Who Cares About Simplicity?

I think appeals to “simplicity” and “Occam’s Razor” are a kind of baby epistemology – maybe a heuristic that has some value for philosophical novices, but a simplification that experts should have gotten past. What matters is the probability of a theory, not “simplicity” per se – but simplicity might have some correlation with probability.

The Amazing Razor

Occam’s Razor tells us something like:

Occam’s Razor: Other things being equal, simpler theories should be preferred over more complex theories.

But why would that be? If you think about it for five seconds, questions like this will probably occur to you: “Why assume the world is simple? Does Occam’s Razor depend on some principle like

For every possible pair of views about a given question, if one view is simpler than the other, the world is a priori more likely to match the simpler view.

?” If so, wow. That’s a pretty sweeping, powerful principle. So, e.g., purely a priori, the probability that the mind is identical to the brain is greater than the probability that the mind is distinct from the brain? Is that what the physicalists who appeal to the “simplicity” of physicalism are saying? Would it also be true that, purely a priori, the probability that cats are planets is greater than the probability that cats are distinct from planets?

There had better be some story about why that’s true. It’d be really interesting to hear what it is, wouldn’t it?

But you’re going to have difficulty finding that story, because the people who go around appealing to Occam’s Razor – using it as a cudgel against theists, dualists, Platonists, free-will believers, moral realists, etc. – rarely have any idea.

Two Cases

Let’s try to understand it. There are certain contexts in which it’s going to be pretty uncontroversial that the simpler theory is more likely correct. Like this:

Case 1: You’re sitting in your apartment, using your only two electrical devices: a desktop computer and a lamp. Both of them suddenly shut off at the same time. You consider two theories, a simple one (S) and a complex one (C):

S: There is a power failure occurring now.

C: The light bulb burned out, and the computer crashed.

Epistemology problem: Comparatively evaluate these, without collecting further evidence.

Answer: S is simpler than C, since S posits a single cause for both pieces of evidence, while C posits two separate causes. S is also more likely to be correct than C, for fairly obvious reasons connected to its simplicity. It’s fairly natural to say that, ceteris paribus, the occurrence of two failures (as in C) is less likely than the occurrence of one failure (as in S).

But is this always true? Definitely not. We can have a case that is very similar to the above, except that the probabilities work out differently.

Case 2: You go away on a sabbatical in Europe for a year. When you leave, for some reason, you leave your only two electrical devices on at home: a desktop computer and a lamp. You return a year later and, as you enter your house, see that both of them are now off. You consider two hypotheses:

S: There is a power failure occurring now.

C: The light bulb burned out, and the computer crashed.

Problem: Comparatively evaluate these, without collecting further evidence.

Answer: Again, S is simpler than C, for exactly the same reason as stated above: it cites a single cause for both data, whereas C cites two separate causes. Nevertheless, this time, C is more likely than S. (If you’re not sure of this, make the time period longer, e.g., ten years.)

Both S and C explain all the data, and I’d be inclined to say they “explain it equally well” – unless, of course, the “goodness” of an explanation just depends on the overall probability of the theory being correct – in which case, we can’t appeal to the quality of explanations in judging which theory to prefer, since that would beg the question. So leaving aside the overall probabilistic assessment of the two theories, it would seem that in this case, S explains the data in Case 2 exactly as well as it did in Case 1; also, C explains the data exactly as well in Case 2 as it did in Case 1. So we should be able to apply Occam’s Razor now and conclude that we should prefer S over C. Right?

But that’s silly; no one would do that.

Replies in Defense of the Razor

Of course there are ways of preserving Occam’s Razor from this otherwise obvious counterexample. Philosophers can pretty much always modify a principle’s interpretation to preserve it from counterexamples.

Outweighed Value

Here’s a first thought. Maybe simplicity has some epistemic value as a virtue of theories, but in Case 2 that value is outweighed, whereas it is not outweighed in Case 1. What would be outweighing the value of simplicity in Case 2? I guess it would be the probabilistic considerations.

How would we distinguish this view from the view that simplicity is just epistemically irrelevant, since sometimes the simpler theory is better, and sometimes it’s worse? Or maybe even complexity is a virtue, but its value is outweighed in Case 1 (by probabilistic considerations) but not in Case 2?

I guess the view would have to be that, once you note that S is simpler than C, you give S an extra little probability boost, or you subtract a little from the probability of C. E.g., you first calculate the joint probability of a light-bulb burnout and a computer crash over the course of a year, then subtract a little bit from that probability, as a sort of “complexity penalty”, before assigning your credence to C.

But that, I take it, is again silly. Of course you don’t do that. You just estimate the probabilities normally – using the conjunction rule, your knowledge of base rates, etc. Once you’ve done that, you’re done; you don’t have to do anything using the notion of “simplicity”.

The Ceteris Paribus Clause

Maybe Occam’s Razor is saved by the “other things being equal” clause. Maybe you can say that other things aren’t equal in Case 2, because in Case 2 (but not in Case 1), there is some other theoretical virtue that differs between the two theories, and that means Occam’s Razor just doesn’t apply to Case 2.

Okay, but what would this other theoretical virtue be? Both theories seem to account for all the data. If true, each predicts the data with equal certainty. Both are equally “conservative” (consistent with the rest of our worldview). Really, there’s only one thing I can see that’s better about theory C: it’s more likely to be true, given your background opinions about the frequency of power failures, light bulb burnouts, and computer crashes.

Suppose we say that: The epistemic virtue of simplicity only comes into play when “other things are equal”, and other things are only equal when the two theories are equally likely to be true. But then what happens – the two theories are equally likely to be true, but somehow, according to Occam’s Razor, you should prefer to believe the simpler one? That’s bizarre. Shouldn’t you attach equal credence to both, since they’re equally likely?

Okay, try again. Maybe we say: simplicity comes into play only when it isn’t the case that the complex theory is equally likely or more likely to be true. Only then do we say the simpler theory is to be preferred. Sure. But that’s trivial: If the more complex theory is not equally or more likely to be true, then the simpler theory is more likely to be true! But it’s also true that if the simpler theory is not equally or more likely to be true, then the more complex theory is more likely. So you could just as well enunciate the following principle:

Plato’s Rogaine: Other things being equal, more complex theories should be preferred over simpler ones.

(The term “Plato’s Beard” has already been taken for another idea.) Sounds silly, but that would be true, given the same sort of interpretive liberties as we’re using to defend Occam’s Razor. When someone brings up Case 1 as a counterexample to Plato’s Rogaine, we just say that other things aren’t equal in Case 1, because there, the simpler theory is more likely.

Conclusion

The important point is that Occam’s Razor is not a useful principle in these cases (and aren't these precisely the sort of cases in which you'd expect the Razor to be useful?). You can do some interpretive maneuvering that lets you say it’s true, but the principle won’t actually help us evaluate theories. We first have to figure out how likely we think each theory is, and arrange our credences accordingly; then, if we like, we can decide which theory, if any, counts as having “the epistemic virtue of simplicity”. But that simplicity doesn’t play any role in generating the probability estimates, so it’s a sort of epistemic epiphenomenon.