
Learning Math - dwaters
I am in my early thirties and honestly, I cant claim to have a good background in math.<p>But, I now have a burning desire to learn it from the ground-up.<p>What are the 'canonical' sources for math, both online and offline? I am lost as to where I should start. I want to have a fundamental, intuitive understanding of it.<p>To clarify, I would not consider it shameful to start at whatever level necessary (even the lowest, if required).
======
iamwil
I'm in my late twenties, and even though I did comm systems as a EE undergrad
(lots of math), I still don't feel like I really have a deep understanding on
probability/stochastic processes, differential equations, and complex
analysis.

www.betterexplained.com has some good tidbits for math.

I've found MIT's opencourseware to be a pretty good help:
[http://ocw.mit.edu/OcwWeb/web/courses/courses/index.htm#Math...](http://ocw.mit.edu/OcwWeb/web/courses/courses/index.htm#Mathematics)

Only the undergraduate courses tend to have video lectures though. The ones on
linear algebra and diff eq are quite good. When I first learned matricies in
high school, the teachers just went through the mechanics of how to manipulate
them and how to calculate a determinant. It wasn't until years later, and when
I started wtching these lectures that it crystallized for me what it actually
meant.

These are monthly lectures on math topics, which have been enlightening.
<http://www.ams.org/featurecolumn/index.html>

If you like exploring, there's this: <http://www.jimloy.com/math/math.htm>

some free online texts
<http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html>

If you haven't had a course before, you can just follow the usual sequence of
courses that high school students and college students take. calculus, multi-
var calc, diff-eq, linear algebra, probabilty. And get a textbook and work
through the problems. If you code, discrete math will probably help somewhere
along the way. Probability/stat for machine learning.

If you've already had the stuff before, It might help just to pick one small
topic in a math field, like gradients in multi-var calc, and just focus on
that for a bit, and inevitably, it'll mention some other math tool that you
don't know about, and just follow your nose and interests.

What I didn't learn until after I finished undergrad is that if you want to
really understand conceptually what things mean in math, and not just how to
manipulate symbols, there's no getting around working on problems paper (or
matlab/mathematica) and just playing with it.

Hope that helps.

~~~
mattculbreth
Wow, a link to Killer Cain's site at Tech. That is awesome. I had Cain's
Calculus classes at 8:00 am for my first year at Tech, and that's a hell of a
way to get welcomed to college. He's still one of best professors I ever had.

------
far33d
This might sound silly - but I've found the best way to learn anything about
math is to start w/ Wikipedia.

Search for a topic you are interested in, like Calculus. Start there. Spend a
few hours reading and clicking through links, finding books that are cited,
etc. If you don't understand something, usually some link will have the
background information you need.

Do this every week or so.

~~~
paulgb
Wikipedia is great for math topics. Also, PlanetMath
(<http://planetmath.org/>) and MathWorld (<http://mathworld.wolfram.com/>) are
both good free online math encyclopedias as well.

------
viergroupie
First of all, kudos on the turnaround. I think the most important thing to
keep in mind is except for a few scattered geniuses, no one is inherently good
at math. I know it sounds like a tired metaphor, but math really is a
different language (which takes a lot of study and practice to achieve
fluency).

If you lack fundamentals, skip the sources and look for the people. Find a
good teacher and take their class. This will probably entail an intense week
of course shopping at a local school. Friendly fellow students can be a huge
boon, at least for me, since I solidify concepts best in conversations with
peers.

If you, for some reason, absolutely cannot take a class in person, I would
encourage you to find a study partner and watch online lectures together. I
really like MIT's Linear Algebra and Differential Equations videos
([http://ocw.mit.edu/OcwWeb/web/courses/av/index.htm#Mathemati...](http://ocw.mit.edu/OcwWeb/web/courses/av/index.htm#Mathematics)),
but but I don't know if those are at your level.

~~~
dwaters
Thanks!

------
gms
The book 'Calculus' by Michael Spivak.

In the same way that SICP transforms you from a high-schooler into a wise
adult when it comes to programming, so too does Calculus when it comes to
maths. If you find the book to be heavy going, then read whatever preliminary
material you need, and go back to it.

Edit: I should also stress that maths requires a fair amount of discipline (a
lot more than programming), so it's really hard to study maths while also
having a day job.

