

Ask HN: Good Reading/Immersion in Mathematics - jawee

I'm interested in getting a good solid basis for study of mathematics. Right now, I am interested in a broad and not necessarily too deep to start with; instead, I want to be acquainted with all of the different fields beyond what I've done in my mathematics classrooms.<p>What books and web pages do you recommend I read, as well as what blogs and podcasts are good to follow to learn more on a constant basis.<p>Thanks!
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zdw
Pick up a discreet mathematics book. Learn logic, set theory, and writing
mathematical proofs (induction, etc.)

I really wish they'd put a class like that right after basic symbolic algebra
in normal school curriculum - it's far more useful in the modern world than
trigonometry.

~~~
kaens
I can't believe no one has mentioned Concrete Mathematics by Knuth & friends
yet.

It's incredibly well-written. Very challenging, yet totally approachable and
accessible to someone with even just a rather basic (even foggy) understanding
of "high-school" math.

~~~
sudhirc
What is the prerequisite for reading this book. Can I read this book
effectively with basic knowledge of high school algebra and some calculus?

~~~
kaens
Yes. I did. It will be _very_ challenging.

I, quite a few times, spent more than four hours with a pencil and paper
making my way through 2-4 pages of material, as since my background was
lacking, there were parts where I had to figure out how they got from A to B.
IIRC, I spent multiple days on a few paragraphs at one point.

That said, it is written how I think math books should be written, is
considered part of the start of Discrete Mathematics, and is conversational,
illuminating, well-layed-out, and approachable despite it's level of
difficulty.

I can't recommend it enough.

Don't view it as something where if you're not making progress you're failing.
Take the time to ensure you understand every bit of what you read before
moving on, and do all the exercises you can muster.

~~~
sudhirc
thanks a lot.

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ecaradec
Potentially related threads :

Math for hackers : <http://news.ycombinator.com/item?id=672067>

Good books on mathematics for somebody who's only taken high school math? :
<http://news.ycombinator.com/item?id=299687>

Recommendation for (re)learning Math Skills :
<http://news.ycombinator.com/item?id=1449799>

------
julietteculver
This is hard to answer without knowing something about your current
mathematical background and more about what you want to get out of learning
some mathematics.

Assuming that you want to learn some 'university-level mathematics', then
you'll really need to be prepared to study and work through problems rather
than just read. Mathematics is an area where it's hard to get breadth of
knowledge without also having at least some depth because things build very
much on each other.

If you really do just want an overview of areas of different areas of
mathematics to whet your appetite, there are books by people like Ian Stewart,
Marcus du Sautoy and Keith Devlin, all worth reading. Just be aware that
reading these is a bit like reading about different programming languages
without ever having written a computer program.

If instead you just want to keep your brain engaged mathematically without
learning more serious mathematics, there are also plenty of recreational
mathematics books out there - Martin Gardner being the name that instantly
springs to mind. On a similar vein, you may also enjoy the books of Raymond
Smullyan which are more focused on logic.

The only really nice non-textbook taster of university-level mathematics that
I have found in Alice in Numberland by Baylis and Haggarty. However it's out
of print so you might have problems getting hold of a copy. It is a lovely
book though if you can get your hands on it.

Otherwise you are looking at textbooks. I'd recommend maybe 'Introductory
Mathematics: Algebra and Analysis' by Geoff Smith as a gentle but rigorous
intro to the basics that I'd expect every maths student to learn at the start
of their degree course. There are lots of alternatives out there too though. I
taught myself lots from Herstein before going to university but that's pretty
heavy going and there are better books out there these days. If you look at
other books, I'd probably suggest getting one on abstract algebra maybe,
covering things like sets/functions and group theory rather than say analysis
or linear algebra to start off with, as it's easier to get into the right
mathematical mindset if you're not distracted by content which you already
have intuitions about.

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sqrt17
"want to be acquainted with all of the different fields" may actually be quite
hard - you can't do complex analysis, differential equations in multiple
dimensions etc. without having a very firm grip on standard analysis
(including all the proofs and definitions that they normally skip in high
school).

If you want to build up your math muscles (as a good preparation for actually
studying maths), you should have a look at some discrete mathematics books
(the one I had was "Discrete Mathematics" by Norman Biggs) as they teach you
to think in terms of proofs.

If you want to get a thorough foundation for non-discrete maths, you should
start with a good (university math) analysis textbook (No idea what's a good
one in English).

Another approach you could take is to take a math book that is targeted at
physicists and EE people - those usually skimp on the proofs and don't contain
enough detail to understand the fundamentals behind it all, but bring you to
the interesting (to physicists and EE people) stuff much quicker than a real
math course would.

