Pillar of Science III: Approximate Models

Science requires Approximations.

Every kind of professional activity changes the way you think.  It rewires your brain so that even when you’re off the job, things start looking a certain way.  For example, to a computer programmer everything looks like an algorithm.  To a teacher, everything is pedagogical.  As a physicist, what goes through my head every day is approximations.

Every time I think about a situation involving black holes, or prove a theorem, or do a calculation, I always have to keep in the back of my mind what kinds of physical effects I’m ignoring, not taking into account.  This habit has leaked out into my thinking about life in general.  Ideas don’t have to be just true or false, instead they can be good approximations in some contexts, and bad approximations in other contexts.

(Partly related: before I started doing physics seriously, I think I had the idea in the back of my head that when I went to grad school, I’d learn how to calculate the really hard problems.  But it turns out there is no way to calculate the answers to the hard math problems.  There are only clever tricks for simplifying hard problems so that they become easy problems.  Frequently, the clever trick is finding some parameter that can be taken to be small, in order to justify some approximation.  The way this works is: first you figure out what happens if the parameter is zero, and then you calculate the tiny effects of it not quite being zero.)

It is impossible for any model of the universe to capture every feature of reality, or else it would be too complicated for human beings to analyze, or to compare to experiment, at the high level of precision demanded by science.  Consequently every scientific theory is applicable only to some limited range of phenomena.  In other words, it isolates some feature of reality which is as free as possible from contaminating influences, and which is simple enough to be either measured experimentally or calculated theoretically (ideally, one can do both, to compare the theory to the experiment).

Therefore, Science consists of a bunch of partly overlapping models which cover different patches of reality. Some of these patches are smaller, and cover very specific situations (e.g. Bernoulli’s principle for fluid dynamics) and others cover a very broad range of situations (e.g. Quantum Electrodynamics, which covers everything related to electricity, chemistry, and light). None of these patches covers everything, and the two broadest patches, Quantum Field Theory and General Relativity, cannot yet be fully reconciled with one another.

One of the implications of this principle is that scientific revolutions seldom result in the complete discrediting of the old well-established theory. The reason is that if the first theory explained a significant patch of data, the new theory can only supercede the old one if it explains all the things the first one explained, and more. Usually this means that the old theory is a limiting situation or special case of the new theory. Thus the old theory is still valid, just in a smaller patch than the new theory. For example, Einstein’s theory of General Relativity superceded Newton’s theory of gravity, but it predicts nearly the same results as Newton in the special case that the objects being considered are travelling much slower than light, and their gravitational fields are not too strong.

Thus the empirical predictions of Newton’s theory are still correct when applied in the proper domain.  However, the philosophical implications regarding the nature of space and time could hardly be more different in the two theories, because the Newtonian theory regards space and time as fixed, immutable, separate entities, while Einsteinian theory regards spacetime as a single contingent field capable of being affected by the flow of energy and momentum through the spacetime.

Philosophers who reason from scientific discoveries should take warning from this: although the empirical predictions of a theory usually survive revolution, the philosophical implications often do not.  Thus our current scientific views on such matters should be taken as somewhat provisional.  On the other hand, it would be even more foolhardy to try to discuss the philosophical nature of space, time, causality, etc. without taking into account the radical changes which Science has made to our naïve intuitions about these concepts.  Some improvement of our thinking is better than none.