The accuracy argument for probabilism

A standard scoring rule argument for probabilism—the doctrine that credence assignments should satisfy the axioms of probability—goes as follows. If s is a scoring rule on a finite probability space Ω, so that s(c)(ω) is the epistemic utility of credence assignment c at ω in Ω, and (a) s is strictly proper and (b) s is continuous, then for any credence c that does not satisfy the axioms of probability, there is a credence p that does satisfy them such that s(p)(ω) is better than s(c)(ω) for all ω. This means that it’s stupid to have a non-probabilistic credence c, since you could instead replace it with p, and do better, no matter what.

Here is a problem with the dialectics behind this argument. Let P be the set of all credence assignments that satisfy the axioms of probability. But suppose that I think that there is some nonempty set M of credence assignments that do not satisfy the axioms of probability but are rationally just as good as those in P. Then I will think there is some way of making decisions using credences in M, just as good as the way of making decisions using credences in P. The best candidate in the literature for this is to use a level set integral, which allows one to assign an expected value E_cU to any utility assignment U even if c is not a probability. Note that E_pU is the standard mathematical expectation with respect to p if p is a probability.

The argument for probabilism assumed two things about the scoring rule: strict propriety and continuity. Strict propriety is the claim that:

1. E_ps(p) > E_ps(c) whenever c is a credence other than p

for any probability p. In words, by the lights of a probability p, then we get the best expected epistemic utility if we make p be our credence.

Now, if I am not convinced by the argument that (1) should hold for any probability p and any credence c other than p, then I will be unmoved by the scoring rule argument for probabilism. So suppose that I am convinced. But recall that I think that credences in M are just as rationally good as the probabilities in P. Because of this, if I find (1) convincing for all probabilities p, I will also find it convincing for all credences p in M, where E_p is my preferred way of calculating expected utilities—say, a level set integral.

Thus, if I am convinced by the argument for strict propriety, I will just as much accept (1) for p in M as for p in P. But now we have:

Theorem 1. If E_p is strongly monotonic for all p ∈ P ∪ M and coincides with mathematical expectation for p ∈ P, and (1) holds for all p in P ∪ M, where M is non-empty, then s is not continuous on P.

(Strong monotonicity means that if U < V everywhere then E_pU < E_pV. The Theorem follows immediately from the Pettigrew-Nielsen-Pruss domination theorem.)

Suppose then that I am convinced that a scoring rule s should be continuous (either on P or on all of P ∪ M). Then the conclusion I am apt to draw is that there just is no scoring rule that satisfies all the desiderata I want: continuity as well as (1) holding for all p ∈ P ∪ M.

In other words, the only way the argument for probabilism will be convincing to me is if my reason to think (1) is true for all p in P is significantly stronger than my reason to think (1) is true for all p in M, and I have a sufficiently strong reason to think that there is a scoring rule that satisfies all the true rational desiderata on a scoring rule to conclude that (1) holding for all p in M is not among the true rational desiderata even though its holding for all p in P is.

And once I additionally learn about the difficulties in defining sensible scoring rules on infinite spaces, I will be less confident in thinking there is a scoring rule that satisfies all the true rational desiderata on a scoring rule.

Instrumental bads

Suppose that a process Q has a chance r of producing some non-instrumentally bad result B, and nothing else of relevance. That fact gives us reason not to actualize Q. But suppose Q is actualized. Is it bad?

Well, if it’s bad, it seems it’s only instrumentally bad. It is no worse to be killed by a well-aimed arrow than by a well-aimed bullet, even though in the case of the well-aimed arrow the process of a deadly projectile’s flight lasts longer. Yet if a process producing a bad result were non-instrumentally bad, it would be worse if it lasted longer.

So we now have four options:

1. Q is always instrumentally bad (whether or not B eventuates)

2. Q is never instrumentally bad

3. Q is instrumentally bad if and only if B eventuates

4. Q is instrumentally bad if and only if B does not eventuate.

Option (4) is crazy. Option (2) destroys the very idea of an instrumental bad. So that leaves options (1) and (3).

If we opt for option (1), then we can have a world that contains instrumental bads without any non-instrumental bads—just imagine that Q obtains, B does not eventuate, and nothing else that’s bad ever happens. This seems a little counterintuitive: instrumental bads are derivatively bad, but how can something be derivatively bad without anything that is non-derivatively bad?