~~~
whacked_new
Have you ever seen one of those REA problem solvers books? They are thick
books full of problems and solutions (and explanations) and are organized from
beginner to advanced.

Do you think this would lead to a more solid foundation (from less
frustration), for self studying, than reading from a thorough but dense text?
I don't know Spivak's Calculus, but some reviewers on Amazon compare it to
Apostol, which I found so abstract, and so unpractical, that I promptly forgot
everything. It is now on my to-read list, but like you said, I won't be
starting until I can dedicate myself to studying it, and now that I have seen
the REA book, I wonder if it would be better to work on that book, as a
refresher and foundation builder.

Oh yeah, dwaters, if you happen to be interested in Apostol, and want a study
buddy, I nominate me.

------
rkts
If you want fundmantal, intuitive understanding I suggest _What is
Mathematics?_ by Courant and Robbins. I got this after pg recommended it in an
essay, and it's wonderful.

------
fauigerzigerk
Steve Yegge says some interesting things about various branches of maths and
how they relate to programming:

[http://steve-yegge.blogspot.com/2006/03/math-for-
programmers...](http://steve-yegge.blogspot.com/2006/03/math-for-
programmers.html)

------
DaniFong
The book _What is Mathematics_ by Courant et. al. is a terrific, read, and
highly recommended from generations of mathematicians. It's one of the few
math books good enough to compel you to read through the whole thing.

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

There's also the Princeton Companion for Mathematics, which isn't out yet but
is available online. It's a wonderful book.

<http://pcm.tandtproductions.com/>

User: Guest Pass: PCM

------
Trep
I remember when our math teacher stood in front of us in High School and told
us math was important because of all the ways we'd use it...

She lied. I think I used the Pythagorean Theorem once, for a cool bookshelf,
five years ago.

Now, I'm learning it all over again. One thing I've found is that if I can
teach something, I understand it better. In that spirit, I'm volunteering with
the local Alternative School, tutoring math. It works out great. I keep myself
about two chapters ahead, the kids are bright and unafraid, and I find myself
motivated ever time I plow into my own book.The best thing is, I spend less
time tutoring the kids than the rest of my morbidly unmotivated 'catch up'
math class spends practicing problem sets. I'm saving time, having fun, and
understanding.

I highly recommend it.

~~~
wallflower
I've found that we don't know how much we don't know until we try teaching it
to someone. Congratulations for choosing to tutor math. Long live SOHCAHTOA.

------
larryfreeman
Hi,

I had the same feeling as you (when I was 38) and I started a blog:
<http://fermatslasttheorem.blogspot.com>

I started with a sketch of the historical approach to Fermat's Last Theorem
(Diophantus, Euler, Gauss, etc.) and I tried to resolve all proofs down to
postulates.

I would recommend finding an area that you are interested in and then finding
a famous problem. My favorite books are:

* Euler: Master of us all

* Euclid's Elements

* Hardy and Wright: Introduction to a Theory of Numbers

* Jonathan Stillwell: Yearning for the Impossible

* Ash and Gross: Fearless Symmetry

* Jean-Pierre Tignol: Galois' Theory of Algebraic Equations

* Edwards: Fermat's Last Theorem/Galois Theory

I've been doing it now for almost three years and it's really worked out well
for me and surprisingly, I am getting around 11,000 unique visitors to my site
every month.

Glad to see all the responses to your comment. :-)

-Larry

~~~
bayareaguy
I love your blog. I used to follow it a few months ago (but then I got busy).
Keep up the good work.

------
asdflkj
Start with mathematical logic, set theory, abstract algebra and number theory,
in that order. Do not follow the usual course of calculus, differential
equations, linear algebra and so on. That is, do learn those things, but later
on.

