This is a classic and exactly what you are seeking for. I think it was originally published in 1962.
https://www.goodreads.com/book/show/405880.Mathematics
https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
http://gen.lib.rus.ec/search.php?req=Mathematics+methods+mea...
Wikipedia summarizes it nicely: "His views on education were particularly anti-Bourbaki." :)
* Problems for Children from 5 to 15: http://jnsilva.ludicum.org/HMR13_14/Arnold_en.pdf
* A mathematical trivium pt. 1: http://www.physics.montana.edu/avorontsov/teaching/problemof...
* A mathematical trivium pt. 2: https://www.latroika.com/mathoman/exos/arnold-exos-math-engl...
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
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.
But we all took that as an indication of our own lack of knowledge and intuition and would just try harder.
When Bob was drawing, it looked amazingly simple. That simplicity invited people into trying painting.
Probably very few could ever draw anything remotely similar in quality to him, though.
Reading good books/having good lecturers certainly helps, but there is no way to replace the work.
I watch with wry amusement how schools constantly try to take the work out of learning. It never works. It's like putting labor-saving machinery in the gym - you'll never get stronger. You gotta put in the sweat.
I don't think Feynman's books are a replacement for a more traditional physics textbook as a student looking to pass a class, become a physicist, or a hobbyist trying to master it, but I do think they're pretty ideal for someone who wants to get much stronger grasp of the concepts than a layman without having to go through the struggle associated with solving problems they'll never actually apply.
http://www.feynmanlectures.caltech.edu/info/exercises.html
Some references to good collections of mathematics problems and solutions would be great for self-study.
If you just read a book and don't work through problems yourself, you simply don't learn enough to do it yourself.
Most books, I make some progress on some problems, get stuck on others, and generally have a good grasp of the overall landscape and where I am lacking; then I go back and reread (and practice) the missing pieces.
But Feynman’s lectures are different in that they make you feel you understand a lot, without really giving you any tools to address things he did not address (and basically only address those things he did address in the same way).
I am not saying they are bad - 25 years later, I still remember (and occasionally use) some of them; most recently insights from the chapter on minimum principles. I am just saying that it only became useful after I already had a good (but not great) grasp of the material from other sources — despite giving that impression when read as introductory.
At a meta level, I think this means we'll never (as a society) be great at teaching, because teachers who make us work make us feel like we're learning less. We prefer (and rate more highly) the professors, like Feynman, who make us feel smart.
Vol 1 was good. Vol 2 was good though overly repetitive, iirc it's 90 percent Maxwell's equations. Vol 3 was unintuitive to me. Still I learned qm from it and made this http://tropic.org.uk/~crispin/quantum/
I'm fairly confident I get qm now, but most of that understanding came from trying to code it in simulation. Which suggests there are better ways to learn than Feynman 3.
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.
It's easier to feel that you know something than to actually know it.
Well said! All students need to keep this in mind.
1. "What Is Mathematics? An Elementary Approach to Ideas and Methods" by Courant and Robbins -- a general book on mathematics in the spirit of Feynman lectures.
2. Strogatz's "Nonlinear Dynamics and Chaos" -- it's a bit narrow in scope (mostly dynamical systems with a little bit of chaos/fractals thrown in) but very good nonetheless.
3. Tristan Needham, "Visual Complex Analysis", beautiful introduction to complex analysis.
4. Cornelius Lanczos, "The Variational Principles of Mechanics" -- this is a physics book, but one of the classics in the subject, and as Gerald Sussman once remarked, you glean new insights each time you read it.
5. Cornelius Lanczos, "Linear Differential Operators" -- an excellent treatment of differential operators, Green's functions, and other things that one encounters in infinite-dimensional vector spaces. This book has some very intuitive explanations, e.g., why d/dx is not self-adjoint (i.e., Hermitian), whereas d^2/dx^2 is.
For chemistry, I would recommend "General Chemistry" by Linus Pauling, even though it's a bit outdated.
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."
[0] https://www.youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJ...
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.
A more formal approach appears in handbooks.[3][4]
[1] Gowers et al., The Princeton Companion to Mathematics. https://press.princeton.edu/books/hardcover/9780691118802/th...
[2] Higham and Dennis, The Princeton Companion to Applied Mathematics. https://press.princeton.edu/books/hardcover/9780691150390/th...
[3] Zwillinger, CRC Standard Mathematical Tables and Formulae. https://www.crcpress.com/CRC-Standard-Mathematical-Tables-an...
[4] Bronshtein, Handbook of Mathematics. https://www.springer.com/gp/book/9783540721222
To me the unique aspect is more the uncompromising intuitionistic approach with little consideration/adaptation for “shallow/correlative thinkers”...
