Modeling space

The obvious model of a Newtonian space is as the set of all triples (x,y,z) of real-numbered coordinates. But the model does not have isotropy that Newtonian space does. It has privileged directions, such as the x-axis, the y-axis and the z-axis. It has privileged coordinates such as (0,0,0). Of course, physical models generally do have properties that aren't found in what is modeled. If I build a model of the solar system out of fruit, the fact that some of the fruit is sweeter need not model any property of the solar system. If I make a model of an ethanol molecule out of sticks and balls, the balls that represent hydrogen atoms differ in their exact mass, and exhibit scratches, in a way that the hydrogen atoms do not.

Nonetheless, even though this is common to all modeling, there really is something a little unsatisfying when the mathematical model does this. Typically when we mathematically model something, we have to abstract or forget on both sides. On the side of what is modeled, the side of the world, we ignore aspects of the physical structure because otherwise things get too complicated. On the side of the model, we ignore aspects of the mathematical structure because they don't, as far as we know, correspond to anything in the physics. Wouldn't it be nice if we could abstract only on one side, that of the world? But some things that would be nice are not an option.

The above remarks do, I think, make Pythagoreanism less plausible. There seems to be structure in the mathematics that models the world that isn't found in the world. This makes it implausible that the world just is composed of the mathematics.

Properties of the model and the modeled

My apologies for yet another technical post that's just notes-to-self.

Quantum Mechanics models the world using a Hilbert space. I wonder what we can say about just how much of the structure of the model is meant to be found in what is modeled. In contemporary mathematics, I guess ultimately any Hilbert space will be a very complex construction out of the empty set. Yet it seems absurd to think that the low-level details of the set-theoretic implementation (say, different ways of constructing the natural numbers out of the empty set) would reflect differences in the world. There are way too many ways to implement these details.

But there will also be differences at higher levels. For instance, there will be cases where the Hilbert space is L²(X), "the space of square-integrable functions" on some set X. I put that in scare quotes, because that's not what L² is, despite often being described so. Rather, it's the space of equivalence classes of square-integrable functions, where two functions are equivalent provided that the set of points where they differ has measure zero. So now we have a question about the model and the modeled. You could think that different members of an equivalence class correspond to different empirically indistinguishable physical states, and the physics simply makes no prediction as to which of the indistinguishable states is exemplified when. Or you could think that each equivalence class corresponds to a single possible physical state. The latter makes for a theory that is simpler and yet seems to give less understanding. It is simpler because it doesn't posit unexplained differences between states. But it seems to give less understanding, because it means that the wavefunction can no longer be seen as an assignment of values to different points in phase space, but rather a more mysterious kind of entity—one modeled as an equivalence class of such assignments.

There may be a third option: There is a privileged member of each equivalence class, and only the privileged member can be physically actualized. This would give us the best of both worlds. We would have a field over phase-space, and no extra indistinguishable physical possibilities. The lack of linear liftings on L²[0,1] makes it a bit harder to realize this hope than one might have wished, but maybe there is still some hope.

Epiphenomenalism and the problem of animal pain

Suppose the following epiphenomenalist thesis is true, at least for non-human animals: qualia do not affect behavior. It's interesting that if this is right, then the argument for atheism from animal pain is seriously weakened. The argument from animal pain contends that God would have reason to prevent many instances of animal pain that he does not in fact prevent. However, we have good reason to think that God's interventions would be targeted and hence minimal. Now a minimal intervention for the prevention of pain is simply to suppress the quale of pain. Given epiphenomenalism, however, suppressing a quale of pain does not affect either behavior or neural state. So if God thus intervened, things wouldn't look any the different. And hence the atheist cannot non-circularly deny that God did intervene to prevent the pain.

Of course, this might be taken to be yet another reason to deny epiphenomenalism.

The argument from partial theodicy

The following would be a superb teleological argument for the existence of God if only we had good reason to accept (1) without relying on theism:

1. Every evil has a theodicy.
2. If every evil has a theodicy, then probably God exists.
3. So, probably God exists.

I can think of two (perhaps not ultimately different) ways of making (2) plausible. First, the best explanation of (1) would be that God exists. Second, that an evil has a theodicy means that it's the sort of thing that God would have a reason to permit if God existed. But it would be very odd if all evils had this hypothetical God-involving property without God existing. It would be a cosmic coincidence.