That suggests we should go for option (3): a process that has a chance of leading to a non-instrumental bad is bad only when the non-instrumental bad eventuates.

But now imagine Molinism is true. Suppose that God knows that Q, if actualized, would not lead to B, even though it has a non-zero chance r of doing so. In that case, the fact that Q has a chance r > 0 of leading to B is no reason for God not to actualize Q. But that something is bad is always a reason not to actualize it. If instrumental bads are an exception for this, then instrumental bads aren’t bads.

Now, I think Molinism is false. But whether (3) is true should not, it seems, depend on whether Molinism is true. So if (3) is false on Molinism, it is simply false.

So we seem to be stuck!

Maybe the right move is this. Fake money isn’t money and merely instrumental bads aren’t bad. This allows us an escape from the Molinism argument. For if merely instrumental bads aren’t bad, there is no problem about the fact that the Molinist God has no reason not to produce them.

Another move might be to say that (3) is true, but disproves Molinism. This doesn’t strike me as right, but maybe it’s defensible.

Until this is resolved, one really shouldn’t be running any arguments that depend on instrumental bads being actually bad.

The intrinsic badness of certain future tensed facts on presentism

It is bad that tomorrow someone will be in intense pain. On eternalism, we can easily explain this: tomorrow’s pain is just as real as today’s. But on presentism and growing block, future pains don’t exist.

Presumably, the presentist and growing blocker will say that the tensed fact of there being an intense pain tomorrow is bad, and this bad tensed fact presently exists.

Is this badness of the future tensed fact about the pain an instrumental or non-instrumental badness? If it’s instrumental, it is not clear what it could be instrumental to. The main candidate (apart from special cases where there is an obvious candidate, such as when the pain leads to despair) is that the fact that there will be a pain tomorrow is instrumental to tomorrow’s pain. But the fact that tomorrow there will be pain won’t cause that pain—otherwise, it would be trivial that every future event has a cause.

So the present badness of there being a pain tomorrow would be non-instrumental. But now imagine two scenarios with finite time lines.

• Scenario A: There is a mindless universe with a day of random particle movement, followed by the formation of a brain which has intense pain for a minute, followed by the end of time.

• Scenario B: There is a mindless universe with a century of random particle movement, followed by the formation of a brain which has intense pain for a minute, followed by the end of time.

Let’s suppose we find ourselves at the last moment of time in one scenario or the other. Then in Scenario A, there was a day of the obtaining of a “future pain fact”, and in Scenario B, there was a century of the obtaining of a “future pain fact”. If a future pain fact is a non-instrumentally bad thing, then there was non-instrumentally bad stuff in Scenario B for a much longer period of time than in Scenario A, and so Scenario B is much worse than Scenario A with respect to future pain. But that seems mistaken: the greater length of time during which there is a future pain fact does not seem any reason to prefer one scenario over another.

Should the A-theorist talk of tensed worlds?

For this post, suppose that an A-theory of time is true, so there is an absolute present. If we think of possible worlds as fully encoding how things can be so that:

1. A proposition p is possible if and only if p holds at some world,

then we live in different possible worlds at different times. For today a Friday is absolutely present and tomorrow a Saturday is absolutely present, and so how things are is different between today and tomorrow (or, in terms of propositions, that it’s Saturday is false but possible, so there must be a world where it’s true). In other words, given (1), the A-theorist is forced to think of worlds as tensed, as centered on a time.

But there is something a little counterintuitive about us living in different worlds at different times.

However, the A-theorist can avoid the counterintuitive conclusion by limiting truth at worlds to propositions that cannot change their truth value. The most straightforward way of doing that is to say:

2. Only propositions whose truth value cannot change hold at worlds

and restrict (1) to such propositions.

This, however, requires the rejection of the following plausible claim:

3. If (p or q) is true at a world w then p is true at w or q is true at w.

For the disjunction that it’s Friday or it’s not Friday is true at some world, since it’s a proposition that can’t change truth value, but neither disjunct can be true at a world by (2).

