I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse casually.
By "casual", I mean something that goes over maybe the history of a certain field and its implications in math and society, going over how it grew and what important contributions occurred at different points. Perhaps even going over the abstract meaning of famous results in the field, or high level overviews of proofs and their innovations. Conversations between mathematicians at the time, stories about how proofs came to be, etc.
I suppose the best place to start would be math history books, but I was curious what else there may be. Are there any good books out there like this? What are your recommendations?
Best Answer
What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level.
But still you may try these books:
Michio Kuga, Galois' dream,
David Mumford, Caroline Series, David Wright, Indra's Pearls,
Hermann Weyl, Symmetry.
Marcel Berger, Geometry revealed,
D. Hilbert and Cohn-Vossen, Geometry and imagination,
T. W. Korner, Fourier Analysis,
T. W. Korner, The pleasures of counting.
A. A. Kirillov, What are numbers?
V. Arnold, Huygens and Barrow, Newton and Hooke.
Mark Levi, Classical mechanics with Calculus of variations and optimal control.
Shlomo Sternberg, Group theory and physics,
Shlomo Sternberg, Celestial mechanics.
All these books are written in a leisurely informal style, with a lot of side remarks and historical comments, and almost no prerequisites. But the level of sophistication varies widely. Also don't miss:
Roger Penrose, The road to reality. A complete guide to the laws of the universe. It is on physics, but contains a lot of mathematics.