Ask HN: Are there books for mathematics like Feynman's lectures on physics?

788 points by pirate_is_back ↗ HN
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

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Calculus by Michael Spivak
seconded. this book is fantastic!
Prelude to Mathematics by W.W. Sawyer was written to give students an overview of modern math concepts beyond algebra. Topics include non-euclidian geometry, linear algebra, projective geometry and group theory. Again, for someone with an understanding of algebra. I enjoyed it and think it's in the spirit of what you're looking for.

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.

I came to add add Introduction to Graph Theory and found it here. I second it! A nice book and I appreciate it's funny introduction as a book designed for liberal arts majors injured by the pedagogy of institutional mathematics (paraphrasing).

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.

Innumerate is infrequently used but it's analogous to calling someone illiterate. Someone that can't do their sums.

Usually saying someone isn't "mathematically mature" ie able to read and use proofs is what you would want to say.

You can do a lot of graph theory without being able to sum integers. No problem. :)
I know you asked for books, but I have to mention the videos of Grant Sanderson (3Blue1Brown) [1].

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

I guess he's pretty popular around here, but if anyone still hasn't seen the channel, please check it out. if I could, I'd vote for this guy to get a nobel prize. seriously. education is critical to our future, and efforts like set an example that I personally consider to be invaluable on the long run. what I mean is that it's easy to recommend videos of people like this, but since it's "just a youtuber", we often fail to reflect more deeply about what's —at least in my opinion— an amazing contribution to humanity.
I agree that he's a great educator. I wonder if a book by him would work as well, though. A lot of his explanations depend on the animations he makes.

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.

I liked Steven Strogatz's books Sync and Infinite Powers.
Seconded, and so is his Nonlinear dynamics and Chaos
I bought Prelude to Mathematics when I was 12, it was the first maths book I bought. That was a very long time ago! I thought it was very good, I don't know of another book quite like it. He produced some other good books as well.
Rudin
Rudin (and baby Rudin) don't really fit the request here, a Feynman-like approach.
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 with our newfound definitions).

It's also a beautiful book.
Yeah, great layout and typesetting. I have a copy mostly for this reason.
Second this. All Spivak's books are gems.
Apostols Calculus was huge for me. Probably the main reason I majored in math.
One of the biggest conceptual moments in my life was taking an intro class to complex analysis in high school (not through my school itself, this was at a nearby university that runs a weekend program for interested applicants). It's probably not a novel intellectual framework for anyone who's spent time learning math from a theoretical point of view, but the guy who taught it opened with an (admittedly ahistorical, but that wasn't the point) tour of the development of numbers, starting with the intuitive case of counting, moving on to algebra and the question of what type of number could satisfy an equation like x + 5 = 2, and so on. He wasn't taking any philosophical position re the question of discovery vs. invention, but merely inviting us to consider the particular case of operations over a set not being algebraically closed and what it looks like to extend that set to support algebraic closure in a rigorously defensible way. Reading Spivak's Calculus, with the way that it starts off its path of inquiry by showing that, beginning with the basis of a totally-ordered field, the axioms at hand don't suffice to demonstrate the existence of e.g. a number that satisfies x^2 = 2, made me feel right at home again. It's like a detective story.
I recommend “What Is Mathematics? An Elementary Approach to Ideas and Methods” by Courant and Robbins. It’s a classic.
Graham, Knuth, and Patashnik's Concrete Mathematics has the same exploratory and informal tone that the Feynman lectures have. It's more about computational math than abstract math.
"Mathematics Can Be Fun" by Yakov Perelman

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...)

ALL books by Yakov Perelman are a must read for every educated person. "Science popularization" at its finest! How i wish modern "authors" wrote books like these in their area of specializations.

Some of his Books;

* Physics for Entertainment Vols I & II

* Algebra for Fun

* Figures for Fun

When I was studying physics, I found Feynman’s books in the library, read them all, and had the feeling I understand everything!

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.

Very much this. I'd recommend using these books as adjunct material. I found them indispensable as an undergrad when I was struggling to shift from a mathematician's rigor-and-proof perspective to a physicist's intuition-and-approximation perspective. However, I don't think I could have come close to passing my QM or E&M courses, even with a mathematical background that was stronger than most of my peers, if I'd only used Feynman to learn the physics.
Exactly, Feynman is a seductive writer, and it is a shock how little you can immediately apply after "understanding" a section. Long ago, they used Feynman's books for my introduction to physics, and it was only after struggling with a problem set that we "knew" the material.

http://www.feynmanlectures.caltech.edu/info/exercises.html

Some references to good collections of mathematics problems and solutions would be great for self-study.