But as I said, (1) is the rub. However, what about this version:

4. Most evils happening to humans have a theodicy.
5. If most evils happening to humans have a theodicy, then probably God exists.
6. So, probably, God exists.

And while we're at it, let's add:

7. If God exists, all evils have a theodicy.
8. So, probably, all evils have a theodicy.

Premise (5) is harder to justify than (2), but I think the reasoning behind (2) still contributes to the plausibility of (5). The best alternative to theism is a form of naturalism, and we just wouldn't expect most evils, or even most evils happening to people, to have a theodicy on naturalism, so our best explanation for why most such evils have a theodicy is that God exists.

I want to say something about why I am restricting (4) and the antecedent of (5) to evils happening to humans. The reason is that we have much better epistemic access to evils happening to humans, and so we are better able to judge of both the magnitude of the evils and the theodicies and lack thereof.

And (4) is much easier to justify than (1). All we need is enough partial theodicies. Plausibly, for instance, many evils—perhaps it's already most evils—are moral evils that are sufficiently non-horrendous that a free will theodicy directly applies to them. Many evils have a good theodicy in terms of the exercise of virtue they enable. And when I reflect on the evils that have befallen me in my life, it's easy to see that I deserve punishment for them all by my sins, and would have deserved a lot more than I got. Granted, I've lived a charmed life, so the applicability of this will be limited. But between freedom, virtue and punishment, it is plausible that the majority of evils happening to people have been covered.

A somewhat different argumentative route is:

9. Most evils happening to humans have a theodicy.
10. The best explanation of (9) is that all evils have a theodicy.
11. So, probably, all evils have a theodicy.
12. If all evils have a theodicy, then probably God exists.
13. At least somewhat probably, God exists.

Finally, there will be first-person versions that make use of a premise like:

14. Every evil (or: most evils) that happened to me has a theodicy.

From unrestricted composition to unrestricted caninity

According to unrestricted composition (UC) for any plurality of things there is a whole that is exactly composed of them. Sider offers a continuity argument for UC. Here's a vivid formulation. Let the Ps be the particles in the even-numbered books on one of my bookshelves. If UC is false, then in the actual world the Ps will be a paradigm case of something that doesn't compose a whole. But there is a world where the Ps compose a dog. And between these two worlds there is a continuous sequence of worlds where the Ps gradually migrate from their every-second-book positioning to their canine positioning. It is absurd to think that suddenly somewhere in this continuous sequence the particles come to compose something. So, Sider concludes, they compose something all along, even in the actual world.

But to a hylomorphist, the argument as I've put it simply fails. There is no world where the Ps compose a dog, since a dog—or any other complex entity—is not composed of matter, but of matter and form. The argument can, however, be reformulated. Say that the Ps materially compose an F provided that the Ps are material and together with some form compose an F. Then the argument gets off the ground. In the actual world, the Ps do not materially compose anything while in the final world they materially compose something. Where along the line do they come to materially compose something?

Now, however, the story is underdescribed. For we have failed to say in which worlds in the sequence there is a substantial form of the dog informing the Ps. Facts about substantial forms should not be assumed to supervene on facts about the arrangement of the particles. There could be zombie dogs that are nothing but heaps of particles looking like a dog. In other words, it's a contingent matter whether a certain kind of arrangement of particles materially composes something—if there is a form informing them, then they compose and if not, not.

Of course, there is a question of explanation: Why is there no form informing the Ps in the actual world but there is one in the the non-zombie dog worlds? But the answers aren't particularly troublesome. Maybe the laws of nature explain that. Maybe God just decides when to create forms and make them inform particles.

However, there is a final move that Sider can make. Instead of asking in which worlds the Ps (materially) compose something, he could ask which arrangements of particles are such that something could be materially composed of the particles in that arrangement. Of course the dog-like arrangement is like that. And the even-numbered-book arrangement is not. So where is the transition in the continuous deformation of the even-numbered-book arrangement into the dog-like arrangement?

This is an interesting question for the hylomorphist. It is closely to the question of what forms there could be (cf. the discussion here and in the Murphy book referenced in the comments there). The hylomorphist could take an unrestricted view. There is a sufficiently wide variety of possible forms and defects that any possible arrangement of matter is compatible with being informed by some form—perhaps defectively. There could be a possible world where something looking just like our even-numbered-book arrangement is a highly defective (it doesn't grow or reproduce) plant.

