
Great books about mathematics - ColinWright
http://wp.kjro.se/2013/12/27/5-insanely-great-books-about-mathematics-you-should-read/
======
tmoertel
Another great book, starting with the basics and then exploring a wide range
of mathematical subjects, is _The Princeton Companion to Mathematics_ :

[http://press.princeton.edu/titles/8350.html](http://press.princeton.edu/titles/8350.html)

It's fun to just flip it open and start reading.

~~~
calvins
This is a wonderful book. A word of caution though: don't get the Kindle
version from Amazon. I thought I'd get it for the Kindle, because the print
version is huge and I sometimes like to read in bed and don't want to lug the
dead tree version I own around, but I had to return it because of the standard
problems with garbled notation (some symbols appearing as a square box or some
other incorrect symbol) that affects every math book I've ever purchased on
Amazon for the Kindle.

There error rate was much lower than in any other math book I've tried, but
still much too high.

------
acak
I really enjoyed the book on Fermat's last theorem by Simon Singh -

[http://www.amazon.com/dp/0802713319](http://www.amazon.com/dp/0802713319)

A long time ago, a friend's mother was complaining to me that (high school)
Math is a dry subject and she doesn't blame her otherwise intelligent son for
not being able get interested and do well in it. I wish I knew of this book
then - I know it cranked up my interest in Math ever since.

~~~
tomsthumb
I read this in a day in high school. It was great, and actually all of his
books are great. The crypto one is really engaging, and a surprisingly easy
while reasonably thorough read. The one about the big bang can get a little
dry in spots, but it's 500 pages and by the end there are more than enough
"wow, that's amazing" moments to make up for it.

------
kriro
"Proofs and Refutations" is a fantastic book, I really enjoy that style.

If someone is looking for a more lightweight introduction to Lakatos I can
highly recommend "For and Against Method" which outlines his "arguments" with
Feyerabend.

If you're not into science theory (imo) Lakatos is basically Popper++ (I think
most people have heard of Popper)

~~~
tokenrove
Something else I just read which is related but lighter is How Mathematicians
Think by William Byers.

------
anaphor
Here is a great one: [http://www.amazon.com/Introduction-Graph-Theory-Dover-
Mathem...](http://www.amazon.com/Introduction-Graph-Theory-Dover-
Mathematics/dp/0486678709)

"A stimulating excursion into pure mathematics aimed at "the mathematically
traumatized," but great fun for mathematical hobbyists and serious
mathematicians as well. Requiring only high school algebra as mathematical
background, the book leads the reader from simple graphs through planar
graphs, Euler's formula, Platonic graphs, coloring, the genus of a graph,
Euler walks, Hamilton walks, and a discussion of The Seven Bridges of
Konigsberg."

------
vincentbarr
Two additional suggestions: 1\. 'A Mathematician's Lament' by Paul Lockhart
([http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...](http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf))

2\. 'How to Solve It' by G. Polya ([http://www.amazon.com/How-Solve-It-
Mathematical-Princeton/dp...](http://www.amazon.com/How-Solve-It-Mathematical-
Princeton/dp/069111966X))

------
nabla9
May I present one of the greatest math books for general audience:
Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N.
Kolmogorov, M. A. Lavrent’ev.

[http://www.amazon.com/Mathematics-Its-Content-Methods-
Meanin...](http://www.amazon.com/Mathematics-Its-Content-Methods-
Meaning/dp/0486409163)

Math books rarely move from the Soviet Union to west, but this did and for
really good reason. Just look at the list of writers included. So far I have
not seen any math books that come even close to this. Reading this book
together with the The Princeton Companion to Mathematics was real treat.

~~~
alok-g
I second the book as well as the above praise for the book as one of the
greatest. I have read several chapters from the book and have loved them all.

Question: I haven't read The Princeton Companion to Mathematics. How would you
compare these two?

~~~
nabla9
The Princeton Companion is more like encyclopedia. Main part of the book is
articles describing 100 or so mathematical concepts in alphabetical orders.
Then it has articles describing major mathematical problems.

------
ivan_ah
For a short, intuitive, practical, and affordable introduction to high school
math and calculus, check out my book: "No bullshit guide to math and physics"
[http://minireference.com/](http://minireference.com/)

For a free alternative, check out:
[http://www.gutenberg.org/ebooks/33283](http://www.gutenberg.org/ebooks/33283)

------
Tycho
Anyone have good reading material about 'how to reason mathematically' on a
basic level? I mean, not going too deep into any specific topic, but how to
get better at interpreting equations and grokking relationships.

