
Classic Mathematics Books for Lifelong Learners - furcyd
https://medium.com/however-mathematics/13-classic-mathematics-books-for-lifelong-learners-7ec2759142da
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ltbarcly3
These sorts of books can be interesting as a sort of 'history of science', but
I find they are very misleading, and can't help but end up as exaggerated
dramatizations of a handful of the personalities involved with virtually no
scientific or mathematical content. I understand that this is probably the
best the authors can do, since actual content would require years of academic
preparation and near full time study on the audience's part to be able to
start to approach most of these topics, but that should be a huge clue that
the essence can't really be boiled down.

If any of the core ideas of these subjects were accessible in any significant
way via just reading a book with no real prerequisites or preparation, people
wouldn't have to spend 4 years of full time study just to get to the point
that _some_ of them are considered to be prepared to start to study them in a
serious way.

There may be some value to science in that these popularizations increase
support for science funding by creating 'fans of science', the people that
read them are no better off or more educated than if they had just read a
romance novel or a western.

TLDR; you aren't learning anything when you read these books, other than a
exaggerated biography (with largely invented stories of conflict and drama) of
some of the scientists.

~~~
g9yuayon
Did you read the book What Is Mathematics? It is certainly not "an exaggerated
biography for some of the scientists".

And the mentioned books, at least some of them, serve different purposes: they
aim to inspire, to motivate, and to offer historical context and intuition.
The last is especially important, as they show people how abstract concepts
emerged from historically concrete endeavors.

When it comes to learning math, you can't take an elite's view. Not everyone
is born Bourbaki dudes or Galois or people like them. Ordinary people like me
don't just fall in love with maths. I was certainly not interested in number
theory as I thought it was too fundamental for me to spend serious time on.
And I'm still not. I was certainly puzzled on why my professors introduced the
concept of functional in linear algebra or quotient groups in algebra or
lattice equations in program analysis or category theory in model checking or
probability space in probability. After all, all I wanted was to learn how to
model the world to be a better programmer. And I was _not_ able to grasp the
abstractions without serious effort. I _needed_ historical context and
motivations to plow through those topics and to enjoy math.

Yes, the elites will love and excel at math for no particular reason. Yet it
is the middle majority like me who will greatly benefit from the mentioned
books and biographies and what not.

~~~
ltbarcly3
I did a rough calculation, and far less than 1% of high school graduates in
the US will be introduced to quotient groups at any point. While you might be
in the middle majority of the people you interact with every day, you are
quite elite compared to the general population, and in no way in any 'middle
majority', and if anything you are doing what I was suggesting is necessary to
appreciate these topics. If you enjoy reading popularizations on top of that,
that's great!

~~~
g9yuayon
Middle majority among those who pursue STEM in college. Not everyone is or
should be in STEM, after all.

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billfruit
Many of the list are pop-maths rather than fundamental texts.

I do see many in software field recommend Chrystal:Algebra an Elementary
Textbook.

I also find Don Knuth, 'Concrete Mathematics' a interesting book for software
people.

~~~
mymythisisthis
Any good recommendations on books that are more than pop-fluff but lighter
than an academic text?

~~~
bgutierrez
I really liked Introduction to Graph Theory by Trudeau. No upper level math
required, but it got rigorous and fun at times. Mostly spatial reasoning.

David Foster Wallace wrote Everything and More which included a lot of math
history, while driving into transfinite numbers and set theory. Kindle
typesetting is generally terrible for math, but was completely readable in
this case. The forward by Neal Stephenson really sets the mood.

~~~
a_wild_dandan
> David Foster Wallace wrote Everything and More

This is far and away my favorite math book. The way DFW can seamlessly
transition from discussing a serious mathematical concept, to the nature of
insanity, to the philosophy of abstraction all within a few pages is
astounding. He somehow manages to do it while maintaining a conversational,
two-folks-at-a-bar tone, teaching me dozens of new word, and being
insightful/funny. Like an English professor sneaked into the math dept.

I didn't know it was available on Kindle. Guess I'm buying it again...

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raintrees
Likely listed elsewhere, from the article for convenience:

Zero: The Biography of a Dangerous Idea - Charles Seife

Measurement - Paul Lockhart

Prelude to Mathematics - W. W. Sawyer

Proofs from The Book - Aigner and Ziegler

The Joy of x - Steven Strogatz

Things to Make and Do in the Fourth Dimension - Matt Parker

What is Mathematics? - Courant and Robbins

A History of PI - Petr Beckmann

An Imaginary Tale - Paul Nahin

e: The Story of a Number - Eli Maor

Imagining Numbers - Barry Mazur

Journey Through Genius - William Dunham

Prime Obsession - John Derbyshire

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frostburg
The Courant-Robbins book is good but I wouldn't actually suggest it for
beginners, it's harder than necessary in many topics.

