Many of the list are pop-maths rather than fundamental texts.
I do see many in software field recommend Chrystal:Algebra an Elementary Textbook.
I also find Don Knuth, 'Concrete Mathematics' a interesting book for software people.
I do see many in software field recommend Chrystal:Algebra an Elementary Textbook.
I also find Don Knuth, 'Concrete Mathematics' a interesting book for software people.
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.
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.
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.
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...
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.
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.
-------
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.
---------------
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.