Gauss, Escher, Bach. Give me math that doesn't read like math!
September 27, 2012 3:55 AM   Subscribe

I have a very strong background in humanities but I've completed three-year undergraduate course in math. Though I passed it without trouble, I feel like traditional textbooks didn't teach make me understand a lot. What are best math resources (books, but not necessarily) that instead of trying to look like PM read more like literature or work on your intuition and talk about big picture implications instead? Something like Godel, Escher, Bach for various areas of mathematics.

What I'm trying to say, I guess, is: I don't work well with text that is trying to be beautiful by being as concise as possible. I need to ramble a bit and explain myself almost everything by way of analogy. And somewhere along the way I'll internalize the concept. Furthermore, I love texts which intermingle hard science with stories, historical and philosophical background.

Here are things I'd like to get better understanding of:
- information theory (plus complexity and all that Chaitin jazz)
- probability theory, stochastic processes, statistics (and bits of finance, actuarial stuff)
- calculus and measure
- algebra, symmetries, group theory
- cryptography
- various math tools used in physics (if I could ever understand math machinery underlying quantum mechanics I'll die a happy man), bits of physics also welcome, of course
...or anything else that's interesting and not hugely specialized, really.

What I don't want: totally entry level, big simplifications.

FYI, I'm studying in Eastern Europe, where (supposedly) there is much more emphasis put on 'going deep' when learning maths (which usually means just biting more than you can chew).
posted by desultory_banyan to Science & Nature (25 answers total) 41 users marked this as a favorite
Unfortunately, those don't exist. By design.

Mathematicians (at least the analysts and algebraists I know) seem to feel that analogy is not the right way of doing things. Instead, there is nothing but the theorem, and the theorem says what it says, and does not say what it does not say. Any analogy would only serve to muddy the waters.

However, Chaitin has bucked that trend, so maybe try any of his books?
posted by pmb at 4:57 AM on September 27, 2012

My experience is limited, but... when I studied a one-semester course in linear algebra in the UK, everything we needed to learn was contained in a 40-page stapled booklet issued by my university. It was very concise, (theorem, proof) basically.

Here in the US, one-semester courses in linear algebra tends to use a textbook of 500+ pages, including *many* pictures, lengthy explanations, stories, lots of color, gazillions of examples, and so on.

If I'm assuming correctly that your Eastern European university uses an approach more like that which I experienced in the UK, you may simply want to explore more "American-style" textbooks. While they tend to get more concise and less colorful the higher the level, I'm convinced that you will be able to find something lengthy enough for your topics of interest.
posted by yonglin at 5:36 AM on September 27, 2012

For the second time in a week I get to recommend the Princeton Companion to Mathematics. The essays in it are written by many different authors in many different styles, and range from the Bourbakisme you're not looking for to the entirely philosophical, and everything in between.

For probability theory, you would really like either of Ian Hacking's great books The Emergence of Probability or The Taming of Chance about the history of the idea that chance events were subject to mathematical law.

For algebra, symmetries, physics, you might like Shlomo Sternberg's Group Theory and Physics; this is more of a traditional textbook, but it teaches group theory starting from crystallography rather than from the traditional axiomatic point of view, and my memory is that it talks about applications to quantum mechanics towards the end.

I too wish there were a hundred books out there like Godel, Escher, Bach, but it's really kind of a singular achievement, isn't it? There's a reason it's the only math book that's ever won -- probably the only math book that ever will win -- the Pulitzer Prize.

That said, as a practicing algebraist, I do indeed hold the view that pmb ascribes to me; to really understand math, as opposed to understanding something about math, requires a willingness to formalize. You ask for books that "work on your intuition" -- a book by J.P. Serre, terse and formal though it is, does work on your intuiton, but only once your intuition has been suitably trained. This training is the work an undergraduate math degree is, or ought to be, doing.
posted by escabeche at 5:48 AM on September 27, 2012 [7 favorites]

Also, I hesitate to recommend a book I haven't read, but James Gleick's The Information was warmly reviewed, and certainly presents itself as exactly the kind of guide you want to information theory and complexity theory.
posted by escabeche at 5:52 AM on September 27, 2012

