Apr 8, 2008
Followup to: The Generalized Anti-Zombie Principle
One generalized lesson not to learn from the Anti-Zombie Argument is, "Anything you can't see doesn't exist."
It's tempting to conclude the general rule. It would make the Anti-Zombie Argument much simpler, on future occasions, if we could take this as a premise. But unfortunately that's just not Bayesian.
Suppose I transmit a photon out toward infinity, not aimed at any stars, or any galaxies, pointing it toward one of the great voids between superclusters. Based on standard physics, in other words, I don't expect this photon to intercept anything on its way out. The photon is moving at light speed, so I can't chase after it and capture it again.
If the expansion of the universe is accelerating, as current cosmology holds, there will come a future point where I don't expect to be able to interact with the photon even in principle—a future time beyond which I don't expect the photon's future light cone to intercept my world-line. Even if an alien species captured the photon and rushed back to tell us, they couldn't travel fast enough to make up for the accelerating expansion of the universe.
Should I believe that, in the moment where I can no longer interact with it even in principle, the photon disappears?
It would violate Conservation of Energy. And the second law of thermodynamics. And just about every other law of physics. And probably the Three Laws of Robotics. It would imply the photon knows I care about it and knows exactly when to disappear.
It's a silly idea.
But if you can believe in the continued existence of photons that have become experimentally undetectable to you, why doesn't this imply a general license to believe in the invisible?
(If you want to think about this question on your own, do so before the jump...)
Though I failed to Google a source, I remember reading that when it was first proposed that the Milky Way was our galaxy —that the hazy river of light in the night sky was made up of millions (or even billions) of stars—that Occam's Razor was invoked against the new hypothesis. Because, you see, the hypothesis vastly multiplied the number of "entities" in the believed universe. Or maybe it was the suggestion that "nebulae"—those hazy patches seen through a telescope—might be galaxies full of stars, that got the invocation of Occam's Razor.
Lex parsimoniae: Entia non sunt multiplicanda praeter necessitatem.
That was Occam's original formulation, the law of parsimony: Entities should not be multiplied beyond necessity.
If you postulate billions of stars that no one has ever believed in before, you're multiplying entities, aren't you?
No. There are two Bayesian formalizations of Occam's Razor: Solomonoff Induction, and Minimum Message Length. Neither penalizes galaxies for being big.
Which they had better not do! One of the lessons of history is that what-we-call-reality keeps turning out to be bigger and bigger and huger yet. Remember when the Earth was at the center of the universe? Remember when no one had invented Avogadro's number? If Occam's Razor was weighing against the multiplication of entities every time, we'd have to start doubting Occam's Razor, because it would have consistently turned out to be wrong.
In Solomonoff induction, the complexity of your model is the amount of code in the computer program you have to write to simulate your model. The amount of code, not the amount of RAM it uses, or the number of cycles it takes to compute. A model of the universe that contains billions of galaxies containing billions of stars, each star made of a billion trillion decillion quarks, will take a lot of RAM to run—but the code only has to describe the behavior of the quarks, and the stars and galaxies can be left to run themselves. I am speaking semi-metaphorically here—there are things in the universe besides quarks—but the point is, postulating an extra billion galaxies doesn't count against the size of your code, if you've already described one galaxy. It just takes a bit more RAM, and Occam's Razor doesn't care about RAM.
Why not? The Minimum Message Length formalism, which is nearly equivalent to Solomonoff Induction, may make the principle clearer: If you have to tell someone how your model of the universe works, you don't have to individually specify the location of each quark in each star in each galaxy. You just have to write down some equations. The amount of "stuff" that obeys the equation doesn't affect how long it takes to write the equation down. If you encode the equation into a file, and the file is 100 bits long, then there are 2100 other models that would be around the same file size, and you'll need roughly 100 bits of supporting evidence. You've got a limited amount of probability mass; and a priori, you've got to divide that mass up among all the messages you could send; and so postulating a model from within a model space of 2100 alternatives, means you've got to accept a 2-100 prior probability penalty—but having more galaxies doesn't add to this.
Postulating billions of stars in billions of galaxies doesn't affect the length of your message describing the overall behavior of all those galaxies. So you don't take a probability hit from having the same equations describing more things. (So long as your model's predictive successes aren't sensitive to the exact initial conditions. If you've got to specify the exact positions of all the quarks for your model to predict as well as it does, the extra quarks do count as a hit.)
If you suppose that the photon disappears when you are no longer looking at it, this is an additional law in your model of the universe. It's the laws that are "entities", costly under the laws of parsimony. Extra quarks are free.
So does it boil down to, "I believe the photon goes on existing as it wings off to nowhere, because my priors say it's simpler for it to go on existing than to disappear"?
This is what I thought at first, but on reflection, it's not quite right. (And not just because it opens the door to obvious abuses.)
I would boil it down to a distinction between belief in the implied invisible, and belief in the additional invisible.
When you believe that the photon goes on existing as it wings out to infinity, you're not believing that as an additional fact.
What you believe (assign probability to) is a set of simple equations; you believe these equations describe the universe. You believe these equations because they are the simplest equations you could find that describe the evidence. These equations are highly experimentally testable; they explain huge mounds of evidence visible in the past, and predict the results of many observations in the future.
You believe these equations, and it is a logical implication of these equations that the photon goes on existing as it wings off to nowhere, so you believe that as well.
Your priors, or even your probabilities, don't directly talk about the photon. What you assign probability to is not the photon, but the general laws. When you assign probability to the laws of physics as we know them, you automatically contribute that same probability to the photon continuing to exist on its way to nowhere—if you believe the logical implications of what you believe.
It's not that you believe in the invisible as such, from reasoning about invisible things. Rather the experimental evidence supports certain laws, and belief in those laws logically implies the existence of certain entities that you can't interact with. This is belief in the implied invisible.
On the other hand, if you believe that the photon is eaten out of existence by the Flying Spaghetti Monster—maybe on this just one occasion—or even if you believed without reason that the photon hit a dust speck on its way out—then you would be believing in a specific extra invisible event, on its own. If you thought that this sort of thing happened in general, you would believe in a specific extra invisible law. This is belief in the additional invisible.
The whole matter would be a lot simpler, admittedly, if we could just rule out the existence of entities we can't interact with, once and for all—have the universe stop existing at the edge of our telescopes. But this requires us to be very silly.
Saying that you shouldn't ever need a separate and additional belief about invisible things—that you only believe invisibles that are logical implications of general laws which are themselves testable, and even then, don't have any further beliefs about them that are not logical implications of visibly testable general rules—actually does seem to rule out all abuses of belief in the invisible, when applied correctly.
Perhaps I should say, "you should assign unaltered prior probability to additional invisibles", rather than saying, "do not believe in them." But if you think of a belief as something evidentially additional, something you bother to track, something where you bother to count up support for or against, then it's questionable whether we should ever have additional beliefs about additional invisibles.
There are exotic cases that break this in theory. (E.g: The epiphenomenal demons are watching you, and will torture 3^^^3 victims for a year, somewhere you can't ever verify the event, if you ever say the word "Niblick".) But I can't think of a case where the principle fails in human practice.
Added: To make it clear why you would sometimes want to think about implied invisibles, suppose you're going to launch a spaceship, at nearly the speed of light, toward a faraway supercluster. By the time the spaceship gets there and sets up a colony, the universe's expansion will have accelerated too much for them to ever send a message back. Do you deem it worth the purely altruistic effort to set up this colony, for the sake of all the people who will live there and be happy? Or do you think the spaceship blips out of existence before it gets there? This could be a very real question at some point.
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