Measuring complications

A simple idea can become a complicated thing to implement.

We have mentioned before, and no doubt you have met in school, the ancient geocentric theory of the universe.  In it, the Earth is stationary at the center of all things.  Each of the planets (a term that includes the Sun and Moon) is carried around the Earth on two uniformly-turning circles, a small one (the epicycle) centered on the rim of the larger one.  Add in another circle for the stars, plus others for technical reasons we won’t go into, and you have a lot of circles.  It can seem quite complicated.  Indeed, a nineteenth-century Astronomer Royal, presenting the theory to a lay audience, felt it necessary to apologize for its complexity, and express astonishment that anyone had ever believed in something so lacking in simplicity.

Other astronomers of the same era said similar things, and the legend grew that the Ptolemaic system had added epicycle after epicycle in a vain attempt to match the actual motions of the planets, until it had finally collapsed under the weight of too many circles.  (This is not at all true, and says more about the conceit of nineteenth-century astronomers than about history or science.)

In its place the Astronomer Royal presented Newton’s laws of motion and gravity.  The diversity of apparent motions becomes a result of a single law that acts on each body the very same way.  Instead of diverse diagrams of circles and tables of painfully-calculated numbers, you can write down a single equation that covers it all.  The principle that the simpler theory is to be preferred as an explanation (sometimes known as Occam’s Razor) seems to act very strongly for Newton.

The fact that this is not necessarily the case is hinted at by that same Astronomer Royal.  In that same series of lectures he regretted that he could not explain how Newton’s equation was actually solved, since it involved more advanced mathematics than his audience could be expected to understand.  In fact the one simple equation is a three-dimensional one (four if you count time), and each dimension must be dealt with.  Worse, the equation for each planet includes each other planet.  The situation is actually quite complex.  Sir John Herschel, a more ambitious popular-science author than the Astronomer Royal, spent a long chapter in his Outlines of Astronomy trying to explain how it was done.  Our astronomer has read it carefully and is not sure he succeeded.

So actually applying the new theory leads to much greater complication.  Does that mean Occam would decide against it?  But as we noted, Newtonian gravity replaces a variety of circles with a single idea.  Conceptually it’s simpler.  It appears one could argue either way, as a matter of philosophy.  Being scientists, of course, we’d rather decide between theories based not on any philosophical Razor, but on which does a better job matching observations.