16 September 2007

hard science, and calculating the area of a circle

The Really Hard Science, by Michael Shermer, from the October 2007 issue of Scientific American. The author makes the claim that the traditional distinction between "hard" and "soft" science is backwards; the social sciences might more reasonably be considered harder. I'm not sure how much I buy this linguistic claim, because "hard" has two antonyms in English, "soft" and "easy", and I suspect there's a reason we call the social sciences "soft sciences", not "easy sciences". The point that all these disciplines are valuable, I support. Shermer points out that modeling a biological or social system is often more difficult than modeling a physical system. We tend to think of the mathematics used by physicists as more complicated than that used by social scientists, but how much of that is a reflection of the subject matter and how much of that is due to the historical alliance between mathematicians and physicists, which means that physicists have tended to know lots of mathematics and, say, sociologists don't know as much? (I struggle with this in teaching calculus; our textbook, like many calculus textbooks, draws a lot of its examples from physics -- centers of mass, moments of inertia, and such -- and the students aren't as likely to be conversant with physics as the authors of the text seem to assume.)

He writes the following:
Between technical and popular science writing is what I call “integrative science,” a process that blends data, theory and narrative. Without all three of these metaphorical legs, the seat on which the enterprise of science rests would collapse. Attempts to determine which of the three legs has the greatest value is on par with debating whether π or r2 is the most important factor in computing the area of a circle.

Which is more important in calculating the area of a circle? I say r2. It's interesting that there is a constant, but the way in which the area of the circle grows when you increase the radius (namely, twice as fast) is more important to me. I may only be saying this, though, because I don't have to actually calculate the area of a circle. The people who made the circular table I'm sitting at certainly care about π to tell them how much raw material they will need.

The "twice as fast" statement there is a bit vague, but I mean it in the sense that if one increases the radius of a circle by some small fraction ε (as usual, ε2 = 0), one increases its area by the fraction 2ε. That is,

π(r(1+ε))2 = π(r2 + 2εr2+ ε2r2) = (πr2)(1+2ε)

where the last equality assumes ε2 = 0. For the most part this is how I think of derivatives -- at least simple ones like this -- in my head. The ε2 = 0 idea is the sort of thing one might see in the proposed "wiki of mathematical tricks" of Tao and Gowers (the real meat of the discussion is actually in the comments on those posts), although perhaps at a slightly lower level than they're aiming at.

6 comments:

Aaron said...

Ooooooh, but here's something cool! For a circle, A = pi*r^2, and P = 2*pi*r, which means A = P^2/(4*pi). But since no closed plane curve of a given perimiter can have a larger area than a circle, the circle is in some sense defined by the relation 4*pi*A = P^2. Can any other curves be defined in this way? I'm guessing not, because the circle is special in that it maximizes A/P^2, but maybe there are other curves that are special in different ways.

Man, I hope I don't read this tomorrow and realize how bleary I must've been when I wrote it. :)

Matt Noonan said...

My absent coblogger Jim pointed out to me that the multiplicative version of the derivative (like the one you computed here) is particularly nice for thinking about some things. In particular, the product says that the %-rate of increase of a product is the sum of the individual % increases, and the quotient rule is the difference of % increases.

Also, it must be that the r^2 is more important since all circles in all geometries grow like r^2, but with a different constant out front :)

Blake Stacey said...

Hey, I work in complex systems: my entire job is to apply physics math to things which are not physics, "reducing" questions in, say, ecology to unsolved problems in quantum field theory.

Anonymous said...

I've always thought that the hard in "hard science" was of the "definite, firm, factual" meaning of the word because soft sciences were more fuzzy if not subjective.

Harald Hanche-Olsen said...

I'd just like to point out a column that's found in every issue of the Annals of improbable research, entitled Soft is hard. It makes the point nicely doesn't it?

Anonymous said...

This has been on my mind for some time..... and I agree with you to some degree.
frokostordning