The problem with math education is that "the basics" (things that I recommend
you start with) are neither easy to understand nor obviously useful in "the
real world". Or at least the latter was true before computer science came
along. But most educational programs were established before CS, so basic math
is regarded as something you don't really need to know. But you do, if your
goal is to understand math, and not to be able to design bridges as soon as
possible.

Now universities are gradually fixing the situation. They still start you off
with calculus and such, but before you go on to more rigorous classes like
Analysis or abstact algebra, they give you a "transition course", which is
essentially a survey of the basics.

~~~
vitaminj
I certainly agree that logic, set theory, etc are the formal bases of
mathematics, but I wouldn't say they are the basics. It'd be like learning the
syntactic rules of grammar before learning words and constructing simple
sentences by rote. Sometimes it's better to have an appreciation of the goals
(which are easier to learn) before embarking on the fundamentals (which are
rigorous but abstract).

When I first started learning set theory, I wondered why this wasn't taught
first since it was so fundamental. It took me a while to realise that I
wouldn't have understood any of it, because you need some measure of number
sense and a moderately well-formed abstract reasoning to appreciate this
stuff.

Throughout my experiences in learning, I've always found that it is a zig-zag
path - learning the superficial or applications, before drilling down to the
fundamentals, and then going back to applications with a new sense of
appreciation and so on. Going from the bottom up sounds to me like a recipe
for losing interest in the subject very quickly.

~~~
asdflkj
I agree about the zig-zag path, but I don't think it necessarily means you
should study calculus before set theory (for example). Only that you shouldn't
delve too deep on your fist pass over set theory. Obviously, AOC independence
proof would not be suitable for high school students. But it doesn't take much
effort to understand what a function is in terms of sets. And don't they teach
Venn diagrams to kids in grade school? That's already set theory.

~~~
smopburrito
along these lines check out: Vector Calculus, Linear Algebra, and Differential
Forms: A Unified Approach by Hubbard and Hubabrd

[http://www.amazon.com/Vector-Calculus-Linear-Algebra-
Differe...](http://www.amazon.com/Vector-Calculus-Linear-Algebra-
Differential/dp/0130414085/ref=sr_1_1?ie=UTF8&s=books&qid=1202163492&sr=8-1)

------
projectileboy
Funny you should ask... I took a lot of math classes all through college, but
at every step I was reaching beyond my grasp, so I never had great
understanding. And then after a few years, I forgot it all. So in my early
thirties, I started over.

Much more important than which text you use is your attitude, and a
willingness to _really_ walk through and understand the proof of a theorem,
and a willingness to work through problems. Having said that, here's what I
did:

Go through the chapter in Feynman Lectures on Physics, Volume I, where he
starts with integers and goes through trigonometry until he winds up at
Euler's Theorem. Do this, and you'll really understand numbers (as well as
algebra and trig).

Then I went through the appendices of my college calculus textbook to pick up
some algebra tricks I had never really learned. (This is a recurring theme,
BTW: you learn a fundamental idea, and then there a bunch of tricks around the
fundamental idea that enable you to actually solve problems. So, to really
"get" math, you need to truly understand the most important fundamental ideas,
_and_ you need to learn some of the problem-solving tricks.)

From here, the school route is to press on to calculus. What's more practical
is to actually learn and understand some probability and statistics.
Especially Bayesian reasoning (<http://yudkowsky.net/bayes/bayes.html>).
Understanding statistics and probability will actually improve your everyday
life. But assuming you still want to press on to calculus...

You need to learn about limits. Actually work through some limit problems. And
then you need to read through the definition of a derivative, and compute some
derivatives by hand, computing the limits. And then you'll really understand
derivatives.

(By the way, when you understand derivatives, you also understand differential
equations. When people take differential equations classes, they're just
learning the bag of tricks used to solve different patterns of differential
equations.)

Now read through the proof of the mean-value theorem until you get it. This
will enable you to understand the fundamental theorem of calculus. And so now
you understand integrals. There's a bag of tricks around solving integrals
which you can learn. At this point you could also start toying around with
Mathematica; you now know just enough to begin appreciating how cool it is.

Once here, most math courses take a little detour and teach some numerical
methods. I wouldn't sweat it too much, although it's a good trick to know that
you can express a lot of different functions (e.g., y = the sine of x) as
algebraic series, because it lets you approximate solutions to problems).

Now learn about vectors and simple vector algebra, which is just enabling you
to generalize your understanding to multiple variables (e.g., z = x^2 + y^2).
This will introduce different flavors of derivatives, as well as some
different flavors of integrals. Just go get the book "Div, Grad, Curl and All
That". You'll need to read a different book to read and understand the theory,
but reading Div, Grad, Curl will give you an intuitive feel, which can be a
big hurdle to getting multivariable calculus.

Before, during, or after your study of "Div, Grad, Curl...", you might want to
learn about matrices, which is a short hand for writing systems of equations
that transform one vector space into another vector space. This is worth
knowing if you really want to understand 3D graphics programming.

And now you know as much math as your average physics or engineering student,
although you should learn about Fourier analysis, because it's fun, and then
you'll understand how your CD player works.

You could quit at this point, and you'd be in pretty good shape, but
everything you've done up to now falls under the heading of "applied math". If
you want to get a taste of what most mathematicians do, you'll need to look at
what's called "abstract algebra". This is actually a ton of fun - just think
of it as a big ol' puzzle: what if you tried doing "math" with stuff other
than numbers? The most general notion is that of a set. And then you can learn
about "groups", which are sets with a little more structure, if you will. And
then you go on to "rings". And then "fields". For all this stuff, go get
Herstein's "Topics in Algebra". It's far and away the best text.

That's as far as I got. I suspect there's a ton of other fun things out there
(number theory? graph and network theory?), but I don't know anything about
it.