But with mathematics, "intuitive" analogies are all in terms of other mathematical objects! You can't build intuition if you don't even know what they trying to abstract over.
In that regards, The Princeton Companion to Mathematics is fantastic because it maps out how the different fields of mathematics are interrelated.
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)
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).
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.
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.
If you have a grasp of algebra and sets this book is an easy read for the curious or mathematically immature.
Edit WhatIsDukkha is correct and their suggestion better reflects what I intended to say.
Usually saying someone isn't "mathematically mature" ie able to read and use proofs is what you would want to say.
Mathematics, Form and Function by Saunders Mac Lane (1986 Springer-Verlag hardcover, ISBN 0-387-96217-4) [2]
[1]: https://www.cut-the-knot.org/books/kac/index.shtml
[2]: https://en.wikipedia.org/wiki/Mathematics,_Form_and_Function
I think this is in the spirit of Feynman's lectures inasmuch as it's not going to bring you to expert-level understanding, but it is going to do a good job giving you some intuitive understanding, which you might then be able to apply to re-studying the material in more detail.
He even takes pull requests, I fixed a few typos.
1. Spivak Calculus
2. Apostol Calculus vol. I and II
3. Courant and Johns, Introduction to Calculus and Analysis vol. I, IIA and IIB
For the more casual computer science or physics major, I'd go with choice 3 which resembles the Feynmann lectures the most. All the rigour of the other two is there but in a more digestible form. It's hard for someone not accustomed to hard maths to digest long proofs, it could give you a bad case of indigestion. It's more a function of patience. Courant and Johns get to the point much more quickly while Spivak and Apostol take their time to do everything thoroughly. Courant and Johns does do everything thoroughly but they are kinder to the reader and delay lengthy rigoourous proofs as long as possible while giving plenty of motivation and intuition.
Also I strongly recommend any books by Ray Smullyan particularly his introduction to mathematical logic. "A Beginner's Guide to Mathematical Logic"
Not really a book that has lectures but it's a great book that covers all popular topics in Mathematics (fun to read).
(Link: https://mirtitles.org/2015/12/07/mathematics-can-be-fun-yako...)
Some of his Books;
* Physics for Entertainment Vols I & II
* Algebra for Fun
* Figures for Fun
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.
I wrote a short review after reading it, which you might be interested in checking out: https://faingezicht.com/articles/2017/10/27/infinity/
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!
It's a zany story, but in that respect it provides a refreshing intro into some math concepts.
In that vein, I also recommend "mathematical mindsets" [2]. A colleague developed a course inspired by this book. Though I only witnessed a tidbit, it radiated with the "new perspective/ new insights / gained understanding" that you'd get from the Feynman lectures.
Sidenote: neither is "now you know how all of maths work", but neither is Feynman thaf (foot physics). More importantly, all of them help you gain a new perspective on things.
[1] eg. https://www.bol.com/nl/p/the-number-devil-a-mathematical-adv...
[2] eg. https://books.google.nl/books/about/Mathematical_Mindsets.ht...
Nobody has mentioned yet "Geometry and the Imagination" by Hilbert and Cohn-Vossen. If there is a Feynman equivalent in math it is certainly this book.
For elementary geometry, the Feynman equivalent is probably "Introduction to Geometry", by H.S.M.Coxeter. Beautifully written, figures on every page, covers all geometric topics (affine, projective, ordered, differential, ...)
For differential geometry, nothing beats "A Panoramic view of Differential Geometry" by Berger. It is a stunning comprehensive overview of the whole field, focused on the meaning and the applications of each part and, strangely for a math book, with no formal proofs. Only the main ideas of the proof and the relationships between them are given, but this allows to fit the whole subject into a single, manageable whole.
I read it myself years ago and it was a great and entertaining way to fill in the gaps from my meager math education.
His explanations of mathematics are the only ones I can think of that have given me the same sort of piercing clarity and insight that one gets from reading Feynman on physics.
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.
>This isn't a popular suggestion (and by that I don't mean to say it's rejected or people don't like it, I just haven't heard it suggested before in this context) but at university for electronic engineering we used K.A. Stroud's Engineering Mathematics. This book is surprisingly little focused on actual applications to engineering, it takes you through calculus by introducing the derivative, for example, and then some linear algebra stuff. But what surprises people is that it starts off with the properties of addition and multiplication - it's that simple. It's a book that starts from zero and takes you very, very far. It won't take you to a mathematician's 100 but it'll take you to any serious engineering undergrad's 100.
https://www.amazon.com/Arithmetic-Paul-Lockhart/dp/067497223...
