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.
Chgg Clou
·
2021-06-04T04:31:47Z (over 3 years ago)