Mathematics: Its Content, Methods and Meaning

I've been relearning trigonometry lately by myself for navigation and astronomy; not for work, just curiosity I guess. One book I've really enjoyed is Heavenly Mathematics by Van Bremmelen. It's a spherical trig textbook, but it's written by a math historian who describes how trigonometry was gradually developed over human history and he discusses its early proofs, methods and applications. I have to confess that the historical approach has really helped me develop a more complete mental picture and appreciation of the math itself. Understanding the "how" and "why" of its development, and seeing the early practical need and implementation for some of this stuff has made the topic a lot more engaging.

Full 3 volumes of Free pdfs on IA:

https://archive.org/details/MathematicsItsContentsMethodsAnd...

This is a great book if you already know good amount of Math. It helps you fit things into a bigger picture. Really appreciate the fact that something like this exists.

The only mathematics books I ever read was textbooks in school but now as adult I want to start from scratch.

This book is really fascinating because it contains a surprising amount of Soviet ideology. The authors repeatedly state that mathematics is posterior to the material world, not prior to it. That is, mathematics is just the observation of regularity in the world, particularly those discovered by people working to create things. Contrast this with the still heavily idealistic world of western mathematics, where mathematicians are more likely to sympathize with the notion that numbers are real things somewhere out there whose structures the real world supervenes upon in some way.

Interesting stuff!

Even though I favor the Soviet view of mathematics personally (I do not think numbers "exist" out there independent of the material world), I think this approach hampers the didactic goals of the text and probably hurt Soviet mathematics as well. The examples in the text are all highly concrete (literally things like rubber mats when discussing curvature). This very down to earth style makes the abstract notions of curvature in other contexts (for example, general relativity) more difficult to grasp, in my opinion.

On the other hand, some people prefer strong, material, examples of mathematical ideas. This book definitely provides that. The section on affine maps in terms of fixing the plane of a surveillance airplane photograph is beautifully concrete.

Where can I find mathematical book titles like this one?

This is one of the best generalist books on mathematics ever published. I highly recommend it.

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