~~~
yummyfajitas
If the original poster is a computer person, I'd recommend some changes to
this list. This is "Math for physics" but "Math for Computer Science" is
rather different, and might be more interesting.

I'd recomment calculus up to integration. Don't worry about integration
tricks, except for integration by parts (the most important formula in
mathematics) and u-substitution. All the other integration tricks are
pointless crap used to fill up time in calc classes.

Vector math is useful if you like either computer graphics or physics, but is
not crucial.

On the other hand, everyone should know probability, even the purest
mathematicians. Just don't try to learn it out of an "Introduction to
Probability and Statistics for Engineers" book, all such books should be
burned. Real/functional analysis would also be useful to better understand
probability.

I'd also suggest combinatorics/graph theory, and perhaps the theory of
automata. That's edging towards computer science, but it is a fundamentally
mathematical topic.

Also, it will be very __slow __going. It's not like picking up another
computer language/framework; it's even harder than Haskell. I've have a Ph.D.
in mathphys/num analysis, but it still takes me a long time to push through an
introductory textbook in a field too far removed from my own. For instance, I
sat through 4 semesters of abstract algebra (3 at the grad level), and I still
don't understand it. Don't get discouraged.

------
mooneater
From your description, its not clear what level you are at, or what level you
want to be at. What do you know now? And what do you want to achieve with
math?

If you are going for the roots of it, after Arithmetic then basic Algebra in a
prerequisite for all further math learning - manipulating and solving
equations using variables. Be very comfy with that before proceeding.

Then try geometry, basic stats, and understand the ideas behind calculus.
Linear algebra.

If you are still interested, look at dicrete math at least to know what it is.
Learn about frequency domain and Fourier analysis, and numerical methods, at
least to know what they are about. All these areas go much deeper than I care
to venture. I think a breadth-first search will arm you with the best
perspective. But as you get further in you see how these different types of
math overlap and combine in various ways.

And that's as far as I ever got =) Like you I still want to learn lots more
math, and I believe that's a life long process.

~~~
dwaters
Thanks!

------
mattculbreth
I've been doing the exercises at <http://projecteuler.net> and that seems a
good way to stay (or get back) in practice. I'm doing these problems in OCaml
so I'm killing two birds with one stone.

~~~
paulgb
Project Euler is great for practice, but it doesn't really give you a well-
rounded math education. The problems focus mainly on number theory and a few
other small areas.

