combination since we as a society have decided that students don't need to learn the so-called "language of science" before learning science.

I can't speak to the practices in American undergraduate textbooks, or just how and in what way the math there has been "dumbed down", but from what I've experienced in my studies, it is usually a question of leaving out lengthy derivations of equations or proofs, in textbooks at least. It is usually trivial to find these in secondary literature, or--certainly for a mathematician--to work them out on one's own, always the best way to develop an understanding of the subject.

Basically, you'd need to supplement the mathematical treatments you find unsatisfactory with the original or advanced ones. It might make for a somewhat slower progress, but hey, if that's what's needed...

4>

Well, I can tell you what we used in my undergraduate physics classes -- these assume you've had a general survey class in physics but not a detailed background in the field.

I'm skipping classical mechanics because it's not very interesting (though the Lagrangian formulation is rather elegant).

For E&M Griffiths Introduction to Electrodynamics is the standard and is very good. (Math: Vector calculus, some DiffEq (primarily, unsurprisingly, the wave equation)).

For quantum, things are a bit trickier. Griffiths has another introductory text, but it may be a bit too introductory for your purposes, since I remember the math being pretty simple (no more than was required for E&M, plus linear algebra); on the other hand what we used in my undergrad course was a grad-level text (Cohen-Tannoudji) and that was a tough slog even with an excellent prof teaching the course. (More than one of my classmates at the time chose to take the graduate course instead because it was easier.) Math in any good undergrad text will be linear algebra and differential equations; in a more advanced treatment you'll get some group theory as well.

I didn't take GR until grad school, in astronomy, but it doesn't presuppose any physical background - there the math is by far the most challenging part, and you may find it rewarding. Schutz's A First Course in General Relativity is the standard. Math: tensors and differential geometry, which he introduces briefly. This hovered just at the edge of my understanding when I took the course; I could manipulate indexes but found an actual intuitive understanding always seemed to be just slightly out of reach.

You've got a campus bookstore, so I'd take a look at what the sophomore physics majors are using and start from there. An alternative approach would be to look at the Feynman Lectures on Physics - these are a little dated in notation and style, but they're aimed at freshman, so they don't assume a physics background. They're aimed at very SMART freshmen, though (at Caltech), so he doesn't pull his mathematical punches.

6>

Well, we learned them simultaneously, starting by covering the basics of vector calculus during the first few weeks of freshman physics - otherwise you're going to require that scientists take an extra three years to learn the math before they can even start in their field, which isn't really reasonable.

alco261, if you'd like a fun example of the Frequentist vs. Bayesian thing for class, here's XKCD from a couple of weeks ago:

XKCD: Frequentist vs. Bayesian

>7 LolaWalser: "from what I've experienced in my studies, it is usually a question of leaving out lengthy derivations of equations or proofs"

This doesn't match my experience. Many science authors are up front in priding themselves on avoiding sophisticated mathematics, making explicit statements to that effect in their prefaces. They're not avoiding the derivations of results; they're avoiding the results themselves, which can't be precisely stated without sufficiently advanced mathematics. Thus, they engage in mathematically-unsatisfying hand-waving. If you're unfamiliar with this phenomenon, I can provide you with any number of examples that you wish.

Speaking of prefaces, here's a passage from the preface of one of the books I mentioned in my original post: "As mathematicians, while researching for this book, we found it very difficult to absorb physical ideas, not only because of eventual lack of rigor--this is rarely a priority for physicists--but primarily because of the absence of clear definitions and statements of the concepts involved. This book aims at patching some of these gaps of communication."

While I'm at it, here's a bit of the preface of Mathematical Foundations of Quantum Mechanics: "The aim of the course was to explain quantum mechanics and certain parts of classical physics from a point of view more congenial to pure mathematicians than that commonly encountered in physics texts. Accordingly, the emphasis is on generality and careful formulation rather than on the technique of solving problems."

These two prefaces express aspirations that may be shared by authors of other science books. The purpose of this thread was to try to track down those books.

>8 lorax:

Thanks for your suggestions.

>9 alco261:

You seem to be saying both that learning the math before the science (a) would destroy interest in the science and (b) is the status quo in your (nevertheless undestroyed) specialization.

Thanks, I've heard lots about the inadequacies of American education over the years.

You mention again "science" books, which I find perplexingly vague (in Europe we don't have the subject of "science", as you seem to in the US, but rather separate subjects, beginning in elementary school with physics, chemistry, biology and progressively adding other more specialised fields, depending on the type of school and degree chosen). If you are looking at something that treats "science" as a great big lump of science-y stuff, then I wouldn't be surprised if sophisticated math is lacking.

As for its lack in specialised textbooks, perhaps it's understandable up to a point, where surveys of whole fields are intended. Background math for every equation in an introductory, general physics course could easily triple the material.

But looking at the quotations you provide, I'm thinking you might also run into a philosophical problem of sorts. Mathematical rigour isn't the cornerstone for much of experimental, empirical, down and dirty scientific research. You talk of mathematics as "the language" of science, but that's a poetical approximation rather than fact.

