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...
But then I tried to solve some final exams from previous years, and realized the feeling is false. These books gave me great intuition - but they made all the math look deceivingly simple, and as a result it is hard to develop the actual problem solving skills and intuition.
I know my experience is not unique - in fact, everyone I know who tried to learn exclusively from Feynman had the same experience.
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", beautiful introduction to complex analysis.
4. Cornelius Lanczos, "The Variational Principles of Mechanics" -- this is a physics book, but one of the classics in the subject, and as Gerald Sussman once remarked, you glean new insights each time you read it.
5. Cornelius Lanczos, "Linear Differential Operators" -- an excellent treatment of differential operators, Green's functions, and other things that one encounters in infinite-dimensional vector spaces. This book has some very intuitive explanations, e.g., why d/dx is not self-adjoint (i.e., Hermitian), whereas d^2/dx^2 is.
For chemistry, I would recommend "General Chemistry" by Linus Pauling, even though it's a bit outdated.
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 Standard Mathematical Tables and Formulae. https://www.crcpress.com/CRC-Standard-Mathematical-Tables-an...
[4] Bronshtein, Handbook of Mathematics. https://www.springer.com/gp/book/9783540721222
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 with our newfound definitions).
Edit: Introduction to Graph Theory by Trudeau is another that I really liked. Very little was applicable to graphs as programmers think of them. Pure math that is easy to grasp and enjoy.
Mathematics, Form and Function by Saunders Mac Lane (1986 Springer-Verlag hardcover, ISBN 0-387-96217-4) [2]
[1]: https://www.cut-the-knot.org/books/kac/index.shtml
[2]: https://en.wikipedia.org/wiki/Mathematics,_Form_and_Function
I think this is in the spirit of Feynman's lectures inasmuch as it's not going to bring you to expert-level understanding, but it is going to do a good job giving you some intuitive understanding, which you might then be able to apply to re-studying the material in more detail.
He even takes pull requests, I fixed a few typos.
1. Spivak Calculus
2. Apostol Calculus vol. I and II
3. Courant and Johns, Introduction to Calculus and Analysis vol. I, IIA and IIB
For the more casual computer science or physics major, I'd go with choice 3 which resembles the Feynmann lectures the most. All the rigour of the other two is there but in a more digestible form. It's hard for someone not accustomed to hard maths to digest long proofs, it could give you a bad case of indigestion. It's more a function of patience. Courant and Johns get to the point much more quickly while Spivak and Apostol take their time to do everything thoroughly. Courant and Johns does do everything thoroughly but they are kinder to the reader and delay lengthy rigoourous proofs as long as possible while giving plenty of motivation and intuition.
Also I strongly recommend any books by Ray Smullyan particularly his introduction to mathematical logic. "A Beginner's Guide to Mathematical Logic"
Not really a book that has lectures but it's a great book that covers all popular topics in Mathematics (fun to read).
(Link: https://mirtitles.org/2015/12/07/mathematics-can-be-fun-yako...)
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.
It's a zany story, but in that respect it provides a refreshing intro into some math concepts.
In that vein, I also recommend "mathematical mindsets" [2]. A colleague developed a course inspired by this book. Though I only witnessed a tidbit, it radiated with the "new perspective/ new insights / gained understanding" that you'd get from the Feynman lectures.
Sidenote: neither is "now you know how all of maths work", but neither is Feynman thaf (foot physics). More importantly, all of them help you gain a new perspective on things.
[1] eg. https://www.bol.com/nl/p/the-number-devil-a-mathematical-adv...
[2] eg. https://books.google.nl/books/about/Mathematical_Mindsets.ht...
Nobody has mentioned yet "Geometry and the Imagination" by Hilbert and Cohn-Vossen. If there is a Feynman equivalent in math it is certainly this book.
For elementary geometry, the Feynman equivalent is probably "Introduction to Geometry", by H.S.M.Coxeter. Beautifully written, figures on every page, covers all geometric topics (affine, projective, ordered, differential, ...)
For differential geometry, nothing beats "A Panoramic view of Differential Geometry" by Berger. It is a stunning comprehensive overview of the whole field, focused on the meaning and the applications of each part and, strangely for a math book, with no formal proofs. Only the main ideas of the proof and the relationships between them are given, but this allows to fit the whole subject into a single, manageable whole.
I read it myself years ago and it was a great and entertaining way to fill in the gaps from my meager math education.
