I’d add V. Arnold’s books on Differential Equations and Geometry to that list.

Wikipedia summarizes it nicely: “His views on education were particularly anti-Bourbaki.” 🙂

I love this one, it’s from 1962 but already had a chapter on

ELECTRONIC COMPUTING MACHINES

§1. Purposes and Basic Principles of the Operation of Electronic Computers

§2. Programming and Coding for High-Speed Electronic Machines

§3. Technical Principles of the Various Units of a High-Speed Computing Machine

§4. Prospects for the Development and Use of Electronic Computing Machines

There is some quality in Russian math books that western authors seem to lack.

Somebody from Russia/Former Soviet Countries from Eastern Europe ABSOLUTELY NEEDS to setup a publishing company (eg. Dover Publications) to bring all of their Science/Engineering books back into print. They will do very well in today’s education market where the emphasis seems to have shifted to huge tomes/useless multiple editions/pretty colouring etc. rather than succinctly presenting the knowledge itself. The Mir (and other) publishers books were huge in many countries of the world and are remembered fondly to this day.

I absolutely agree about the quality of Russian math books. “Bronshtein and Semendyayev” and “Abramowitz and Stegun” come to my mind.

I doubt that a publishing company bringing back those titles into print would be very successful. One thing is that many of these titles are mostly of historical value. Who needs a book with mathematical tables nowadays? Many of the books are still available as used books too.

The most compelling argument though is that they are easily available on libgen (like this very post proves). So in a sense the publisher you wish for already exists, just not in a the form you probably thought of.

EDIT: Oops, I just learned that Milton Abramowitz and Irene Stegun are actually Americans.

I disagree with you on the publishing front. Mathematics is fundamental and timeless (what do you mean “many of these titles are mostly of historical value”?) They may need some trivial editing (though i would much prefer that they be published as they were with a note explaining the historical aspects) but otherwise they were information dense and succinct with an eye to Applications. They were all excellent across the board. They were directly responsible for educating a lot of poor people in many countries due to their very low cost and affordability. I would say this was one of the biggest successes of the Soviet ideology i.e. the education of the masses in Science & Technology fields. Current day Russians/Eastern Europeans/Central Asians can justifiably be very proud of this part of their History.

Much of “modern” textbooks are full of excessive verbiage obscuring the essentials, “pretty printing” disguised as “easy comprehension” and a racket for the publishers to make money. Why in the world do i need so many editions of books containing Mathematics which has not changed in centuries? Why do they cost an arm and a leg? Education is as fundamental as Health services and both should be affordable in service of the population.

So again, somebody setup a publishing company (eg. Dover Publications) and bring ALL the forgotten books from the Soviet era back into print 🙂

> Mathematics is fundamental and timeless

Absolutely and you certainly have a point with what you wrote. May opinion is more along the lines of: “the content is still as valuable as it ever was but the presentation is not.”

Take one of the examples I mentioned. Abramowitz and Stegun is a collection of mathematical tables. If you needed to calculate the sine of a value, would you rip out your chuffed copy of Abramowitz and Stegun or would you use your calculator?
Even for the more obscure tables there is probably nothing in the book that isn’t in Mathematica. If I really needed to look into the book for some reason I would be too lazy find my copy, given that online versions[1] as well as extended and improved versions[2] are just a few mouse clicks away.

Now, a book of mathematical tables is like an extreme example but I still feel the same sentiment for all my old math books. Why bother with a physical copy if I have a searchable online version right at my fingertips? When i comes to the books from the Soviet era I guess libgen has them all and I think most people would not buy a physical copy anyway.

You are very wrong here. It is the very presentation in those books viz; succinct and concise, no frills approach, high information density and with an eye to applications which makes them so valuable today. It is the best way of Science teaching distilled from the brains of a whole lot of smart people.

I am not sure why you are fixated on one book of tables. It is irrelevant in the broader scheme of things. For example none of the Mir books that i have, have anything to do with pre-calculated tables other than a few appendices.

There are a huge swath of students across the world who do not have the same access to technology as we do. Printed books are still the norm amongst the majority of students in the world. Printed books will also outlast any Digital media presentation of books due to its simplicity and robustness i.e no problems like DRM, unreadable extinct formats, availability of good ereaders, health aspects, etc (there is a whole lot more i can elaborate here).

