
Ask HN: Exploring math without a teacher? - glebm
The ability to understand the beauty of math requires rigorous study. However, most people do not have access to the kind of training pure math requires.<p>Many of my friends easily get interested in the subject when they are shown beautiful proofs and constructs, but I am at loss when they ask for resources to learn more, since we had never used books in our studies. During class, the teacher would show the art of math by proving theorems and building constructs together with the students using only the whiteboard.<p>I am looking for books or online courses on that are suitable for beginners (while still formal and not dumbed down) and emphasize the beauty and love for math, on the following subjects:<p>* Intro to sets, mappings, boolean logic, predicate theory<p>* Number theory<p>* Rings, fields, groups, vector spaces, Galois theory, etc<p>* Set theory, measure theory, functional analysis<p>* Combinatorics, graph theory<p>* Language theory, lambda calculus<p>* Other subjects that are not in this list<p>Books rigorously building the field entirely from ground up (axioms) with detailed looks at all the important proofs with multiple versions and highlighting relations to other fields would be best. It is good if the book has exercises in the form of main story lemmas and side story &#x2F; related proofs.<p><i>Please share your favorite pure math works below.</i>
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picomancer
Fraleigh's abstract algebra text [1]. Sipser's _Introduction to the Theory of
Computation_ [2] is also excellent, and possibly of special interest to HN.

[1] [http://www.amazon.com/First-Course-Abstract-Algebra-
Edition/...](http://www.amazon.com/First-Course-Abstract-Algebra-
Edition/dp/0201763907/ref=sr_1_1)

[2] [http://www.amazon.com/Introduction-Theory-Computation-
Michae...](http://www.amazon.com/Introduction-Theory-Computation-Michael-
Sipser/dp/053494728X/ref=sr_1_2)

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thejteam
If you are looking for an accessible book on Measure Theory, I like "Lebesgue
Integration on Euclidean Space" by Frank Jones. A good undergraduate level
textbook, problems integrated into the text instead of at the end of the
chapter. I wish I hadn't given my copy away now.

For number theory I liked "Number Theory and its History" by Oystein Ore. Best
part is it is cheap. Good intro to number theory, it was accessible to me in
high school (I was advanced, but no genius) and I learned things that I never
even encountered in college as a math major.

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tokenadult
Almost any book by John Stillwell

[http://www.amazon.com/John-
Stillwell/e/B001IQWNS2/ref=sr_tc_...](http://www.amazon.com/John-
Stillwell/e/B001IQWNS2/ref=sr_tc_2_0?qid=1382393789&sr=1-2-ent)

meets your requirement of "It is good if the book has exercises in the form of
main story lemmas and side story / related proofs." I like Stillwell's books a
lot for readability and starting from the basics.

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Grothendieck
Real analysis lectures of Francis Su at Harvey Mudd:

[http://www.youtube.com/playlist?list=PL0E754696F72137EC](http://www.youtube.com/playlist?list=PL0E754696F72137EC)

These could go with the either the book by W. Rudin or (haven't read) A.
Browder. These are not easy for a complete beginner; maybe the lectures can
provide some motivation. "Understanding Analysis" by S. Abbott is another
rigorous and much easier, but very good, introduction.

"Advanced calculus" by Shlomo Sternberg ... a work of profound beauty. This
was the book used at Harvard in the 60s for the best freshman students, but it
begins in a slow yet deep way with sets, logic, linear algebra, calculus,
metric spaces... It's my favourite book on calculus on manifolds.
[http://www.math.harvard.edu/~shlomo/](http://www.math.harvard.edu/~shlomo/)

Some freely available books by Robert Ash:
[http://www.math.uiuc.edu/~r-ash/](http://www.math.uiuc.edu/~r-ash/) \- in
particular, his misleadingly-named "Complex variables" (with W.P. Novinger) is
a short, rigorous book on complex analysis.

"Naive Set Theory" by Paul Halmos is now available for something like $12.

(I have not read the following books.)

For number theory, maybe the book by George Andrews? It's very elementary and
very cheap, and looks top notch.

I'd like to read "The Cauchy-Schwartz master class" (on inequalities) but
haven't purchased it yet.

There are many books on combinatorics and graphs by the Hungarian school.
Probably deserving special attention for discrete math are "Concrete
mathematics" by Knuth et al. and "Analysis of algorithms" by Sedgewick and
Flajolet (distinct from Sedgewick's "Algorithms").

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greenlakejake
For number theory try this college textbook: A Friendly Introduction to Number
Theory

The first six chapters are online so you can see whether you like the
approach:
[http://www.math.brown.edu/~jhs/frintch1ch6.pdf](http://www.math.brown.edu/~jhs/frintch1ch6.pdf)

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X4
I'm on a similar path, but I decided to learn it through a MOOC. Why don't you
also join a MOOC? I'm not sure where and how to do this atm, so suggestions to
do this in Germany are welcome. I'll do a master's degree in a field of Maths
with a MOOC.

Btw. we learned most of the stuff above in CompSci throughout the first three
semesters. If you ask if it's worth the money: "To be honest, all I did was
learn the "script" and "Wikipedia"." You can do better at home. In some cases
I needed more info, when Wikipedia went mad about unnecessary "details".

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glebm
Thank you for these great suggestions, please keep making them! Later I'll
publish them as a coherent list here:
[http://blog.glebm.com/2013/10/21/exploring-math-without-a-
te...](http://blog.glebm.com/2013/10/21/exploring-math-without-a-teacher.html)

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impendia
"Elementary Number Theory" by Underwood Dudley is great.

Also, I enthusiastically second tokenadult's recommendations of all of
Stillwell's books, which has quite rightly been voted to the top.

