18 February 2008

The back of the universe

Brent Yorgey is working on a book intended to introduce high-school-level students to mathematics through hands-on problem solving. (His working title is The Math Less Traveled.) In his preface, he writes:
If you're having trouble with a problem, don't look at the answer -- put it down and come back later. This is how 'real' mathematics works: there's no 'Back of the Universe' where mathematicians can go look up the answer when they can't solve a problem!
If there were a "Back of the Universe", of course, that still wouldn't be practical, because it would be really far away.

Some might argue that the "Back of the Universe" is in fact the universe itself, at least in problems that have some interpretation in terms of real-world things. But just like the back of the textbook, physical experiment only gives answers, not solutions. If you throw a ball into the air and see that its trajectory is parabolic, that's something -- but experiment can't generally tell you why something is true.

The other wonderful thing about "real mathematics" is that, unlike for homework, you don't have to do all the problems; I believe I recently came across something stating that if a research mathematician solves one out of every one hundred problems they look at, they're doing well. This counterbalances the fact that real problems are much harder than textbook problems. (I wish I could cite this claim; it's on one of the many "advice for graduate students in math" web pages out there.)

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