Nonetheless, there is a remaining problem. While the even-numbered-book arrangement may be apt for materially composing a defective plant, it's surely inapt for materially composing a dog. So there will seem to be a discontinuous transition between those arrangements that can and those that cannot materially compose a dog. One answer here is that "dog" is vague. This doesn't fit with traditional Aristotelian views, though, on which all dogs have an exactly similar form, and so one could meaningfully ask about the range of arrangements that could be informed by a form that's exactly like that. But perhaps the Aristotelian can yield some ground here. Another answer would be unrestricted canine composition: any material arrangement could materially compose a dog, albeit a highly defective one. I am somewhat drawn to this strange view. Yet is it that strange? I think I can imagine a dog continuously deforming into the even-numbered-book arrangement but where rather than dying the dog comes to be more and more defective. I am dualist enough that I can even imagine the dog being conscious throughout the process.

Guessing strategies and causal finitism

Suppose that during an infinite past a fair die was rolled every day, and that this game will end in a year. You know all the outcomes of the past rolls. Before each roll, you are asked whether you think the roll will come up six. If you answer correctly, you get a dollar. Otherwise, you lose a dollar.

There is an obvious strategy: Always guess "No." Then out of six rolls, on average, you will win five times and lose once, so you will on average make about 67 cents per roll. Here's a very reasonable claim:

1. Guessing "no" is the optimal strategy for an agent that does not have foreknowledge of the future.

But it turns out that, given the Axiom of Choice, there is a strategy that beats this, a strategy guaranteed that you will win infinitely often and lose at most finitely often, and hence that gives you a long-run average of a dollar per roll, rather than the measly 67 cents of our above strategy. The strategy is to use a variant of the solution to the fourth hat puzzle here. For the technically minded reader I'll sketch the strategy below.

But (1) is obviously true: it's clear that whenever you are being asked to guess, you should say "no", and surely that's the best policy. So (1) is both true and false on the above assumptions (including the assumptions needed to make the alternate strategy go). And hence I think we should reject the possibility of knowing the outcomes of a backwards-infinite sequence of coin tosses. And the best way to do that is to embrace causal finitism: to deny that anything (say, your current knowledge) can depend on infinitely many events.

For the technically minded reader, here's the strategy. Consider the set of all backwards-infinite sequences of die rolls. Say two sequences are equivalent if they differ in only finitely many places. For any equivalence class E of sequences, choose a member f(E) (by the Axiom of Choice). Now whenever you're asked to make a guess, you already know all but finitely many of the items in the actual world's sequence of rolls. So you know which equivalence class E the actual world's sequence will fall into. So you guess according to f(E). And since the actual world's sequence differs from f(E) in only finitely many places, you're right all but finitely often.

Natural law and participation in God

According to Natural Law, the right thing to do is that which accords with one's nature. But what if something really nasty accorded with one's nature? This is, of course, akin to the objection to divine command theory from the question "What if God commanded something really nasty?" Both theories can give the same answer: "That's just impossible." God couldn't command something really nasty and there just are no possible natures of rational beings that require such nastiness. As far as that goes, this is fine, though at this point in the literature there are two more steps in the dialectic to think about.

I want to, however, consider a side-step. Why is it impossible? One could think this is just a brute and unexplained impossibility, but that is unsatisfactory intellectually. Even apart from the Principle of Sufficient Reason, we don't like brute facts that look like too much of a coincidence. And it looks like too much of a coincidence that all of the nasty cases are impossible. We want an explanation.

The divine command theorist has a pretty immediate explanation. We're talking about God's commands, and necessarily God is perfectly good or, if one prefers, perfectly loving. (Of course, those divine commands who want to define the good, and not just the obligatory, in terms that involve divine choices cannot give this answer. But so much the worse for that version of divine command theory.)

I think the Natural Law answer can be similar. A nature is an essential (in the medieval sense, maybe not the modal sense) mode of participation in God. It's impossible for a rational being's essential mode of participation in God to require nastiness, because of the nature of God. (Why is God's nature that way? Maybe here we have a brute necessity. A single brute necessity is much less problematic than a whole slew of them. Or maybe we can talk of God's perfection here.)