Yeah, he was like that in person as well. When I was an undergraduate, he would "teach" a seminar on Tuesday afternoons called "PhysX" where you could go and ask any question you wanted. He'd go up to the blackboard and extemporaneously write things down and explain things in such a way that thought you really understood. But when you got back to your room and tried to replicate the chain of reasoning, there were always pieces missing or leaps that you now couldn't make. (It felt like the Star Trek Episode, "Spock's Brain".)

But we all took that as an indication of our own lack of knowledge and intuition and would just try harder.

I had a teacher in the University who took some courses taught by Feynman and had the same experience. He even tried to record some of the lectures, but the result was similar. While he was listening, he felt like everything was very clear. But as soon as he stopped the tape, nothing made sense anymore.
The funny thing is, if I remember his autobiographies correctly, he wouldn't let anyone else get away with that. I think he said that if he didn't understand something, he would always ask about it.
I also had a prof who took Feynman’s classes. He told us Feynman was a tough grader - he didn’t give partial credits, you either solved a problem or you didn’t.
Bob Ross -- the painter analogy comes to mind.

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.

I daresay you would have the same experience with any other book.

If you just read a book and don't work through problems yourself, you simply don't learn enough to do it yourself.

I agree in general, but Feynman’s books are different. I usually read textbooks cover to cover; then go on to solve the hardest problems I can find at that level (sometimes they are from the book, sometimes from different sources).

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.

Physics is a bit like that anyway though, you can't learn it by reading or listening to anyone, you have to solve things yourself from scratch, usually multiple times.

Reading good books/having good lecturers certainly helps, but there is no way to replace the work.

> 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.

This is the real weakness of most MOOCs. Most people are not disciplined enough to do nearly enough of the real work themselves without some sort of outside incentive. At least not until they have several years of practice.
I think that's fair for trying to master physics or to become a physicist. But if you're looking for an intuitive understanding of the concepts or relearning it, which seems to be what the OP is looking for, Feynman-style seems like the optimal type of book.

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 Caltech (home of Feynman) in the 1970s, his books were not used as the main texts in physics classes, but as supplements.
Came here to say this. The professors I've learned the most from were the ones who weren't awesome at explaining things. I didn't understand them and had to struggle through the material. That struggle made the material stick more. I think ideally you want a professor who's 80% good at explaining things, but leaves enough gaps and says things just confusingly enough that you have to engage your brain. Feynman was too clear, which allowed my brain to coast.

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.

disagree here. you're judging society on the most naive ranking a student would give. in an ideal world you can optimize for the perfect amount of understanding and imperfect leaps for each student to best address long term understanding and success.
You're right that in principle it's possible. But say you were a great teacher, and knew that clear teaching was worse than imperfect teaching. Could you actually make your lectures less clear on purpose?
Personally I thought the Feynman lectures were ok but with room for improvement.

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 a very good start. From there I think the most productive thing anyone could do is make a very thorough study of Classical Mechanics. People underestimate how much a thorough knowledge will help them. Start with an easy book and work your way up. Goldstein and Laundau are excellent intermediate level choices. For a beginner I think Jakob Schwichtenbergs "No Nonsense Classical Mechanics" could work or Leonard Susskind's "Theoretical Minimum Classical Mechanics" . Personally, I really liked Jakob's book. You'll need a friend or a study group online to help you when you get stuck. Classical mechanics is very serious physics and I regard a thorough foundation in say Hamiltonian Mechanics as a solid achievement. a sure sign someone could go on and learn E&M, Statistical mechanics, Quantum mechanics, Relativity and Gauge theories. For a semi advanced book if you know some advanced maths try Spivak's Classical Mechanics and anything by V. Arnold.
People want intuitive explanations because it seems the easiest. It is, that's the problem. Learning is supposed to be painful. You have to get your hands dirty.

It's easier to feel that you know something than to actually know it.

Learning requires a lots of false starts, traps, etc. Once one has mastered, he/she can provide an intuitive explanation (a path through the wild forest that learning is).
After doing the hard learning, you can lecture your intuitive mental model you have. But it's difficult to install that mental model into a beginner's mind. Often the intuitions are illusory mnemonics for the deeper understanding, which if you never learned in the first place would just point to nothing. You have to do the hard learning to arrive at the "intuitive" mental model.
While i see where you are coming from; i feel that you are putting the cart before the horse. While Intuition by itself is not enough, it should absolutely be the first thing you should focus on before doing the hard work through rigor and formalisms. The former can be "grasped" while the latter needs "practice and applications". This is how Science itself developed (a good example is Faraday vs. Maxwell's approaches). Intuition/Rigor are analogous (in a certain sense) to Theory/Practice. You need both, each amplifying the other's effects at various stages.

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]”

"Hard work" is not just rigor and formalism. Hard work is going through a lot of intuitive models that turn out to be false. If seen this way, the comment you respond to, makes sense.
> It's easier to feel that you know something than to actually know it.

Well said! All students need to keep this in mind.

