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Expanding my pure math knowledge?
September 21, 2012 9:38 AM   Subscribe

Can anyone recommend some advanced (pure) math books to learn new subjects? Also nonfiction books about math that are good reads?

I was a math major in college, and focused on (and enjoyed) the pure math side. I took a lot of Number Theory classes and some analysis, but somehow entirely missed all topology and abstract algebra courses. When I search for good recommendations for books to learn stuff about these areas I seem to only find recommendations of 600-800 page textbooks which are comprehensive, or Topology for dummies. Is there anything in the middle? I'm fine with it going less in depth than a textbook (I can always go get the textbook afterwards if I want), I'd just like a book that is more geared towards a strong math background. Also feel free to recommend any pure math subject since that's how I've tagged the question, but personally I am most interested in abstract algebra and topology.

Also, I'd like to hear about any general nonfiction math books that you found interesting, it seems hard to sort them from the textbooks on Amazon.
posted by DynamiteToast to Education (23 answers total) 18 users marked this as a favorite
Herstein's Topics in Algebra is a classic.
posted by Nomyte at 9:48 AM on September 21, 2012

Also, more related to analysis, but have you read Spivak's Calculus on Manifolds? It covers a very interesting generalization of vector calculus and not all undergraduate programs teach this aspect.
posted by Nomyte at 9:53 AM on September 21, 2012

For the former, I know you said you did number theory, but Hardy and Wright's An Introduction to the Theory of Numbers is a lovely book, whether you know the subject or not. Do not confuse it with Hardy's A Course of Pure Mathematics, which is an obsolete analysis textbook.

For the latter, I almost hate to mention Douglas Hofstadter's Gödel, Escher, Bach because I assume anyone interested in math will already have read it, but just on the tiny chance that you haven't... it's fantastic.

James Newman's four-volume set, The World of Mathematics, contains many wonderful papers over the entire range of math, from Aristotle to Gödel.
posted by ubiquity at 9:54 AM on September 21, 2012 [1 favorite]

abstact algebra and topology are both vast topics... many of the things you learn in a class in either are of purely technical interest and useful only for very specific areas. so, it's hard to make recommendations without knowing what specifically, very specifically, you are interested in.

but, since you've never had *any* abstract algebra, I've always sort of liked Michael Artin's Algebra as an undergrad textbook. I'm not sure what's in the 2nd edition, but hopefuly it's not any worse than the first. if you want to learn more number theory, you have to decided which you prefer: Galois Theory or Algebraic Geometry (or both really)

if you are interested in "geometric" flavors of topology, opening a standard textbook can be exceedingly dull... to the point that I'm not sure what to recommend. but then, books with a more geometric flavor tend to require a bit of outside knowledge i.e. differential geometry.

(also, don't open Spivak's Calculus on Manifolds, it's a destroyer of souls... (see his 5 volume attempt to write great american novel monograph on differential geometry))
posted by at 10:02 AM on September 21, 2012

BTW, Newman, who invented the terms googol and googolplex, and Hoffstadter are linked, in that Newman's book inspired Hoffstadter to take up the study of mathematical logic and eventually write his own book. Furthermore, Newman's book contains a brief explanation of Gödel's incompleteness theorem, which Newman later expanded (with Ernest Nagel) into the book Gödel's Proof (1958), which Hofstadter later expanded in a second edition in 2002.
posted by ubiquity at 10:02 AM on September 21, 2012 [1 favorite]

David Foster Wallace wrote a book on infinity called Everything and More: A Compact History of Infinity that might interest you.
posted by artdesk at 10:09 AM on September 21, 2012

(Singer/Thorpe is a standard geometric introduction to topology... but it's not an easy road)

I always recommend Bamberg and Sternberg's A Course in Mathematics for Students of Physics. It's full of neat mathematics and physics, but it's not written in a "theorem/proof" style so it won't actually look like "pure math" even though it has bunches of it...
posted by at 10:24 AM on September 21, 2012

Armstrong's Basic Topology isn't half bad. Despite being more towards the epic tome end, I think Munrkes's Topology is fairly accessible.

In the world of undergrad abstract algebra textbooks, I kind of have a soft spot for Beachy and Blair over the more standard Artin. There's more material in Artin, though.