I recommend it so often that I almost feel like I'm selling it, but Tom Korner's The Pleasures of Counting does a great job of this for a wide variety of fairly "basic" areas of mathematics, and his Naive Decision Making delves a bit more deeply into probability and game theory. If you like those, he has some more heavy duty textbooks that also have a lot of interesting anecdotes and asides.
posted by crocomancer at 6:14 AM on September 27, 2012

Several popular favorites in this recent question were described as "intuitive," or at least not completely conforming to the pattern of theorem-proof-corollary. I'm curious about the Needham book on complex analysis myself.
posted by Nomyte at 6:53 AM on September 27, 2012

As I said in last week's thread that Nomyte references, Visual Complex Analysis is a super example of advanced math being taught largely through pictures and appeals to intuition. I loved it.
posted by dfan at 8:18 AM on September 27, 2012 [1 favorite]

An Imaginary Tale: The Story of Sqrt(-1) by Paul J. Nahin
"e": The Story of a Number by Eli Maor
posted by Captain Chesapeake at 8:23 AM on September 27, 2012 [1 favorite]

I suggest two books by John Derbyshire : Prime Obsession and Unknown Quantity

Both are good but (I think) Prime Obsession is the better of the two and probably comes closer to the standard of hard science with historical background that you are looking for.
posted by metadave at 8:45 AM on September 27, 2012

I apologise in advance if this is too simple for you but I believe that, by the nature of the subject it tackles, that is not possible: The Infinite Book by John Barrow.
posted by HopStopDon'tShop at 9:07 AM on September 27, 2012

I found this column by Steven Strogatz to be pretty interesting and useful

Also, not for math, but for physics, there is Mr. Tompkins in Wonderland.
posted by taltalim at 9:20 AM on September 27, 2012

I think there's been a column/blog in the NYTimes trying to basically explain math. I haven't thought of it in a year or two though, so no idea whether it's still running.
posted by acm at 10:14 AM on September 27, 2012

The best way might be to use one of those advanced textbooks and then do a LOT of problems. That's what helps make math intuitive to me.
posted by schroedinger at 11:22 AM on September 27, 2012

Is David Foster Wallace's 'Everything and More' something that would appeal? It sounds like the writing style might suit but I have no idea whether the content would be hardcore enough? Disclaimer: I've not read it (yet) and am unfortunately not heavily mathsy.
posted by pymsical at 2:06 PM on September 27, 2012

pmb, to clarify my position, you're obviously right, but what I was trying to say, I guess, was that my experience of mathematics so far has been mostly "hey, here's theorem X that I now understand... now what?" - you can know a book about Fourier transforms by heart, yet remain absolutely clueless about the most basic engineering concepts that actually make use of them. There's knowing your stuff and there's knowing and I doubt if I can explain that better.

My favourite example of that would be Bayes Theorem, where it takes 5 minutes to understand the concept, yet you can ponder endlessly about that little equation and it's practical uses (head down to if you don't believe me). My courses in mathematics have, so far, been of the former variety.

Thanks for all recommendations, browsing through all that stuff now. I'll mark best answers when I had a chance to browse through some of them. And as schroedinger says, doing a lot of problems is always a good method. But sometimes working at something at a different angle can save you hours or weeks of struggling with a concept.
posted by desultory_banyan at 3:34 PM on September 27, 2012

Oh, forgot to add: escabeche, I read Gleick's Information. It's a great, great read. Not very useful in this context (not a lot of actual math inside), but certainly got me fascinated in Shannon and his work.
posted by desultory_banyan at 3:59 PM on September 27, 2012

The World According to Wavelets is a great little book, especially since you already mentioned Fourier and information theory. I wrote some more about it, about a decade ago. (Man, that makes me feel old...)
posted by mbrubeck at 4:12 PM on September 27, 2012

Princeton Companion is awesome.

Newman World of Mathematics is slightly less awesome but cheap.

The ones I have most enjoyed are:

Courant and Robbins What is Mathematics?

Hilbert Foundations of Geometry

Hamming Methods of Mathematics.

