Introductory inspiring math books
Bjorn Edstrom
I'm a programmer who's thinking of studying mathematics instead of
computer science (for various reasons). I haven't studied math since
high school though, and everything I learned there was "how", not
"why". I have always been fascinated by math though, and it's really
something I want to become good at. It feels wiser to get a math
degree than a comptuer science degree too, because I really don't want
to work as a programmer (programming is mainly a hobby, although a
hobby I've studied quite seriously the past years).
Before I enroll somewhere, I thought I would read one or more books
about mathematics at higher-than-high-school-level to see if it's for
me. Basically, what I'm looking for is interesting and inspiring math
books aimed at new university students that covers some subject of
math from the beginning, not assuming you have prior knowledge of the
field. I don't mind reading huge tomes, books thousands of pages or
more is fun.
Ultimately, if it exists, It would be really nice with a book that
covers a little of everything, like mathematical notation, calculus,
algebra, set theory, discrete math, topology and god knows what. :-)
It would also be nice with a book (or books) that cover high school
math (with high school math I mean branches of math touched in high
school, not math at high-school level) more rigorously, as a
preparation for higher studies.
Sorry if I can't be more specific than that.
Cheers!
B
Rob
this" and "do that". I am now a masters student in mathematics, and
throughout my entire undergraduate degree, I kept thinking "Ok,
seriously, math can't get any more complicated, right? I mean, what
more is there?" I believed that right from the time I finished high
school, up to about last year, when I realized how little I know, and
how vast (and I mean VAST) math is. To give you an idea, here is an
area of math I find specifically interesting, and it's sub-topic-areas,
each of which is incredibly VAST in results, conjectures, problems, and
questions.
Graph Ramsey Theory, small subtopic of
Ramsey Theory, small subtopic of
Graph Theory, small subtopic of
Discrete Mathematics, small subtopic of
Math in general.
I don't have any book recommendations, but I do erge caution. :) Math
is not something to be entered into lightly. :)
Gottfried Helms
in it, much inspiring, only I don't know, whether it is
already (ever) translated from german to english.
In german its title is "modern mathematics" in a series,
which I'd translate something like "the fun/satisfaction
of discovering/digging for insight"
of Herbert Meschkowski. Its a convenient paperback, covering
basic mathematical facts (and problems) in the historical context,
backed with original letters of famous mathematicians discussing
that current mathematical problem - a real source of
insight for the beginner.
I would like to see it edited in english, too.
Gottfried Helms
Narcoleptic Insomniac
> Let me point out one thing: high school math is not
> math. :) It's "do this" and "do that". I am now a
> masters student in mathematics, and throughout my
> entire undergraduate degree, I kept thinking "Ok,
> seriously, math can't get any more complicated,
> right? I mean, what more is there?" I believed that
> right from the time I finished high school, up to about
> last year, when I realized how little I know, and
> how vast (and I mean VAST) math is.
>
> [...]
No doubt you've probably heard this quote, but it's
definately worth repeating:
"Finally I am becomming stupider no more"
~Paul Erdos' self suggested epitaph.
I find comfort knowing that the second most prolific
mathematician ever felt this way. ^_^
Regards,
Kyle
Narcoleptic Insomniac
> Hi there,
Hello.
> I'm a programmer who's thinking of studying mathematics
> instead of computer science (for various reasons).
Like...?
> I haven't studied math since high school though, and
> everything I learned there was "how", not "why". I have
> always been fascinated by math though, and it's really
> something I want to become good at. It feels wiser to
> get a math degree than a comptuer science degree too,
> because I really don't want to work as a programmer
> (programming is mainly a hobby, although a hobby I've
> studied quite seriously the past years).
I went to MSOE for software engineering for about two
months before I dropped out and decided to pursue
mathematics instead.
