I'm slowly reading through Feynman's Lectures on Physics and I find myself wondering, is there an analogous book (or books) for math?
Mathematical equivalent of Feynman's Lectures on Physics?
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11By this, do you mean a good approach to physics given through sweeping motions, appeals to intuition, and a lack of focus on historical precedence in favor of developing patterns for modern thought? If that's the case, I recommend Feynman's Lectures on Physics. Or do you mean for calculus books, which are sort of the bread and butter of math? If that's the case, then no. I don't. Really though, what do you mean? – 2011-09-06
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2Although I'm sure some readers will be familiar with Feynman's Lectures, perhaps you could elaborate on what it is about them that you're searching for. Easy to follow? Witty? Deep?? – 2011-09-06
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3By this, I mean a good approach to mathematics given through sweeping motions, appeals to intuition and a lack of focus on historical precedence in favor of developing patterns of modern thought. :-) Seriously, I thought that was very well-said. – 2011-09-06
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0What strikes me about Lectures on Physics is its conceptual orientation. It is easy to read in that I don't need a pen and paper at hand to work through details, but I find it necessary to regularly stop and contemplate his ideas and examples. Content-wise, it is not at the level of a calculus book. My impression is that it covers much of what would constitute an undergraduate education in physics. – 2011-09-06
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2There will be much praise for the people who can write a three-volume series of books which covers ‘an undergraduate education’ in mathematics, let alone ‘modern mathematics’. (The same goes for physics really. A lot has happened in the 50-or-so years since the Feynman lectures, but undergraduate mathematics barely even reaches the early 20th century!) – 2011-09-06
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4I didn't mean to imply The Lectures were equivalent to an undergraduate education in physics, but rather, it covered a similar amount of material, had a similar breadth. This impression might be quite wrong. Obviously it isn't a replacement for such an education, nor does it cover the material at the same level of depth. Of course, I'd be happy to find a book that covered modern mathematics through, say, 1950. – 2011-09-06
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0@Zhen Lin "..but undergraduate mathematics barely even reaches the early 20th century!" I think its not meant to be! It is supposed to build strong conceptual abilities and problem solving skills rather than feed us with as much mathematics as we can take in 3 or 4 years.Is'nt it? – 2011-09-06
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0@Dinesh: That's a very applied point of view. I think mathematics is a subject worthy of study in its own right, and I feel that at the end of a good undergraduate course students should have a decent chance of understanding the latest research. This, mutatis mutandis, was the motivation behind Feynman's lectures. – 2011-09-06
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0@Zhen Lin : I certainly think that mathematics is worthy of study in its own right. However, I think it is wildly unrealistic for a student with an undergraduate education to understand most of the latest research. I'm a professor and have been doing this a long time, but despite going to many a colloquia I rarely understand the latest research in areas other than those in which I specialize. – 2011-09-06
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0@Adam: I understand this as well, having finished three years of full-time mathematics education but still finding most things impenetrable. But I think this is something that could be changed—which is what I meant by ‘should’—after all, four years is a long time and one imagines that it should be possible to develop at least one specialism in that time. – 2011-09-06
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1@Zhen Lin : What evidence do you have that this could be changed? Math is huge and it takes an enormous amount of time to get to most of its frontiers. Do you have some kind of magic wand that will make it easier? In a related question, why the focus on understanding the "latest research"? Who cares when a result was proved, as long as it is beautiful and gives you pleasure? When I want to learn math for fun (as opposed to doing research), I usually spend my time reading papers from the '50's and '60's... – 2011-09-06
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7http://www.physicsforums.com/archive/index.php/t-199223.html has a detailed discussion on the same topic – 2011-09-06
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0@Adam: I largely agree, but (a) I am planning to go into research, and (b) the field I'm interested in—topos theory—didn't even really _exist_ until the 1960s. So _I_ necessarily have to catch up to the latest mathematics. As for evidence, no, I don't have much evidence to suggest that it is possible, beyond anecdotes of what _some_ undergraduates have achieved with their own efforts. (Some of the top users here on MSE and on MathOverflow are still undergraduates!) – 2011-09-06