Alternately, we might limit the propositions true at a world to those expressible in B-language. But if our A-theorist is a presentist, then this still leads to a rejection of (3). For on presentism, the fundamental quantifiers quantify over present things, and the quantifiers of B-language are defined in terms of them. In particular, the B-language statement “There exist (tenselessly) dinosaurs” is to be understood as the disjunction “There existed, exist or will exist dinosaurs.” But if we have (3), then worlds will have to be tensed, because different disjuncts of “There existed, exist or will exist dinosaurs” will hold at different times. A similar issue comes up for growing block.

So on the most popular A-theories (presentism and growing block), we have to either allow that we inhabit different worlds at different times or deny (3). I think the better move is to allow that we inhabit different worlds at different times.

Mill on injustice

Mill thinks that:

1. An action is unjust if society has a utility-based reason to punish that actions of that type.

2. An action is wrong if there is utility-based reason not to peform that action.

Mill writes as if the unjust were a subset of the wrong. But it need not be. Suppose that powerful aliens have a weird religious view on which dyeing one’s hair green ought to be punished with a week in jail, and they announce that any country that refuses to enforce such a punishment as part of the criminal code will be completely annihilated. In that case, according to (1), dyeing one’s hair green is unjust. But it is not guaranteed to be wrong according to (2). The pleasure of having green hair could be greater than the unpleasantness of a week in jail, depending on details about the prison system and one’s aesthetic preferences.

The problem with (1), I think, is that utility-based reasons to punish actions of some type need have little to do with moral reasons, utilitarian or not, against actions of that type.

The three big mysteries of the concrete world

There are three big mysterious aspects of the concrete world around us:

• the causal

• the mental

• the normative.

The three mysteries are interwoven. Teleology is the domain of the interplay of the causal and the normative. And the mental always comes along with the normative, and often with the causal.

There is no hope of reducing the normative or the mental to the causal. Some have tried to reduce the normative to the mental, either via relativism (reducing to the finite mental) or Plantingan proper functionalism (reducing to the divine mental), neither of which appears particularly appealing in the end. I’ve toyed with reducing the mental to the normative, but while there is some hope of making progress on intentionality in this way, I doubt that there is a solution to the problem of consciousness in this direction.

Theism provides an elegant non-reductive story on which the three mysterious aspects of concrete reality are all found interwoven in one perfect being, and indeed follow from the perfection of that perfect being.

I wonder, too, if there is some way of seeing the three mysteries as reflective of the persons of the Trinity. Maybe the Father, the ultimate source of the other persons, is reflected in causality. The Son, the Logos, in the mental. And the Spirit, the loving concord of the Father and the Son may be reflected in the normative. But such analogies can be drawn in many ways, and I wouldn’t be very confident of them.

Necessity and the open future

Suppose the future is open. Then it is not true that tomorrow Jones will freely mow the lawn. Moreover, it is necessarily not true that Jones will freely mow the lawn, since on open future views it is impossible for an open claim about future free actions to be true. But what is necessarily not true is impossible. Hence it is impossible that Jones will freely mow the lawn. But that seems precisely the kind of thing the open futurist wishes to avoid saying.

Two difficulties for wavefunction realism

According to wavefunction realism, we should think of the wavefunction of the universe—considered as a square-integrable function on R³ⁿ where n is the number of particles—as a kind of fundamental physical field.

Here are two interesting consequences of wavefunction realism. First, it seems like it should be logically possible for the fundamental physical field to take any logically coherent combination of values on R³ⁿ. But now imagine that the initial conditions of the wavefunction “field” are have it take a combination of values that is not a square-integrable function, either because it is nonmeasurable or because it is measurable but non-square-integrable. Then the Schroedinger equation “wouldn’t know” what to do with the wavefunction. In other words, for quantum physics to work, given wavefunction realism, we need a very special initial combination of values of the “wavefunction field”. This is not a knockdown argument, but it does suggest an underexplored need for fine-tuning of initial conditions.

Second, the solutions to the Schroedinger equation, understood distributionally, are only defined up to sets of measure zero. In other words, even though the Schroedinger equation is generally considered to be deterministic (any indeterminism in quantum mechanics comes in elsewhere, say in collapse), nonetheless the solutions to the equation are underdetermined when they are considered as square-integrable fields on R³ⁿ—if ψ(⋅,t) is a solution for a given set of initial conditions, so is any function that differs from ψ(⋅,t) only on a set of measure zero. Granted, any two candidates for the wavefunction that differ only on a set of measure zero provide the exact same empirical predictions. However, it is still troubling to think that so much of physical reality would be ungoverned by the laws. (There might be a solution using the lifting theorem mentioned in footnote 6 here, though.)

