Why infinitesimals are too small to help with infinite lotteries: Part I

Let N be the natural numbers 0, 1, 2, 3, .... Let [n] be the first n natural or hypernatural numbers (the hypernaturals are an extension of the naturals that include infinite "natural" numbers; the reciprocals of such infinite naturals are infinitesimal). Thus, [5] = {0,1,2,3,4}.

Suppose we want to model a fair lottery on N by using hyperreal infinitesimals. (What I say should extend to other constructions of infinitesimals.) Let the relevant probability be P₁. Thus, P₁({n})=u for some infinitesimal u>0. Various paradoxes follow: P₁ is non-conglomerable (a corollary of this), this non-conglomerability can be made to spill out into other domains of investigation (see this) and we can't define expected values without violating domination (Paul Pedersen has proved this this summer).

But nevermind the paradoxes. I want to get at one thing that is fundamentally wrong about this approach: no matter which infinitesimal u we've picked, u is infinitely too small.

Here's why. If u>0 is infinitesimal, 1/u is an infinite hyperreal. Let K be an infinite hypernatural number such that K<<1/u (for instance, let K be the hypernatural closest to the square root of 1/u), where we say that a<<b for positive real or hyperreal numbers a and b if and only if b/a is infinite (i.e., a is infinitely smaller than b). Note that then u<<1/K. Now consider a second fair lottery, this time on the set [K]. It is clear that for the second lottery the probability of getting any particular outcome P₂({n}) for n in [K] should be 1/K, if we are dealing with hyperreal probabilities. None of the paradoxes follow if we do this [as long as we deal only in internal functions and internal partitions --note added later]: this assignment is both intuitive and stands up to scrutiny.

But now let n be any (finite) natural number. Then P₁({n})=u<<1/K=P₂({n}). But this is unacceptable, because P₁ correctly represents a fair lottery on N and N is a proper subset of [K] (since K is infinite, we have n<K for all n in N). Thus, u is supposedly the individual outcome probability for a lottery on N, but u would be infinitely too small for the individual outcome probabilities in the case of a lottery on the (much) larger state space [K]. Thus, a fortiori, u is infinitely too small for representing the individual outcomes of a fair lottery on N.

And this is true for any infinitesimal u. So the lesson is that the individual outcome probabilities for a fair lottery on N must be bigger than every infinitesimal. But they must also be smaller than every positive real number, since otherwise they will add up to more than 1. So there is nothing for them to be. Such a lottery is, simply, not a probabilistically coherent concept.

Infinitesimals are too small

I am getting to the following intuition as to why it's not appropriate to resolve zero-probability problems, like the problem of the probability of a particular number being picked in the case of an infinite lottery. The intuition is that the infinitesimals are too small to do the job: infinitesimals are too small to bridge the gap between zero and one, even if you have infinitely many of them. Wish I had time to go into why I am getting this intuition, but I need to grade assignments on mathematical induction instead.

Infinity and probability

You are one of a countably infinite number of blindfolded people arranged in a line. You have no idea where in the line you are. Each person tosses a independent fair die, but doesn't get to see or feel the result.

Case 1: What is the probability you tossed a six?

Obviously 1/6.

Case 2: You are informed that, surprisingly, all the even-numbered people tossed sixes, and all the odd numbered people tossed something else. What is the probability you tossed a six?

It sure seems like it's no longer 1/6. For suppose it's still 1/6. Then when you learned that all the even-numbered people tossed sixes and the odd-numbered ones didn't, you thereby received evidence yielding probability 5/6 that you were one of the odd-numbered ones. But surely you didn't. After all, if there were ten people in the line and you learned that all but the tenth tossed a six, that wouldn't give you probability 5/6 that you were the tenth!

But as long as infinitely many people tossed six and infinitely many didn't (and with probability one, this is true), there is always some ordering on the people such that relative to that ordering you can be correctly informed that every second person tossed a six. That puts the judgment in Case 1 into question.

Moreover, why should the alternating ordering in Case 2 carry any weight that isn't already carried by the fact which you know ahead of time (with probability one) that there are equally many sixes as non-sixes?