> Before I enroll somewhere, I thought I would read one
> or more books about mathematics at higher-than-high-
> school-level to see if it's for me. Basically, what I'm
> looking for is interesting and inspiring math books
> aimed at new university students that covers some
> subject of math from the beginning, not assuming you
> have prior knowledge of the field. I don't mind reading
> huge tomes, books thousands of pages or more is fun.
I believe all mathematics students learn the following
piece of information quickly - there's only so much
mathematics you can *read*, the rest you have to actually
*do* to learn.
Moreover, most undergraduate/graduate mathematics texts
aren't mean to be "read". Depending on the author,
subject, and how much caffine you've had, I dare say
you're doing great if you can digest a dozen pages in an
hour.
> Ultimately, if it exists, It would be really nice
> with a book that covers a little of everything, like
> mathematical notation, calculus, algebra, set theory,
> discrete math, topology and god knows what. :-)
There is a course at my university aimed at incomming
freshman called Survey of Mathematics. Guess what the
name of the text book is! : - P
'Survey of Mathematics with Applications' by Angel,
Abbott, and Runde:
http://www.amazon.com/gp/product/0321112504/103-0383731-1240628?v=glance&n=283155
Personally, I have not taken this course, nor have I
studied from this text. However, the text is laying
around in the tutoring center where I work. Looking
through the table of contents it covers: sets, logic,
number theory, basic algebra, basic linear equations,
geometry, probability, and some other stuff.
Even if you decide not to go with this exact textbook,
I'd suggest browsing around for other texts that include
"survey" or "introduction to..." in their titles.
Good Luck,
Kyle
Stephen J. Herschkorn
>Before I enroll somewhere, I thought I would read one or more books
>about mathematics at higher-than-high-school-level to see if it's for
>me. Basically, what I'm looking for is interesting and inspiring math
>books aimed at new university students that covers some subject of
>math from the beginning, not assuming you have prior knowledge of the
>field. I don't mind reading huge tomes, books thousands of pages or
>more is fun.
>
>Ultimately, if it exists, It would be really nice with a book that
>covers a little of everything, like mathematical notation, calculus,
>algebra, set theory, discrete math, topology and god knows what. :-)
>It would also be nice with a book (or books) that cover high school
>math (with high school math I mean branches of math touched in high
>school, not math at high-school level) more rigorously, as a
>preparation for higher studies.
>
Courant and Robbins, What is Mathematics?
Stewart, Concepts of Modern Mathematics
The latter is a particularly gentlle introduction. Unfortunately, it
does not give suggestions for further reading, but you can always look
for that on the Internet (e.g., by asking here.)
Go to amazon.com or your local well-stocked bookstore to look for
similar books.
Also possibly of interest to you:
Davis and Hersh, The Mathematical Experience
Newman, ed., The World of Mathematics: to peruse, not to read cover to
cover.
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
Stephen Montgomery-Smith
I think that it is rather difficult to find many books that you ask
about, because I can see that they will be difficult to write. The
problem is that to really do high level math, you need to know a lot of
background that requires many classes. So to give a flavor of what this
is like to someone without the background is necessarily going to be an
incomplete experience.
Look at the book by Timothy Gowers "A Very Short Introduction to
Mathematics" - I think you will find it interesting.
As an aside, I knew Tim Gowers at college. While I cannot describe us
as very close friends, our relationship definitely went beyond the
superficial. He later solved a rather important but technical problem
in Banach Space Theory. Shortly after that I had a conversation with
him, where I "advised" him that he should get out of this technical
mumbo jumbo, and solve some real problems that would get him attention
(I was thinking about the Navier-Stokes equation - this was before it
became one of the Clay problems).
Anyway six months later it was announced that he had received the
Field's Medal (regarded as the Math equivalent of the Nobel Prize). And
I think that he knew he was going to receive it while I was "advising"
him. How embarressing for me! (Well I am not sufficiently embarressed
to not relate the story.)
I have not met him since he received the Field's Medal. My guess is
that he would smile if I reminded him of this story.
