
Ask HN: Are there books for mathematics like Feynman's lectures on physics? - jagira
I have started re-learning college level Physics and am thoroughly enjoying Feynman&#x27;s Lecture on Physics. Are there similar books available for Mathematics (&amp; Chemistry) - books that are fundamental and easy to read?
======
nabla9
"Mathematics : Its Content, Methods and Meaning" by A. D. Aleksandrov, A. N.
Kolmogorov ,M. A. Lavrent'ev. (3 Volumes)

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.goodreads.com/book/show/405880.Mathematics)

[https://www.amazon.com/Mathematics-Content-Methods-
Meaning-V...](https://www.amazon.com/Mathematics-Content-Methods-Meaning-
Volumes/dp/0486409163)

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

~~~
rramadass
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.

~~~
weinzierl
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.

~~~
rramadass
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 :-)

~~~
weinzierl
> 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.

[1]
[http://people.math.sfu.ca/~cbm/aands/intro.htm](http://people.math.sfu.ca/~cbm/aands/intro.htm)

[2] [https://dlmf.nist.gov/](https://dlmf.nist.gov/)

~~~
rramadass
> but the presentation is not

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 :-)

------
beagle3
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.

~~~
joelaaronseely
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.

~~~
gdsimoes
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.

~~~
AlexCoventry
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.

~~~
cellularmitosis
“That which I do not understand, I cannot create”

------
m31415
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.

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.

~~~
newprint
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.

~~~
rrss
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.

------
anton_tarasenko
In terms of depth and breadth, the Princeton companions get close to
Feynman.[1][2]

A more formal approach appears in handbooks.[3][4]

[1] Gowers et al., The Princeton Companion to Mathematics.
[https://press.princeton.edu/books/hardcover/9780691118802/th...](https://press.princeton.edu/books/hardcover/9780691118802/the-
princeton-companion-to-mathematics)

[2] Higham and Dennis, The Princeton Companion to Applied Mathematics.
[https://press.princeton.edu/books/hardcover/9780691150390/th...](https://press.princeton.edu/books/hardcover/9780691150390/the-
princeton-companion-to-applied-mathematics)

[3] Zwillinger, CRC Standard Mathematical Tables and Formulae.
[https://www.crcpress.com/CRC-Standard-Mathematical-Tables-
an...](https://www.crcpress.com/CRC-Standard-Mathematical-Tables-and-
Formulas/Zwillinger/p/book/9781498777803)

[4] Bronshtein, Handbook of Mathematics.
[https://www.springer.com/gp/book/9783540721222](https://www.springer.com/gp/book/9783540721222)

~~~
bjornsing
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”...

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

~~~
adrianmonk
I'd use 3 words to describe Feynman's style: clarity, accessibility, and
fascination.

~~~
amelius
This description made me think of the "Mathologer" channel on YouTube.

------
jordigh
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).

~~~
oldgradstudent
It's also a beautiful book.

~~~
frostburg
Yeah, great layout and typesetting. I have a copy mostly for this reason.

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

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

~~~
rsapkf
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.

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

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

~~~
iamaelephant
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.

~~~
elorant
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.

------
bgutierrez
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.

~~~
agentultra
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.

~~~
WhatIsDukkha
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.

~~~
bjornsing
You can do a lot of graph theory without being able to sum integers. No
problem. :)

------
nicklaf
Mathematics and Logic by Mark Kac and Stan Ulam (1992 Dover paperback, ISBN
978-0486670850) [1]

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](https://www.cut-the-
knot.org/books/kac/index.shtml)

[2]:
[https://en.wikipedia.org/wiki/Mathematics,_Form_and_Function](https://en.wikipedia.org/wiki/Mathematics,_Form_and_Function)

~~~
devmacrile
+1 on Mathematics, Form and Function

------
jlebar
I am very much enjoying Evan Chen's "An Infinitely Large Napkin".
[https://venhance.github.io/napkin/Napkin.pdf](https://venhance.github.io/napkin/Napkin.pdf)

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.

~~~
alok-g
Looks very good. Thanks for the suggestion.

------
Mugwort
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"

------
KingCobra
"Mathematics Can Be Fun" by Yakov Perelman

Not really a book that has lectures but it's a great book that covers all
popular topics in Mathematics (fun to read).