So there is an explanatory gap that Natural Law points to, and bringing in God closes that explanatory gap. Are there other ways of closing that gap? Maybe. One would be a heavily Platonic theory on which natures are modes of essential participation in the Form of the Good. The Platonism here would be more like Plato's own Platonism than our more anemic contemporary Platonism. The participation relation would not be exemplification as in contemporary Platonism, but something ontologically meatier, more like the participation in the theistic version of Natural Law.

In any case, the question of why something nasty couldn't be required by one's nature points towards serious metaphysics.

Mathematics and intellectual humility

The discipline where we have the greatest consensus of certainty is mathematics. Yet that discipline is also pretty close to being the one where we have least consensus as to what we're talking about. This should make us collectively humble.

Deep Thoughts XL

The unknown is not very well known.

[My son wrote this on my board while bored in my office.]

Rational fickleness

The ideal rational agent, it seems, would respond to evidence instantly. After all, you should believe in accordance with your evidence, so when you have new evidence your beliefs should not fall behind. This means that the ideal rational agent's degrees of belief will oscillate a fair amount, since we constantly get bits of evidence in various directions.

But humans do not operate in this way. Our degrees of belief tend to update fairly slowly, and tend to be free of small oscillations this way and that way. Moreover, while our commonsense evaluations criticize the person whose beliefs lag behind her evidence too much as being closed-minded, we do not praise the person whose beliefs constantly oscillate in sync with the changing body of evidence. On the contrary, we are apt to think him fickle and unsteady.

Now I don't want to overstate the above point. Well-confirmed beliefs do tend to stay well-confirmed when new evidence comes even in an ideal rational agent, and there is a granularity in our beliefs that doesn't allow us to distinguish between, say, 99.999% confidence and 99.997% confidence. But nonetheless, it seems to me that our attitudes do not favor the instantaneously evidence-responsive and hence sometimes rapidly oscillating degrees of belief found in the ideal rational agent. It's as if we expected people to pass their evidential support through a moving average low-pass filter.

Are we simply mistaken in wanting greater belief steadiness from people? Shouldn't people respond instantly to evidence?

Yes and no. While I think we should apportion our beliefs to the evidence (with evidence very broadly construed), probably we only really count as having a piece of evidence when we've evaluated its impact. It takes time to do that. Moreover, because of limited resources, we will evaluate several pieces of evidence at a time in connection with a given question: we just do not have the leisure to evaluate each piece of data exactly when it first becomes possible to do so. But if we evaluate several pieces of evidence together, then this produces an effect very much like that of a moving average. Of course, once the evidence has been evaluated, belief should follow instantly. However, when someone's beliefs oscillate too much, that is a sign that either he is a really quick thinker—and few are like that—or that he is failing to evaluate evidence with the carefulness that is called for.

Further, while belief should instantly follow the evaluation of evidence, the behavior of a rational agent may not look like it follows. There is a cost to switching our behavior to a new track, which can make it rational to keep our old behavior—maybe only temporarily, until we are able to find a way to switch at lower cost—even if it wasn't the behavior which we would have adopted had we had our new beliefs. If I have done form A of exercise over the years, and the latest research says that B is healthier, then it can still make sense to do A because of the costs of buying new equipment, acquiring new habits, etc. So people's behavior will, and quite rationally so, seem to lag behind the changing evidence, and will appear to smooth out oscillations in the evidence. I say "seem" and "appear", because in fact the continuation of the old behavior may be quite in keeping with the new beliefs once the switching costs are considered. Thus, the fickle agent whose behavior changes too much with incoming evidence is rightly to be criticized for not paying enough attention to switching costs.

Of course, there are times when behavior needs to switch much faster. If I get evidence that I have been acting unjustly, then switching costs become irrelevant: I must refrain from injustice no matter the cost. And there is, all other things being equal, a value to being an agile agent, one able to change behaviors quickly without much cost when new evidence comes. But only all other things being equal: there are values in habits that can override the value of agential agility.

Two kinds of fungibility

I was teaching Jennifer Whiting's "Impersonal Friends" this morning—I love that piece—and I was going into the usual distinction between the fungible and nonfungible. I generally illustrate this with heirlooms. While money is fungible, the ring inherited from great grandmother is not: if the ring were swapped out for another just like it, it wouldn't be as good.

