I’m looking for a book that contains an overview about the different fields of mathematics; how they relate to each other; some of the key concepts and unsolved problems…etc
Would there be any single comprehensive source (book or web) that serves this function?
Here’s a good math website:
Thanks, I already know about this great resource since it was Weisstein’s effort.
But it is like an encyclopedic resource, not the well-threaded overview that I’m looking for…
Mathematics and its History by John Stillwell, published by Springer Verlag, on sale as part of the Yellow Sale 2003 until December.
I don’t know of anything current, but Kasner and Newman’s Mathematics and the Imagination would be right up your alley if it weren’t 60 years old. It’s still a worthwhile read if you can find it.
Until something better pops up, have a look at this site.
You can’t get moe up to date than The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, by Keith J. Devlin, an introduction to the seven pullzes whose solutions will earn a $1,000,000 prize.
The books of Ivars Peterson are a good popular introduction to most areas of math. The Mathematical Tourist is subtitled “Snapshots of Modern Mathematics” and has a 2001 revision. He has a series of others out.
Mathematics Today, edited by Lynn Arthur Steen, is a series of essays on various branches of math but it’s from 1980 and out of print.
You could also try one of the huge number of books that have come out in recent years on problems like Fermat’s Last Theorem, the Reimann Hypothesis and Goldbach’s Conjecture, because solving those gets you into an enormously varied amount of math, and it obviously must be presented in a way that ties them all together.
Isaac Asimov wrote several books that are easy to follow.
amore Thanks, looks promising and in line.
ultrafilter That’s a good site. Luckily, my uni. library has Kasner, so I’ll take a look.
Exapno The Clay Institute problems sound interesting, but I suppose those come later on in my reading, or do they? The Peterson seems to be what I’m looking for, alongwith the Stillwell work.
Thanks to all.
I second heartily any Asimov math book. E.g., “Science, Numbers, and I.” Many of his F&SF collections contain a lot of math lore, including history.
As for “The Mil. Problems” book. One of the problems isn’t even a Math problem, it’s a Computer Science problem. Stupid, stupid, stupid.
The Mathematical Experience by Davis and Hersh was a phenomenal book. I recall that reading it gave me the sense that I truly understood some of the deeper issues and linkages within mathematics. And, it was a relatively ‘easy’ read that covered a lot of the history and ideas of mathematics. Truly a remarkable work.
I also forgot to mention Bell’s Men of Mathematics, which is a standard.
Much as I love Asimov, his books are on numbers and not on mathematics. Mind you, they’re great for numbers. But that’s a whole different differential than math.
Ditto on Mathematical Experience. I didn’t include it because it’s another older work.
And the new edition isn’t really an update:
[quote]
The addition of exercises and problems converts the 1981 edition into a textbook for math courses for liberal arts students and future secondary school math teachers, and courses in the appreciation of mathematics.
I found many universal concepts that were applied in several fields of higher math (i.e. triangle inequality). But then again, branches of higher math tend to overload definitions, a basis in the topological sense has no relation to a basis in the vector space sense.
I can see a connection, but it’s not like they’re the same object. It never bothered me.
It never bothered me, it’s just that if someone asks, “What’s the definition of a basis in math?”. You couldn’t answer without refining the question.
You might try Fantasia Mathematica and the follow-on, * The Mathematical Magpie*. Both are anthologies of articles and odds-and-ends written by an assortment of writers on the subject of mathematics. I found these two books quite helpful when trying to explain difficult math concepts to students.
That’s like saying that Goldbach’s Conjecture isn’t a math problem, it’s a number theory problem, or that squaring the circle isn’t a math problem, it’s a geometry problem.
And if you want something that’ll make you go “whoa”, while also leading you deeply into some fundamental mathematics, try Hofstadter’s Gödel, Escher, Bach: An Eternnal Golden Braid. But be warned, it’s a very dense book.
Chronos
I was about to pick up that book 5 years ago. Since, being a looong book, I asked for opinions. I was told that the author posits a theory of consciousness out of all the material in the book presented beforehand. Then I read some reviews that basically claimed to point out fundamental flaws in that theory. So, I didn’t read it then. So, is that the focus of GBE and is the theory presented, flawed?
P.S. I know it won a Pulitzer and probably contains a lot of useful insights nonetheless.
A couple of my favorites along the lines of what you seem to be looking for:
Concepts of Modern Mathematics by Ian Stewart, and
The Language of Mathematics by Keith Devlin
Both are fairly modern, lively (i.e. not dry and textbooky), and aimed at the general reader.
Umm, Hofstader is a long winded bore. Asimov could have covered the same material in less than 100 pages. There’s a reason his SA column didn’t last long. Do not waste your time on G-E-B.
And Asimov did write about Math, quite a lot.
Number Theory is a branch of Mathematics. Computer Science most definitely isn’t.