------
mattmaroon
I highly recommend starting with Theory of Poker by David Sklansky for basic
prob/stats knowledge. It makes a good primer since it's all real-world
applications and I know of multiple universities and corporations that use it
as such.

------
sgoraya
I really found the following blog helpful for various math topics

[http://betterexplained.com/articles/how-to-develop-a-
mindset...](http://betterexplained.com/articles/how-to-develop-a-mindset-for-
math/)

In addition, I have found myself reviewing 'Discrete Mathematics w/
Applications' by Epp on numerous occasions;

My .02 - review logic / discrete math & move from there - The fundamentals of
math logic and reasoning have been very important for me - I always felt this
type of course should have been taught before a lot of the other subjects like
Calc / Diff Equations etc.

Good learning!

------
mark-t
The only way to learn math is to do math, and you should learn to do math the
same way you learn to program: solve problems that you find interesting.

Personally, I like to do contest problems. You can find tons and tons of them
on John Scholes's website: <http://www.kalva.demon.co.uk/> but they tend to be
on the hard side. The AIME is possibly the easiest on there, and those are the
sort of problems where the average high school only has one student every
decade or so that can solve any.

There are contests for junior high students and less advanced high school
students, too, but I can't find the problems online. There are books, though.
You might want to look at the AMC 8, 10, and 12.

Or, you know, make some up. Think of a problem you've always been curious
about, and try to solve it. There are forums online that can offer guidance.
For example, I work on <http://math2.org/mmb/> .

------
arvid
You really need to learn Math by doing it there really is no other way. Books
are really just references and guides and can give you good problems to work
from. Either find a friend who will study with you or get direction from a
math professor.

I do not know Spivak's Calculus but his advanced books (by Publish or Perish)
are excellent, So I assume his calculus book is also. Especially Calculus on
Manifolds and A Comprehensive Introduction to Differential Geometry. Anyone
who wants to understand calculus on higher dimensions should read Calculus on
Manifolds.

If you want to learn from the masters and you have the confidence, audacity
and intelligence. I would suggest Fundamentals of Abstract Analysis by Andrew
Gleason and Geometry and the Imagination by David Hilbert.

Just a warning. These books are for people adept at mathematics and are
willing to spend hours on a page or two. If you are not, then avoid these
books.

------
hugh
If you want to go to the fundamentals, I'd suggest The Principles of
Mathematics by Bertrand Russell. It won't tell you much about the specific
sub-branches of mathematics, but it's a good introduction to mathematics the
way mathematicians see it, as opposed to mathematics the way high-school
students (or teachers, for that matter) see it.

To give you an idea of the flavour of it, here's the opening sentence:

"Pure Mathematics is the class of all propositions of the form p implies q,
where p and q are propositions containing one or more variables, the same in
the two propositions, and neither p nor q contains any constants except
logical constants."

It's way out of copyright, so there's an online version:

[http://fair-use.org/bertrand-russell/the-principles-of-mathe...](http://fair-
use.org/bertrand-russell/the-principles-of-mathematics/)

------
jonp
"1089 and all that" by David Acheson is charming and very readable. It covers
a variety of different types of maths but more importantly get across what
maths is really about. I've bought it for friends and family with a wide range
of maths backgrounds (graduate mathematicians to not-since-school-forty-years-
ago).

Also "Alice and Numberland", Baylis and Haggarty; and "The Foundations of
Mathematics", Stewart and Tall. These are both pitched somewhere between high
school and university level and bridge the gap well.

------
tx
I find it helpful to just explore the areas of math that "stand in the way"
when I am studying something practical. After all, math is rarely a target,
math is a language that you use to solve a problem.