Stephen
Bjorn Edstrom
wrote:
>Let me point out one thing: high school math is not math. :) It's "do
>this" and "do that". I am now a masters student in mathematics, and
>throughout my entire undergraduate degree, I kept thinking "Ok,
>seriously, math can't get any more complicated, right? I mean, what
>more is there?" I believed that right from the time I finished high
>school, up to about last year, when I realized how little I know, and
>how vast (and I mean VAST) math is.
Yeah, I am (painfully) aware of this, the concious/unconcious
competence/incompetence thing. Same thing with programming, even after
you've programmed for 10 years there are things to learn. You don't
know this after many years, but it's true. I am sure there are many
math geeks in here who have studied math for a couple of years and
they think they know what they are doing, when in reality they don't,
like everyone else. :-)
I do think I will become a better programmer by studying mathematics
than computer science though, math keeps coming back to me in more
ways than I feel comfortable with handling in an "engineer-sense". I
am for instance coding a software synthesizer now. So I had to learn
digital signal processing. And then I had to learn about fourier
transforms and how to use it in practice. Or, once I had to write a
graphics library, and how to use knowledge from math guys in practice.
And this continues, and the more I code the more I just simply have to
accept some math as "true". This is tedious, It'd be more fun and
enlightening to derive and prove things yourself. It feels math is
much more than a "tool" to "use". And heck, If I can learn math, I can
learn anything! Personal challange, too! :-)
>To give you an idea, here is an
>area of math I find specifically interesting, and it's sub-topic-areas,
>each of which is incredibly VAST in results, conjectures, problems, and
>questions.
>
>Graph Ramsey Theory, small subtopic of
>Ramsey Theory, small subtopic of
>Graph Theory, small subtopic of
>Discrete Mathematics, small subtopic of
>Math in general.
>
>I don't have any book recommendations, but I do erge caution. :) Math
>is not something to be entered into lightly. :)
Graph theory is fun, but I have only really "studied" it in a computer
science sense (like, coding a shortest path from an algorithm textbook
:-) ). I have a graduate friend who's working with NP-hard problems
related to "hypergraphs". Some day, I want to understand what he's
doing.
So, I think I understand the big picture what I'm getting myself into.
I lack the necessary details to get started though, and this is where
the books come handy.
Cheers!
B
David Park
only high school algebra but has real mathematics. He also has a second book
"Mathematics and Its History" that is more advanced but doable.
Also, if possible, I would suggest you get Mathematica and try to learn its
'functional programming' and 'rule base programming', which are a very
mathematical way of thinking. A good CAS can be very useful provided you use
it as a mathematician and push computer science into the background. It
would go well with "Numbers and Geometry".
David Park
dj...@earthlink.net
http://home.earthlink.net/~djmp/
"Bjorn Edstrom" <b...@bjrn.se> wrote in message
news:9sljs19ocj4o3qf91...@4ax.com...
Stephen Montgomery-Smith
> Also, if possible, I would suggest you get Mathematica and try to learn its
> 'functional programming' and 'rule base programming', which are a very
> mathematical way of thinking. A good CAS can be very useful provided you use
> it as a mathematician and push computer science into the background. It
> would go well with "Numbers and Geometry".
I would probably recommend the opposite. I mean, if you can hold of
Mathematica, you might find it fun to mess about with. But it is so
expensive that I don't think that the experience will make it in any way
worth it. I don't think that it will help you learn any real
mathematics the way mathematicians do it.
john_r...@sagitta-ps.com
Bjorn Edstrom wrote:
> Hi there,
>
> I'm a programmer who's thinking of studying mathematics instead of
> computer science (for various reasons). I haven't studied math since
> high school though, and everything I learned there was "how", not
> "why". I have always been fascinated by math though, and it's really
> something I want to become good at. It feels wiser to get a math
> degree than a comptuer science degree too, because I really don't want
> to work as a programmer (programming is mainly a hobby, although a
> hobby I've studied quite seriously the past years).