Concrete mathematics is great but not something you could suggest to, for
example, a biologist or medical doctor that wants a deeper understanding of
some aspect of mathematics. It's (explicitly) meant for computer science.

~~~
kayuri
I've read it in high school and was truly fascinated with which elegance and
simplicity the book approached some of the most complex math problems. Of
course it just scratched the surface but partially because of this book I
chose the Math Department. And our school teacher recommended Courant-Robbins
to all the students in a class, many of which actually joined Math Dept too.
So, it might be not just me. :)

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soVeryTired
For anyone with an undergrad maths background, I'd recommend 'Fearless
Symmetry' and 'Elliptic Tales'. They would be tough going for someone without
a maths background, but they give a view to the forefront of some active areas
of research in maths.

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mbrudd
Before buying a copy of _Prime Obsession_ , please learn more about the
author:
[https://en.wikipedia.org/wiki/John_Derbyshire](https://en.wikipedia.org/wiki/John_Derbyshire)

I bought and read his book some time ago, and now I regret supporting him with
my purchase.

~~~
yters
> There have of course been some blots on the record, but I don't see how it
> can be denied that net-net, white Europeans have made a better job of
> running fair and stable societies than has any other group.

If colonialism, WW2 and USSR are 'blots' he might have to reexamine his
definition of 'blot'.

White supremacists hold themselves up as people willing to speak difficult
truths. But their logic tends to be pretty question begging. It's like
atheists who claim religion is the root of violence, all the while ignoring
the genocidal atheists regimes of the 20th and 21st centuries.

~~~
p1esk
Which group did better than white Europeans?

~~~
yters
The groups that didn't start global wars or genocidal regimes that wiped out
hundreds of millions of people?

Another odd thing is that these white supremacists act like Judeo-Christian
culture (which is what they are really praising) sprang fully formed from the
Caucasian race. However, the white Europeans were just warring tribes until
they were colonized by the Romans. And, Judeo-Christian culture was founded by
Jews (Jesus was a Jew), whom the white supremacists tend to dislike. It'd be
interesting if they could name some significant aspect of Judeo-Christian
culture that came entirely from white Europeans. White supremacists come off
as bumbling ethnocentric historical revisionists.

------
amai
I can recommend
[https://en.wikipedia.org/wiki/God_Created_the_Integers](https://en.wikipedia.org/wiki/God_Created_the_Integers)

"God Created the Integers: The Mathematical Breakthroughs That Changed History
is an anthology, edited by Stephen Hawking, of "excerpts from thirty-one of
the most important works in the history of mathematics.""

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sonabinu
I would recommend "Fermat's Enigma" by Simon Singh. It's is book that goes
into the heart of how mathematics is sometimes elusive to the most sincere
pursuit and at the same time yields itself to a creative mind that chases it
with passion.

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jeff76
Any books on how to do proofs, that also includes solutions?

~~~
rguzman
How to Solve It by Polya.

~~~
emgee_1
This.