(Link: [https://mirtitles.org/2015/12/07/mathematics-can-be-fun-
yako...](https://mirtitles.org/2015/12/07/mathematics-can-be-fun-yakov-
perelman))

~~~
rramadass
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

------
Beldin
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.

[1] eg. [https://www.bol.com/nl/p/the-number-devil-a-mathematical-
adv...](https://www.bol.com/nl/p/the-number-devil-a-mathematical-
adventure/1001004000849661/)

[2] eg.
[https://books.google.nl/books/about/Mathematical_Mindsets.ht...](https://books.google.nl/books/about/Mathematical_Mindsets.html?id=1_KlCgAAQBAJ&printsec=frontcover&source=kp_read_button&redir_esc=y)

------
mturmon
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.

------
metastart
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.

~~~
avyfain
+1 on DFW's book. It's odd, and really it's more a discussion on the history
of modern mathematics than an in depth explanation of the concepts themselves,
but it is a great read.

I wrote a short review after reading it, which you might be interested in
checking out:
[https://faingezicht.com/articles/2017/10/27/infinity/](https://faingezicht.com/articles/2017/10/27/infinity/)

~~~
sah2ed
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!

------
enriquto
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.

------
DrPhish
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.

[https://www.goodreads.com/book/show/383087.Mathematics](https://www.goodreads.com/book/show/383087.Mathematics)

------
oliveshell
I know you asked for books, but I have to mention the videos of Grant
Sanderson (3Blue1Brown) [1].

His explanations of mathematics are the only ones I can think of that have
given me the same sort of piercing clarity and insight that one gets from
reading Feynman on physics.

[1]:
[https://www.youtube.com/c/3blue1brown](https://www.youtube.com/c/3blue1brown)

~~~
slx26
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.

------
math_phd_to_dev
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.

~~~
wbhart
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.

------
claudiawerner
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.

------
Jun8
It's not wide-ranging but Terence Tao's _Solving Mathematical Problems_
([https://books.google.com/books/about/Solving_Mathematical_Pr...](https://books.google.com/books/about/Solving_Mathematical_Problems_A_Personal.html?id=ZBTJWhXD05MC))
makes you understand mathematical thinking better than any other book I have
seen at that level.

------
ughitsaaron
This may be a bit too elementary, but I recently read “Arithmatic” by Paul
Lockwood and found it easy, provocative, and fascinating.

[https://www.amazon.com/Arithmetic-Paul-
Lockhart/dp/067497223...](https://www.amazon.com/Arithmetic-Paul-
Lockhart/dp/0674972236)

~~~
mxyxpt1k
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.

~~~
ughitsaaron
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.

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

It is called "Who is Fourier: A Mathematical Adventure".

I was tremendously surprised by this unusual gem of a book. It covers the
range from basic arithmetic to logarithms, trigonometry, calculus to fourier
series.

[https://www.amazon.com/Who-Fourier-Mathematical-
Adventure-2n...](https://www.amazon.com/Who-Fourier-Mathematical-
Adventure-2nd/dp/0964350432)

~~~
mkadlec
And it's only $975!

~~~
rramadass
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](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.

------
chefschef
Mary Boas wrote one of my fav books targeting folks in the physical sciences:
[https://www.amazon.com/Mathematical-Methods-Physical-
Science...](https://www.amazon.com/Mathematical-Methods-Physical-Sciences-
Mary/dp/0471198269)

~~~
UncleSlacky
Stroud's "Engineering Mathematics" is better for my money.

------
jonbarker
More general purpose problem solving but written by a mathematician and one of
the best books I've ever read!
[https://en.wikipedia.org/wiki/How_to_Solve_It](https://en.wikipedia.org/wiki/How_to_Solve_It)

~~~
Mugwort
Agreed. Hard for some people to appreciate but so true.

------
ivan_ah
If you're looking for something very basic (high school and calculus), you can
check my book No Bullshit Guide to Math & Physics:
[https://www.amazon.com/dp/0992001005/](https://www.amazon.com/dp/0992001005/)
Extended preview here:
[https://minireference.com/static/excerpts/noBSguide_v5_previ...](https://minireference.com/static/excerpts/noBSguide_v5_preview.pdf)

There is also the No Bullshit Guide to Linear Algebra
[https://www.amazon.com/dp/0992001021/](https://www.amazon.com/dp/0992001021/)
Extended preview:
[https://minireference.com/static/excerpts/noBSguide2LA_previ...](https://minireference.com/static/excerpts/noBSguide2LA_preview.pdf)

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/](https://live.sympy.org/) and this is a short
tutorial that explains the basic commands
[https://minireference.com/static/tutorials/sympy_tutorial.pd...](https://minireference.com/static/tutorials/sympy_tutorial.pdf)

------
mazsa
The Best of All Possible Worlds: Mathematics and Destiny by Ivar Ekeland,
[https://www.goodreads.com/book/show/407320.The_Best_of_All_P...](https://www.goodreads.com/book/show/407320.The_Best_of_All_Possible_Worlds)
It is definitely not like Feynman's Lectures still one of the best books I've
ever read.