As I was teaching, though, I realized that that's too quick. Suppose that in the first place great grandmother instead had a different but similar ring and it was passed down through the generations to us. That would make no difference to anything that matters to us. So the ring is broadly fungible: it can be swapped for another ring with relevantly similar historical properties. The same seems true of all heirlooms.

But are persons fungible in the same way? Here's a thought. If something rightly matters a great deal to me it matters objectively at least somewhat. It rightly matters a great deal to me that I exist. It thus rightly matters a great deal to me that all of my history wasn't swapped for that of another similar individual. Therefore it matters objectively at least somewhat. Hence I am objectively not fungible even in that broader sense.

Betting on paradoxical sets

Suppose that a point z will be uniformly randomly chosen on the surface of a sphere S and you are asked to place bets as to which set z is in. Then, plausibly:

1. If two sets A and B are equivalent under rotations about the center of the sphere, you should accept this offer: get three dollars if z is in A and pay two dollars if z is in B.

But now consider a paradoxical decomposition of the whole sphere S, by a version of the Banach-Tarski Paradox[note 1]. In this, the sphere is partitioned into two subsets C and D, each of which can be decomposed into a finite number of subsets that can be rotated to form the whole sphere. Applying (1) to each set in the decomposition of C and its rotation, you will accept a sequence of deals that adds up to:

2. If z is in C, you get three dollars and if z is in S you pay two dollars.

Repeating this with D's decomposition, you get a sequence of deals that adds up to:

3. If z is in D, you get three dollars and if z is in S you pay two dollars.

But of course if z is in C or D, it is also in S, and if it's in S, then it's in exactly one of C or D. It follows that the deal adds up to:

4. No matter what, you get three dollars and you pay four dollars.

So, repeated application of (1) yields an unacceptable conclusion.

One might say that this is an artifact of the fact that there is no finitely additive rotation-invariant probability measure on the sphere. But I think the above formulation is a little bit more telling. I make no reference to probabilities here. All I assume is (1), which is a very intuitive rationality judgment, namely that when one has two equivalent scenarios, one should accept an unequal bet between them that is in one's favor.

What to conclude? One conclusion might be that a single application of (1) is fine, but the sequence of applications needed to yield (4) is not.

My own conclusion, however, is that it is metaphysically impossible to have a betting scenario like the above. But why not? What's wrong with it? Well, one possibility is that space is necessarily discrete, but that doesn't seem very plausible to me.

My own preference, however, is to conclude that it is impossible to have anything causally depend on whether a random point (or a particle or the like) is in one of these weird sets that are found in the paradoxical decomposition of the sphere. Why is that? I think it's because it would in effect be a violation of causal finitism, the thesis that no event can causally depend on infinitely many things. But the full story here requires significant amounts of work to complete.

"Ręka" and "hand"

I've been thinking about a curious issue in translation, which is not that uncommon. In most ordinary contexts, the Polish "ręka" and the English "hand" would be interchangeable in the sense that where a speaker of one language would use the one, the speaker of the other would use the other. Where the English-speaker talks of having something in his hand, the Polish-speaker talks of having it in his ręka, and so on. But the two terms are not synonymous. In non-medical Polish, "ręka" refers to the whole of the upper limb (though in medical Polish, it refers just to the hand), while the English "hand" refers only to the area from the wrist to the fingertips. The Polish term referring to the exact same part of the body as the English "hand" does is "dłoń", but the word is significantly less used than "ręka" (as per Google hits in .pl sites, say), and in many ordinary contexts using "dłoń" for "hand" would make for awkward translation. Conversely, to translate the Polish "ręka" as "arm", which would refer to the same part of the body (I am assuming that the arm includes the hand), would in most cases lead to awkwardness as well. It sounds funny to talk of picking up one's phone with one's arm, and so on.

Thus, it seems that these are cases where the natural translation from one language to the other does not in fact preserve the truth conditions. One can pick up one's phone with one's ręka without picking it up with one's hand (say, use the crook of the elbow), even though in the context of picking up a phone one would translate "ręka" as "hand", unless it was obvious from the context that the hand wasn't the part of the arm that was being used.

Maybe what is happening here is that when a sentence asserts a proposition p and implicates a stronger proposition q, we feel no qualms about using a translation that asserts q, or vice versa. To say in Polish that one picked up one's phone with one's ręka implicates the stronger proposition that one did this with one's hand, since if one had picked it up in the unusual way with the crook of the elbow, say, we would have expected the speaker to mention this. (This is a case where the usual Gricean presumption that one will use an equally brief but more precise term in place of a less precise one is defeated by the fact that the more precise and equally brief term "dłoń" is also less commonly used.) So one translates the implicature rather than the assertion.

