Ask HN: Are there books for mathematics like Feynman's lectures on physics?
I have started re-learning college level Physics and am thoroughly enjoying Feynman's Lecture on Physics. Are there similar books available for Mathematics (& Chemistry) - books that are fundamental and easy to read?
179 comments
[ 3.5 ms ] story [ 226 ms ] threadEdit: 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.
If you have a grasp of algebra and sets this book is an easy read for the curious or mathematically immature.
Edit WhatIsDukkha is correct and their suggestion better reflects what I intended to say.
Usually saying someone isn't "mathematically mature" ie able to read and use proofs is what you would want to say.
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.
[1]: https://www.youtube.com/c/3blue1brown
Maybe if schools and colleges had off-the-shelf software that was 80% as good at making explanatory animations as his Python lib is we would see a huge boost in the understanding of maths.
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
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).
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...)
Some of his Books;
* Physics for Entertainment Vols I & II
* Algebra for Fun
* Figures for Fun
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.
http://www.feynmanlectures.caltech.edu/info/exercises.html
Some references to good collections of mathematics problems and solutions would be great for self-study.
But we all took that as an indication of our own lack of knowledge and intuition and would just try harder.
When Bob was drawing, it looked amazingly simple. That simplicity invited people into trying painting.
Probably very few could ever draw anything remotely similar in quality to him, though.
If you just read a book and don't work through problems yourself, you simply don't learn enough to do it yourself.
Most books, I make some progress on some problems, get stuck on others, and generally have a good grasp of the overall landscape and where I am lacking; then I go back and reread (and practice) the missing pieces.
But Feynman’s lectures are different in that they make you feel you understand a lot, without really giving you any tools to address things he did not address (and basically only address those things he did address in the same way).
I am not saying they are bad - 25 years later, I still remember (and occasionally use) some of them; most recently insights from the chapter on minimum principles. I am just saying that it only became useful after I already had a good (but not great) grasp of the material from other sources — despite giving that impression when read as introductory.
Reading good books/having good lecturers certainly helps, but there is no way to replace the work.
I watch with wry amusement how schools constantly try to take the work out of learning. It never works. It's like putting labor-saving machinery in the gym - you'll never get stronger. You gotta put in the sweat.
I don't think Feynman's books are a replacement for a more traditional physics textbook as a student looking to pass a class, become a physicist, or a hobbyist trying to master it, but I do think they're pretty ideal for someone who wants to get much stronger grasp of the concepts than a layman without having to go through the struggle associated with solving problems they'll never actually apply.
At a meta level, I think this means we'll never (as a society) be great at teaching, because teachers who make us work make us feel like we're learning less. We prefer (and rate more highly) the professors, like Feynman, who make us feel smart.
Vol 1 was good. Vol 2 was good though overly repetitive, iirc it's 90 percent Maxwell's equations. Vol 3 was unintuitive to me. Still I learned qm from it and made this http://tropic.org.uk/~crispin/quantum/
I'm fairly confident I get qm now, but most of that understanding came from trying to code it in simulation. Which suggests there are better ways to learn than Feynman 3.
It's easier to feel that you know something than to actually know it.
Here is a neat communication from Faraday to Maxwell on receiving one of Maxwell's paper;
“Maxwell sent this paper to Faraday, who replied: "I was at first almost frightened when I saw so much mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well." Faraday to Maxwell, March 25, 1857. Campbell, Life, p. 200.
In a later letter, Faraday elaborated:
I hang on to your words because they are to me weighty.... There is one thing I would be glad to ask you. When a mathematician engaged in investigating physical actions and results has arrived at his conclusions, may they not be expressed in common language as fully, clearly, and definitely as in mathematical formulae? If so, would it not be a great boon to such as I to express them so? translating them out of their hieroglyphics ... I have always found that you could convey to me a perfectly clear idea of your conclusions ... neither above nor below the truth, and so clear in character that I can think and work from them. [Faraday to Maxwell, November 13, 1857. Life, p. 206]”
Well said! All students need to keep this in mind.
If you find it intuitive, you find it correct and the brain doesn't change your neural structure; because why would it? No neural structure change -> you didn't learn anything.
If you find it difficult, do mistakes, can't get the correct answer -> your brain start to change its neural structure to be able to resolve these problems -> you learn.
>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.
[0] http://calculusmadeeasy.org/1.html
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. The...
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.
http://www.feynmanlectures.caltech.edu/I_toc.html
It is... very average looking. Did something happen here?
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
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.
Ouch! Boas is maybe not as inspiring as Feynman. But when you see a copy on someone's bookshelf. It tends to be just as dog-eared and spine-cracked as Surely You're Joking
Another resource I just thought of. While not a textbook per se. Math competition problems from previous years can be very stimulating ;)
https://www.maths.cam.ac.uk/undergrad/pastpapers/past-ia-ib-...
https://kskedlaya.org/putnam-archive/
But, [2] turned out to be kind of a dud. It was not really fun to browse, and I wasn't sure who it was directed to. The articles that I sampled read like they were intended for academic applied math folks, rather than introductions for interested outsiders. It's a huge book, so YMMV, and has been very well-reviewed by high-profile and well-qualified academics (like Steven Strogatz) but I spent a couple evenings with the book and could not recommend.
In any event, it's not like Feynmann's lectures! It's an encyclopedia.
TLDR: "it was good for someone, but it was not the book I wanted".
(PS: recommending the CRC tables is an odd thing, this is also nothing like Feymann's lectures)
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.
[1] https://www.math.columbia.edu/~woit/wordpress/?p=154
Feynman's Lectures are much more complete in that sense, even though as other comments on this thread note, the reader may not be able to use the learning to solve practical problems without going beyond Feynman's lectures.
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...
The book is quite good. It is written like a "Manga" book and hence has tons of drawings to help develop intuition for the concepts. It is written by a group of ordinary people with help from Scientists (a quirky club named Transnational College of Lex from Japan - https://en.wikipedia.org/wiki/Hippo_Family_Club ) and thus is very accessible. Highly recommended for High school students and above.
Note that the same group has also published two other books in the same vein; a) What is Quantum Mechanics b) What is DNA; both of which are also highly recommended.
For actually learning GR, I prefer Wald.