The web of science Check out Dave Winer's post on his weblog about going into applied math/computer science rather than pure math. He notes that his advisor-and most mathematicians generally-believe that pure math is the highest form of intellect. Dave-after 26 years of experience and hindsight-disagrees. He says pure math is simpler than applied math (he goes on about how it's hard to communicate math to non-math people). I don't know about all this. I took math up to linear algebra (easy) & multivariable calculus (not as easy-at least for me). Though more than the typical college student-it's a lot less than the "mathy" types. I found math kind of hard at a certain point, somewhere in the vague space between required math classes for chemistry majors (varieties of basic calculus and differential equations) and elective math classes for science majors (upper level calculus and differential equations-though linear algebra was pretty easy). Not because it was theoretically hard-it was theoretically easy (no unpredictable lab work), but the basic concepts just didn't bubble up through my brain like organic molecular conformations or balancing complex redox equations. When I would look at equations and problem sets in chemistry (and to some extent physics & biology) I could stare for about 15 minutes at a question and the answer or solution would jump out at me. I rarely found this happening in my later math classes. I had hit a "math wall" as I liked to call it (though if I'd take one upper division statistics class that I'd heard was pretty easy-I still could have gotten a minor in math). Some people encountered that their freshman year taking calculus-and they'd have to switch out of the sciences altogether. Natural science types only needed a basic level of math fluency-I found that I used very little of the power of differential equations or nuances of multivariate calculus in the upper division physical chemistry classes that I took-though those classes were prerequisites. E.O. Wilson was once asked on The Charlie Rose Show what social science had achieved. His response was not much-and his explanation was that it was "real hard." But by this-Wilson meant that political science and sociology, and to a lesser extent economics, are difficult to model mathematically because of their complexity. But let's be frank, does anyone believe that the average political scientist is a deeper thinker than the average physicist? Math is theoretically easy-yeah-but it always turns out to be hard for the average human. I'll end on a whimsical note. Biologists defer to chemists-who defer to physicists-who defer to mathematicians-who defer to God. A scientist says 4.00 (+/-.01), a mathematician says 4 and an engineer says between 3 and 5, but will call it 6 to be on the safe side. Joel adds: I was a somewhat-mediocre math major as an undergrad. I took all the courses and even got good grades in most of them, but I never really grokked what was going on. Unable to think of anything else to do with myself, I applied to lots of grad schools, was summarily rejected from most of them, and moved to Seattle to get my PhD in [pure] mathematics. We used to refer to the attitude Winer describes as "math imperialism" -- the belief that anyone who could do math was doing math. Contrapositively, anyone who wasn't doing math was unable to. It wasn't prevalent, but there were several professors with such an attitude. (By the end of the first quarter, the grad students generally had too little self-esteem to look down on anyone.) I never picked up this attitude. I was a surprisingly good student, but after two years I decided I didn't want to be a mathematician and left. Winer writes:

Pure math is a solo thing, very introspective, it's wonderful to see what your mind is capable of -- but unlocking power in other people's minds is much more of a challenge, and imho more gratifying.

I think he only gets it half right. Pure math is a solo thing. The sorts of problems I was working on were the type of things that only a couple of hundred people in the world would understand, let alone care about. And in the final analysis, the isolation and impracticality were enough of a turn-off to drive me away. I agree with Winer that "unlocking power" is more gratifying, though gratification is a personal matter -- Andrew Wiles found working in his attic for 7 years gratifying. But I disagree that "unlocking power" is "more challenging." Math is Hard. Those two years of math grad school were the most intellectually challenging time of my life, and I [dogmatically] expect them to stay that way no matter what I end up doing. I actually switched to math after being a dismal failure at "unpredictable" lab work in physics and chemistry, so I understand where Razib is coming from when he asserts that math is "theoretically easy," but it's not, unless you use "easy" to mean "involves no lab work." Once you get to the level of math which involves writing abstract proofs, you have to develop a very deep and unnatural mathematical intuition. Unless you're the next Ramanujan, a necessary condition is to work tons and tons of problems. And as far as I can tell, no one knows what a sufficient condition is. The flaw in reasoning (on both sides) is the belief that something challenging is somehow more worthwhile as a result. (I seem to remember the phrase "intellectually unimpeachable" being tossed around a fair amount back in the day.) I'm going back to school to study "Social Science" (mostly economics and some political science), and I expect it [possibly wrongly] to be substantially less difficult than my stint in math grad school. But I also expect it to be incomparably more rewarding, since I'll be working on problems which really interest me, and whose resolution lots of people care about. Razib (again) People do what they are well suited to and what they like. Though I received a degree in biochemistry-I found lab work something I didn't have a passion for (though the questions being asked in scientific endeavors were often fascinating to me-the mundane day to day work of running gels or running a drip over a period of hours was less glamorous). Today I do most of my "work" in IT-related tasks-coding, debugging and consulting. I say "work" because I actually have a lot of fun tackling these questions. To follow Joel's point-I find some of the arguments behind Scholasticism very difficult to comprehend-but that doesn't mean it's more worthwhile than something that comes easier to me-like say intellectual history (which requires more attention to detail than following serpentine streams of logic that the former demands). To each his own. On an interesting note, it turns out that mathematicians do tend to defer to God more than other scientists.