~~~
lovemath
My wife gave me this great book for Christmas:

    
    
      Love and Math: The Heart of Hidden Reality
      by Edward Frenkel
    

It begins with the author's struggle to learn the math behind quantum physics
in spite of cold-war era soviet educational obstacles and leads bit by bit
into the Langlands program, drawing connections between group theory, number
theory and harmonic analysis.

It's definitely my favorite book of 2013.

[http://www.amazon.com/Love-Math-Heart-Hidden-
Reality/dp/0465...](http://www.amazon.com/Love-Math-Heart-Hidden-
Reality/dp/0465050743)

~~~
auctiontheory
Yup. My thought as I read _Love and Math_ a couple of months ago was "this
would be great to give to a high school senior who is wondering whether to
continue with mathematics." If you truly love doing math, you'll feel it when
you read this book.

------
emilga
> I’ve been asked over and over for good books about mathematics for a
> layperson, someone who hasn’t taken advanced courses in university and is
> more simply interested in learning about what math is, and some of the more
> interesting historical figures and results from mathematics.

The book you want is "What is Mathematics?" [0] by Courant and Robbins.

[0]
[http://books.google.no/books?id=_kYBqLc5QoQC&printsec=frontc...](http://books.google.no/books?id=_kYBqLc5QoQC&printsec=frontcover&hl=no&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)

------
alok-g
Does someone have a recommendation for a book that get deeper into mathematics
by teaching actual mathematics rather than teaching about it? Many of the
books I have come across leave me with more questions than answers [1].

I am interested in pure mathematics mainly including logic, set theory,
category theory, etc.

[1] An excellent example not mentioned by others here is also The Road to
Reality by Roger Penrose. The sections on mathematics are very good, but
ultimately leave the topic open-ended too soon.

~~~
jfarmer
What are you looking for that isn't simply a math textbook? Honestly, that's
how mathematicians learn "actual" mathematical subjects, too.

If you're comfortable with calculus as a subject, for example, and want a
"pure mathematics" approach, I recommend Michael Spivak's _Calculus_. If
you've never worked through a pure math textbook from start to finish, that's
a good start.

There's not that much interesting in the three subjects you listed — logic,
set theory, and category theory — that doesn't depend on other subjects or a
prior level of mathematical maturity. Category theory was originally invented
to solve and categorize problems in algebraic topology, for example.

For example, I rather like mathematical logic and model theory, but you're
going to have a rough time if you don't have a visceral, intuitive
understanding of countability arguments and at least a handful of subjects
you'd be reasoning "about." Unless you know the standard model of arithmetic,
for example, how can you think about non-standard models? Without that,
important results like the Löwenheim–Skolem theorem will likely seem
contextless.

~~~
alok-g
Good points. I have already gone through several levels of engineering
mathematics, so do understand differential and integral calculus well. I am
assuming that is that Spivak's book is about -- please correct me if I am
wrong; Amazon is not showing a preview of the book.

You are right in bringing countability arguments into the picture; I
understand them only to some level. I would love to read a book that gives it
a formal treatment. The following has been great for example:

[https://news.ycombinator.com/item?id=6838917](https://news.ycombinator.com/item?id=6838917)

Another one showed up on HN recently that I am still to read in full:

[https://news.ycombinator.com/item?id=6966695](https://news.ycombinator.com/item?id=6966695)

Thanks

Finally, cannot help but mention in praise, Colin Wright here on HN has been a
good help before in clearing some of my doubts on the subject.

~~~
jfarmer
Spivak's _Calculus_ starts with a set of 13 axioms which characterize the real
numbers and then derives all the results you're familiar with in calculus.
It's rigorous in the mathematical sense, so if you've never worked through a
rigorous math textbook before then this might be a good start since you're
familiar with the underlying material.

Here are some exercises to give you a sense of the flavor. If you find these
exercises trivial then the textbook might not be for you. If you find them
hard, well, welcome to math! :)

These are all before we get to any "calculus." Here "function" means a
function of the real numbers.