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1@Zhen: Dear Zhen, This discussion is a litle off-topic, but let me just highlight your use of the expression *their own efforts*. Speaking as someone who has taught undergraduates and graduate students, some of them very strong, for many years, I can say that it is not realistic, as an institutional goal, to reach the frontiers of research in mathematics in an undergraduate education. (I presume the same is true of physics, byt the way.) However, nothing stops you from trying to learn more mathematics yourself. The number of students who are ready and able to teach themselves mathematics – 2011-09-06
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0... at a significantly faster rate than a standard undergraduate curriculum is small (too small to design a university curriculum around, I think), but I agree that those who are in this group should be encouraged. Regards, – 2011-09-06
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0+1 to Matt E. One should point out to Zhen Lin that the top undergrad users on MSE and MathOverflow (eg Qiachu and Akhil) are not at all normal undergraduates. At this point, they are basically graduate students in all but name. Certainly when I was their age I was not nearly as advanced, and when I started graduate school (about 15 years ago) I was completely ignorant of "current research". And most of my peers (at a top 5 department) were similar. – 2011-09-06
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0I think that the most significant features of Feynman's treatment of the subject are his unconventional approach (RPF's analyses are often unique, where other texts are similar to each other) and clarity of explanation from first principles. I am very fond of mathematics, and am likewise looking for a mathematical work with a similar level of non-conventionality in topic choice in addition to treatment. – 2011-09-07
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0As an aside, if you like Feynman's "Lectures on Physics", I'd also highly recommend QED (again on Physics) which is a small but remarkable book. On a different subject is his "Lectures on Computation", which is another very unconventional treatment, also highly recommended. – 2011-09-07
17 Answers
The Princeton Companion seems to me to be an attempt to achieve a similar mixture of depth, accuracy, content, motivation, and context. However, because math is a different kind of subject, this is a very different kind of book.
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2I haven't completely read either of these two books (I doubt there's many people that have), but I've dipped into both of them quite a lot, and my impression is that Feynman is more "elementary" in some ways than Princeton. – 2011-09-08
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0This is available in Taiwan in English and called "The Mathematics Book". It's really good. The exposition is reorganized to be chronological instead of thematic. – 2013-02-20
I tend to agree with Adam-the sheer scale and difficulty level of most mathematics beyond the level of basic calculus would make a book like this almost impossible to write. I think the closest anyone's ever come to writing the kind of book you're suggesting is Kolomogrov, Alexandrov and Laverentev's Mathematics:Its Content, Methods And Meaning. This 3 volume overview-originally in Russian-attempts to give an overview of all mathematical fields for students without much background-only some high school algebra,geometry and calculus is needed. Admittedly,though-in the Soviet Union in the 1960's, most of these students had stronger backgrounds then most of today's undergraduates in America! It's currently available in Dover paperback-I think you'll find it worth a serious look.
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7One reviewer at Amazon explicitly compares this book to the Feynman lectures. – 2011-09-06
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0Thanks for the reply! I wasn't aware of this book. I will definitely take a look. – 2011-09-06
As for me, Vladimir Arnold's writing style is sometimes similar to Feynman's style. For instance, Arnold's Ordinary Differential Equations may be appealing to those, who appreciate Feynman's lectures. I can also recommend the following books:
- V. I. Arnold, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics)
- V. I. Arnold, Lectures on Partial Differential Equations
- V. I. Arnold, A. Avez, Ergodic Problems of Classical Mechanics (Advanced Book Classics)
Tristan Needham's Visual Complex Analysis has sometimes been compared to Feynman's Lectures.
"...it is comparable with Feynman's lectures in Physics. At every point it asks 'why' and finds a beautiful visual answer. ...I believe that this book can make every student understand and enjoy complex analysis. If its methods could be applied in teaching more generally, mathematics would become a flourishing subject" -- NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY
It's much more specific in scope then Feynman. But it remains the best written math text book I've read.
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0+!. I don't agree that it's comparable to Feynman's lectures. But Needham's book is deservedly a classic;it's a book both mathematicians and physicists can benefit greatly from working through. – 2011-12-18
Joe Harris's textbooks often remind me of Feynman's style, in that they frequently omit details, and may cause the casual reader to think he knows more than he does, but do a wonderful job conveying the most important points of the theory from an expert's perspective. I am thinking here of Representation Theory, Moduli of Curves and Algebraic Geometry: A First Course. (Since this is rather mixed praise, let me add that Harris and Feynman are among my favorite authors; just that the reader needs to be vigilant about filling in the gaps.)