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.
It is called "Who is Fourier: A Mathematical Adventure".
I was tremendously surprised by this unusual gem of a book. It covers the range from basic arithmetic to logarithms, trigonometry, calculus to fourier series.
https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2n...
The book is quite good. It is written like a "Manga" book and hence has tons of drawings to help develop intuition for the concepts. It is written by a group of ordinary people with help from Scientists (a quirky club named Transnational College of Lex from Japan - https://en.wikipedia.org/wiki/Hippo_Family_Club ) and thus is very accessible. Highly recommended for High school students and above.
Note that the same group has also published two other books in the same vein; a) What is Quantum Mechanics b) What is DNA; both of which are also highly recommended.
There is also the No Bullshit Guide to Linear Algebra https://www.amazon.com/dp/0992001021/ Extended preview: https://minireference.com/static/excerpts/noBSguide2LA_previ...
Both come with a review of high school math topics, which may or may not be useful for you, depending on how well you remember the material. Many of the university-level books will assume you know the high school math concepts super well.
One last thing, I highly recommend you try out SymPy which is a computer algebra system that can do a lot of arithmetic and symbolic math operations for you, e.g. simplify expressions, factor polynomials, solve equations, etc. You can try it out without installing anything here https://live.sympy.org/ and this is a short tutorial that explains the basic commands https://minireference.com/static/tutorials/sympy_tutorial.pd...
Firstly, much of mathematics is symbolic and any description of equations in an intuitive style is unnecessarily verbose if it abandons the symbolic approach, essentially taking one back to descriptions like those used in ancient Greece before the invention of algebra, e.g. "and the third part of the first is to the second part of the first as the fourth part of the area is to the square on the gnomon".
The second reason is that an intuitive style supposes that one can answer natural questions that might arise, in an order that they are likely to arise in the mind of the student. Often the natural questions are much more difficult to answer mathematically, or the answers are not known.
The third reason is that concepts have arisen historically for non-obvious reasons, or reasons only known to experts with far more knowledge than the reader is expected to have, or the originator of the ideas did their best to obscure their motivation. This makes it extremely hard to motivate certain concepts naturally (intuitively) since such motivations are simply not known. For example, it is not hard to motivate solvable groups through a study of solubility of polynomial equations. But it is much harder to motivate the related concept of nilpotent groups, where the true motivations lie far deeper in the theory than the concepts themselves.
The fourth reason is that it is a massive effort to come up with good examples. Even the best textbook authors often struggle to come up with accessible examples for the concept they are trying to explain. Often, good examples require a really broad knowledge of mathematics that goes way beyond the narrow field being taught. Examples end up being very artificial, and neither intuitive nor typical, as a result.
Don't get me wrong. If someone told me something like the Feynman lectures existed for mathematics, I would salivate and spend a lot of money to acquire them. But having experimented with many styles of writing notes for myself on mathematics over the years, I well appreciate how hard, or perhaps impossible the task would really be. Of course there are some oases in mathematics where such an intuitive approach is possible.
My personal take is that good linear algebra books at any level are great "tours of mathematics". Start with Strang and never stop. In a few years you'll be balled up with Kreyszig scribbling proof attempts in receipts, flaming unkempt hair and everyone around you will think you're weird but you'll be so, so happy.
[0] https://www.amazon.com/gp/aw/d/0486652416/
[1] http://alvand.basu.ac.ir/~dezfoulian/files/Numericals/Numeri...
Something that might be close would be the survey _Mathematics: Its Contents, Methods and Meaning_ by Aleksandrov, Kolmogorov et al. https://www.goodreads.com/book/show/405880.Mathematics
For actually learning GR, I prefer Wald.
https://www.amazon.com/Tour-Calculus-David-Berlinski/dp/0679...
And there's this book: "Conceptual mathematics" by Lawvere and Schanuel. It's unlike any other mathematics text I have found. Fundamental and easy to read: yes. Also leads up to some deep ideas in an intuitive way.
Nathan Carter's 'Visual group theory' also seems an interesting experiment, if you are interested in that part of mathematics, though I have not read it.
My longest problem has been I have no idea what is going in the formula or fundamental questions like, "why is there a square root there". It is hard to describe my issue, but I've been very horrible with math anyways. Can't do gas station math anyways.
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.
http://www.feynmanlectures.caltech.edu/I_toc.html
It is... very average looking. Did something happen here?
https://www.amazon.com/Playing-Infinity-Mathematical-Explora...
A lesser known one that isn't quite as comprehensive is a little Dover tome by Mendelson: Number Systems and the Foundations of Analysis. It starts off with the (abstract) natural numbers, and from there develops (parts of) real and complex analysis, using a categorical point of view throughout.