There are zillions of equations in all of these but all of these authors seem to abide by a method of going slow and feeding the reader by tiny spoonfuls with helpful asides about which parts deserve especial appreciation for the quasi-divine wonder therein. I think that is what you are looking for. First Courant and Robbins. Second Hilbert. Third Hamming. You should be able to get cheap copies of these books if they are not in your local library.
posted by bukvich at 9:43 PM on September 27, 2012 [1 favorite]

Have you looked at any applied math books aimed towards people in the physical sciences? They're lighter on proofs but heavier on real-world problems and applicability towards physics, chemistry, and engineering concepts. I like Boas's "Mathematics for the Physical Sciences" as it covers a LOT of topics, but may not be applied enough for your tastes.
posted by schroedinger at 11:31 AM on September 28, 2012

I have read Knuth's "Surreal Numbers" (see the link in mail's comment, above) and it is delightful and is just what you are looking for. (I'd completely forgotten about it; thanks for the reminder, mail!)
posted by mbrubeck at 5:52 PM on September 28, 2012

"various math tools used in physics (if I could ever understand math machinery underlying quantum mechanics I'll die a happy man), bits of physics also welcome, of course"

I'd suggest two books: J. J. Sakurai's Modern Quantum Mechanics, 2nd ed. -- not Advanced QM, his other well-known textbook, which is a step towards QFT. Sakurai is very, very dense, but you can read the first three or four chapters, work every problem and you will understand how Dirac notation works and how it helps you get most of the way toward solving a problem for free, as it were.

I also like PAM Dirac's Principles of Quantum Mechanics. It was published in 1930 or so, but (I've only read the second edition) still seems to explain QM beautifully. These are good places to start with quantum, because they approach the field sort-of axiomatically -- here are states, here are operators that act upon them, here are generators of certain effects, etc. -- rather than messing around with too much of the physics side of things. Too many undergraduate QM texts start by trying to get you to understand some odd quirks about a particular experiment, and then try to smear the math on the backside of that, instead of putting it out front. I think you can explain almost all of QM with nothing more than a Stern-Gerlach experiment.

For a good handbook to all college level mathematics, I always recommend Riley, Hobson, and Bence's Mathematical Methods for Physics and Engineering. It's great. It's not precisely what you're after, but it benefits from its exhaustive depth and a surprisingly clear style. Beware, however, that it is still a math textbook, and while a very good one, it is perhaps not necessarily what you wanted.
posted by samofidelis at 3:47 PM on September 29, 2012 [1 favorite]

Problems for Mathematicians Young and Old. This is a really fun book by the great Paul Halmos. It's not intended to teach you about a specific topic. Instead, through the use of problems, Halmos helps to think a bit more deeply about seemingly straightforward mathematical problems. The coverage is excellent, with problems from combinatorics, calculus, probability, analysis, algebra, and measure theory among many others. The catch is that it seems to be out of print. But if you can find it, I think it would be worth the effort.

Road to Reality. I haven't finished reading it, but I'm enjoying it so far. Similar to yourself, I was interested in an intuitive and broad discussion of physics without holding back on the requisite mathematics. Has exercises.
posted by Bokmakierie at 6:19 PM on September 29, 2012 [2 favorites]

Thanks everyone! Few things I managed to check out so far:

- Princeton Companion is the best. It's not exactly what I was looking for (won't teach you anything in depth), but I tried about a dozen different topics, familiar and not, and each time walked off very content with what I've learned or new perspective I've got on things.

- Penrose's 'Road to Reality' - I've actually read about 200 pages of it before. It's good: it checks all the boxes I wanted: he writes clearly and it's quite interesting and not overly dense.
Still, high-level physics use some serious math (topology, for starters), so I'm giving it a pass for now - I'm pretty sure you won't be able to understand most of the concepts Penrose's writes about without extra resources.

- Working on Visual Complex Analysis along with my regular complex analysis course. So far the book looks interesting, but I'll reserve my judgment when I'm further into it, because so far it's been simple stuff.

- Courant & Robbins - I used it once, when learning analysis, it was okay. Might get back to it to have a look at linear algebra and topology parts, so if I do, I'll post impressions.

Thanks once again!
posted by desultory_banyan at 4:39 AM on November 10, 2012

update: not that deep into it (constantly getting sidetracked) but Visual Complex Analysis generally kicks ass and is a great example of what I was looking for

Code Book by Singh is a good introduction to cryptography - not for mathematicians, but in spirit of some other good recommendations in here
posted by desultory_banyan at 4:31 AM on January 2, 2013

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