However, none of these books attempts anything like an overview of all of math.
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0I've never seen this description for Harris's books before. This makes me more inclined to read them! – 2011-09-06
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0Thanks for the reply, David! Since some are claiming that the sheer scope of a "Lectures on Mathematics" would make it infeasible, I was thinking of asking the related question of Feynman-style exposition for particular areas of math. – 2011-09-06
I am surprised that no one has mentioned the book "What is Mathematics?" by Richard Courant.
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1No votes for Richard Courant? Do I have to say more about this book? Now I am puzzled. – 2011-11-16
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0I just started reading this work and I think you don't have to say more but just say Albert Einstein's words on this piece - "A lucid representation of the fundamental concepts and methods of the whole field of mathematics". – 2015-05-18
Although it is quite expensive to buy (since it is out of print), perhaps you could borrow MacLane's Mathematics: Form and Function from a library. I found it to be a beautiful overview of mathematics and interconnections between topics you may have seen.
First 16 chapters from Penrose's "Road to reality" could be quite close. He starts from fractions and goes to calculus on manifolds, group theory, complex analysis, Rieman geometry, Lie algebras, etc. All on ~350 pages! Would be great if he could spend the rest 700 pages on math alone. That could be something comparable to Feynman's lectures...
Another one (similar in style, not popularity) would be a small book by Lars Garding "Encounter with Mathematics" - quite advanced topics described in a pleasant manner - a nice relief after so popular dry "definition, theorem, proof" approach.
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0"Road to reality" is incredibly dense for someone just starting out. I recommend having knowledge in Calculus and Geometry to really get the breadth of what he is talking about. – 2014-06-21
This should really be a comment, but I don't have the reputation. I don't think any such book exists. In fact, I don't think that such a book is possible. There are two reasons for this.
Math, even at the undergraduate level, is much bigger than physics. It's not that it is impossible for anyone to understand everything that is taught to undergrads -- I certainly feel comfortable teaching any undergraduate-level course in my university. Rather, there are an enormous number of topics (calculus, geometry, linear algebra, abstract algebra, topology, partial differential equations, combinatorics, probability, etc) each of which has its own pattern of thought. At some point in your mathematical life, you will start to view them as one subject, but I don't think there is a way to teach undergraduates the foundational materials without having the topics fragment. A book that tried to describe all of them would be just too disjointed and incoherent.
You can learn a lot of physics without getting your hands dirty too much (via informal thought experiments, easy calculations, etc). This is basically the pattern in Feynmann's book -- it's all intuition and (almost) no detail. Math, however, doesn't work that way. You can't learn math without getting down to the details in a serious way. I guess you could tell a fun story, but the students would learn nothing from it.
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0Downvoted for pessimism, eh? Care to explain your vote? – 2011-09-06
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2As I've said before here, I like "What is Mathematics?" by Courant and Robbins and "Mathematics for the Million" by Hogben. Both of these are available in reasonably priced (less than US $20) paperback editions. – 2011-09-06
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3@marty, those are good books, but do they really do for Math what Feynman does for Physics? – 2011-09-06
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0Yes "What is mahematics" by Courant and Robbins can be the best fit. – 2011-09-06
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1"What is mathematics" is a fine book, but it not quite analogous to the Feynman lectures, which cover things in far more depth and insight. This is not a criticism of Courant and Robbins -- I don't think that the subject is such that you could write an analogous book. As an illustration, I know professors of physics who still occasionally flip through the Feynmann lectures to get inspired. I know no professional mathematicians who do the same for Courant-Robbins (or, for that matter, any other "popular" or "semi-popular" math books). – 2011-09-06
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2"Math, even at the undergraduate level, is much bigger than physics." I think you have a very narrow view of physics if you believe this to be true. Of course Feynmann doesn't explain all physics, this would be impossible... – 2011-09-06
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2-1 if I had the reputation here. `Math is much bigger than physics`. Physicists have to learn all those math fields (usually with less rigor) and apply them to (newtonian mechanics, special relativity, statistical mechanics, thermodynamics, electrodynamics, quantum mechanics, general relativity, optics, electronics, solid state physics, nuclear physics, quantum field theory, etc) each with its own pattern of thought. Its possibly true that math is typically more sequential than physics, and that its much easier to give a meaningful survey course of physics with insights than math. – 2011-09-06
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6@drjimbob - Obligatory XKCD reference: [Fields arranged by purity](http://xkcd.com/435/) – 2011-09-06
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4@Mark: Obligatory Abstruse Goose reference: [Prerequisites](http://abstrusegoose.com/272) (Keep clicking, there are 7 pages.) – 2011-09-06
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0Excellent post, especially point 1) – 2016-01-29
I don't know if these are in english, but in Spain (and Russia) there are a collection of books of the URRS editorial called "Lecciones de Matemática" (Math lessons) of a russian mathematician called V. Boss which covers a lot of advance and modern mathematics in a fresh way not going into the particular details, but more centering on the intuition about each mathematical topic with its structure -something like giving perspective about the mathematical topic-.
