There's a style of "popular" books about math and physics that involves actually learning some of the math and physics. They don't just talk ABOUT the subject, they try to convey some OF it.
In the math books, many theorems are stated without proof. So you have to learn enough to understand the statement of the theorem, but you don't have to wade through the proof. Some of the theorems are illustrated by simple examples.
A couple of such math books are "Elliptic Tales" and "Fearless Symmetry" both by Ash and Gross. Elliptic Tales is about elliptic curves, an important but difficult part of mathematics. Fearless Symmetry is their "popularized" introduction to algebraic number theory, including the Wiles-Taylor proof of Fermat's Last Theorem. I wrote an Amazon review of Fearless Symmetry on 6/4/13.
A physics book like that is Road to Reality by Roger Penrose. I was obsessed with this book for months, I did solved all the exercises that looked challenging except for one or two, and I learned a LOT from it. It's about mathematical physics.
I put "popular" in quotes because most people would find such books too difficult and challenging, and it requires thought and commitment to understand them. Road to Reality could be more challenging than a physics textbook, just because he doesn't explain the concepts and equations rigorously - so you have to figure out the rigorous meaning, if you want to fully understand.
But I find them interesting. I'm too stressed and my mind has been too fuzzy with allergies to read a "real" math book, containing the proofs and all the technicalities. These are math lite. Anyone enjoyed other books like this?
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I have The Road to Reality, and am about a hundred pages in. I wish I had someone to work through it with!
There's an online forum for discussing Road to Reality exercises. There used to be a partial solution set, many of the solutions I wrote :)
I have a partial solution set. I haven't been updating it with solutions from the people on the forum, though. Someone else may have a more up to date version.
ps Exercise 27.16 in Road to Reality asks you to show that a connected 3-space that is isotropic around 2 separate points p and q, is homogeneous.
This isn't true! Counter-example: if the space is the 3-sphere, p is a point on the sphere, and you have a matter distribution that is a function only of distance from p, then the matter distribution is also isotropic around the antipodal point q.
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