Backwards causation, the A-theory and God

Suppose:

1. There are tensed facts.

2. If F is a contingent fact solely about physical reality that does not depend on creaturely free choice, then God can effectually will an exact duplicate of F.

Assumption (1) is a central claim of the A-theory of time, in fact form. Assumption (2) is a hedged consequence of omnipotence, formulated to take into account the possibility of uncreatable Platonic entities and the essentiality of origins.

Add:

3. Backwards causation is impossible.

We now have a problem. Let B be the tensed fact that the Big Bang occurred billions of years ago. This is a contingent fact solely about physical reality that does not depend on creaturely free choice. So, by (2), God can effectually will an exact duplicate of B. But an exact duplicate of B would still be a tensed fact about what happened billions of years ago. And to will such a fact about the past would be backwards causation, contrary to (3).

Note how the problem disappears if we don’t have tensed facts. For then all we have is an untensed fact such as that the Big Bang occurs at t₀, and God can will that without backwards causation, whether God is in time (e.g., he can then will it at t₀) or outside time.

I personally don’t have a problem with backwards causation. But a lot of A-theorists do.

I suppose what the A-theorist should do is to replace (2) with:

4. If F is a contingent fact solely about physical reality that does not depend on creaturely free choice, then God can effectually will a perhaps re-tensed exact duplicate of F.

Divine temporalism once again

I’m thinking about my recent argument against divine temporalism, the idea that God has no timeless existence but is instead in time, and time extends infinitely pastwards.

Here’s perhaps a simple way to make my argument go (I am grateful to Dean Zimmerman for suggestions that helped in this reformulation). If infinite time is a central feature of reality, as the temporalist says, then one of the most fundamental things for God to decide about the structure of creation is which of these three is to be true:

1. Nothing gets created.

2. There is creation going infinitely far back in time.

3. There is creation but it doesn’t go infinitely far back in time.

But without backwards causation, a temporal God cannot decide between (2) and (3). For at any given time, it’s already settled whether (2) or (3) is the case.

Now, it seems that the temporalist’s best answer is to deny the possibility of (2). We don’t expect God to choose whether to create square circles, and so if we deny the possibility of (2), God only needs to choose between (1) and (3).

But there are two issues with that. First, creation going infinitely far back in time is the temporalist’s best answer to the Augustinian question of why God waited as long as he did before creating—on this answer (admittedly contrary to Christian doctrine), God didn’t wait.

Second, and perhaps more seriously, there is the question of justifying the claim that (2) is impossible. There are four reasons in the literature for thinking that in fact creation has a finite past:

1. Big Bang cosmology

2. Arguments against actual infinity

3. Arguments against traversing an actually infinite time

4. Causal finitism.

None of these allow the temporalist to justify the impossibility of creation going infinitely far back in time. Big Bang cosmology is contingent, and does not establish impossibility. And if the arguments (ii) and (iii)
are good reasons for rejecting an infinite past of creation, they are also good reasons for rejecting divine temporalism, since divine temporalism would require God to have lived through an actually infinite time. And (iv) also seems to rule out divine temporalism. For suppose that in fact creation follows an infinite number of days without creation. During that infinite number of days without creation, on any day we could ask why nothing exists. And the answer is that God didn’t decide to create anything. So the emptiness of the empty day causally depends on God’s infinitely many decisions in days past not to start creating yet, contrary to causal finitism.

Predictions and Everett

Imagine this unfortunate sequence of events will certainly befall you in a classical universe:

1. You will be made to fall asleep.

2. Upon waking up, you will be shown a red square.

3. You will be made to fall asleep again.

4. While asleep, your memory will be reset to that which you had in step (1).

5. Upon waking up, you will be shown a green triangle.

6. You will be made to fall asleep for a third time.

7. While asleep, your memory will be reset again to that which you had in step (1).

8. Upon waking up, you will be shown a green circle.

9. You will then be permanently annihilated.

Questions:

10. How likely is it that you will be shown a green shape?

11. How likely is it that you will be shown a red shape?

The answers to these questions are obviously: one and one. You will be shown a green shape twice and a red shape one, and that’s certain.

