Simple becomes difficult

We ponder how the same scientific task gets harder with time.

Science generally gets more exact, closer to real observations, over time.  But it also gets more difficult to do, we note.  Consider, as an example, calculating the positions of the planets.  This may not be the most urgent of tasks, but it does have a long history, which makes it useful for our purposes.

The Babylonians (by which we mean a few studious and advanced mathematicians of that nation) had a system for predicting where the planets might be seen in the sky.  They had noticed that each moved against the stars, completing a full circle in a characteristic time.  But they also noticed that this movement did not happen at a steady rate.  They came up with fast and slow sectors, allowing them to say where each might be found at a given date.  By our standards it was crude, but served to direct their gaze to the proper area.  And it only required a certain level of arithmetic skill.

The Greeks were concerned not only with what happened in the sky, but why and how.  With a more complicated system, as in the finished form set out by Ptolemy, an astronomer could predict planetary positions with almost all the accuracy one needed in the days before telescopes.  To do this, however, one needed a familiarity with geometry and trigonometry.  Such a level of mathematical skill was rare in the ancient and medieval world.  However, any reasonably competent graduate of Precalculus today should have it.  We’re not saying that most High School students could master the technical details of the Ptolemaic system, much less produce it from scratch.  We do assert that they have the mathematics to follow it, if given a clear presentation, and could perform any of the necessary calculations if called upon.

When Kepler replaced combinations of circles with ellipses, the problem became much harder mathematically.  He was a master of calculation, in the days before calculus or even logarithms, and his methods are not now much taught.  But we point out that one necessary step, the solution of “Kepler’s Equation,” cannot be done in any closed form, even now; one must iterate.  It’s a good task for properly supervised digital computers.

Newton had to invent a whole new field of mathematics, Calculus, to deal with planetary motion.  Using it, one can show that two point masses (or spheres) moving under gravity must describe a conic section: ellipse, circle, parabola or hyperbola.  We are now into advanced undergraduate mathematics, though some students won’t see this particular demonstration until grad school.  But if one adds a third mass, the problem becomes impossible.  That is, there is no general solution once we go beyond two spheres.  There are families of solutions for various approximations (where the third body has negligible mass, for instance), but we can’t even tell what shape the solution takes in general.  Nineteenth-century astronomers had methods to produce excellent approximations for the Solar System, but each required a great deal of work.

Once we go to General Relativity, even the two-body problem can’t be solved.  One can calculate an orbit if one of the bodies has negligible mass, but the situation of two comparable objects has to be turned over to computers.  Even approximations of the entire universe are easier.

Strictly speaking, the problem cannot even be posed in Quantum Mechanics.  QM doesn’t include gravity at all.  We suppose something might be contrived by way of an imposed spherical potential function, and perhaps it has even been done.  But it’s a matter of using a certain mathematical apparatus for something it’s not designed to do, and it might not work.

So there’s a point in learning, and using, a discarded theory.  If you use a better one, a simple problem could become impossible.