Oh, and if you hang out on Youtube, be sure to watch the catsters - this is
category theory, presented by actual working mathematicians, at an accessible
level (and with a cute UK accent too).

~~~
rcthompson
Is there dependency graph of mathematical fields somewhere? That would be
useful.

~~~
sandGorgon
i'm not able to dig it up right away - but just a couple of days back there
was a blog post by someone who gave a long, very comprehensive listing of
topics that take you all the way from basic to postgraduate level math.

It was organized in a dependency graph kind of way.

------
acangiano
You may find this useful: <http://math-blog.com/mathematics-books/>
(DISCLAIMER: It's my site.)

~~~
anatoly
I like this list; thank you for taking the time to compile and annotate it.

------
Darmani
It may be helpful if you state your mathematical background, though I'm
guessing that, if you had taken any proof-based math course, you wouldn't be
asking this.

I recommend The Art of Problem Solving I and II. On the one hand, they're
intended for (mathletic) middle and high-schoolers. On the other hand, some of
their problems are quite challenging, and much of the material therein is what
my school teaches in its intro discrete math courses since very few students
learned it in middle and high school.

<http://www.artofproblemsolving.com/Store/contests.php>

~~~
jawee
I'm currently a junior in high school. I am doing some summer work in math and
considering doing it as a degree in college, but so far I have only taken
basic Algebra, Geometry, Trigonometry, and currently in Statistics (as well as
in a math-focused Computer Science track). I'm looking at Calculus I next
year.

------
vosper
I love Calculus Made Easy by Silvanus Thompson (updated by Martin Gardner).

~~~
wazoox
This is an absolute must. You can get it free (the original version) here :
<http://www.gutenberg.org/ebooks/33283>

------
ubasu
Here's my list:

Basic

1\. Chapter Zero (Carol Schumacher) 2\. Naive Set Theory (Paul Halmos)

Linear Algebra

1\. Finite-Dimensional Vector Space (Paul Halmos) 2\. Linear Algebra Done
Right (Steven Axler)

Real Analysis

1\. Real Mathematical Analysis (Charles Pugh) 2\. Introduction to Analysis
(Maxwell Rosenlicht)

Algebra

1\. A first course in abstract algebra (John Fraleigh)

These books are better for self-directed reading compared to some of the
classics like Rudin or Herstein. These should keep you busy for a while.

------
barik
Related to this, I've found most text books aren't the best way to learn the
material since they don't actually provide answers to solutions. You get the
theory but little followup practice to apply your understanding of that
theory.

Are there any good texts that are more problem workbook style? One example
that comes to mind is "Exercises in Probability", or some of the 3000 Problems
books as published by Schaum.

------
corey
I'm working through Calculus by Michael Spivak. If you want a thorough
knowledge of calculus(who doesn't?) then I wholeheartedly recommend it. While
it's more deep than broad, it will surely help you learn to start thinking
more like a mathematician.

<http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896>

------
J_McQuade
In my view, the best place to start for a good grounding in rigorous
mathematics is Velleman's 'How to Prove It'

As for a good broad overview of many areas, the title that springs to mind is
'the nature of mathematical modelling' by Gershenfeld, though you'd better
have some decent maths experience before tackling that one - it can be tough-
going, but is refreshing in its breadth and clarity.

------
gary201147
If youre looking to learn mathematics as a tool rather than an end goal the
list above seems far too abstract. A good foundation in analysis is probably
as abstract as you'll need for a majority of applied fields. For computational
science and learning you need to know (albeit very well):

linear algebra (strang, trefethen, golub and van loan) optimization (nocedal,
bertsekas) probability (rice, casella & berger, grimmett) statistical learning
(tibshirani, bishop)

A good free online book was recently an HN topic:
<http://news.ycombinator.com/item?id=1738670>

------
sz
Check out The Unapologetic Mathematician "blath":

<http://unapologetic.wordpress.com/>

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tgflynn
You might want to have a look at the "Princeton Companion to Mathematics".
It's goals seem close to your "broad but not too deep" objective.

------
leif
Linear Algebra Done Right, Axler. Don't learn Linear Algebra without it.

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tychonoff
[http://topics.nytimes.com/top/opinion/series/steven_strogatz...](http://topics.nytimes.com/top/opinion/series/steven_strogatz_on_the_elements_of_math/index.html)

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jpcosta
I really like this one: <http://academicearth.org/subjects/mathematics>

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drmoldawer
I'd definitely recommend Godel, Escher, Bach by Doug Hofstadter. Not just
about mathematics, but fascinating.

Also, anything by Martin Gardner.

~~~
Darmani
GEB is excellent as a pop-science/pop-philosophy book, but (like most pop-sci
books) it's terrible for actually learning any subject. You need more than one
proof every 200 pages.

~~~
kaens
Yup. It gave me a nice intro to formal systems / number theory iirc. Intro
only though, and it was only a small part of the book.

