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Marcin Jendrzejewski science forum beginner
Joined: 23 Jun 2006
Posts: 1

Posted: Fri Jun 23, 2006 10:55 am Post subject:
Recommendations for learning more mathematics



Hello everyone,
Hopefully this won't be too rambling, or worse, vague. I've just
finished a computer science degree(my first degree). Like most people
I've come across a handful of maths over the years, however what has
often has bothered me is that I don't really have a good understanding
of maths.
Now I would like to rectify this. I don't have any particular reason for
doing so other than I would like to learn and understand more. I don't
expect to become a maths god, but just something I can do a night or two
a week to get a better understanding of maths and maybe to have a better
understanding of the maths behind topics I'm interested in (say physics,
computing or linguistics, although this isn't an exhaustive list).
My basic question is whether there is a guide of recommended branches of
mathematics that are best learned before others? A sort of dependency
graph of maths topics. I don't mind going over the basics again(even if
it seems childish), because for a variety of reasons I don't think I
learned all of them thoroughly. (I think part of this is that I can't
exactly point and say "I know this topic").
I realise that mathematics isn't a certificate that once you know how to
add two numbers, and do a few other things,that you can say "I know
maths". But I think that's the point, I want to learn how to think more
formally and precisely, presumably in a way conducive to learning and
understanding maths (and maybe extending ideas for my own needs). And
what I think I'm getting at, is that I'm wondering is whether there's a
list of topics that form a solid background for toppling more advanced
maths ? I can easily go and pick up a book and work at trying to
understand a topic, but I'm just wondering if there's some basic topics
that appear frequently enough that would help other topics.
So like how algebra shows up all over the place; there's various things
to learn about algebra to get oneself into the right frame of mind to
understand other topics. Or how when covering a new algebra, the
identity laws, associativity etc. laws are discussed to describe the
"primitive operations" so that you can reason out the rest of the topic
(or understand why certain things work and others don't).
I hope I've been clear enough. Please let me know if you have any
ideas.Thanks in advance.
Marcin 

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Frederick Williams science forum addict
Joined: 19 Nov 2005
Posts: 97

Posted: Fri Jun 23, 2006 4:15 pm Post subject:
Re: Recommendations for learning more mathematics



Marcin Jendrzejewski wrote:
Quote:  My basic question is whether there is a guide of recommended branches of
mathematics that are best learned before others? A sort of dependency
graph of maths topics.

Logical dependency isn't the same as pedagogical dependency. Which do
you want?
This might be of interest: http://www.ams.org/mathweb/

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CELKO science forum beginner
Joined: 04 May 2005
Posts: 23

Posted: Sat Jun 24, 2006 12:03 pm Post subject:
Re: Recommendations for learning more mathematics



Thre was a good series of books and vidoes from COMAP that give a
survey of modern mathematics. Look for the title FOR ALL PRATICAL
PURPOSES. 

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Frederick Williams science forum addict
Joined: 19 Nov 2005
Posts: 97

Posted: Sat Jun 24, 2006 1:17 pm Post subject:
Re: Recommendations for learning more mathematics



Marcin Jendrzejewski wrote:
Quote: 
... I can easily go and pick up a book and work at trying to
understand a topic, ...

That may be the best thing to do, but only if you're "picking up" the
books at a library so that you can be sure they suit you before you
spend your hardearned money ion them.
Let's pretend that mathematics divides into algebra, analysis, geometry
and number theory (such a division is of course a matter of
administrative convenience not a matter of logic) then you might like to
try:
for algebra, Birkhoff & Mac Lane's Survey of Modern Algebra,
for analysis, Burkill's A First Course in Mathematical Analysis,
for geometry, Coxeter's Introduction to Geometry,
for number theory, Hardy & Wright's An Introduction to the Theory of
Numbers.
If you want to tackle a topic that will introduce you to a bit of
everything, then you might like to try:
McKean & Moll's Elliptic Curves.
That probably requires some preparation in complex analysis, in which
case you might like to try:
Ahlfors's Complex Analysis.
These just reflect my tastes in topics and books. Ymmv.
If you want to approach mathematics via things that connect with
computing then such subjects as:
Logic, Recursive function theory (especially via register machines),
Modal logic;
Numerical mathematics;
Cryptography, Number theory
come to mind.
Don't forget collateral reading in history (Kline's Mathematical Thought
from Ancient to Modern Times) and problems (Polya & Szego's Problems and
Theorems in Analysis).
For properly informed guides to study, don't forget sci.math where
professional teachers of mathematics may be accosted.

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zachary.hahn@gmail.com science forum beginner
Joined: 28 Jun 2006
Posts: 1

Posted: Wed Jun 28, 2006 8:56 pm Post subject:
Re: Recommendations for learning more mathematics



Speaking from my personal experience as a math major I'd recommend
first gaining a solid knowledge of calculus. While the knowledge you
learn by studying calculus may not apply to some of the fields you
study later on it is certainly good practice to establish mathematical
rigor, not to mention a solid knowledge of limits and series is
important. Even if you've taken calculus classes in college for your CS
degree it might be worthwhile to study a more mathematicaly dense book
on the subject. One in which the proofs are done out in a bit more
detail than being handed to you as "Axioms" of mathematics. Furthermore
its a very accesible subject with thousands of very well written books.
After that I'd recommend picking up a good real analysis book
("Principles of Mathematical Analysis  by Walter Rudin). It's a fairly
expensive book for it's size (around $130) so I wouldn't recommend
buying it unless you have great interest. You may be lucky to find it
in the library, especially a university library. You'll also want to
study a detailed a rigorous construction of the real and complex number
lines, beginning with the natural numbers. My professor constructed
these from his own notes and so I have no book to recommend to you. I
believe most discrete mathematics books cover the subject in varying
detail (use the library!)
The sky is the limit with studying analysis, but I'd atleast recommend
covering up to a rigorous definition of continuity and all it's
implications and going as far as constructing Riemann calculus. After
that you can continue on and study Lebesque theory or move into complex
analysis.
If you want to go further I'd certainly recommend studying Abstract
Algebra (Group, Ring, Field, Galois Theory, etc...), but you may want
to study pointset topology first. I studied real analysis then group
theory, then topology and ring theory. To me pointset topology seems
to be somewhat of a stepping stone between analysis and group theory
(as you move up into algebraic topology, but don't worry about that
right now). I found the Springer series texts in undergraduate
mathematics to be good (Basic Topology  by Armstrong), and will cover
all the basics (topological axioms, openness, closedness, topological
continuity, compactness, connectedness, HeineBorel theorem, Hausdorff
spaces, metric spaces, homeomorphisms, etc...)
Then you can go into Abstract Algebra (and linear algebra). After
you've done all that you can go off in all sorts of different
directions. Certainly check out the Mathematical Atlas which was posted
above. It's a great resource and has overviews and links to texts for
almost every subject.
This is merely the path I've taken in my study of mathematics, and it's
worked fairly well for me. But by no means is it the best. Hope this
helps and best of luck to you. If you're serious keep your nose to the
grindstone. It won't come easy at first, but I'm sure you'll enjoy the
knowledge you gain.
Zach 

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