1\. Let f be a function that satisfies the conclusions of the Intermediate
Value Theorem. Prove that if f takes on each value _only once_ then f is
continuous. Generalize this to the case where f takes on each value only
finitely many times.

2\. Prove that if n is even, then there is no continuous function f which
takes on every value exactly n times.

3\. A set A of real numbers is said to be sense if every open interval
contains a point of A. Prove that if f is continuous and f(x) = 0 for all
numbers x in a dense set A then f(x) = 0 for all x.

4\. Find a function which is continuous at every irrational point and
discontinuous at every rational point (and prove it as such)

Spivak's _Calculus_ is used as a first-year calculus textbook at lots of
schools, so if you find the above even a little challenging or strange-seeming
then I'd recommend going through the book.

The last chapter of the textbook is a rigorous construction of the real
numbers from the rationals using Dedekind cuts (referenced in the first link).

~~~
impendia
> Spivak's Calculus is used as a first-year calculus textbook at lots of
> schools

Umm... where? Not at Stanford, where we used a mainstream, much easier book.
So does Princeton. Harvard is famous for having developed a "touchy-feely"
calculus book.

Perhaps abroad? It is typical of calculus courses in the US that the students
come with fairly weak backgrounds, and a major purpose is to expose and patch
holes in the students' backgrounds in algebra and trigonometry.

~~~
jfarmer
If you took me to mean the "default" first-year calculus textbook then yes,
that's not common. But it's definitely aimed at first-year college students,
or at least people who haven't had prior exposure to rigorous mathematical
thinking. Compare the style to, say, Spivak's _Calculus on Manifolds_ to see
what I mean.

(Edit: I just read your HN bio and know you know the stylistic differences,
etc. Sorry!)

Spivak is the first-year Honors Calculus textbook at my alma mater, the
University of Chicago. Harvard is also famous for having the most difficult
first-year math classes that use even more advanced textbooks like Rudin's
_Principles of Mathematical Analysis_.

My HS background in mathematics was definitely "weak," too. My senior year was
the first year my school district ever offered calculus of any stripe in its
entire history and I still managed to handle Spivak my first year of college.
I took the AP Calculus test on my own and got a 4/5\. It's not _that_ crazy.

------
dmoo
An enjoyable fiction that may help your kids get interested in maths is The
Parrot's Theorem reviewed on a maths fiction site
[http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumbe...](http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf123)

------
jamestomasino
Your History of Mathematics is great, but I prefer the two volume work by D.E.
Smith. Here's volume 1: [http://www.amazon.com/History-Mathematics-Vol-Dover-
Books/dp...](http://www.amazon.com/History-Mathematics-Vol-Dover-
Books/dp/0486204294/ref=pd_cp_b_0)

------
thejteam
I read "The Mathematical Experience" when I was in high school. Unfortunately,
I do not remember many details of the book (for me, "when I was in high
school" is approaching 20 years ago) except I recall that I read it over and
over again and thought it was excellent.

~~~
ericssmith
I agree with the author about History of Mathematics and Journey Through
Genius, so I did what he said and bought The Mathematical Experience
immediately.

------
swotavator
of all the technical things I have learned over the years, higher mathematics
(say calc 1-3 and diff. eqns.) is the topic I most regret brain dumping. I am
not so much interested in the history, but can anyone recommend a streamlined
primer?

~~~
rahimnathwani
The first few chapters of The Theoretical Minimum. There is a book and also a
free set of online lectures.

------
cdixon
Great list. I would also add:

What is Mathematics? [http://www.amazon.com/Mathematics-Elementary-Approach-
Ideas-...](http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-
Methods/dp/0195105192)

------
ekm2
#6. Euclid's _Elements_.

~~~
thejteam
I'll second that with caveats. "The Elements" requires quite a bit of
commentary to make sense and most editions seem to aspire more to
translational accuracy than mathematical understanding. But if you can get
past that it is the singular "must read" book in mathematics.

------
idoescompooters
Awesome. Just what I've been looking for!

------
_sabe_
It's easy to find good programming e-books, but e-books about mathematics are
harder to find. Does anyone know if there's a equivalent of O'reilly for math-
books?

~~~
tokenrove
Springer-Verlag has ebooks but they're incredibly expensive, unfortunately.