Now consider a variant story where personal identity is not maintained in sleep. Perhaps each time in sleep the person who fell asleep will be annihilated and replaced by something that is in fact an exact duplicate, but that isn’t identical with the original according to the correct metaphysics of diachronic personal identity. (We can make this work on pretty much any metaphysics of diachronic personal identity. For example, we can make it work on a materialist memory theory as follows. We just suppose that before step (1), you happen to have three exact duplicates alive, who are not you. Then during the nth sleep cycle, the sleeper is annihilated, and a fresh brain is prepared and memories will be copied into it from your nth doppelganger. Since these memories don’t come from you, the resulting brain isn’t yours.)

And in the variant story, let’s ask the questions (10) and (11) again. What will the answers be? Again, it’s easy and obvious: zero and zero. You won’t be shown any shapes, because you will be annihilated in your sleep before any shapes are shown.

Now consider Everettian branching quantum mechanics. Suppose there is a quantum process that will result in your going to sleep in an equal superposition of states between having a red square, a green triangle and a green circle in front of your head, so that upon waking up an observation of the shape will be made. Now ask questions (10) and (11) again.

I contend that this is just as easy as in my classical universe story. Either the branching preserves personal identity or not. If it preserves personal identity, the answer to the questions is one and one. If it fails to preserve personal identity, the answer to the questions is zero and zero. The only relevant ontological difference between the quantum and classical stories is that in the quantum stories the wakeups might count as simultaneous while in the classical story the wakeups are sequential. And that really makes no difference.

In none of the four cases—the classical story with or without personal identity and the branching story with or without personal identity—are the answers to the questions 2/3 and 1/3. But those are in fact the right answers in the quantum case, contrary to the Everett model.

Now, one might object that we care more about decisions than predictions. Suppose that you have a choice between playing a game with one of two three-sided fair quantum dice:

• Die A is marked: red square, green triangle, green circle.

• Die B is marked: green square, red triangle, red circle.

And suppose pain will be induced if and only if the die comes up red. Which die should you prudentially choose for playing the game? Again, it depends on whether personal identity is preserved. If not, it makes no difference. If yes, clearly you should go for die A on the Everett model—and that is indeed the intuitively correct answer. But the reason for going for die A on the Everett model is different from the reason for going for it on a non-branching quantum mechanics. On the Everett model, the reason for going for die A is that it’s better to get pain once (die A) rather than twice (die B).

So far so good. But now suppose that you’ve additionally been told that if you go for die A, then before you roll A, an irrelevant twenty-sided die will be rolled. (This is a variant of an example Peter van Inwagen sent me years ago, which was due to a student of his.) Then, intuitively, if you go for die A, there will be twenty red branches and forty green branches on Everett. So on die A, you get pain twenty times if personal identity is preserved, and on die B you get pain only twice. And so you should surely go for die B, which is absurd.

One might reasonably object that there are in fact infinitely many branches no matter what. But then on the no-identity version, the choice is still irrelevant to you prudentially, while on the identity version, no matter what you do, you get pain infinitely many times no matter what you choose. And that doesn’t work, either. And if there is no fact about how many branches there will be, then the answer is just that there is no fact about which option is preferable on the identity version, and on the no-identity version, indifference still follows.

This is all basically well-known stuff. But I like the above way of making it vivid by thinking about classically sequentializing the story.

Against divine temporalism

I stipulate that:

0. According to pure divine temporalism, God is a being in time without a timeless existence all of whose decisions are made at moments of time.

I will argue that on plausible assumtions divine temporalism is incompatible with divine creative libertarian freedom.

First, we need this:

1. If pure divine temporalism is true, time has no beginning in the sense that before every moment of time, there was an earlier moment.

This is because everyone agrees that God is eternal. If there were a moment that had no moment before it, then according to pure divine temporalism, that moment would be God’s first moment of existence, without any timeless existence prior to or beyond it, and that is just incompatible with divine eternity. At that first moment it would be correct to say that God has just appeared.

One might object by saying that the first moment has infinite duration, and so it was an infinitely long changeless state. This is difficult to understand. An infinitely long changeless state seems like a timeless state more than anything else. In any case, if the point is pressed, I will simply stipulate that I don’t allow for moments like that.