~~~
rramadass
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.

------
wbhart
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.

------
thanatropism
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.

------
pcvarmint
Hamming, R.W., _Numerical Methods For Scientists and Engineers_ [0] [1]

[0]
[https://www.amazon.com/gp/aw/d/0486652416/](https://www.amazon.com/gp/aw/d/0486652416/)

[1]
[http://alvand.basu.ac.ir/~dezfoulian/files/Numericals/Numeri...](http://alvand.basu.ac.ir/~dezfoulian/files/Numericals/Numerical.Methods.For.Scientists.And.Engineers_2ed_Hamming_0486652416.pdf)

------
cassalian
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

------
harshreality
Feynman was a unique teacher of course, and pure math isn't quite as easy to
teach intuitively.

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](https://www.goodreads.com/book/show/405880.Mathematics)

------
auntienomen
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.

~~~
tobinfricke
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.

~~~
auntienomen
True, but the question was about books that are like the Feynman Lectures. MTW
has similar tone and breadth, and a similar reputation for being useless for
learning.

For actually learning GR, I prefer Wald.

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

------
hos234
I liked Steven Strogatz's books Sync and Infinite Powers.

~~~
falcor84
Seconded, and so is his Nonlinear dynamics and Chaos

------
dempedempe
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).

------
master_yoda_1
All of Statistics: A Concise Course in Statistical Inference
[https://www.amazon.com/All-Statistics-Statistical-
Inference-...](https://www.amazon.com/All-Statistics-Statistical-Inference-
Springer/dp/1441923225)

------
DoreenMichele
I enjoyed "A tour of the calculus," at least the half I read.

[https://www.amazon.com/Tour-Calculus-David-
Berlinski/dp/0679...](https://www.amazon.com/Tour-Calculus-David-
Berlinski/dp/0679747885/ref=asc_df_0679747885/?tag=bingshoppinga-20&linkCode=df0&hvadid={creative}&hvpos={adposition}&hvnetw=o&hvrand={random}&hvpone=&hvptwo=&hvqmt=e&hvdev=m&hvdvcmdl={devicemodel}&hvlocint=&hvlocphy=&hvtargid=pla-4583795260672248&psc=1)

------
mathgenius
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.

------
prof_mm
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.

------
threespice
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.

~~~
rramadass
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.

------
justin66
Here's a question about the Feynman Lectures. I remember looking at the
digitized text a few years ago, perhaps right after they made the digital copy
freely available, and thinking the typesetting was pretty great. Looking at it
today:

[http://www.feynmanlectures.caltech.edu/I_toc.html](http://www.feynmanlectures.caltech.edu/I_toc.html)

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

------
chx
By the end of this book reaches college levels. There's nothing better to show
how interesting mathematics, to look at it with wonderment. Our education
system is to blame.

[https://www.amazon.com/Playing-Infinity-Mathematical-
Explora...](https://www.amazon.com/Playing-Infinity-Mathematical-Explorations-
Excursions/dp/0486232654)

------
vector_spaces
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

~~~
sriram_malhar
This was a good read, thank you.

------
mike00632
"Number Theory and its History" (1948) by Øystein Ore is considered a classic.

[https://www.amazon.com/Number-Theory-History-Dover-
Mathemati...](https://www.amazon.com/Number-Theory-History-Dover-
Mathematics/dp/0486656209/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=&sr=)

------
mikorym
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.

------
cottonseed
I asked this on stackoverflow:
[https://math.stackexchange.com/questions/62190/mathematical-...](https://math.stackexchange.com/questions/62190/mathematical-
equivalent-of-feynmans-lectures-on-physics)

------
AareyBaba
I suggest this "Map of mathematics" as a starting point that gives you a
reasonable birds-eye view of the field.
[https://www.youtube.com/watch?v=OmJ-4B-mS-Y](https://www.youtube.com/watch?v=OmJ-4B-mS-Y)

------
ganzuul
"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.

------
xvilka
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.

------
hasitseth
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.

------
fano
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.

------
mjcohen
I've always liked "Mathematics for the Million" by Lancolet Hogben.

