I came to a realization years ago that when it comes to math/science/computers, if I just read about it, I’m not really learning anything. I don’t learn unless I’m participating. Math books without _lots_ of exercises probably don’t really impart that much, although they can be a fun distraction if you need something to read on an airplane.

In my country we have a saying: la matemática entra por la mano, no por los ojos -- mathematics is absorbed with your hands, not with your eyes.

I agree. Some things only stick by repeated application. This becomes more relevant the one older gets.

Depending on your level of math ability you can often learn a lot by reading textbooks and proofs very carefully and closely (my opinion)

It depends on your definition of learning. If by learning you mean remembering how one author defined and proved various things then sure. If by learning you mean gaining the ability to do what the author did except for new problems, then at some point you will have to grapple with the fact that coming up with new ideas is a very different mental exercise from recalling things you've read.

A supporting factor in the necessity of exercises for learning to actually do the math is that authors often hide important technical tricks in the exercises.

Of course, a sufficiently intelligent person could learn anything by reading tea leaves and then deriving on their own whatever it was they wanted to learn, so in a sense you can learn math by doing anything.

I disagree. Generally speaking students aren't able to teach themselves mathematics on their own until they hit the graduate level. That's not a hard rule, it just depends on mathematical maturity.

But everyone I know who is able to teach themselves math without lectures or a professor still does the exercises in the textbook. Math isn't a spectator sport, it's really important to do active learning.

This depends on the textbooks. It is very, very hard to find a textbook that is (a) well written, (b) suitable for self-study and (c) affordable.

Idk, I tried to self study the first two chapters of a very hated textbook (Principles of Mathematical Analysis). If you’d have asked me if I understood the material I would have said no. But by the time I took the actual class on that material I made high As on all the exams and could call out mistakes my professor made on metric sets with counter examples.

You can't hardly learn anything by just reading or watching it.

Any good recommendations on books that are more than pop-fluff but lighter than an academic text?

I really liked Introduction to Graph Theory by Trudeau. No upper level math required, but it got rigorous and fun at times. Mostly spatial reasoning.

David Foster Wallace wrote Everything and More which included a lot of math history, while driving into transfinite numbers and set theory. Kindle typesetting is generally terrible for math, but was completely readable in this case. The forward by Neal Stephenson really sets the mood.

> David Foster Wallace wrote Everything and More

This is far and away my favorite math book. The way DFW can seamlessly transition from discussing a serious mathematical concept, to the nature of insanity, to the philosophy of abstraction all within a few pages is astounding. He somehow manages to do it while maintaining a conversational, two-folks-at-a-bar tone, teaching me dozens of new word, and being insightful/funny. Like an English professor sneaked into the math dept.

I didn't know it was available on Kindle. Guess I'm buying it again...

Warning: These books are kind of all over the place as far as technical prerequisites go, but they all satisify being heavier than pop-sci and lighter than a textbook. :)

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Aleksandrov et al - Mathematics: Its content, Methods, ad Meaning. Overview of undergrad curriculum by a number of prominent russian mathematicians.

Levi - The mathematical Mechanic. Uses various physical intuitions and scenarios to solve mathematical problems.

Pask - Magnificent Principia. Tour of newtons principia that includes both newtons geometric language and modern vector calculus version.

Stillwell - Elements of Mathematics. Discusses some important classical problems in the history of mathematics taken from 8 subjects: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, logic. Wow!

Yandell - The Honors Class: Hilbert's problems and their solvers. Lots of mini biographies with some information describing each of Hilbert's famous problems.

Lawvere & Schanuel- Conceptual Mathematics. Some basic notions of category theory, geared towards teaching you Lawvere's clarification and generalization of the incompleteness theorems.

Cormen - Algorithms Unlocked. Pop Science version of some of the textbook.

Nahin - Inside interesting Integrals. Lots of fun integrations!

Nahin - He also wrote some books about euler and imaginary numbers which ive partially read, they seemed good.

Pierce - An introduction to information theory; symbols signals and noise. sounds and looks like a dover textbook but is a non-fake pop science book.