One of my favorite parts in the latter:
“What is our intuitive understanding of the natural numbers? Surely this being the firmest of all our mathematical ideas, should have a definite, transparent meaning. Let us examine a few attempts to make this meaning clear:
(1) The natural numbers may be thought of as symbolic expressions: 1 is |, 2 is ||, 3 is |||, 4 is ||||, etc. Thus, we start with a vertical stroke | and obtain new expressions by appending additional vertical strokes. There are some obvious objections with this approach. First, we cannot be talking about particular physical marks on paper, since a vertical stroke for the number 1 may be repeated in different physical locations. The number 1 cannot be a class of all congruent strokes, since the length of the stroke may vary; we would even acknowledge as a 1 a somewhat wiggly stroke written by a very nervous person. Even if we should succeed in giving a sufficiently general geometric characterization of the curves which would be recognized as 1’s, there is still another objection. Different people and different civilizations may use different symbols for the basic unit, for example, a circle or a square instead of a stroke. Yet, we could not give priority to one symbolism over any of the others. Nevertheless, in all cases, we would have to admit that, regardless of the difference in symbols, we are all talking about the same things.
(2) The natural numbers may be conceived to be set-theoretic objects. In one very appealing version of this approach, the number 1 is defined as the set of all singletons {x}; the number 2 is the set of all unordered pairs {x, y}, where x =/= y; the number 3 is the set of all sets {x, y, z} where x =/= y, x =/= z, y =/= z; and so on. Within a suitable axiomatic presentation of set theory, clear rigorous definitions can be given along these lines for the general notion of natural number and for familiar operations and relations involving natural numbers. Indeed, the axioms for a Peano system are easy consequences of the definitions and simple theorems of set theory. Nevertheless, there are strong deficiencies in this approach as well.
First, there are many competing forms of axiomatic set theory. In some of them, the approach sketched above cannot be carried through, and a completely different definition is necessary. For example, one can define the natural numbers as follows: 1 = {∅}, 2 = {∅, 1}, 3 = {∅, 1, 2}, etc. Alternatively, one could use: 1 = {∅}, 2 = {1}, 3 = {2}, etc. Thus, even in set theory, there is no single way to handle the natural numbers. However, even if a set-theoretic definition is agreed upon,it can be argued that the clear mathematical idea of the natural numbers should not be defined in set-theoretic terms. The paradoxes (that is, arguments leading to a contradiction) arising in set theory have cast doubt upon the clarity and meaningfulness of the general notions of set theory. It would be inadvisable then to define our basic mathematical concepts in terms of set theoretic ideas.
This discussion leads us to the conjecture that the natural numbers are not particular mathematical objects. Different people, different languages, and different set theories may have different systems of natural numbers. However, they all satisfy the axioms for Peano systems and therefore are isomorphic. There is no one system which has priority in any sense over all the others. For Peano systems, as for all mathematical systems, it is the form (or structure) which is important, not the “content”. Since the natural numbers are necessary in the further development of mathematics, we shall make one simple assumption:Basic Axiom There exists a Peano system.“
Elliott Mendelson, Number Systems and the Foundations of Analysis
https://www.amazon.com/Number-Theory-History-Dover-Mathemati...
Not to be taken literally, of course. But there is some truth in that. If you are an engineer it makes sense to skim all kinds of math books. If you are a mathematician then I would say rather look for something that gels well with your personality and run with it.
I'm currently learning group theory, matrices, and graph theory.
Goes well with the Art of Problem Solving site/books for practice.
- Concepts of Modern Mathematics - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...
- Methods of Mathematics Applied to Calculus, Probability, and Statistics - https://www.amazon.com/Methods-Mathematics-Calculus-Probabil... (all books by Richard Hamming are recommended)
- Calculus: An Intuitive and Physical Approach - https://www.amazon.com/Calculus-Intuitive-Physical-Approach-...
For a Textbook reference, the following are quite good;
- Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences - https://www.amazon.com/Mathematical-Techniques-Introduction-... (easy to read and succinct)
- Mathematics for Physicists: Introductory Concepts and Methods - https://www.amazon.com/Mathematics-Physicists-Introductory-C...
For General reading (all these authors other books are also worth checking out);
- Mathematics, Queen and Servant of Science - https://www.amazon.com/Mathematics-Queen-Servant-Science-Tem...
- Mathematics and the Physical World - https://www.amazon.com/Mathematics-Physical-World-Dover-Book...
- Mathematician's Delight - https://www.amazon.com/Mathematicians-Delight-Dover-Books-Ma...
