How can you foretell if a problem is one whose solution admits a simple mathematical description?
Kindly see the emboldened phrase below. What kind of problems is the author referring to?
THE UNREASONABLE EFFECTIVENESS OF CLASSICAL GEOMETRY
For Apollonius and the Greek geometers, ellipses were conic sections: surfaces obtained by slicing a cone along a plane. Kepler showed (although it took the astronomical community some decades to catch on) that the planets traveled in elliptical orbits, not circular ones as had been previously thought. Now, the very same curve arises as the natural shape enclosing heights of parents and children. Why? It’s not because there’s some hidden cone governing heredity which, when lopped off at just the right angle, gives Galton’s ellipses. Nor is it that some form of genetic gravity enforces the elliptical form of Galton’s charts via Newtonian laws of mechanics.
The answer lies in a fundamental property of mathematics—in a sense, the very property that has made mathematics so magnificently useful to scientists. In math there are many, many complicated objects, but only a few simple ones. So if you have a problem whose solution admits a simple mathematical description, there are only a few possibilities for the solution. The simplest mathematical entities are thus ubiquitous, forced into multiple duty as solutions to all kinds of scientific problems.
The simplest curves are lines. And it’s clear that lines are everywhere in nature, from the edges of crystals to the paths of moving bodies in the absence of force. The next simplest curves are those cut out by quadratic equations,* in which no more than two variables are ever multiplied together. So squaring a variable, or multiplying two different variables, is allowed, but cubing a variable, or multiplying one variable by the square of another, is strictly forbidden. Curves in this class, including ellipses, are still called conic sections out of deference to history; but more forward-looking algebraic geometers call them quadrics.† Now there are lots of quadratic equations: any such is of the form
$Ax^2 + Bxy + C y^2 + D x + E y + F = 0$
for some values of the six constants A, B, C, D, E, and F. (The reader who feels so inclined can check that no other type of algebraic expression is allowed, subject to our requirement that we are only allowed to multiply two variables together, never three.) That seems like a lot of choices—infinitely many, in fact! But these quadrics turn out to fall into
- You could also make a case for curves of exponential growth and decay, which are just as ubiquitous as conic sections. † Why they are called quadrics as opposed to quadratics is a nomenclatural mystery I have not managed to penetrate.
three main classes: ellipses, parabolas, and hyperbolas.*
*There are actually a few extra cases, like the curve with the equation $xy = 0$, which is a pair of lines crossing at the point (0,); these are considered "degenerate" and we will not speak of them here.
Ellenberg, How Not to Be Wrong (2014), pages 307-8.