>
> Before I enroll somewhere, I thought I would read one or more books
> about mathematics at higher-than-high-school-level to see if it's for
> me. Basically, what I'm looking for is interesting and inspiring math
> books aimed at new university students that covers some subject of
> math from the beginning, not assuming you have prior knowledge of the
> field. I don't mind reading huge tomes, books thousands of pages or
> more is fun.
>
> Ultimately, if it exists, It would be really nice with a book that
> covers a little of everything, like mathematical notation, calculus,
> algebra, set theory, discrete math, topology and god knows what. :-)
Perhaps the biggest problem you'll find is that most undergrad maths
books are not self-contained, and even less so graduate text books,
despite their authors' earnest declarations in the preface.
The problem is that to be completely self-contained vastly increases
the length of a book, and taking that to its logical conclusion you
end up with something like the Bourbaki series (which would be worth
a glance BTW, as you say you don't mind reading huge tomes - They
look a bit intimidating at first glance, but are quite readable once
you get stuck into them.)
To minimize the frustration and wasted time this interdependence
entails, it helps to plan your study and work up from the most
fundamental areas. So I'd recommend the following, in the order
listed:
"Topics in Algebra", Herstein
Covers basic set theory, abstract algebra (groups,
rings, and algebraic fields), and linear algebra
"An Introduction to Grobner Bases", by Ralf Froberg
Constructive introduction to ring theory - maybe
just study the first two or three chapters.
"Algebra" by Serge Lang
Similar coverage to Herstein's book, but also covers category
theory and homology. You won't get far these days without
knowing the basics of these, even increasingly in physics
and computer science (certainly the category theory).
"Topology" by Klaus Janich (Springer)
Good, chatty but rigorous, introduction to topology
(Analysis books - I'll leave others to provide some references.
One would be "Analysis" by Serge Lang)
"A Handbook of Fourier Theorems" by D C Champeney
Very useful reference for analysis related to Fourier
Theory and much more - Little exaggeration to say it
is a "secret weapon" for anyone who wants to become
an expert in all this in a short time! [*]
"Constructive Combinatorics" by Stanton & White (Springer)
There are dozens more books one could have substituted for
any of these, which perhaps other readers may suggest.
But where you go from there depends on your interests, e.g.
applied maths v. pure maths. For example, I didn't mention
any probability or statistics books; but perhaps this is
something that interests you.
[*] I only wish there were more "rigorous survey books" like
this, as another big problem studying maths books is getting
bogged down in a mass of fussy details and losing sight of
the big picture. Most authors make little or no effort to
sketch the main aims of the theory, or unsolved problems
and challenges still remaining.
Lastly, I should mention the excellent survey articles and
book reviews in the Bulletin of the AMS. Most of these may
be a bit over your head (as they are over mine!) But they
give a very good idea of the "state of play" in various
areas of maths. Also, these days some interesting survey
and expository papers also appear in www.arxiv.org
(and also of course on numerous web sites).
Bjorn Edstrom
wrote:
>I would recommend "Numbers and Geometry" by John Stillwell. It presupposes
>only high school algebra but has real mathematics. He also has a second book
>"Mathematics and Its History" that is more advanced but doable.
>
>Also, if possible, I would suggest you get Mathematica and try to learn its
>'functional programming' and 'rule base programming', which are a very
>mathematical way of thinking. A good CAS can be very useful provided you use
>it as a mathematician and push computer science into the background. It
>would go well with "Numbers and Geometry".
I have experience with purely functional programming languages
(Haskell) and some of the theory behind (eg. currying, lambda calculus
etc), as well as a few other functional languages that allow side
effects (Scheme etc). I can't say I use these languages for real world
purposes, but I don't regret learning them. Scheme is the most
beautiful language I know of. I have never used Mathematica though,
but the rule based programming seems interesting ("A language that
doesn't affect the way you think about programming, is not worth
knowing" --Alan Perlis).