My understanding is that you can't learn without going through mistakes first.

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.

Reposting a comment I wrote a while ago, and may be appealing given you're learning 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.

"Calculus Made Easy" by Silvanus P. Thompson (1910). It is availably freely online via the Gutenberg project and many other forms too. Chapter 1 is probably the best mathematics chapter I have ever read [0]. In two paragraphs, it beats most other calculus books.

[0] http://calculusmadeeasy.org/1.html

Came here to tell this. IIRC, a newer edition of the book by Martin Gardner was published in 1998 with some notable updates. I read the newer one when I was in high school and as with all other Martin Gardner books, this was an absolute gem to learn and understand calculus.
Came to say the same. CMD+F didn't disappoint.
The closest I'm aware of is What Is Mathematics? by Courant, Robbins, and Stewart -- starts off developing the natural numbers, goes onto number theory, analysis, complex numbers, set theory, projective geometry, non-euclidean geometry, topology, calculus, optimization, and some chapters on recent developments (as of its republishing in 1996, book was originally published in 1941).

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...

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Mathematics is typically approached in a different way than physics. But there are some books that offer a similar perspective to what Feynman tried to achieve, IMHO. I would recommend to look at works by John Stillwell, for example 'Elements of mathematics' or 'Mathematics and its history'.

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.

Here's a question about the Feynman Lectures. I remember looking at the digitized text a few years ago, perhaps right after they made the digital copy freely available, and thinking the typesetting was pretty great. Looking at it today:

http://www.feynmanlectures.caltech.edu/I_toc.html

It is... very average looking. Did something happen here?

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 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

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”...

This. I think everyone in this topic is missing the point of what makes the Feynman Lectures unique.
I'd use 3 words to describe Feynman's style: clarity, accessibility, and fascination.
This description made me think of the "Mathologer" channel on YouTube.
It's easy to gain physical intuition because you can often explain one physical phenomenon in terms of another physical phenomenon that you have much more real life experience with.

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.

Good list! I'd also add Mary Boas' Mathematical Methods in the Physical Sciences. A standard textbook for incoming students across disciplines and very accessible
We used that book for a course and I found it among my less favourite ones. its been a few years since I used it, but I remember it shallow and uninspiring. not trying to start an argument here, maybe just an outlier opinion since this seems a standard textbook.
It was one of my course books as well, but I think it's aimed at the American market and style of learning/presentation. I much preferred Stroud's "Engineering Mathematics" which was a course book for engineers at my university (I studied physics).
I purchased [2], having enjoyed Nick Higham's other book (a treatise on matrix computations), and knowing how well-received [1] was.

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)

The Princeton Companions to Mathematics and Applied Mathematics are beautiful to leaf through at the library. They're also hardcore heavy-weight (physically) and unlikely to be read twice, so don't buy them.

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.

Penrose's 'Road to Reality' [1] is a kind primer on where the math comes from, as it applies to physics. Kind of a philosophical walkthrough of how math applies to physics. It is nowhere near as concise as Feynman's lectures, but it does complement them pretty well, while getting more into the math, and why the math is needed to describe various aspects of physical reality.

[1] https://www.math.columbia.edu/~woit/wordpress/?p=154

As it happens I bought both Road to Reality and Lectures on Physics at the same time, about 14 year ago. I read and re-read Lectures on Physics but I was never able to finish Road to Reality. I have kept my aging copy and hope to one day get through it, but it's a MUCH more difficult read, at least in my opinion.
It definitely is quite difficult, but it's also very inspiring in finding various topics to read more. I must have bought at least 20 books from the vast bibliography in the appendix of the book.
I have a love-hate relationship with that book. It elegantly shows how to think about the topics, but it also left me with many unanswered questions to which I could never find answers with any amount of Internet searching (even though those answers are known to the humankind, buried in a set of complex topic-specific books).

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.

My recommendation below is not the equivalent of a Feynman's series for math, but one that is pegged much lower, for someone interested in basic remedial math.

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...

And it's only $975!
Oh No! That is an unfortunate link from Amazon where somebody is trying to hustle money. You don't need the 2nd edition since there is no change from the 1st which you can get for $10+ from many sites.

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.

Feynman's PhD advisor, John Wheeler, together with Charles Misner and Kip Thorns, wrote a textbook on general relativity called Gravitation. It's gigantic, 1200 pages, and its tone is similar to the Feynman lectures. And it is at least partially a math textbook, as it includes a fairly complete introduction to Riemannian geometry.
I think even among professionals in general relativity, MTW has developed a reputation as being too dense and old-fashioned, certainly so for a beginner. If you're first approaching GR, Sean Carroll's notes or books are much more approachable.
True, but the question was about books that are like the Feynman Lectures. MTW has similar tone and breadth, and a similar reputation for being useless for learning.

For actually learning GR, I prefer Wald.