But if the judgment in Case 1 is wrong, then were we to find out we are in an infinite multiverse, that would undercut our probabilistic reasoning which assumes we can go from intra-universe chances to credences, and hence undercut science. Thus, a scientific argument that we live in an infinite multiverse would be self-defeating.

I don't exactly know what to make of arguments like this.

A way of grading

I've settled on this method for grading weekly undergraduate papers. I write short comments in the body of the paper. For longer comments, I write a circled letter in the margin of the hard copy, and type a comment for that letter in a file of comments on the week's papers (e.g., "a. This sentence is incomprehensible" in the comments file and a circled "a" in the relevant place in the paper). I can copy and paste comments if the same ones are deserved. I also type overall comments and a grade in that file. I then copy and paste the typed comments into email as soon as I've graded each paper, and send to the student. The hard copy, which explains which letters refer to what, is handed back next class.

A nice thing about this method is that students get a grade and some comments faster, though not all comments will be clear without the context of the paper. I also keep the major comments on my computer which is good for letters of recommendation. Another nice thing is that I no longer waste paper printing comments, as I used to. A fully electronic workflow, I suppose, would be a further step, but I am not sure I am ready for it.

Desire and rationality

I claim:

1. Either (a) one cannot desire what one does not believe to be worth having or (b) it is possible to have no reason, even prima facie, to bring about something one desires.

In either case the view that desires are non-doxastic reason-giving states is false.

A simple case is where you are told you will avoid great punishment if you induce in yourself a desire to eat mud, which you antecedently know there is no point to doing. You then induce in yourself this desire. If (a) is false, you should be able to do this while maintaining your belief that there is no point to the action. But you do not gain a reason to do the action in this case: you only have a reason to desire it, not a reason to do it.

Objection: There is inner turmoil when you have an unsatisfied desire for eating mud.

Response: This isn't a reason to eat mud, as much as a reason to end having a desire to eat mud. Eating mud is sometimes a means to this, but not always. We could imagine a situation where the only ways available to eat mud are ways that wouldn't make the desire go away (e.g., you know you have a psychological quirk that makes the desire not go away, or maybe you know an alien will ensure that as soon as you swallow mud, you start believing you're eating chocolate, so you will never form the belief that you have eaten mud, though you will have).

Three kinds of instrumental rationality

There are three levels of instrumental rationality, in order of decreasing thickness:

1. Rationality with regard to genuine ends that one has. Some of these ends may be self-given and others may be ends that one has independently of what one desires and pursues.
2. Rationality with respect to what one desires or what are goals of one's pursuits.
3. Rationality with respect to arbitrary states of affairs. Thus, if a student fails an exam, that is rational with respect to the state of affairs of getting a low grade in the course, whether or not that state of affairs is one the student pursues, desires or should pursue or desire.

The thinnest option does not make actions even be prima facie rational. My sticking a pin in my nose is instrumentally rational in the third sense with respect to creating pain in myself, but is not even even prima facie rational. The third option only makes actions prima facie conditionally rational, provided that the state of affairs is one that is at least prima facie rational to pursue.

More controversially, I think the same is true of the middle option. That I pursue E and C appropriately promotes E only makes it even prima facie rational to pursue C when pursuing E is at least prima facie rational. That I have set myself to pursue a goal does not automatically make that goal be a genuine end of mine. And what I said about pursuit goes over, even more controversially, for for desires. So the middle option only gives conditional prima facie rationality: pursuing C is prima facie rational provided that pursuing E is.

One might think: "provided that pursuing or desiring E is." But there may be cases where desiring E is rational but pursuing E is not even prima facie rational. Suppose you will kill me unless I desire to step on a point-up tack. I form this desire quite rationally, but this rational desire does not give me a reason to step on the tack, given that I continue to believe that the action is not worth pursuing.

It is only the first kind of instrumental rationality that is a genuine form of rationality, that makes actions at least prima facie rational. In fact, the thinnest and medium options don't have any normativity to them at all: they just tell us about causal and logical connections between events in the world (thinnest) or events in the world and mental states (medium).

Two desiderata for preaching about hell

1. Hell needs to be presented in such a way that nobody would be willing to go there.
2. It needs to be shown that hell is an expression of divine love.

Nonconglomerability and hyperreal-valued probability

Note added later: This follows by applying Theorems 3.1 and 3.2 of Schervish et al. to the standard part of P.