I think I belong in some grey zone. I barely remember the trigonometry
relationships from high school, or the calculus. On the other hand
(unlike most high school students), I have picked up loads of bits and
pieces from programming (contests). I know more about practical graph
theory than my friends who study computer science for example. I have
coded a radix-2 FFT using inline assembly. I know that you can make
Miller-Rabins primality test non-random if you use the bases 2, 7 and
61, if you are only interested in testing numbers smaller than 32 bit.
I have used Eulers totient function. And I often read mathworld just
to find interesting things to code. I don't tell you this to sound
impressive, but rather because I am really lost here and I'm looking
for books at the right level. Most people learn math before
programming, not the other way around.
Many people have recommended I should read Knuth's Concrete
Mathematics, and it's on my to-buy list, but I think it's a better
idea to read/study something more general about mathematics first,
rather than a math book especially for programmers (which will
probably give me some wrong ideas about what math is about).
I will check out the "Numbers and Geometry", thanks for the
suggestion! "The Mathematical Experience", "Concepts of Modern
Mathematics", and "What Is Mathematics?" seems like other interesting
books too!
Thanks!
B
Bjorn Edstrom
Oh, this was a nice list, thank you! All of these books have excellent
reviews at Amazon, and they are very inexpensive too!
Thanks!
Bill Dubuque
Be sure to see the earlier thread [1].
Very beautiful is: Numbers and Figures by Rademacher and Toeplitz.
among other classics mentioned there. One of my posts there includes
a list compiled by the eminent Russian mathematician Vladimir Arnold.
--Bill Dubuque
[1] http://google.com/groups?threadm=y8zwuu99dls.fsf%40nestle.ai.mit.edu
Bill Dubuque
>
> Be sure to see the earlier thread [1].
>
> Very beautiful is: Numbers and Figures by Rademacher and Toeplitz.
> among other classics mentioned there. One of my posts there includes
> a list compiled by the eminent Russian mathematician Vladimir Arnold.
>
Numbers and Figures was republished by Princeton and Dover
with the title: The Enjoyment of Math [2]
See also [3] for Kac & Ulam: Mathematics and Logic, and others.
Both are highly stimulating reading for a budding mathematician
Also helpful are online reviews, excerpts and content searching
available at Amazon, books.google.com, etc.
bookfinder.com may help you locate good deals on used books.
Enjoy!
--Bill Dubuque
[2] http://www.amazon.com/exec/obidos/tg/detail/-/0691023514
http://store.doverpublications.com/0486670856.html
[3] http://google.com/groups?threadm=y8zlmdr...@nestle.ai.mit.edu
http://www.amazon.com/gp/product/0486670856/qid=1137392531
Christopher J. Henrich
<b...@bjrn.se> wrote:
Here are some favorites of mine:
What is Mathematics, by Courant & Robbins, new edition with addenda by
Ian Stewart.
Geometry and the Imagination, by Hilbert & Cohn-Vossen. The German
original is called Anschauliche Geometrie. I think the English
translation is more readily available.
An Adventurer's Guide to Number Theory, by Richard Friedburg.
--
Chris Henrich
http://www.mathinteract.com
God just doesn't fit inside a single religion.
Herman Rubin
>> Be sure to see the earlier thread [1].
Many set theory books or logic books are self-contained,
because these subjects are rarely taught at an earlier
level. For example, Godel's _Consistency of the Continuum
Hypothesis_ is effectively self-contained, although except
for the first part, it is quite difficult reading.