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Boyer - The History of Calculus and its Conceptual Development. A bit old, and spends too much time on formalism in the 18th and 19th century, there might be something better out now.

Weyl - Philosophy of Mathematics and Natural Science.

Reichenbach - The philosophy of Space and time.

Klein - Elementary mathematics from the advanced viewpoint. these were written at the beginning of the 20th century so are slightly outdated, but kline was extremely forward-looking in his point of view.

Courant & Robbins - What is Mathemaics. Another classic.

Hilbert & Cohn-Vossen - Geometry and the Imagination. The gold standard of popular mathematics writing. A beautiful book.

Regarding Principia, noted astronomer Subramanian Chandrashekhar, had written "Principia for the Common Reader", which has commentary in modern language, while yet using only the same geometric method that Newton used. While I have read only a small portion of it, it was very fascinating.

Somewhat dated but...I really like "A Logical Approach to Discrete Math" by Gries and Schneider. It uses an algebraic approach to introduce logic then works its way into number theory, graphs, and sets. There are a lot of good exercises and the topics are highly relevant for CS.

The content is so valuable that it deserves a year in high school in my opinion, though that is a minority view where I live.

I like this book: https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-N...

Jeremy Kun, who writes the "Math ∩ Programming" blog,[1] recently published a book, "A Programmer's Introduction to Mathematics,"[2] that seems to leverage the conceptual knowledge programmers already have to help them learn serious math. There also are sections in the book on math culture which seems interesting, something I haven't seen elsewhere.

[1] https://jeremykun.com/

[2] https://pimbook.org/

Man you just hit the nail on the head.

In a similar vein, I love documentaries and I learned much as a child that way. I find that they are way too basic now and often repeat things I've seen/known for years. This is needed for other kids of course, but as an adult the only stuff is really basic or so advanced that it is actual work to make it through (tough after a rough week at work).

One of my favorite examples of finding the middle ground is in dealing with SETI. I love the subject and have seen numerous documentaries on it over the years that cover what they're doing from 50,000 feet. Paul Davies is the lead at SETI and I read his book "The Eerie Silence" and it was really awesome. It perfectly found the middle ground between a documentary which has to be ok for children and a textbook on the subject. Paul Davies is kind of to SETI what Alan Kay is to computing in some ways as someone with deep experience in the field and lots of philosophical thoughts on the subject. His writing is really clear though (I love Kay, but don't have a strong enough computing, historical computing, math, or biology background to keep up with a lot of his writing). Two others that stand out to me are the classic "A Brief History of Time" by Stephen Hawking that blew my mind as a kid and the equally good, but lesser known "Black Holes and Time Warps" by Kip Thorne which made an excellent follow up book that gives a good high level appreciation which would likely be helpful for anyone planning to study those subjects in college at a rigorous mathematical level. Kip's book is also funny. There is a picture of Stephen Hawking delivering him a years subscription to a magazine (I think it might have been lewd in nature) for losing a bet to Kip over whether Cygnus A would have a black hole I think. As a highschool student I wrote to him with some career questions and he was kind enough to respond! Max Tegmark has a book "Our Mathematical Universe" which covers two parts: one on why we might be part of a simulation, and the first part which is like a condensed cosmology lesson from A to Z. He had a big part in several big discoveries and hearing the play by play is gripping. He was apparently a pretty gifted coder, so he got tasked with a lot of the data analysis and therefore he and his first wife were among the first to see the actual results of these big experiments. Code by Petzold is also an amazing tour de force if you want to understand computers.

What field of mathematics interests you? Or you want some digestable intro to all of them? In that case I would recommend Kevin Houston "How to Think Like a Mathematician".

“An Introduction to the Theory of Numbers” by Hardy and Wright.

The author may have just chosen what's in stock at Amazon, given that all the links are Amazon affiliate links (tag=alikayaspor0b-20).

Would also recommend 'Surreal Numbers' by Don Knut hand 'The Book of Numbers' by Conway and Guy.