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0I looked around for this book but couldn't find anything. Do you know the author's full first name? – 2011-09-06
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0I am really unable to find the complete first name, really it does not appear in any source I have -even in the books it dos not appear-. – 2011-09-06
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2After more digging than I thought would be necessary, I finally uncovered [this page](http://www.easternowl.com/books/publishers/publisher19941.html?detailed=boss&x=0&y=0), which leads me to believe the author's name is [Valeriĭ Boss](http://www.easternowl.com/books/authors/author184615.html). – 2011-09-06
If you are looking for a treatment of a mathematical subject which is unconventional, then I'd recommend "Concrete Mathematics" by Graham, Knuth & Patashnik [ISBN-13: 978-0201558029] which is reasonably far-reaching, and perhaps "On Numbers and Games" by Conway [ISBN-13: 978-1568811277] which is specifically concerned with the theory of numbers and number-like entities. Both of these are very 'elementary', in the sense of operating from first principles.
If, on the other hand, you're looking for breadth-of-coverage, then consider the enormous dictionary "CRC Concise Encyclopedia of Mathematics" by Weisstein [ISBN-13: 978-1584883470] (also on-line http://mathworld.wolfram.com/; new shorter edition forthcoming), which is very good to "dip into", but would not really be suitable for cover-to-cover reading.
Another option, available both in paper and on-line is "NIST Handbook of Mathematical Functions" [ISBN-13: 978-0521192255] or "NIST Digital Library of Mathematical Functions" http://dlmf.nist.gov/. Obviously, this work is primarily concerned with various special functions, including those of trigonometry and combinatorics.
There are also various handbooks of mathematics, which arrange the material by topic, but with very little discussion. I do not have a strong view on which of these is best, as various options have their own merits, but perhaps that of Bronshtein & Semendyayev [ISBN-13: 978-3540621300] has relatively large breadth and contains much more explanatory text than is typical for a handbook.
This is a small start, but in my (somewhat unqualified estimation) fabulous. It's lecture notes of a real analysis course given by Fields Medal winner, Vaughan Jones. They are elegant, self-contained, and beautifully typed up by an anonymous student. Here is a link to download them:
I would recommend Stephen Hewson's "A Mathematical Bridge"; it is similar in tone to Feynman's lectures. While it's perhaps not as comprehensive (500 pages vs 1500), Hewson manages to cover a very impressive range of topics (all the "highlights" of an undergrad math course).
His explanations of the major concepts are the best I've read anywhere, and it does a good job of giving the reader a sense of what the major fields of mathematics are and how they relate to one another.
I.M Gelfand's books on trigonometry,algebra,functions and graphs and calculus of variations(and much more) are comparable to Feynman Lectures. He has even stated his effort to write a book like feynman's in the book's preface. I strongly recommend the books. You can search the books in amazon for user reviews.
I want to mention two resources:
Differential Geometry Reconstructed by Alan U. Kennington which is freely available and a video lecture series by Fredric Schuller called "Geometric Anatomy of Theoretical Physics".
'John Baez's lectures on Mathematics'. You can find something similar in the notes of John Baez on the blog by the same author. It's a super nice and laid back approach. But it is still arduous in some points. And it requires the reader a certain mathematical maturity. Analogous to the Feynman's Lectures on Physics ask.
Can't believe no one has (yet) mentioned George Polya's incomparable How to Solve It. It's very dissimilar to Feynman in that it covers very few (if any) specific subfields of mathematics - but it's very similar in that it attempts to give the student an understanding of how to approach the discipline, and to build their intuition so they can grapple with problems in the field.