His explanations of mathematics are the only ones I can think of that have given me the same sort of piercing clarity and insight that one gets from reading Feynman on physics.
>This isn't a popular suggestion (and by that I don't mean to say it's rejected or people don't like it, I just haven't heard it suggested before in this context) but at university for electronic engineering we used K.A. Stroud's Engineering Mathematics. This book is surprisingly little focused on actual applications to engineering, it takes you through calculus by introducing the derivative, for example, and then some linear algebra stuff. But what surprises people is that it starts off with the properties of addition and multiplication - it's that simple. It's a book that starts from zero and takes you very, very far. It won't take you to a mathematician's 100 but it'll take you to any serious engineering undergrad's 100.
https://www.amazon.com/Arithmetic-Paul-Lockhart/dp/067497223...
It is called "Who is Fourier: A Mathematical Adventure".
I was tremendously surprised by this unusual gem of a book. It covers the range from basic arithmetic to logarithms, trigonometry, calculus to fourier series.
https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2n...
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...
Firstly, much of mathematics is symbolic and any description of equations in an intuitive style is unnecessarily verbose if it abandons the symbolic approach, essentially taking one back to descriptions like those used in ancient Greece before the invention of algebra, e.g. "and the third part of the first is to the second part of the first as the fourth part of the area is to the square on the gnomon".
The second reason is that an intuitive style supposes that one can answer natural questions that might arise, in an order that they are likely to arise in the mind of the student. Often the natural questions are much more difficult to answer mathematically, or the answers are not known.
The third reason is that concepts have arisen historically for non-obvious reasons, or reasons only known to experts with far more knowledge than the reader is expected to have, or the originator of the ideas did their best to obscure their motivation. This makes it extremely hard to motivate certain concepts naturally (intuitively) since such motivations are simply not known. For example, it is not hard to motivate solvable groups through a study of solubility of polynomial equations. But it is much harder to motivate the related concept of nilpotent groups, where the true motivations lie far deeper in the theory than the concepts themselves.
The fourth reason is that it is a massive effort to come up with good examples. Even the best textbook authors often struggle to come up with accessible examples for the concept they are trying to explain. Often, good examples require a really broad knowledge of mathematics that goes way beyond the narrow field being taught. Examples end up being very artificial, and neither intuitive nor typical, as a result.
Don't get me wrong. If someone told me something like the Feynman lectures existed for mathematics, I would salivate and spend a lot of money to acquire them. But having experimented with many styles of writing notes for myself on mathematics over the years, I well appreciate how hard, or perhaps impossible the task would really be. Of course there are some oases in mathematics where such an intuitive approach is possible.
My personal take is that good linear algebra books at any level are great "tours of mathematics". Start with Strang and never stop. In a few years you'll be balled up with Kreyszig scribbling proof attempts in receipts, flaming unkempt hair and everyone around you will think you're weird but you'll be so, so happy.
[0] https://www.amazon.com/gp/aw/d/0486652416/
[1] http://alvand.basu.ac.ir/~dezfoulian/files/Numericals/Numeri...
Something that might be close would be the survey _Mathematics: Its Contents, Methods and Meaning_ by Aleksandrov, Kolmogorov et al. https://www.goodreads.com/book/show/405880.Mathematics
https://www.amazon.com/Tour-Calculus-David-Berlinski/dp/0679...
And there's this book: "Conceptual mathematics" by Lawvere and Schanuel. It's unlike any other mathematics text I have found. Fundamental and easy to read: yes. Also leads up to some deep ideas in an intuitive way.
Nathan Carter's 'Visual group theory' also seems an interesting experiment, if you are interested in that part of mathematics, though I have not read it.
My longest problem has been I have no idea what is going in the formula or fundamental questions like, "why is there a square root there". It is hard to describe my issue, but I've been very horrible with math anyways. Can't do gas station math anyways.
http://www.feynmanlectures.caltech.edu/I_toc.html
It is... very average looking. Did something happen here?
https://www.amazon.com/Playing-Infinity-Mathematical-Explora...
A lesser known one that isn't quite as comprehensive is a little Dover tome by Mendelson: Number Systems and the Foundations of Analysis. It starts off with the (abstract) natural numbers, and from there develops (parts of) real and complex analysis, using a categorical point of view throughout.