Finally, and most important, research is beginning to show that we retain/understand less when using ebooks/ereaders than when we read a printed book. This is very much true of technical books (borne out by my own experience) where you need concentrated attention with body and mind. For example, we intuitively jump back and forth across pages, use our fingers as book marks, subconsciously create spatial maps of what we are reading etc. all of which have no analogues with current day ereaders. Cognitive Science is still trying to figure out how best to use modern technology. So don’t throw away your old Maths books just yet 🙂

I remember when one of my professors told me how AVL trees were invented by mathematicians in the Soviet Union. It was in a paper titled “An algorithm for organizing information”. It was a side project for them, and it’s humbling to hear how simple the idea was to geniuses.

Hey, that’s a neat fact! I did know that AVL stood for “Adelson-Velskii and Landis” but not that they were Soviet-era Mathematicians.

I can second this. An amazing volume. If you want more details, I’d stick with the Soviet math school, which was amazing in it’s pedagogical soundness and user friendly texts. Anything from the EMS — Encyclopedia of Mathematical Sciences — is worth picking up.

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.

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.

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.

Isn’t that true for any physics or math texts? You need to do problems. That’s why there were exercise books to accompany Feynman’s lecture.

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.

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

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.

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.

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.

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.

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.

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]”

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

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

At Caltech (home of Feynman) in the 1970s, his books were not used as the main texts in physics classes, but as supplements.

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”…

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.

This. I think everyone in this topic is missing the point of what makes the Feynman Lectures unique.

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)

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.

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.

“Nonlinear Dynamics and Chaos – Steven Strogatz, Cornell University” is available[0] on youtube as a series of 25 lectures. (From 2014.)

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.”

Just a note, “Visual Complex Analysis” is a terrible book to learn complex analysis. Proofs are very iffy, akin to sketches of proofs. With that said, it is a excellent supplement to another std. complex analysis textbook.

Is it just the normal difference between e.g. an engineer’s approach to real analysis and a mathematicians (but complex analysis swapped in), or something else?

I can think of a lot of fields where a decent grasp of complex analysis concepts would be very helpful even without being able to do rigorous proofs.

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).

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.

Apostols Calculus was huge for me. Probably the main reason I majored in math.

“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.

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.

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.

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.

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.

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.

I would recommend “The Number Devil” [1] for children (from 11 on) and adults alike.

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.

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.

I think that those who are down-voting people suggesting Rudin are missing the forest for the trees. The level is not as low as Feynman’s lectures, but if you want a text that has the hallmarks of a true master at work, Rudin fits the bill. I personally always felt that what made Feynman’s lectures what they are were, was that the man actually understood the subjects on such a deep level that his mind was able to hone in on those little simple thought experiments and ways of looking at things that gave you huge insight into what he was lecturing on. Rudin comes at analysis with the same level of understanding and that’s why Real & Complex and Baby Rudin are still the gold standard for analysis texts. It is impossible to convey the sense of elegance that mathematicians speak about until you’ve seen it and been blown away by it. Rudin will do that and leave you speechless, he’s the master. I can tell you that my first encounter with Real & Complex leaving me thinking that Rudin was on a fundamentally higher plane than any author I had read before, it was like watching a magician. If you’re looking for a real Feynman in mathematics, IMHO Rudin is your man.

When I did analysis at university, I read Rudin instead of the recommended text. When the exam came around, I tried to generalise all the problems on the exam and gave answers which did not impress the person marking it, as they looked nothing like the standard solutions. As a result, I very nearly failed analysis. I wasn’t particularly upset, as I used to consider my exam grade as a reflection of how well the course was taught, rather than a grading of my own comprehension. This is not a level of arrogance I would recommend to students looking to pass their exams.

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.

There are some really good suggestions here. I would like to add that learning calculus properly is of the utmost importance. It’s no exaggeration when I say that this is unquestionably the single best, most profitable action you can take. Any one of these books can change your life.

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”

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

All the books cited here by Arnold, Spivak, and Lanczos are extraordinarily good.

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.

If you’re looking for a book that’s both easy and stimulating to read, but that discusses a lot of mathematics in reasonable detail, I highly recommend the novelist David Foster Wallace’s Everything and More: A Compact History of Infinity.

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.

OT: Interesting blog you’ve got there.

I’ve seen Donella Meadows’ Thinking in Systems book recommended here a few times before, but your review really pulled the trigger for me, so thanks!