Suppose P is a finitely-additive hyperreal-valued probability on the natural numbers such that P assigns infinitesimal value to each natural number. While groggy from a cold plus baby-feeding at night, I've been trying to prove that P is nonconglomerable in the following strongish sense: there is a partition A₁,A₂,... of the naturals, an event E, and real numbers a<b such that P(E|A_i)<a for all i but P(E)>b. Thus, if we are trying to figure out if E is true, and we plan to observe the A_i, we know ahead of time that no matter which A_i we observe, our probability for E will go down. (This, of course, violates van Fraassen's reflection principle.) This is likely already known, but I couldn't find it on the Internet.

My proof sketch starts by dividing into two cases. Either the standard part of P takes on infinitely many values or not. If it takes on infinitely many values, then the standard part of P is a merely finitely additive measure that takes on infinitely many values, and so by the 1984 Schervish et al. theorem, it is nonconglomerable, and hence so is P in my strongish sense. If, on the other hand, the standard part of P takes on finitely many values, then P is a convex linear combination, with real coefficients, of a purely infinitesimal finitely additive signed measure and a finite number of indicator functions of free ultrafilters. And then with a bit of work one can construct a counterexample to conglomerability, I think. (I am not completely sure which extensions of the reals this works for, but it does work for the hyperreals.)

DIY stuff

In case anybody is curious what hobby stuff I've been doing lately, here are two of my recent instructables.

Bow sight:

Vibrating bassinet:

Spatializing time

This photography is rather interesting.

A principle about induction and explanation

Here's an intuition I have. Suppose that I somehow knew that a dozen of boxes have appeared ex nihilo for no cause (not even a stochastic one) in my office. I open half of them and each one was purple inside. Do I have good reason to think that the others are also purple inside? As long as I hold on to my knowledge that there is no explanation of the boxes' presence and character, I think not. It is rather like when I get heads six times in a row when tossing a fair coin—as long as I get to hold on to my knowledge that the coin is fair, I have no reason to think subsequent tosses will be heads.

This suggests to me that induction requires that the cases we do induction over be non-brute, that they have explanations. But not just any explanations will do. The cases need to have a common type of explanation. If one box was materialized in my office by aliens, and another was delivered by my best friend, and another coalesced from the drippings in a leaky ceiling, and so on, then I don't get to do induction across the cases.

Thus:

1. That all observed Fs are Gs gives me knowledge by induction that all Fs are Gs only if there is a common type of explanation as to why each F is a G.

Normally, when all observed Fs are Gs, that gives us reason to think that there is a common explanation, say that Fness is a natural kind that includes Gness. But when there is no such common explanation, then even if all Fs are Gs, we don't know it—we have Gettiered knowledge.

Inductive inferences across kinds

I observe some ravens, and they are all black. This gives me good reason to think all ravens are black. This is an inductive inference within a natural kind. One might have this picture of the inductive inference here: observing the ravens, we learn something about the appropriate-level universal raven that they fall under. One might then think that all inductive inference is like this: We observe instances of a genuine, non-gerrymandered natural kind K, and conclude that the kind is such-and-such.

But I don't think this is all that happens. Here are a few other kinds of cases.

1. From the fact that octopi behave in some ways like we do, we infer that they are conscious. But there is no biological taxon K that contains both octopi and humans such that we have good reason to think that all Ks are conscious. The lowest level taxon containing octopi and humans is the subregnum Bilateria and we have little reason to think all Bilateria are conscious. We might seek for a natural kind that isn't a taxon, like critters that exhibit apparently intelligent behavior. But that's a gerrymandered kind. We might try for a non-gerrymandered kind, like critters that exhibit intelligent behavior, but then we would have to have reason to think that octopi exhibit intelligent behavior rather than merely apparently intelligent behavior, and our problem would return.
2. We have good reason to think that all life on earth descends from a single ancestor. But organism on earth isn't a natural kind.
3. We can do induction within artificial kinds. That all the pens that I have observed have ink in them gives me reason to think all pens have ink. But pen isn't a natural kind.

Does this matter? Maybe. (I think a theist may have a better explanation of why induction not-within-a-kind works than a naturalist. But the thoughts here are inchoate.)