I wonder, though. Maybe cases like this are evidence that the distinction between implicature and assertion is artificial. This would have the important consequence that the wrongness of implicating contrary to one's mind, or at least intentionally doing so, is the same sort of thing as lying. I don't want to embrace that consequence in general. I think false implicature is qualitatively less morally problematic than lying.

Wanting to be even more sure

We like being sure. No matter how high our confidence, we have a desire to be more sure, which taken to an extreme becomes a Cartesian desire for absolute certainty. It's tempting to dismiss the desire for greater and greater confidence, when one already has a very high confidence, as irrational.

But the desire is not irrational. Apart from certain moral considerations (e.g., respecting confidentiality) a rational person does not refuse costless information (pace Lara Buchak's account of faith). No matter how high my confidence, as long as it is less than 100%, I may be wrong, and by closing my ears to free data I close myself to being shown to have been wrong, i.e., I close myself to truth. I may think this is not a big deal. After all, if I am 99.9999% sure, then I will think it quite unlikely that I will ever be shown to have been wrong. After all, to be shown to be wrong, I have to actually be wrong ("shown wrong" is factive), and I think the probability that I am wrong is only 0.0001%. Moreover, even if I'm wrong, quite likely further evidence won't get me the vast distance from being 99.9999% sure to being unsure. So it seems like not a big deal to reject new data. Except that it is. First, I have lots of confident beliefs, and while it is unlikely for any particular one of my 99.9999%-sure beliefs to be wrong, the probability that some one of them is wrong is quite a bit higher. And, second, I am a member of a community, and for Kantian reasons I should avoid epistemic policies that make an exception of myself. And of course I want others to be open to evience even when 99.9999% sure, if only because sometimes they are 99.9999% sure of the negation of what I am 99.9999% sure of!

So we want rational people to be open to more evidence. And this puts a constraint on how we value our levels of confidence. Let's say that I do value having at least 99.9999% confidence, but above that level I set no additional premium on my confidence. Then I will refuse costless information when I have reached 99.9999% confidence. I will even pay (perhaps a very small amount) not to hear it! For there are two possibilities. The new evidence might increase my confidence and might decrease it. If it increases it, I gain nothing, since I set no additional premium on higher confidence. If it decreases it, however, I am apt to lose (this may requiring tweaking of the case). And a rational agent will pay to avoid a situation where she is sure to gain nothing and has a possibility of losing.

So it's important that one's desire structure be such that it continue to set a premium on higher and higher levels of confidence. In fact, the desire structure should not only be such that one wouldn't pay to close one's ears to free data, but it should be such that one would always be willing to pay something (perhaps a very small amount) to get new relevant data.

Intuitively, this requires that we value a small increment in confidence more than we disvalue a small decrement. And indeed that's right.

So our desire for greater and greater confidence is indeed quite reasonable.

There is a lesson in the above for the reward structure in science. We should ensure that the rewards in science—say, publishing—do not exhibit thresholds, such as a special premium for a significance level of 0.05 or 0.01. Such thresholds in a reward structure inevitably reward irrational refusals of free information. (Interestingly, though, a threshold for absolute certainty would not reward irrational refusals of free information.)

I am, of course, assuming that we are dealing with rational agents, ones that always proceed by Bayesian update, but who are nonetheless asking themselves whether to gather more data or not. Of course, an irrational agent who sets a high value on confidence is apt to cheat and just boost her confidence by fiat.

Technical appendix: In fact to ensure that I am always willing to pay some small amount to get more information, I need to set a value V(r) on the credence r in such a way that V is a strictly convex function. (The sufficiency of this follows from the fact that the evolving credences of a Bayesian agent are a martingale, and a convex function of a martingale is a submartingale. The necessity follows from some easy cases.)