One of my favorite parts in the latter:
“What is our intuitive understanding of the natural numbers? Surely this being the firmest of all our mathematical ideas, should have a definite, transparent meaning. Let us examine a few attempts to make this meaning clear:
(1) The natural numbers may be thought of as symbolic expressions: 1 is |, 2 is ||, 3 is |||, 4 is ||||, etc. Thus, we start with a vertical stroke | and obtain new expressions by appending additional vertical strokes. There are some obvious objections with this approach. First, we cannot be talking about particular physical marks on paper, since a vertical stroke for the number 1 may be repeated in different physical locations. The number 1 cannot be a class of all congruent strokes, since the length of the stroke may vary; we would even acknowledge as a 1 a somewhat wiggly stroke written by a very nervous person. Even if we should succeed in giving a sufficiently general geometric characterization of the curves which would be recognized as 1’s, there is still another objection. Different people and different civilizations may use different symbols for the basic unit, for example, a circle or a square instead of a stroke. Yet, we could not give priority to one symbolism over any of the others. Nevertheless, in all cases, we would have to admit that, regardless of the difference in symbols, we are all talking about the same things.
(2) The natural numbers may be conceived to be set-theoretic objects. In one very appealing version of this approach, the number 1 is defined as the set of all singletons {x}; the number 2 is the set of all unordered pairs {x, y}, where x =/= y; the number 3 is the set of all sets {x, y, z} where x =/= y, x =/= z, y =/= z; and so on. Within a suitable axiomatic presentation of set theory, clear rigorous definitions can be given along these lines for the general notion of natural number and for familiar operations and relations involving natural numbers. Indeed, the axioms for a Peano system are easy consequences of the definitions and simple theorems of set theory. Nevertheless, there are strong deficiencies in this approach as well.
First, there are many competing forms of axiomatic set theory. In some of them, the approach sketched above cannot be carried through, and a completely different definition is necessary. For example, one can define the natural numbers as follows: 1 = {∅}, 2 = {∅, 1}, 3 = {∅, 1, 2}, etc. Alternatively, one could use: 1 = {∅}, 2 = {1}, 3 = {2}, etc. Thus, even in set theory, there is no single way to handle the natural numbers. However, even if a set-theoretic definition is agreed upon,it can be argued that the clear mathematical idea of the natural numbers should not be defined in set-theoretic terms. The paradoxes (that is, arguments leading to a contradiction) arising in set theory have cast doubt upon the clarity and meaningfulness of the general notions of set theory. It would be inadvisable then to define our basic mathematical concepts in terms of set theoretic ideas.
This discussion leads us to the conjecture that the natural numbers are not particular mathematical objects. Different people, different languages, and different set theories may have different systems of natural numbers. However, they all satisfy the axioms for Peano systems and therefore are isomorphic. There is no one system which has priority in any sense over all the others. For Peano systems, as for all mathematical systems, it is the form (or structure) which is important, not the “content”. Since the natural numbers are necessary in the further development of mathematics, we shall make one simple assumption:Basic Axiom There exists a Peano system.“
Elliott Mendelson, Number Systems and the Foundations of Analysis
https://www.amazon.com/Number-Theory-History-Dover-Mathemati...
Not to be taken literally, of course. But there is some truth in that. If you are an engineer it makes sense to skim all kinds of math books. If you are a mathematician then I would say rather look for something that gels well with your personality and run with it.
I'm currently learning group theory, matrices, and graph theory.
- Concepts of Modern Mathematics - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...
- Methods of Mathematics Applied to Calculus, Probability, and Statistics - https://www.amazon.com/Methods-Mathematics-Calculus-Probabil... (all books by Richard Hamming are recommended)
- Calculus: An Intuitive and Physical Approach - https://www.amazon.com/Calculus-Intuitive-Physical-Approach-...
For a Textbook reference, the following are quite good;
- Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences - https://www.amazon.com/Mathematical-Techniques-Introduction-... (easy to read and succinct)
- Mathematics for Physicists: Introductory Concepts and Methods - https://www.amazon.com/Mathematics-Physicists-Introductory-C...
For General reading (all these authors other books are also worth checking out);
- Mathematics, Queen and Servant of Science - https://www.amazon.com/Mathematics-Queen-Servant-Science-Tem...
- Mathematics and the Physical World - https://www.amazon.com/Mathematics-Physical-World-Dover-Book...
- Mathematician's Delight - https://www.amazon.com/Mathematicians-Delight-Dover-Books-Ma...