Sometimes a wikipedia article (along with sources at the bottom) is enough,
sometimes I'd need to buy a book, like in case with statistics. Amazon book
reviews are usually very helpful and good math books are quite expensive
(easily in $100+ range) but can be had for a fraction of original price when
bought used.

~~~
dwaters
Thanks!

------
doubleplus
Thanks for posting this... bookmarking it for the answers. I'm early-thirties,
too, and while I finished the first year of calculus (10 years ago!), I
screwed up and got a PoliSci degree. Now I'm looking at maybe going back for
CS, but I need to relearn trig and calc so I can finish the physics and diff
equations prereqs. It would be nice to not have to spend a year retaking the
classes.

------
iratsu
One thing I would recommend looking at is Category theory. MarkCC has several
blog posts about category theory at the Good Math Bad Math blog
(<http://scienceblogs.com/goodmath/goodmath/category_theory/>, in general i
would also recommend looking at MarkCC's other blog posts as a source of math
inspiration).

Wikipedia also has a nice article about category theory.
(<http://en.wikipedia.org/wiki/Category_theory>), and I will also recommend 2
books on the subject:

First is, of course, Category Theory for the Working Mathematician by Saunders
Maclane. It is an excellent book, but I must also warn that it is not an easy
book and probably requires a fair bit on mathematical maturity.

Second is Basic Category Theory for Computer Scientists by Benjamin Pierce. It
is much more of an introductory book and provides many examples in Computer
Science which are useful to those doing theoretical CS.

------
giardini
Before digging into the various books others have suggested, you would do well
to read "Where Mathematics Comes From" by George Lakoff and Rafael Nunez:
[http://www.amazon.com/Where-Mathematics-Comes-Embodied-
Bring...](http://www.amazon.com/Where-Mathematics-Comes-Embodied-
Brings/dp/0465037712/ref=pd_bbs_sr_1/103-1216395-3355860?ie=UTF8&s=books&qid=1202144892&sr=1-1)

That book explains the origins and understanding of the basic items of
mathematical analysis: infinity, sets, classes, limits, the epsilon-delta of
calculus and alternatives, infinitesimals, etc. The explanation is from the
viewpoint of psychological understanding. It details how we build up a
scaffolding of tools (starting with basic counting) sufficient to slay the
dragons of modern physics and mathematics.

------
william-newman
"But, I now have a burning desire to learn it from the ground-up. What are the
'canonical' sources for math, both online and offline?"

It'd be easy to spend multiple lifetimes studying math, so you'll have to set
some priorities. Applied vs. pretty, pragmatic vs. rigorous, discrete vs.
continuous, and various subfields within "applied," e.g. So presently, when
you have a better idea what your priorities are, you'll probably want to pose
a variant of the question again.

(E.g., not "what are the 'canonical' sources for math" but something as
specific as "what are the 'canonical' sources for math leading up to what I'd
need to understand X" where X is something like "the cryptanalysis of the Data
Encryption Standard" or "the proof of Fermat's last theorem [good luck:-]" or
"why people think Y's work was important" where Y is Galois or Hilbert or
Ramanujan or Noether or Erdos or Matiyasevic or whoever.)

Meanwhile, if you just want to see what the fuss is about before trying to
formulate a more specific question, I can recommend any of four kinds of
samplers.

1\. For about 80-90% of ways of analyzing the physical world, one really wants
to know calculus. Get _A Concept of Limits_ (cheap from Dover), the three most
promising calculus books from your local library (and/or webbed tutorials),
and a basic dealing-with-the-physical-world book which assumes you know
calculus (e.g., just about any serious physics text, or _The Art of
Electronics_, or something acoustics or signal processing or whatever). Keep
fiddling with them, and doing exercises as necessary, 'til the pieces fit
together.:-| Expect it to be quite a lot of work --- by my estimate, freshmen
and sophomores at Caltech in the 1980s generally spent at least 250 hours on
it, sometimes more like 1000. And it will probably be much easier if, like
them, you can arrange to get at least 1 hour of feedback every 20 hours of
study from someone who already understands the stuff.

2\. For anything in computers, getting familiar with the basic math of
reasonably serious algorithms is really useful. I, like many people, like
_Introduction to Algorithms_. Get it and study it; understand at least a
representative number of chapters. My estimate is that this is a lot easier
than option #1, maybe five times easier. It isn't anywhere near as big a
hammer for dealing with the physical world, but it can be extremely handy for
dealing with the software world.

3\. If you want to see what all the fuss is about in some representative areas
of less-physical, less-computer-y math, I know of two Dover books which try to
drag you from advanced high school math to a famous math result. _Abstract
Algebra and Solution by Radicals_ drags you through (the modern, cleaned up
and rigorous version of) Galois' proof that there is no closed-form formula
for solving polynomials of fifth order. _Computability and Unsolvability_
drags you up to Matiyasevic's proof that Hilbert's tenth problem is insoluble.
Working through either of them in detail would be a lot of work, almost
certainly more than you want to do if your interest turns out to lie in
something else like graph theory or algorithms or topology or statistics. But
you could probably learn a lot about roughly how things are done merely by
skimming either of them a few times. (And if just seeing broadly how things
are done is your priority, you might prefer _AAaSbR_, since showing broadly
how things are done seems to be one of its priorities too.)

4\. Peter Winkler's newish (2004) _Mathematical Puzzles_ book is also very
good and very math-y and well worth looking at as a sort of inspiration.
However, if you ever get tempted to think that the extreme elegance of puzzle
solutions is representative of how math gets done, look back at section 3
before jumping to conclusions.

"I am lost as to where I should start. I want to have a fundamental, intuitive
understanding of it."

My closest thing to a literal answer to that would be: read _AAaSbR_. Like it
very, very much.:-) Like it _so_ much, in fact, that you are motivated to
really study something like _Algebra_ by MacLane and Birkhoff (which is like a
big watershed in which _AAaSbR_ is but one stream). After you get your mind
around a good chunk of that (enough that you feel no great fear of an open-
book exam composed of exercises from your choice of 20% of the chapters, say),
do some variant of the calculus stuff I described in section 1 to see how
abstract math ties into the stuff people analyze in the physical world. But I
doubt in fact this is what you want. I suspect it'd be more than a full-time
year of work for most people. And even if you had the time and energy, well
before you finished I think you'd probably prefer to stop studying the
foundational stuff so deeply and start to climb up some shortcut to some
application or subspecialty.

Incidentally, mooneater's advice "algebra [...] Be very comfy with that before
proceeding" is good... but note that it's referring to a high school algebra
which has rather different priorities from something like what MacLane and
Birkhoff mean. Don't try to follow mooneater's advice by going to a university
library, taking down a book titled "Algebra," and running away screaming "math
is not for me." I learned a lot of useful math, did my Ph. D. on quantum
mechanical Monte Carlo simulations, and only understand a little of MacLane
and Birkhoff (but have looked parts of it in order to try to understand a
little bit about "categories" and some other stuff, and would consider more
time spent understanding it to be time well spent).

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jrnewton
I'm in the same boat, right now working towards calculus, running through an
algebra review. The big problem I have with some of the "classic" texts (and
wikipedia/mathworld) is that they go way over my head very quickly, usually
due to my lack of knowledge regarding the nomenclature and symbols of math. Or
that I lack the basic knowledge to understand why an equation can transform
from state a -> b. I've found that the Barron's College Review series (on
algebra and calculus) to be easy to digest. I'm also using Algebra &
Trigonometry by Sullivan as a reference for issues that are fuzzy in the
Barron's book. I also plan on visiting sets, graphs and logic.

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fugue88
I have noticed that learning math from lectures (and similar means) versus
learning math by reading a math book are very different skills.

Learning from lectures may be very good to get a handle on the basics. You may
learn the facts faster, and keep motivation up.

But to learn beyond the basics, especially areas that are less likely to be
covered at your local community college, you should learn how to learn from a
book (this _is_ different than learning from books in other subjects).

Perhaps, if you take some lectures, you may want to read a chapter or two
ahead of the lecture, to get used to reading a math book as a primary source.

Good luck! Math is beautiful!

------
Kaizyn
There's an author Morris Kline who wrote a number of fairly useful books on
mathematics. Particularly worth mentioning are Mathematics for the
Nonmathematician and Calculus: An Intuitive and Physical Approach. The former
traces through the history of mathematics and places each of the math
discoveries in their proper context. The latter covers most of the topics
covered in the first couple calculus classes. What makes Kline's approach
valuable is how he grounds the abstract math topics in practical examples and
in ways making them more accessible than your typical textbooks.

------
simplegeek
There are just so many sub-branches? Can you please specify one, e.g. Number
Theory, Graph Theory, and etc? Are you looking for resources to learn
theoretical computer science?

