Monday, January 14, 2008

Occam's Razor: Useful is better

See my previous post on Occam's Razor, which described the interpretation I see most often. This post describes my preferred interpretation (which may or may not be a common interpretation).

A scientific hypothesis is a way of explaining a phenomenon. But the purpose isn't just to answer the question "why?" A useful hypothesis makes predictions. Why does the sun rise every morning? Because the Earth rotates. The reason this answer is important is not because it tells us about ultimate "why"s. The reason that it's important is that it makes predictions. For example, we can predict an equatorial bulge. We can predict the height of a geosynchronous orbit.

There can be lots of hypotheses to explain any one phenomenon. The way we decide between them is that we compare the predictions they make. We find an experiment under which hypothesis A predicts X, and hypothesis B predicts Y. If we observe X, we prove A (provided that we didn't forget a hypothesis C that also predicts X).

So what happens when we have two hypotheses that always make the same predictions? This occurs occasionally in physics. For example, there is an alternative hypothesis to "The Earth rotates". Perhaps the entire universe revolves around us. Under General Relativity, you can consider a rotating reference frame by changing around a few laws. Though it's not obvious without studying General Relativity, this hypothesis makes the exact same predictions as the hypothesis that the Earth rotates. Therefore, we can never choose one of these hypotheses over the other.

So why is it that we say the Earth rotates, and not that the universe rotates around us? It is because of Occam's Razor. Occam's Razor states that given two hypotheses with identical predictions, we should use the one that's easier to use. Neither hypothesis is more true than the other, but one is definitely more useful. If we wanted to send a satellite into orbit, we must choose one of the hypotheses to make our calculations. The calculations are much simpler if we choose the hypothesis that the Earth rotates. Physics is complicated enough without our help. Note that the most "useful" hypothesis may depend on the situation. For example, I doubt that when you commute, you take into account that your destination is slowly rotating around the Earth.

All of this was based on the premise that the only purpose of explaining things was to make predictions. But I lied--this is not always the case. Sometimes, due to basic human curiosity, we really are interested in ultimate "why"s. What if you think our "ultimate purpose" hinges on a question? For example, perhaps you think that the universe rotating around us means our planet is special. Well, Occam's Razor does not necessarily apply in this case, since usefulness is not a concern. You are absolutely free to choose whichever makes you happiest. And whenever you need to make predictions, you can temporarily switch to the simpler hypothesis.

But this raises a question. If two hypotheses are indistinguishable, why should either of them make you happier than the other?