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"Mathematics : Its Content, Methods and Meaning" by A. D. Aleksandrov, A. N. Kolmogorov ,M. A. Lavrent'ev. (3 Volumes) This is a classic and exactly what you are seeking for. I think it was originally published in 1962. https://www.goodreads.com/book/show/405880.Mathematics https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
There is some quality in Russian math books that western authors seem to lack.
In terms of depth and breadth, the Princeton companions get close to Feynman.[1][2] A more formal approach appears in handbooks.[3][4] [1] Gowers et al., The Princeton Companion to Mathematics. https://press.princeton.edu/books/hardcover/9780691118802/th... [2] Higham and Dennis, The Princeton Companion to Applied Mathematics. https://press.princeton.edu/books/hardcover/9780691150390/th... [3] Zwillinger, CRC Standar…
But is really breadth and depth what makes Feynman’s approach to physics unique? To me the unique aspect is more the uncompromising intuitionistic approach with little consideration/adaptation for “shallow/correlative thinkers”...
But with mathematics, "intuitive" analogies are all in terms of other mathematical objects! You can't build intuition if you don't even know what they trying to abstract over.
In that regards, The Princeton Companion to Mathematics is fantastic because it maps out how the different fields of mathematics are interrelated.
Spivak's Calculus. It's "just" calculus... but it's also everything else leading up to it. It's a wonderful book, written in a very engaging style, and it shows you how mathematicians think and how they play. It shows you why we have proofs, why things go wrong, and all that had to happen before we came up with a definition of derivatives and integrals that we're happy with (and of course, all of the things we can do…
It's also a beautiful book.
For mathematics, I would recommend: 1. "What Is Mathematics? An Elementary Approach to Ideas and Methods" by Courant and Robbins -- a general book on mathematics in the spirit of Feynman lectures. 2. Strogatz's "Nonlinear Dynamics and Chaos" -- it's a bit narrow in scope (mostly dynamical systems with a little bit of chaos/fractals thrown in) but very good nonetheless. 3. Tristan Needham, "Visual Complex Analysis", b…
From the description:
"This course of 25 lectures, filmed at Cornell University in Spring 2014, is intended for newcomers to nonlinear dynamics and chaos. It closely follows Prof. Strogatz's book, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering."
The mathematical treatment is friendly and informal, but still careful. Analytical methods, concrete examples, and geometric intuition are stressed. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.
A unique feature of the course is its emphasis on applications. These include airplane wing vibrations, biological rhythms, insect outbreaks, chemical oscillators, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory. The theoretical work is enlivened by frequent use of computer graphics, simulations, and videotaped demonstrations of nonlinear phenomena.
The essential prerequisite is single-variable calculus, including curve sketching, Taylor series, and separable differential equations. In a few places, multivariable calculus (partial derivatives, Jacobian matrix, divergence theorem) and linear algebra (eigenvalues and eigenvectors) are used. Fourier analysis is not assumed, and is developed where needed. Introductory physics is used throughout. Other scientific prerequisites would depend on the applications considered, but in all cases, a first course should be adequate preparation."
[0] https://www.youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJ...
There is also the No Bullshit Guide to Linear Algebra https://www.amazon.com/dp/0992001021/ Extended preview: https://minireference.com/static/excerpts/noBSguide2LA_previ...
Both come with a review of high school math topics, which may or may not be useful for you, depending on how well you remember the material. Many of the university-level books will assume you know the high school math concepts super well.
One last thing, I highly recommend you try out SymPy which is a computer algebra system that can do a lot of arithmetic and symbolic math operations for you, e.g. simplify expressions, factor polynomials, solve equations, etc. You can try it out without installing anything here https://live.sympy.org/ and this is a short tutorial that explains the basic commands https://minireference.com/static/tutorials/sympy_tutorial.pd...
https://www.amazon.com/Arithmetic-Paul-Lockhart/dp/067497223...
It's probably the best book on mathematics I've read. It's not a textbook the way the Feynman lectures are, but it's stimulating and a good read. Other books mentioned like Visual Complex Analysis or Courant's book are dry and take a lot of effort to get through. Some of the older books mentioned may be great (I've found many older textbooks much clearer than more recent ones), but I personally haven't read them so I can't make a recommendation there.
You can also check YouTube videos/courses e.g. one I found great was MIT Professor Gilbert Strang's Linear Algebra course -- his videos are easy to follow, stimulating and clear.
"Mathematics : Its Content, Methods and Meaning" by A. D. Aleksandrov, A. N. Kolmogorov ,M. A. Lavrent'ev. (3 Volumes) This is a classic and exactly what you are seeking for. I think it was originally published in 1962. https://www.goodreads.com/book/show/405880.Mathematics https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
Wikipedia summarizes it nicely: "His views on education were particularly anti-Bourbaki." :)