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.

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’m currently reading “Mathematics: From the Birth of Numbers
by Jan Gullberg” with my 3 sons (16, 12 and 10), and by taking it slow (one minor number per day) with lots of work together its helping build things up for them from first principles.

I read it myself years ago and it was a great and entertaining way to fill in the gaps from my meager math education.

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.

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.

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.

Did you mean Paul Lockhart? His book, “Measurement” is also good. He starts with some simple concepts, like length, and goes on to develop Geometry and then Calculus while encouraging the reader to consider various questions along the way.

Yes, Lockhart! This is why you shouldn’t post with jet lag.

He uses a similar approach in “Arithmatic”. He begins the book by describing different base number systems used throughout human history (the way different civilizations did “counting”). He does that in order to argue that a number itself shouldn’t be confused with its representation.

I might check out “Measurement” next! Thanks for the recommendation.

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.

More of a general dive into tons of different topics can be found in: “What Is Mathematics? An Elementary Approach to Ideas and Methods” by Richard Courant & Herbert Robbins

For some reason, Ivar Ekeland’s books are not that well known. I myself came across his book Mathematics and the Unexpected by random chance while browsing the Maths Section in the Library. I was hooked and set about getting/reading all three of his “popular maths” books; a) Mathematics and the unexpected b) The broken dice, and other mathematical tales of chance c) The best of all possible worlds: Mathematics and destiny.

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.

For chemistry, check out “General Chemistry” by Linus Pauling. Pauling had the same passion for chemistry that Feynman had for physics. He wrote General Chemistry with that passion, and it shows. It’s a really engaging introduction to chemistry. (Replete with exercises).

I recommend “What Is Mathematics? An Elementary Approach to Ideas and Methods” by Courant and Robbins. It’s a classic.

It’s extremely difficult to write a mathematics textbook in the intuitive style. There are some reasons for this.

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.

I am also been interested for the longest time – even with a degree in Mechanical Engineering – to understand mathematic like Calculus or the equations you see in neural networks papers.

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.

Try, Try again until you “get” it. Seriously, Maths should be read by everybody in the spirit of playing a game i.e. with no pressure and in a relaxed frame of mind. There are Concepts, Objects and various Rules interconnecting them. And somehow these “games” turn out to be useful in explaining the real world.

Get some school/college textbooks (high school level onwards) and some “popular maths” books and start reading. Once something catches your fancy you can dive deep as needed. You are studying to gain “understanding” and not to get through an exam or prove something to somebody.

I believe mathematics as a field is really suffering because there is not much in the way of “Feynman”-style books. But people like John Baez and John Conway have countered this trend somewhat. You should definitely try reading anything by these two. Conway does tend to be a bit too brief at times.

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.

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.

To paraphrase my supervisor “Mathematicians don’t read, we write.”

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.

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:

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

Always. Robin Hartshorne’s “Geometry: Euclid and Beyond” is the best book of its kind. It does Euclidean geometry with Hilbert’s axioms and cleans up some of the loose ends of Euclid’s classical treatment. Hartshorns book also covered the 5th postulate very thoroughly and non-Euclidean geometry. Hyperbolic geometry is treated axiomatically. He also has a nice treatment of axiomatic projective geometry which you can download for free. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.475…

“Number: The Language of Science: A Critical Survey Written for the Cultured Non-Mathematician” really opened the field up to me and did away with some of the misconceptions I had incurred in school.

I’m currently learning group theory, matrices, and graph theory.

There is a second problem too – many really good books available only as hardcover volumes. So if you have a Kindle/iPad/whatever and want to save the trees and your own precious living place, you have to limit the reading to PDF and ePub.

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. 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

Calculus And Analytic Geometry
by Ross L. Finney and George B. Thomas
is a good introduction to basic calculus. I understand you want something covering everything in math. Courant is the best one and others have written about it a lot.

Yes that’s a very good one. So many editions – with Einstein raving about its skill in bringing Math to the masses in a preface. Now it’s almost unknown.

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.

Are there any equivalent of these books for kids? something 9-10 year olds can read and get interested in Math (or not afraid of it).

“Mathematical Omnibus: Thirty Lectures on Classic Mathematics ”
Very accessible and covering a broad range of topics.

Rudin (and baby Rudin) don’t really fit the request here, a Feynman-like approach.