Comparing Empirically Identical Theories
April 2023
I recently read a relatively new Slate Star Codex post titled Building Intuitions on Non-Empirical Arguments in Science in which Scott Alexander asks a very interesting question: if the observational implications of two theories are identical, then how do we pick which one to believe in? Scott introduces this dilemma via a famous example: on the one hand we have the theory of paleontology and on the other we have the theory of the paleontologist-Satan. Paleontologist Satan wants to fool us into thinking that God did not create the universe. So he hides fake fossils all across the Earth in such a way that it would cause the humans to think that there were creatures like dinosaurs millions of years before Adam and Eve. In other words, the theory of the paleontologist-Satan makes the exact same predictions as the theory of paleontology; the only difference being that propositions of the first start with "To fool us, Satan did ___." Due do this effect, no experiment can tell us which one is false. All the observable implications of the two are identical, and the non-observable implications are, well, unobservable1. So how do we pick which one to believe in?
Having introduced this dilemma, Scott then goes over some of the various unsatisfactory solutions before presenting the standard rationalist argument: the theory of paleontology is simpler, so it is better. This invocation of Occam's razor is not entirely obvious—it's certainly not obvious to William of Occam. But I do think that it makes sense. The paleontologist-Satan theory is the paleontology theory plus some of other stuff. This already makes it less likely than the normal theory of paleontology. In probability parlance, if the paleontologist theory is $\Pi$ and the paleontologist-Satan theory is $\Sigma$, then Bayes' theorem states that $$P(\Sigma | \Pi) = \frac{P(\Pi | \Sigma)P(\Sigma)}{P(\Pi)}.$$ But since we know that the paleontologist-Satan theory already has the paleontologist theory built in, then $P(\Pi | \Sigma) = 1$. So $$P(\Pi)P(\Sigma|\Pi) = P(\Sigma).$$ And since all of these numbers are smaller than $1$, then $P(\Pi) > P(\Sigma)$. As such, probability theory tells us that the simpler theory—paleontology—is more likely to be true than the more complex theory of the paleontologist-Satan.
Well that settles it, right? I don't think so. This whole probability gibber jabber is perfectly true, but it's not especially relevant. If I intend to actually use paleontology to do something, it makes no difference if I use one or the other. And since paleontology has very few practical applications, lets use a different example.
Religious thinkers that want to remain compatible with science and keep believing in miracles often make the case that nothing happens without God's direct involvement. If cotton is lit after being put into contact with a candle, it's not because of the candle that the cotton catches fire; but rather because God willed the candle to be lit in this particular instant. In this view, if nature seems to follow certain laws, then it is only because God is consistent rather than arbitrary. Though every once in a while, when a miracle is necessary, God will break his own rules and will something other than what we have come to expect.
Suppose that Solomon is of this view and that—for whatever reason—he wants to build a space temple. After asking God very nicely if He could intervene and help his prophet out, and being rejected, he calls a meeting of the Royal Aeronautics and Space Agency of Judah. In this meeting, Solomon and his biblical friends use the theory of God-Likes-General-Relativity—GLGR for short—to make their plans for the space temple. The conclusions of GLGR are identical to the conclusions of Einstein's theory of general-relativity—hereafter known as GR. The only differences between GLGR and GR is that the propositions of GLGR start with "God seems to really like ___" and that the theory is considered inapplicable to events before or during creation.
Since the theory of GLGR has identical results to GR, then Royal Aeronautics and Space Agency of Judah is successful in putting Solomon's space temple into orbit. Right at the moment that the Judeans are about to start celebrating their great achievement, you time travel to Jerusalem and frantically shout at Solomon: "Wait! But the GLGR theory is more complicated that the theory of general relativity, so you should believe in GR instead of GLGR!"
Solomon isn't too happy to see you again, your last conversation was somewhat annoying. But since he learned lots of biology from you, he decides to give you a second chance to see what you have to say. He is very disappointed once you start going over Bayesian statistics. Just as you're about to finish your proof of Bayes' theorem, Solomon interrupts and says "well that's cool and all, but why should I use GR over GLGR?" And you were just about to get to it, despite the fact that GR and GLGR have the exact same conclusions, GR is still more likely to be true because of Bayes' theor— "So... GR can't improve the lives of my subjects better than GLGR?" And after a long pause, you're forced to say no. Solomon asks again: "why should I pick GR?"
By this point you're beginning to realize that Solomon's criterion for judging empirically identical theories is very different from yours. You're concerned about which theory is more likely to be true, Solomon's been thinking about the practical implications of each theory. But now you have some questions of your own to ask Solomon: does he prefer GLGR to GR? And if so, why does he? His response is that he does indeed prefer GLGR to GR, and he does so because of his religious beliefs. That doesn't really help you, so you ask him about why he believes in his religion. If he doesn't execute you for blasphemy, then he'll sigh. You're the $n$th time traveler that's asked him this and he's tired of it. His prepackaged response is that he has been in communication with God—being a prophet and all—and due to this he is absolutely convinced of all the doctrines of Judaism. He, however, cannot give you these pieces of empirical evidence because he can't convince God to talk to you nor will you take his word for it. "And besides," Solomon continues, "my religion is obviously the best! The people who follow its rules rigorously are prosperous, just like my kingdom. We just put a space temple into orbit in 940 BCE!"2
Solomon has indirectly answered your question. The reason he prefers GLGR is because of his religion, and he believes in his religion because he believes that it improves his life and the lives of those around him. Though you may disagree with his assessment about religion, you must admit that his pragmatic approach is quite sound. Judging a theory based on how much it improves you and your life is a much more relevant criterion than "more likely to be true." After all, theories were made for man, not man for the theories.
Satan is assumed to not show his face around the Earth after the Jesus incident. Read Paradise Lost for more information.
He knows about the Christian era because of the sheer number of time travelers he has to deal with.