~~~
novalis78
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.

------
slowhand09
Anything by Metin Bektas. [https://www.amazon.com/s?i=digital-
text&rh=p_27%3AMetin+Bekt...](https://www.amazon.com/s?i=digital-
text&rh=p_27%3AMetin+Bektas)

------
meieo0
Mathematics for the Nonmathematician
[https://www.amazon.com/dp/0486248232](https://www.amazon.com/dp/0486248232)

------
seshagiric
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).

~~~
hackermailman
Wildberger has lectures on math for k-6
[https://m.youtube.com/user/njwildberger/playlists](https://m.youtube.com/user/njwildberger/playlists)

Goes well with the Art of Problem Solving site/books for practice.

------
zhamisen
"Mathematical Omnibus: Thirty Lectures on Classic Mathematics " Very
accessible and covering a broad range of topics.

------
AlexCoventry
I greatly enjoyed Spivak's _Calculus_.

------
bluishgreen
Mathematics, form and function. Saunders Mclane

------
aportnoy
Calculus by Michael Spivak

~~~
dilap
seconded. this book is fantastic!

------
ethagnawl
_How to Bake Pi_ by Eugenia Cheng

------
rramadass
In addition to all the other good suggestions, the following are recommended
(have not seen these mentioned so far);

\- _Concepts of Modern Mathematics_ \- [https://www.amazon.com/Concepts-
Modern-Mathematics-Dover-Boo...](https://www.amazon.com/Concepts-Modern-
Mathematics-Dover-
Books/dp/0486284247/ref=sr_1_1?crid=214QSPLG829VY&keywords=concepts+of+modern+mathematics&qid=1571974129&s=books&sprefix=concepts+of+modern+ma%2Cstripbooks-
intl-ship%2C1448&sr=1-1)

\- _Methods of Mathematics Applied to Calculus, Probability, and Statistics_
\- [https://www.amazon.com/Methods-Mathematics-Calculus-
Probabil...](https://www.amazon.com/Methods-Mathematics-Calculus-Probability-
Statistics/dp/0486439453/ref=cm_cr_arp_d_product_top?ie=UTF8) (all books by
Richard Hamming are recommended)

\- _Calculus: An Intuitive and Physical Approach_ \-
[https://www.amazon.com/Calculus-Intuitive-Physical-
Approach-...](https://www.amazon.com/Calculus-Intuitive-Physical-Approach-
Mathematics/dp/0486404536/ref=cm_cr_arp_d_product_top?ie=UTF8)

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-...](https://www.amazon.com/Mathematical-Techniques-Introduction-
Engineering-Physical/dp/0199282013/ref=dp_ob_title_bk) (easy to read and
succinct)

\- _Mathematics for Physicists: Introductory Concepts and Methods_ \-
[https://www.amazon.com/Mathematics-Physicists-
Introductory-C...](https://www.amazon.com/Mathematics-Physicists-Introductory-
Concepts-Methods-
ebook/dp/B07N48G4KD/ref=sr_1_1?keywords=mathematics+for+physicists&qid=1571974732&s=books&sr=1-1)

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...](https://www.amazon.com/Mathematics-Queen-Servant-Science-
Tempus/dp/155615173X/ref=sr_1_1?keywords=mathematics+queen+and+servant&qid=1571977505&s=books&sr=1-1)

\- _Mathematics and the Physical World_ \-
[https://www.amazon.com/Mathematics-Physical-World-Dover-
Book...](https://www.amazon.com/Mathematics-Physical-World-Dover-
Books/dp/0486241041/ref=sr_1_1?keywords=mathematics+and+the+physical+world&qid=1571977524&s=books&sr=1-1)

\- _Mathematician 's Delight_ \- [https://www.amazon.com/Mathematicians-
Delight-Dover-Books-Ma...](https://www.amazon.com/Mathematicians-Delight-
Dover-Books-
Mathematics/dp/0486462404/ref=sr_1_1?keywords=mathematician%27s+delight&qid=1571977538&s=books&sr=1-1)

------
rq1
Real and Complex Analysis by Walter Rudin.

------
mathmania
Rudin

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

------
sacrificedcapon
Not as encompassing as Feynman's lectures on physics, but Euclid's Elements is
an excellent mathematical text.

~~~
Mugwort
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...](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.475.3070&rep=rep1&type=pdf)