To me, as to many working scientists, math is an important tool, and a necessary foundation for many of our skills, but it is simply inadequate as a primary mode of communication, as it is also (in our time at least) often inadequate in describing complex real phenomena (which doesn't stop us from using it--only be aware "rigour" may have different meaning in, say, analysis, and biological modelling).

>12 LolaWalser:

Thanks for your reply. I'm looking for books addressing specific areas of science (such as the 3 positive examples I gave in my original post), not addressing science in general. This would include even the social sciences. (I will be teaching a course on social choice theory for our math majors next semester, and I hope to produce a textbook aimed at mathematicians in the process, myself.)

Certainly there is an imperfect fit between mathematics and reality (just as there is an imperfect fit between English and reality). More rigor is not, in general, what mathematicians like myself want out of our science reading. It's for the language of math to be used when it would clarify things for readers like us. If a PDE would explain the basics of your theory, it would be great for us if you just put it in.

I haven't intended to demand that all science books for non-specialists be written this way. I'd just like to know which ones have been.

cpg, Penrose's Road to Reality came to my mind while reading your initial post, and I see it's in your library -- is this anything like what you're looking for or are there specific ways it misses the mark? (I haven't looked at that book for many years and no longer have it handy.)

There is a subfield of "mathematical physics" that at its best clarifies the physical interpretation of mathematical language through rigor/care with the formalism. The previous sentence is my attempt to boil down the introduction to Thirring's Classical Mathematical Physics, and sounds a bit like what you write in #13. As a specific example, this introduction mentions that hand-wavings about "infinitely small quantities like 'infinitesimal variations' ... disappear and are replaced more precisely with mappings of the tangent spaces."

I used Thirring's book and Spivak's Calculus on Manifolds as supplements in an undergraduate reading course in general relativity. (The main text was Foster & Nightingale's Short Course in General Relativity, which I believe is comparable to Schutz.) Thirring turned out to be too sophisticated for me at that time (as an upper-level undergrad with a major in math as well as physics), though I may just not have worked on it as carefully as I should have.

I can follow lorax and list other textbooks I've found to be reasonably rigorous, but first I think it's useful to mention the "spiral staircase" analogy to physics education (at least in the US): in pre-college, undergraduate and graduate physics courses, students repeatedly revisit the same subjects, each time with greater mathematical sophistication (which is further developed and extended in the process). So although it's the higher-level textbooks that tend to present the most mathematical rigor, these may still be accessible to non-specialists with the necessary technical skills. Even advanced textbooks should be reasonably self-contained and not rely on the reader having a detailed background in the subject. ("Should" is always a magic word, of course.)

As a specific example, both Jackson's Classical Electrodynamics (a widely-used graduate text), and the Griffiths text lorax mentioned, start with Coulomb's law. The main difference I see, with both books open in front of me, is that Jackson assumes the reader knows vector calculus and doesn't spend a chapter introducing vector analysis. I would guess that the main downside of more advanced texts from your perspective may be that they go into more detail on specific applications than you might need or want to see. Jackson simply covers a lot more than Griffiths, so you would need to skip through it to skim off the cream -- but the cream is there.

Classical mechanics, if done well, can introduce a lot of the machinery also used (or adapted) by quantum mechanics. Most undergraduate texts don't do it well; during my upper-level classical mechanics course, I used a copy of Goldstein's Classical Mechanics from the library, which covered the same material as the official text (and more), but better.

For quantum mechanics, I am more fond of the text from my undergraduate course (Townsend's Modern Approach to Quantum Mechanics) than that from grad school (Quantum Mechanics: Fundamentals by Gottfried and Yan). Just this week I did my first teaching, running a graduate Mathematical Physics course whose professor was out of town at a workshop. I took most of the material for my lectures from Townsend (showing how associated Laguerre polynomials and spherical harmonics pop up in hydrogenic atoms), with some supplements from Jackson and from Quantum Mechanics: Non-Relativistic Theory by Landau and Lifshitz. (The Landau and Lifshitz Course of Theoretical Physics is classic, but may be a bit dated and may assume a lot of the reader.)

Finally, for general relativity, I recently audited a graduate course in addition to the undergraduate reading course I mentioned above. This course was primarily based on Carroll's Spacetime and Geometry, which I preferred to the alternate text, General Relativity by Wald.

(There are oodles of quantum field theory books, and I see you mention one I'm not familiar with, but these typically assume a solid background in quantum mechanics and special relativity.)

>15 daschaich:

Thanks very much for your suggestions. The Road to Reality is a weird duck. In the preface, Penrose says "I have not shied away from presenting mathematical formulae", but he also makes clear that among the readers for whom the book is intended are those "who claim that they cannot manipulate fractions". It was certainly ambitious of Penrose to try to address such a wide audience. My bookmark informs me that I got stalled at page 153, over 200 pages before the physics really begins. Unfortunately, I have some sort of neurotic aversion to skipping ahead in books, so simply jumping over the math I already know is probably not a viable option. Penrose suggests that his mathematically-inclined readers "may find that there is something to be gained from [Penrose's] own perspective" on topics they already know, so maybe I should give him another shot.

PS. I've heard (but not checked myself) that older editions of Jackson's book assume greater mathematical literacy, and as a result cover some topics more elegantly.

"Weird duck" is a good description of what I remember of Road to Reality.