Personally, knowing that Tim Gowers had something to do with it is recommendation enough. A gifted expositor and brilliant mathematician, I'm ordering this today.
Recommend you also read his on-line stuff as listed above. Tim is one of the clearest and most insightful writers I know, as well as being thoroughly nice to and tolerant of lesser mortals such as myself.
If you want to see some sample articles first, you can login to http://pcm.tandtproductions.com/ with U/P : guest/PCM - he put this information up on his blog a year ago perhaps. Look under 'Resources/Sample Articles'.
I enjoyed reading most of ANKoS, but it is a very different book from the book under review. I trust a collection of articles edited by Timothy Gowers as a guide to the latest carefully considered ideas in mathematics much more than I trust Wolfram's interesting but idiosyncratic personal account of the foundations of mathematics.
Here is a link to links to multiple reviews of A New Kind of Science. Some of the reviews are by quite eminent mathematicians.
Cosma Shalizi's review is particularly good because (1) Shalizi knows a lot about the stuff Wolfram is writing about and (2) it is called "A rare blend of monster raving egomania and utter batshit insanity".
I'm sure. I read 'emperor's new mind' and loved it for the most part. I have quite a bit of a backlog of books that are earlier in the queue before I get to it.
Which reminds me, we should consider the semantics of starting a science/math/etc book club dedicated to important books and concepts.
Another great book by the same: The Emperor's New Mind. Covers everything from Turing machines to the Mandelbrot set to Quantum Mechanics. An amazing read.
I should probably read one of his books, seeing as he was at my graduation ceremony and got his honorary professorship. (from what I remember about his speech there wasn't much more than the usual "inspirational" stuff, though)
I've got it, have tried to read parts of it, and have found it mostly too terse to follow. Maybe it's different if you're attempting to read it cover to cover. I bought it mostly as a quick-reference for the parts of physics I'd not had much exposure to.
A book in a similar vein is "A Grand Unified Tour of Theoretical Physics", by Ian Lawrie. While I admire Penrose's book, I believe Lawrie's is written at a more consistent level and is probably more accessible to many (though not all) people.
the author of this comment and book recommendation is an accomplished professional physicist of no little reputation. Apparently he does not want to toot his own horn here on this site, so I'll do it for him.
As well as Timothy Gowers (who is, yes, an absolutely first-rate mathematician and a very good expositor too), the PCM has a bunch of contributions from Terry Tao (blog at http://terrytao.wordpress.com/; http://terrytao.wordpress.com/?s=princeton+companion+to+math... to find pre-production versions of several of his PCM articles) who is also a Fields Medallist, a very good expositor, and a blogger.
In my comment I offered a clarification/suggestion on how you can use karma points to show your appreciation for a link you like. Your comment did not add any value.
This book looks awesome. Mathematics is so vast that as a student its hard to know what classes you should take, or even what the map of the territory looks like. That is one of the reasons I got frustrated with my pure math BS.
The Russian school put out some pretty massive volumes, like
A very good physics book is 'Visual Quantum Mechanics' and its sequel, 'Advanced Visual Quantum Mechanics'. Don't let the name fool you, this is not a picture book, it's a real mathematical physics books, but with the clearest exposition of QM I'm aware of. It's called "Visual" because of the accompanying Mathematica visualizations.
Page 338, in connection with the "elementary" proof of the prime number theorem. (I searched for "Selberg" using Amazon's search-inside feature, because I don't know how that feature treats accents. But, actually, it turns out that it treats them sensibly; searching for "Erdos" turns up 17 matches.)
I didn't believe you, but I've been and checked. Even though he was supervised by Bollobas (Erdos Number 1) he appears never to have co-authored with him, an unusual situation. So it seems that my Erdos Number is smaller than his - perhaps I should offer to co-author a paper with him. I'd get the kudos of co-authoring with a Fields Medal winner ( http://en.wikipedia.org/wiki/Fields_Medal ) and he'd lower his Erdos Number to 3.
I never by a first-edition math book. The second-edition will have tons of changes due to valuable feedback, along with typically the biggest wave of corrections of any increment in edition.
It's a collective action problem. I'm just giving the my Ayn Rand-esque solution to the problem, whereby I look out for myself and end up unintentionally screwing everyone else. =(
If you want to boggle your mind this Christmas I suggest you read GEB. It's not just Math, but Philosophy, AI, programming, logic, human mind, etc. I started reading it last year, and I'm still reading it this year, probably will be next year, year...
I second this recommendation, though I suspect most people at HN have already read it. I can also recommend two other books by Hofstadter: Metamagical Themas, and Fluid Concepts and Creative Analogies
Last night I went to sleep early, so I approved the comments during my morning break. I moderate comments in order to prevent spam and idiots. For example, the only comment on this post that was rejected was along the lines of "You are gay because you read math". I refuse to publish stuff like that.
I'm not a math guru by any means. I got as far business cal in college. However, I have always been fascinated with math (and intensely wished it would come easier for me.)
That being said, if I wanted to slowly build up a mathematical foundation (maybe half hour a day), would buying a book like this be right place to start? Or should I be looking elsewhere?
Yeah, this book is great even if you don't have a formal math background. As somebody who didn't even finish high school I'm finding this book very easy to learn from because it is so well written.
It certainly depends on what you're interested in. This is more of an encyclopedia of mathematics than a textbook. If you want an broad idea of many different fields of mathematics and the history behind them, presented in a well-written and relatively elementary style, this is the book for you. However, the problem with being relatively elementary is that most of the articles merely skim the surface of whatever subject they are describing.
If you want to actually be able to solve problems using math (the fun part, for most amateur mathematicians), a textbook would be a better bet. If you stopped math at business calculus and want a great introduction to the calculus that mathematicians use, I wholeheartedly recommend Michael Spivak's Calculus as a next step.
My favorite maths book is "little rudin" - Principles of Mathematical Analysis by Walter Rudin - you can just spend hours reading the same page over and over ...
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and the newer stuff is on his blog:
http://gowers.wordpress.com/
Here is a link to links to multiple reviews of A New Kind of Science. Some of the reviews are by quite eminent mathematicians.
http://shell.cas.usf.edu/~eclark/ANKOS_reviews.html
Which reminds me, we should consider the semantics of starting a science/math/etc book club dedicated to important books and concepts.
Lots of good book recommendations below, folks. Keep scrolling.
The Russian school put out some pretty massive volumes, like
http://www.amazon.com/Mathematics-Its-Content-Methods-Meanin...
but I found those to be a little too caught up in the proposition/proof cycle to be useful as a guide to the uninitiated.
One oddity: I keep looking for a mention of Paul Erdos in the book, and haven't found one.
http://en.wikipedia.org/wiki/Erdos_number
http://www.nationmaster.com/encyclopedia/William-Timothy-Gow...
Thanks to Gowers et al for this fabulous book!
If it's truly a good book, I don't mind buying an updated 2nd edition.
http://www.shallowsky.com/blog/science/fibonautilus.html
That being said, if I wanted to slowly build up a mathematical foundation (maybe half hour a day), would buying a book like this be right place to start? Or should I be looking elsewhere?
If you want to actually be able to solve problems using math (the fun part, for most amateur mathematicians), a textbook would be a better bet. If you stopped math at business calculus and want a great introduction to the calculus that mathematicians use, I wholeheartedly recommend Michael Spivak's Calculus as a next step.