Hell and Auschwitz

The oldest Holocaust survivor, Antoni Dobrowolski, who went to Auschwitz as punishment for teaching young poles has died at 108. The article quotes him as saying that Auschwitz was "worse than Dante's hell".

My initial reaction was that this is surely an overstatement. But a moment's reflection suggests that Dobrowolski is correct, at least as concerning hell itself (I won't comment on Dante's hell, since I am no Dante scholar). Hell is a place that upholds the dignity of its inmates by acknowledging their autonomous choice for evil, giving them justice and limiting their downward moral slide, while the concentration camps aimed at the destruction of autonomy and dignity. It is a terrifying thought that we humans can produce something worse than hell.

But at the same time, we have to remember that in a choice between hell and Auschwitz, we should choose against hell. So perhaps hell is worse? Or maybe we need to distinguish: in itself, in some sense, Auschwitz is worse, but hell also guarantees lack of union with God, while Auschwitz is compatible with union with God, just as the Cross was.

Nonexplanatory Platonic entities

Benacerraf-style arguments that numbers couldn't be any particular collection of abstract entities (say, some particular set-theoretic construction) because there is a multitude of other constructions that could play the same role will fail if numbers play an explanatory role in the world. And one can imagine metaphysical views on which they play even a physical explanatory role. For instance, charge and mass play an explanatory role, indeed perhaps a causal one, in the world. But a Platonist could think that to have a charge or mass of x units (in the natural respective unit system) is to be charge- or mass-related to the number x. In other words, such determinables are relations, whose second relatum must be a number, and their determinates are cases of that relation for a fixed second relatum.

Now, one can still construct a relation to some set-theoretic isomorph of the numbers that structurally functions just like charge. For instance, if f is an isomorphism from the abstracta relata of charge to some abstract Ss, then we could say that a is related by charge* to y, where y is one of the Ss, precisely when a is related by charge to an x such that y=f(x). But there will be a matter of fact as to whether it is charge or charge* that explains the motion of particles. Surely they both don't—that would be a bogus case of overdetermination.

The point generalizes to other cases of Platonic entities that play an explanatory role—not necessarily a physical one—in the world. For instance, propositions might explain the co-contentfulness of sentences. An isomorph of the system of propositions could play some of the same roles for us, but it would not in fact explain the co-contentfulness of sentences. Compare this case. There is an isomorphism between legal US voters and some set of social security numbers. We can then construct a relation voting* between numbers and candidates such that n votes* for c if and only if the voter with social security number n votes for c. But while one could use facts about voting* to organize our information about elections, it is facts about voting—an action performed by persons, not social security numbers—that in fact explain election outomes.

That said, I think this approach will still tell against the standard set-theoretic constructions of numbers in two ways. First, it will tell against any particular construction. For how likely is it that this construction is the right one? Second, it will tell against anything like the set-theoretic constructions being the numbers. For it seems really unlikely that having a charge of three units is anything like a matter of being related in some way to the set {∅, {∅}, {∅, {∅}}}. So this approach is most plausible if numbers are some kind of sui generis entities.

But, on the other hand, the Benacerraf argument could apply against Platonic entities that play no explanatory role but are merely introduced for our convenience of expression. On some views, possible worlds are like that.

An argument for evolution and an argument for supernatural beings

Consider:

1. Many instances of F have explanations fitting an evolutionary explanatory schema.
2. There are no instances of F that we have good reason to think lacking an explanation fitting an evolutionary explanatory schema.
3. So, probably, all instances of F have an explanation fitting an evolutionary explanatory schema.

For instance, F can be biological diversity or non-initial biological complexity. Now compare:

4. Many instances of G have explanations fitting an agential explanatory schema.
5. There are no instances of G that we have good reason to think lacking in an explanation fitting an agential explanatory schema.
6. So, probably, all instances of G have an explanation fitting an agential explanatory schema.

For instance, G can be orderly complexity or usefulness or value. But there are instances of orderly complexity, usefulness and value with the property that if they have an agential explanation, they have an agential explanation involving supernatural beings. The orderly complexity, usefulness and value in the laws of nature is like that, for instance.

I am not inclined to think either of the arguments above very strong, however.