You might want to take a look through the Chicago undergraduate math bibliography.
posted by hoyland at 10:30 AM on September 21, 2012 [1 favorite]

Have you read Flatland?
posted by Chocolate Pickle at 10:51 AM on September 21, 2012

The Princeton Companion to Mathematics was written with, essentially, you in mind. (Disclosure: I know this because I was tangentially involved in the preparation of the book and wrote a couple of the number theory articles into it.)
posted by escabeche at 10:58 AM on September 21, 2012 [2 favorites]

Volker Runde's A Taste of Topology is a short (192 pages) rigorous introduction to point-set topology, suitable for a first course in topology for upper-level undergrads in an honors program (and who therefore know what it is to prove things and already have something of a mathematician's attitude toward what counts as an interesting and natural question). Disclosure: I know Runde; he's a professor at my school.

Be warned about DFW's Everything and More: it's full of serious mathematical errors, so much so that I wonder if he was pulling some literary unreliable narrator trick that I'm not sophisticated enough to understand. (See Michael Harris's review (PDF) in the AMS Notices.)
posted by stebulus at 12:26 PM on September 21, 2012

If you can read French, I'd recommend Bourbaki. Even if you can't because it uses a tiny enough vocabulary that we used it in my undergrad topology course.
posted by Obscure Reference at 12:27 PM on September 21, 2012

I should mention that I was referring to his Topologie générale.
posted by Obscure Reference at 12:29 PM on September 21, 2012

Simon Singh's book Fermat's Last Theorem is fascinating and very readable.
posted by meronym at 1:04 PM on September 21, 2012 [1 favorite]

I enjoyed Gamma: Exploring Euler's Constant by Julian Havil.
posted by hattifattener at 1:40 PM on September 21, 2012

I second Escabeche's recommendation of The Princeton Companion to Mathematics. It is an absolutely stunning book that covers essentially every modern topic of importance in pure math (and some in applied math too), and does it in a wholly accessible way. It also provides a degree of rigor and specificity missing from many pop-nonfiction books about math, which often avoid actually discussing the real math out of fear of scaring the reader.

I wouldn't really recommend Bourbaki as a light/intro read; many of the Bourbaki books are necessary references and very worthwhile, but the Bourbaki group intentionally avoids intuition in favor of exacting rigor in exposition. This is great if you already have an idea of the lay of the land in a given topic and want to find a certain fact, but not so good for light/fun reading.
posted by Frobenius Twist at 2:29 PM on September 21, 2012

Symmetry: An Introduction to Group Theory and Its Applications by McWeeny.

Teach Yourself Mathematical Groups by Barnard, an easy non-rigorous read.

Another vote for Havil's Gamma.
posted by phliar at 2:32 PM on September 21, 2012

A Tour of the Calculus.

If Flatland is your cup of tea, there is also Alice in Flatland (my son says it is arguably related to topology). Number Devil is fun but might be a bit on the basic side.

Not a book and they don't explicitly explain set theory, but Pit Droids basically teaches set theory. Probably out of print. Might be able to pick it up on ebay or something.
posted by Michele in California at 2:50 PM on September 21, 2012

Introduction to Metric and Topological Spaces is a slim and readable textbook, which you might like.

The Mathematical Experience is a good thought-provoking read.
posted by philipy at 3:29 PM on September 21, 2012 [1 favorite]

I've heard really good things about Korner's "Fourier Analysis". It includes some history and biography stuff and discussion, as well as actual math.
posted by leahwrenn at 6:05 PM on September 21, 2012

For topology, the standard intro text is the book by Munkres. As I recall, it's actually a very fun read.
posted by dragonfruit at 6:07 PM on September 21, 2012

Run, do not walk, and pick up Complex Visual Analysis by Tristan Needham. Probably the finest textbook I've ever read.
posted by dfan at 6:12 PM on September 22, 2012 [1 favorite]

I plan to give a lot of these a try. A friend had Godel, Escher, Bach, and I'm gonna borrow it when she's done. Also, I was at a book store and someone recommended Everything and More so I picked that up as well. Thanks!
posted by DynamiteToast at 5:31 PM on October 21, 2012

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