Now, add this:

2. Every contingent feature of creation not even partly due to creaturely indeterministic activity was decided on by God with God having had the possibility of deciding otherwise. (Divine creative libertarian freedom)

Next, add some plausible claims:

3. The fact N that there was a moment of time before which there were no stars obtains.

4. The fact N is a contingent feature of creation not even partly due to creaturely indeterministic activity.

5. There is no backwards causation.

6. Time is linearly ordered: for any distinct moments of time t₁ and t₂, one is earlier than the other.

Finally:

7. For a reductio ad absurdum, assume pure divine temporalism.

What do we have? Well, our assumptions imply that God at some time decided on N while yet having the possibility of deciding to the contrary. But prior to any past time t₁, the fact N was already in place. History by time t₁ already made it be the case that there was a time before which there were no stars. So if there is no backwards causation, at no past time t₁ did God have the possibility of making N not be true. It was always already too late! But divine creative libertarian freedom requires that possibility.

Objection 1: The fact N does not actually obtain. We live in a sequential multiverse and before every time there were already stars in our universe or another.

Response: In that case, let S be the following contingent feature of creation: it was always the case that there already had been at least one star. I.e., for any past time t, there was a time t′ < t at which there had already had been at least one star. And an argument similar to the above goes through with S in place of N. At any past time, it was already too late to make S true, because history at that time was sufficient to make it be the case that prior to every time there was a star.

Objection 2: Fact N is made true by an infinite conjunction of facts such as that in year n there were no stars, in year n − 1 there were no stars, in year n − 2 there were no stars, and God unproblematically makes each of these facts true while having the power not to make it true.

Response: This objection is basically a rejection of (2). It says that some facts (even among the ones that aren’t due to creaturely indeterminism) aren’t freely decided on by God, but are instead consequences of other facts freely decided on by God. This reminds one of the Principle of Double Effect: God need not intend all the consequences of what he intends. He intends N_n, there not being stars in year n, as well as N_n − 1, and N_n − 2, and so on, but doesn’t intend their joint consequence N. I think this is a powerful objection. I don’t want to rule out the possibility of such a thing. But N is a morally unproblematic and structurally central part of the arrangement of reality. It seems very plausible that even if we reject (2) in general, we should accept it in the special case of morally unproblematic and structurally central parts of the arrangement of reality. Otherwise, God isn’t really in charge of creation.

What is a material thing?

Here's a theory: a material thing is something that has or is a causal power that is not a mental causal power. Variant: that is not a rational causal power.

Boltzmann brain blackouts

Some cosmological theories lead to the worrisome conclusion that most people with present brain states like ours are Boltzmann brains—random aggregations of molecules in space that came together to form a brain in a little bubble of oxygen. Usually when people talk about Boltzmann brains, they talk of how this induces a sceptical problem for the theory that generates them. Thinking about Boltzmann brain issues that way leads to messy epistemological questions such as whether we get to simply assume that we have hands, and the like. Moreover, if there is evidence for the cosmological theory, then that becomes evidence for Boltzmann brains, which then undermines the evidence for the cosmological theory, and that’s all a mess.

Here is how I suggest we think about what happens when a cosmological theory T leads to a Boltzmann brain issue. The vast majority of Boltzmann brains—even ones with brain states like ours—are short-lived. Their bubble of oxygen dissipates in the absence of gravity, and after a brief moment of hypoxia they die. So think of the point this way. If a cosmological theory predicts a large ratio of Boltzmann brains to ordinary evolved brains, then the theory makes an empirical prediction: in a moment you are extremely likely to start blacking out. So just do the experiment: wait a moment and see if you’re blacking out. If you’re not, then you’ve got very strong disconfirmation of the cosmological theory, and you’re done with it. You don’t have to worry about self-defeat, Moorean questions about whether you have two hands, or anything deep like that. (And if you are blacking out, then if it’s a Boltzmann brain related blockout, you’ll be dead in a moment. If you do come back to, and not in the afterlife, that’s massive evidence against the theory again, but now you should see a doctor about your blackout problem.)

In fact, you don’t even have to wait: on cosmological theories that generate too many Boltzmann brains, you should expect to already be starting to black out—because most of the Boltzmann brains will be extremely short-lived.

Objection: There will be long-lived Boltzmann brains, too.