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winchling
_Proofs from the Book_ looks superb but does anyone know of another
introduction to proofs or compendium of simple proofs suitable for young
mathematicians?

~~~
dragon96
I'm a big fan of Art and Craft of Problem Solving by Zeitz.

The main downside is that there isn't an official solution manual, so checking
your work takes a little more work. But one of the main upsides is that there
isn't an official solution manual, so you actually have to work through the
problems. :)

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jim_bailie
As a primary guide to the subject, I would recommend "The Road to Reality" by
Roger Penrose.

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molteanu
Better read Euclid's Elements and then you'll know if you like maths or not.

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turingspiritfly
Why not The Princeton Companion to Mathematics?

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tzs
Some suggestions, aiming for books that will teach you some interesting math
rather than teach you _about_ some interesting math like most of the books on
that list do (except for "Proofs From the Book" and "What is Mathematics?",
which would be on my list below if they weren't already in the submitted
list). The following range all over the place in prerequisites, from things
you could probably do with just middle school algebra to things that probably
need early college level.

"Challenging Mathematical Problems with Elementary Solutions" by Yaglom and
Yaglom, Volume 1 and 2.

Volume 1 contains 100 problems from probability and combinatorics. Volume 2
contains 74 problems from a variety of areas including points and lines,
lattices of points in the plane, topology, convex polygons, distribution of
objects, nondecimal counting, theory of primes. Complete solutions are
included for each problems, as well as hints.

Available from Dover so relatively inexpensive but good quality. Here are the
Dover links, but of course they are available from Amazon and the other usual
places. I'm linking to Dover because that will have the most complete
description.

[http://store.doverpublications.com/0486655369.html](http://store.doverpublications.com/0486655369.html)

[http://store.doverpublications.com/0486655377.html](http://store.doverpublications.com/0486655377.html)

\----------------------------------------

"Three Pearls of Number Theory" by Khinchin. One of Khinchin's former students
was seriously wounded in WWII, and to pass the time during his long recovery
in the hospital he wrote to his old professor and asked if he had anything
mathematical to study to pass the time.

Khinchin wrote back with three problems in elementary number theory that had
recently been solved by people who were not a "great number theorist".
Khinchin gave his former student the proofs along with guidance, examples,
clarifications, and notes to help understand them.

Dover link:
[http://store.doverpublications.com/0486400263.html](http://store.doverpublications.com/0486400263.html)

Review at MAA: [https://www.maa.org/press/maa-reviews/three-pearls-of-
number...](https://www.maa.org/press/maa-reviews/three-pearls-of-number-
theory)

\----------------------------------------

"The Enjoyment of Math" by Rademacher and Toeplitz. The MAA review has a good
summary:

> This is a serious math book that has minimal prerequisites: geometry and
> college algebra, but no trig or calculus. It contains 28 largely independent
> chapters that solve a variety of famous and difficult math problems, mostly
> in the areas of plane geometry and number theory. The problems include:
> Fermat’s last theorem for exponent 4, unique factorization in number fields,
> a number of geometrical maximization problems including several versions of
> the isoperimetric problem, some transfinite numbers, the 5-color map
> coloring theorem, and the arithmetic mean - geometric mean inequality.
> There’s no analysis per se in the book, but several topics depend on the
> analytic ideas of continuity and variation.

> This book was first published in German in 1930 and in English in 1957 as
> The Enjoyment of Mathematics, and is still in print today in both languages.
> This implies that there is still an audience for it, but it is hard to
> imagine exactly what this audience is. The book was developed out of a
> series of public lectures and was intended as a “popular math” book. While
> it is very clear and well-written, the reasoning in all the chapters is very
> intricate (especially in the geometric problems), and the book is much more
> difficult than anything that appears in popular math books being written
> today. It’s also too difficult for a math appreciation text. The modern
> (2000) Preface to the German edition suggests that the book is suited for
> bright high-school students who are hungry for learning, and maybe this is
> its real audience today

[https://www.maa.org/press/maa-reviews/the-enjoyment-of-
math](https://www.maa.org/press/maa-reviews/the-enjoyment-of-math)

[https://www.amazon.com/gp/product/B07DMWX5FC/](https://www.amazon.com/gp/product/B07DMWX5FC/)

\----------------------------------------

Anneli Lax New Mathematical Library is a whole series of books described
thusly at the AMS site:

> Featuring fresh approaches and broad coverage of topics especially suitable
> for high school and the first two years of college, the volumes in this
> series are an excellent source of enrichment material for teachers and
> students. Good mathematical reading with lively exposition.

[https://bookstore.ams.org/nml](https://bookstore.ams.org/nml)

I read "Ingenuity in Mathematics" by Honsberger in high school and it was
good. Kind of like "The Enjoyment of Mathematics" but a lot easier.

A lot of books in this series can be good stepping stones to more advances
books. For example, Olds "Continued Fractions" could be a reasonable read
before then reading Khinchin's "Continued Fractions". The latter is available
from Dover and is about 1/3 the price of the Olds book, so personally I'd
start with Khinchin, and if it turns out a simpler intro is needed then I'd
get Olds.

This is a good point to toss in a note about Dover. They like to take older
books, often out of print, get the rights to them, and publish a relatively
inexpensive but high quality paperback edition. The difficulty level ranges
from classic elementary intro texts to advanced material for practicing
mathematicians. (And not just math...they do this for physics, chemistry, and
various other fields of science and engineering).

If you are interested on some math topic and want a book on it, it is usually
a good idea to have a look at the Dover catalog to see if they have something
about that at the level you are looking for.

\----------------------------------------

"A Book of Abstract Algebra" by Pinter, available as a Dover edition.

[http://store.doverpublications.com/0486474178.html](http://store.doverpublications.com/0486474178.html)

The usual undergraduate abstract algebra stuff: groups, rings, fields, the
impossibility of the classic Greek duplicating the cube and trisecting the
angle problem, Galois theory and solvability by radicals.

What sets this book apart is that although it is rigorous and proves nearly
everything, it takes things in smaller steps than a lot of other books, and
has a lot of well chosen exercises that further cement the material, often by
applying it to some interesting practical area. The exercises are grouped into
sections, each of which focuses on a particular concept from the chapter, or
develops and proves interesting things. One or two exercises from each of
these sections usually has a solution given.

Only about $12 at Amazon. If you haven't done much proof-based math before
this could be a good first proof-based book.

------
dankle
Behind paywall