One can easily tell which are NOT self-contained.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
rip pelletier
> Before I enroll somewhere, I thought I would read one or more books
> about mathematics at higher-than-high-school-level to see if it's for
> me. Basically, what I'm looking for is interesting and inspiring math
> books aimed at new university students that covers some subject of
> math from the beginning, not assuming you have prior knowledge of the
> field. I don't mind reading huge tomes, books thousands of pages or
> more is fun.
good question, and you've gotten a lot of different answers. let me
throw in my two cents: my favorite introductory books. amidst all the
"first books" in mathematics, these are some of the outstanding
introductions, in my opinion.
calculus, volume 1 by apostol (it was the freshman text at caltech).
it proves what it can, and that's quite a bit. as a result, it has the
flavor of a true math course. my studies would have much been easier if
i'd entered as a freshman, rather than as a sophomore having to play
catch up.
differential & integral calculus: the 2 volume set by courant. never
used it for a class, but i've read my copy, and i like it; can't say
whether it's a good reference, but it reads well.
linear algebra & its applications, by strang. not the indispensible
reference (that's halmos' finite dimensional vector spaces), but another
marvelously readable introduction.
abstract algebra, by fraleigh. (a few topics in the 1st edition have
disappeared from the latest ones, but it's still a fine introduction.)
survey of modern algebra, by birkhoff & maclane. not to be confused
with algebra by maclane & birkhoff. it studies the archtypical algebraic
structures, but makes it clear that what they're saying about the field
of polynomials, for example, applies to all fields.
topics from the theory of numbers, by emil groswald.
intro to differential geometry, by barrett o'neill. head and
shoulders over any other intro.
undergraduate topology by kasriel. not sure how much you need before
this becomes readable, but it's a wonderful trip from the vector space
R^n to normed linear spaces and then metric spaces and then to
topological spaces. those generalizations, when i saw them, took my
breath away. vector spaces of functions? oh my god, yes!
>
> Ultimately, if it exists, It would be really nice with a book that
> covers a little of everything, like mathematical notation, calculus,
> algebra, set theory, discrete math, topology and god knows what. :-)
> It would also be nice with a book (or books) that cover high school
> math (with high school math I mean branches of math touched in high
> school, not math at high-school level) more rigorously, as a
> preparation for higher studies.
someone else will have to answer this, because i do not own the
following book, but i have read here that serge lang has a book called -
perhaps - basic mathematics, which is precisely high school math done
right.
two "second" books in math.
a course in mathematics for students of physics, (2 volumes) by
bamberg & sternberg. vol 1 is worth having just for the treatment of
thin lenses using matrices. a survey book, but it probably requires some
familiarity with calculus & differential equations.
applied analysis, by lanczos. don't know that it's of interest - it's
probably too specialized at this point - but he's readable. i buy his
books on principle because i like his style & treatment.
and then there's:
fourier analysis, by korner. despite the title, this is a wide-ranging
book. it includes, for example, 4 chapters on fraudulent statistics.
let me stop here, for now.
vale,
rip
--
NB eddress is r i p 1 AT c o m c a s t DOT n e t
gwl...@nukove.com
one for about $10). A thoroughly amusing ride through 2,500 years of
mathematics, even for a person such as myslf who received a PhD nearly
30 years ago. The book not only covers the great theorems but points
out what nutcases many mathematicians were. I would teach a college
pro-seminar from it if I still taught.
Stephen Montgomery-Smith
For a racy (although apparently rather inaccurate) history of Math,
there is the book by Bell "Men of Mathematics."
Stephen
Karl M. Bunday
inspiring math books:
> someone else will have to answer this, because i do not own the
> following book, but i have read here that serge lang has a book called -
> perhaps - basic mathematics, which is precisely high school math done
> right.
I own Basic Mathematics by the late Serge Lang,
http://www.amazon.com/gp/product/0387967877/
which I learned about from the Web site of Professor Hung-hsi Wu at Berkeley.
The book is quite good, and a good review of "high school" mathematics from a
higher perspective. In that same genre are the various books by Israel M.
Gelfand, e.g. his delightful Algebra
http://www.amazon.com/gp/product/0817636773/
and the follow-up books in the same series.
I will heartily second the recommendation for any book by John Stillwell, any
book by Ian Stewart, and the books by Apostol or Courant mentioned earlier in
this thread.
--
Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345
Learn in Freedom (TM) http://learninfreedom.org/
remove ".de" to email