Response: Sure. But for entropic reasons they will be much less common than the short-lived ones. You might, of course, worry that in many of these cosmological scenarios there are infinitely many Boltzmann brains, and infinitely many are short-lived and infinitely many are long-lived, and you can’t say that the short-lived ones are more common. The short-lived ones will be more common in a “typical” large finite region, but overall we just have infinity. Now, if you are worried about this—and I think you should be—then that worry already applied at the beginning of the story when you looked at the ratio of ordinary to Boltzmann brains, because there will be infinitely many of each on such a cosmological theory, and the formulation of the problem that I gave at the beginning, namely that Boltzmann brains greatly outnumber ordinary brains, is inaccurate. (I think if you do have this worry, then the theory has another problem, namely that probabilistic reasoning makes no sense in a world described by the theory. That is a kind of sceptical and self-defeat problem, but of a different nature.)

My point in this post is modest: if you want to say that Boltzmann brains greatly outnumber ordinary brains, then instead of thinking deep stuff about self-defeat of theories and scepticism, you should just think of the theory that generates this prediction as falsified by future observation.

What I think is wrong with Everettian quantum mechanics

One can think of Everettian multiverse quantum mechanics as beginning by proposing two theses:

1. The global wavefunction evolves according to the Schroedinger equation.

2. Superpositions in the global wavefunction can be correctly interpreted as equally real branches in a multiverse.

But prima facie, these two theses don’t fit with observation. If one prepares a quantum system in a (3/5)|↑⟩+(4/5)|↓⟩ spin state, and then observes the spin, one will will observe spin up in |3/5|^2=9/25 cases and spin down in |4/5|^2=16/25 cases. But (roughly speaking) there will be two equally real branches corresponding to this result, and so prima facie one would expect equally likely observations, which doesn't fit observation. But the Everettian adds a third thesis:

3. One ought to make predictions as to which branch one will observe proportionately to the square of the modulus of the coefficients that the branch has in the global wavefunction.

Since Aristotelian science has been abandoned, there has been a fruitful division of labor between natural science and philosophy, where investigation of normative phenomena has been relegated to philosophy while science concerned itself with the non-normative. From that point of view, while (1) and (less clearly but arguably) (2) belong to the domain of science, (3) does not. Instead, (3) belongs to epistemology, which is study of the norms of thought.

This point is not a criticism. Just as a doctor who has spent much time dealing sensitively with complex cases will have unique insights into bioethics, a scientist who has spent much time dealing sensitively with evidence will have unique insights into scientific epistemology. But it is useful, because the division of intellectual labor is useful, to remember that (3) is not a scientific claim in the modern sense. And there is nothing wrong with that as such, since many non-scientific claims, such as that one shouldn’t lie and that one should update by conditionalization, are true and important to the practice of the scientific enterprise.

But (3) is a non-scientific claim that is absurd. Imagine that a biologist came up with a theory that predicted, on the basis of their genetics and environment, that:

4. There are equal numbers of male and female infant spider monkeys.

You might have thought that this theory is empirically disproved by observations of a lot more female than male infant spider monkeys. But our biologist is clever, and comes up with this epistemological theory:

5. One ought to make predictions as to the sex of an infant spider monkey one will observe in inverse proportion to the ninth power of the average weight of that sex of spider monkeys.

And now, because male spider monkeys are slightly larger than females, we will make predictions that roughly fit our observations.

Here’s what went wrong in our silly biological example. The biologist’s epistemological claim (5) was not fitted to the actual ontology of the biologist’s theory. Instead, basically, the biologist said: when making predictions of future observations, make them in the way that you should if you thought the sex ratios were inversely proportional to the ninth power of the average weights, even though they aren’t.

This is silly. But exactly the same thing is going on in the Everett case. We are being told to make predictions in the way you should if the modulus squares of the weights in the superposition were chances of collapse. But they are not.

It is notorious that any scientific theory can be saved from empirical disconfirmation by adding enough auxiliary scientific hypotheses. But one can also save any scientific theory from empirical disconfirmation by adding an auxiliary philosophical hypothesis as to how confirmation or disconfirmation ought to proceed. And doing that may be worse than obstinately adding auxiliary scientific hypotheses. For auxiliary scientific hypotheses can often be tested and disproved. But an auxiliary epistemological hypothesis may simply close the door to refutation.

To put it positively, we want a certain degree of independence between epistemological principles and the ontology of a theory so that the ontology of the theory can be judged by the principles.