~~~
dwaters
Simplegeek, thanks for the reply. I believe I am not at a level where I can go
depth-first into a certain sub-branch. I want to start with the basics.

~~~
simplegeek
Alright. Well, I would suggest you to take a look at Martin Gardner's
work([http://www.amazon.com/s/ref=nb_ss_gw/102-4845988-2776161?url...](http://www.amazon.com/s/ref=nb_ss_gw/102-4845988-2776161?url=search-
alias%3Dstripbooks&field-keywords=Martin+Gardner&x=0&y=0)).

I think Martin Gardner did the same for Mathematics that Jon Bentley did for
computer programming (or vice versa :o)). His books are fun to read. Some of
the puzzles will be difficult for you, at first, but once you get the ball
rolling you will be hooked. There are couple of usenet groups that you will
find helpful while finding for hints for the solutions(notice that I didn't
say ask for solutions on those usenets). Please also read the following

 _\- "How to solve it" by George Polya.

_ \- "How to prove it" --hmmm, cannot recall the author name.

As others suggested, learn Algebra, Trigonometry, Calculus, Discrete
Mathematics and etc. After that, try to settle on a sub-field and focus on
that for at least 10 years. Another thing that you can do is to try to talk to
some professors at a university nearby and tell them you can do some research
as a volunteer (10-15 hours/week). I think you will find at least one
professor interested in this idea out of 100. Don't give up, this can work.
There was this Nobel Laureate at University of Utrecht and he has a very good
collection of pointers on background information that a theoretical physicist
should possess. I'm sorry I cannot recall his name. So good luck. I know if
you will persist you will have a lot of fun doing it.