This line of thought now has a connection with the theory of scoring rules. A scoring rule measures our inaccuracy—it measures how far we are from truth. If a proposition is true and we assign credence r to it, then the scoring rule measures the distance between r and 1. Particularly desirable are strictly proper scoring rules. Now for any (single-proposition) scoring rule, we can measure the agent's own expectation as to what her score is. It turns out that the agent's expectation as to her score is a continuous, bounded, strictly concave function ψ(r) of her credence r and that every continuous, bounded, strictly concave function ψ defines a scoring rule such that ψ(r) is the agent's expectation of her score. (See this paper.) This means that if our convex value function V for levels of confidence is bounded and continuous—not unreasonable assumptions—then that value function V(r) is −ψ(r) where ψ(r) is the agent's expectation as to her score, given a credence of r, according to some strictly proper scoring rule.

In other words, assuming continuity and boundedness, the consideration that agents should value confidence in such a way that they are always willing to gather more data means that they should value their confidence in exactly the way they would if their assignment of value to their confidence was based on self-scoring (i.e., calculating their expected value for their score) their accuracy.

Interestingly, though, I am not quite sure that continuity and boundedness should be required of V. Maybe there is a special premium on certainty, so V is continuous within (0,1) (that's guaranteed by convexity) but has jumps—maybe even infinite ones—at the boundaries.

Individual and group interest, and infinity

There are infinitely many people. A random process causes each one to independent develop a cancer, either of type A or of type B. The chance that a given individual develops a type A cancer is 9/10 and the chance that she develops a type B cancer is 1/10. It is not possible to diagnose whether an individual has type A or type B cancer. There are two drugs available, either of which—but not both, because they are toxic when combined—could be distributed by you en masse to all of the infinitely people. There is no possibility of distributing different drugs to different people—the logistics only make it possible for you to distribute the same drug to everyone. Drug Alpha cures type A cancer but does not affect type B, and drug Beta cures type B cancer but does not affect type A.

What should you do? Clearly, you should distribute Alpha to everyone. After all, each individual is much more likely to have type A cancer.

But now suppose that an angel reveals to everyone the following interesting fact:

• (F) Only finitely many people have type A cancer.

You're very surprised. You would have expected infinitely many to have type A cancer and infinitely many to have type B cancer. But even though F is very unlikely outcome—indeed, classically it has zero probability—it is possible. So, what should you do now?

The obvious answer is that you should distribute Beta to everyone. After all, if you distribute Alpha, finitely many people will be cured, while if you distribute Beta, infinitely many will be. Clear choice!

But not so fast. Here is a plausible principle:

• (I) If you're choosing between intrinsically morally permissible options X and Y and for every relevant individual x, option X is in x's best interest, then option X is the best option to choose.

But there is an argument that it is in every individual's interest that you distribute Alpha to her. Here's why. Let x be any individual. Before the angel's revelation of F, it was clearly in x's best interest that she get Alpha. But now we have all learned F. Does that affect what's in x's best interest? There is a very convincing argument that it does not. Consider this proposition:

• (F_x) Among people other than x, only finitely many have type A cancer.

Clearly, learning F_x does not affect what is to be done in x's best interest, because the development of cancer in all the patients is independent, so learning about which cancers people other than x have tells us nothing about x's cancer. To dispute F_x is to buy into something akin to the Gambler's Fallacy. But now notice that F_x is logically equivalent to F. Necessarily, if only finitely many people other than x have type A cancer, then only finitely many people have type A cancer (one individual won't make the difference between the finite and the infinite!), and the converse is trivial. If learning F_x does not affect what is to be done in x's best interest, neither should learning the equivalent fact F. So, learning F does not affect what is in x's best interest, and so the initial judgment that drug Alpha is in x's best interest stands.

Thus:

1. Necessarily, if I is true, then in the infinitary case above, you should distribute Alpha.

But at the same time it really was quite obvious that you should save infinitely many rather than finitely many people, so you should distribute Beta. So it seems we should reject I.

Yet I seems so very obviously true! So, what to do?

There are some possibilities. Maybe one can say deny I in cases of incomplete knowledge, as this one is. Perhaps I is true when you know for sure how the action will affect each individual, but only then. Yet I seems true without the restriction.

A very different suggestion is simply to reject the case. It is impossible to have a case like the one I described. Yet surely it is possible for the outcome of the random process to satisfy F. So where lies the impossibility? I think the impossibility lies in the fact that one would be acting on fact F. And the best explanation here is Causal Finitism: the doctrine that there cannot be infinitely many things among the causal antecedents of a single event. In the case as I described it, the angel's utterance is presumably caused by the infinitary distribution of the cancers.