I’ve been asked variants of this question a number of times. So here’s a thorough answer for everyone.
First, some well-written books on topics that interest me:
- Roger Penrose's The Road to Reality
- Cosma Shalizi's blog bactra.org
- John Baez’s blog
- Matthen’s tumblr and demonstrations.wolfram.com
- John Stillwell’s stroll through the historical development of mathematics
- Wikipedia. As bad as the writing is, a lot of graduate students have clearly put in many hours explaining the concepts and connections among algebra, geometry, topology, logic, discrete-maths, and analysis. If you want to join me in my quest to edit the mathematics sections of the World Encyclopedia into readability and Strunk+White-compliance, that would be awesome. :)
- Jan Koenderink's book about weird shapes
- Tristan Needham's book about complex numbers
- When you’re ready to take it to the next level and really understand stuff, watch Gilbert Strang’s lectures on linear algebra (MIT OCW).
- Tony Robbin’s book about the fourth dimension
- Bill Lawvere & Steve Schanuel’s book about categorial thinking.
Second, some personalisation advice: Read anything mathematical that talks about something you’re interested in.
- If you are into weather / fluids, check out MIT OCW’s courses from the atmospheric science department (free pdf’s).
- If you’re into the philosophy of quantum mechanics, try Itamar Pitowsky's book or the Stanford Encyclopedia of Philosophy (it will get into hilbert spaces soon enough).
- I haven’t read it, but Doug Hofstadter’s Gödel Escher Bach seems to have inspired a lot of people.
- Are you into puzzles? computer graphics? metaphysics? linguistics? liquids? animals? music theory? morality? bargaining theory? economic growth? interpersonal relationships? politics? the function of cells? systems theory? how the body moves? There are mathematical takes on all of those.
I’ve seen speech-pathology students with bad-mathematics syndrome take off like a fish in water when they do maths that relates to what they know—building sawtooth waves, just intonation, EQ’s. You know, ear stuff.
I recommend starting from what you know and using mathematics as a bridge to other things. (“Hey, I didn’t realise audio engineering gets me trigonometry for free! And from there I can go on to land surveying.”)
Third, get used to reading s l o w l y. I’m naturally a slow reader so this wasn’t a big adjustment for me. But people who are used to breezing through a paperback in an afternoon are often dismayed when they can’t do the same with mathematics. Expect mathematics books to take 10, 20, 30 times as long per page—or more—as regular English reading. You’re expected to re-read passages, turn back to refer to earlier definitions, grab pen & paper and play with a few examples yourself, and possibly cross-refer to other books / Wikipedias.
In fact, the speed/depth tradeoff is part of what makes mathematics pleasurable. After 2-3 years of reading news I’m typically left with a shallow, jerky, shiftless sensation. After 2-3 years of reading mathematics, I literally see the world differently—in a good way. I feel I’ve developed something worthwhile that makes my mind a more interesting place to be in, rather than a jumble of chattering trifles. So don’t worry if you only get through 2-3 pages at a sitting. Not every page will take 20 minutes to comprehend, but some single pages need to slosh around in your brain for days or months. Don’t worry about it. That’s normal. Go slow; go deep.
Finally: I try to write for who didn’t do the whole calc 1-2-3-4, ode, real analysis thing. In an ideal world, my writing would come out well enough that an artist or writer with no mathematical confidence could parse it and be inspired with a new thought-shape.
That’s for the original posts.I also try to make people aware of things that are well-known in my circles but not well-known in general. For example, did you know that some guy was actually able to define complexity?
There are three ways to explore this blog:
- http://isomorphismes.tumblr.com/tagged/economics groups together posts.
- http://isomorphismes.tumblr.com/search/quasicrystals lets you search.
- http://isomorphismes.tumblr.com/archive lets you browse the site backwards from today.
- http://isomorphismes.tumblr.com/archive/2010/12 lets you browse the site backwards from December 2010.