~~~
sigstoat
"how to prove it" is by Velleman

------
kobs
Art Of Problem Solving Forum -
<http://www.artofproblemsolving.com/Forum/index.php>

------
tokipin
recently i found a very good collection of lectures:

<http://www.youtube.com/profile_play_list?user=nptelhrd>

a couple are math-specific, but they keep adding videos so there may be more
later on

------
kajecounterhack
this might sound a little...retarded...

but honestly math.com is a great site. I've used it for enrichment and if you
start at a topic that seems useful, it can be a great place to learn. Of
course books are good for practice problems and all...

------
kashif
At your level, you may want to go through the video lectures - "Joy of
Mathematics" Google for it.

PS: There is some excellent advice here and I intend to make good use of it.
Thanks everyone.

------
anewaccountname
Journey Through Genius is a pretty good way to go "ground up".

------
ptn
Try Euclid. Though difficult (which is not that much of a problem given that
you are not a complete beginner), it's all there.

------
curi
You don't have to say "thanks" 6 times.

~~~
curi
Wow. <3

~~~
curi
This is interesting. No one gives any reason I'm wrong about what I said. I
don't quite see the purpose of downmodding a post that's already on the bottom
and greyed out. But I guess no one is going to explain their reasons.

~~~
iamwil
Generally, people that pervade hacker news are pretty sensitive to cultivate a
high signal to noise ratio, probably because a lot of us have the 'slippery
slope' mentality, and seen what can happen if you give and inch, if Reddit and
Digg are any indication.

In the guidelines: "Resist complaining about being downmodded. It never does
any good, and it makes boring reading. "

If your comments don't add any information to the topic at hand, it usually
gets downmodded. Exceptions seem to be if they're wise-ass or snarky comments.

~~~
curi
How come people who want a high signal to noise ratio, downmod a short,
helpful comment which lets someone know not write "thanks" 6 times? In other
words, I was asking someone to improve the signal to noise ratio.

As far as getting downmodded for my additional comments, I expected that and
don't mind. I have read the rules. BTW, I do think people here are far too
inclined to downmod stuff they dislike without refuting it.

~~~
rms
I upmodded every post in your comments history.

<3 downmodders.

~~~
curi
lol. did you read them?

