Wednesday, March 17, 2010

Another argument against actual infinity

I am actually kind of suspicious that there is a subtle problem with my conditional probabilities in the following argument. It's a rather complex argument. Start with this observation. Suppose I know that there are N people in the universe, and ten minutes ago, a random process independently occurred to each of the N people, bestowing upon each the unobservable property Q, with probability p. How likely is it, given this information, that I have Q? The answer is obvious: p. But now suppose that I learn some additional information: I learn exactly how many people now have Q and how many don't. Presumably, approximately pN have Q and (1-p)N don't, but I now have better than an approximation—I have an exact number. Let's say, K people have Q. So now my probability that I have Q is K/N. Observe that p has now dropped out, because the information I have supercedes the information that involves p. For instance, maybe N=10, and in one world where p=1/4, three people have Q, and in another world where p=1/3, also three people have Q. If I am in the first world, my probability for having Q should be 3/10, and in the second, likewise.

The above illustrates a general principle: If I know the actual distribution at t1 of the property Q in the population, my best estimated probability for having Q depends on that actual distribution, and not on the history of how the members of the population got to have Q. Different historical processes could produce the same distribution.

Now, suppose an actual infinity is possible, and so there are countably many people in the population. If this is possible, it is also possible that some process at t0 independently bestowed Q on some of the people, with probability p strictly between 0 and 1. Suppose this is all I know. Then I ought to assign probability p to the claim that I have Q. But by the above principle, if I were to learn what the distribution of Q in the population is, I should use that distribution to estimate the probability of my having Q, instead of using information about p. But I do know the distribution of Q in the population is—even without actually observing anything. For I know that countably infinitely many members of the population have Q and countably infinitely many members of the population lack Q. (This works best if the persons are otherwise indiscernible, or differ in respect of properties that have no ordering or topology or significance to them.) At least, the probability that this is so is 1, and that's surely good enough for knowledge. But now here is the funny thing: this fact about the distribution is independent of p. Whatever the value of p is, the distribution would almost surely (i.e., with probability one) be infinitely many Qs and infinitely many non-Qs. So, by the above trumping principle, regardless of the value of p, I ought to assign the same probability (1/2? undefined?) to my having Q. But it is obvious that I ought to assign p. So we have a contradiction.

Here's a perhaps clearer way to run the argument. Two processes independently bestowed the properties Q and R, bestowing Q with probability 1/10 and R with probability 9/10, on all the members of an infinite population. I now know that in the population, there now are infinitely many Qs and infinitely many non-Qs. This is exactly the same distribution as the distribution of Rs. If my probabilities should depend on the distribution of Qs and Rs, as the trumping principle says they should, it follows I should assign the same probability to my having Q as to my having R. But plainly this is false—it is nine times as likely that I would have R than that I would have Q. Hence we have a contradiction.

15 comments:

Heath White said...

This stuff is really not my forte, but here are two thoughts:

1. It is surely _possible_ that a finite number of people are assigned Q, while an infinite number are assigned non-Q. I don't know how that affects matters.

2. One conclusion might be that all probabilities on infinite domains are identical. That is, suppose Pr(x has Q) = Pr(x has R), for all Q and R, when x is a member of an infinite set. This might not be any stranger than facts like infinity minus one equals infinity, or infinity divided by two equals infinity.

Alexander R Pruss said...

Heath:

1. It is possible, but it has probability zero.

2. That may be, but it is weird. It means that if I toss a die in an infinite universe, I should not assign probability 1/6 to its landing on 1.

Alexander R Pruss said...

Another thought: Suppose all probabilities on an infinite population are the same. That is incompatible with quantum physics (on probabilistic interpretations). And if we accept quantum physics, we get the claim that we don't have an infinite number of objects in our world. And that itself is interesting.

Alexander R Pruss said...

A different version of this argument has been posted on prosblogion.

enigMan said...

To begin with, the trumping principle may well be false. It all depends upon one's interpretation of probability theory (and upon one's favourite probability theory), but intuitively the trumpting principle is false. If you know that there was a probability of p that Q we bestowed upon you, then that it the chance that you have Q. Now, if you find out that everyone has Q, then you know that you have Q, and so the probability is not p but 1; and similarly if K = 0. But in all other cases, it does not matter what K is. Why should it? K does not affect your chance of having Q. We may use K to estimate p when we are not told what p is, but if we know what p is then that trumps knowledge of K, surely?

Suppose I fairly toss a coin. Is it heads or tails? Is the chance not 50% of heads? Suppose you find out that, of all those tossing coins in this area today only a third of them got heads. Does that affect that 50%? But in another area over this week two thirds got heads! Surely the trumping factor is that as I tossed the coin there was a 50% chance of heads. It is the process that determines the individual chance (at least on the single-case propensity interpretation of objective probabilities).

enigMan said...

Still, there are lots of other paradoxes of the infinite, and if the natural numbers do not form a set, but are rather indefinitely extensible, then it may well be that one of them plausibly proves it. Suppose, for example, that one is sure that by gathering things one at a time no one could go from having nothing to having aleph-null things (i.e. one for each natural number) without at any time having any other numbers of things than zero or aleph-null. Then one might think that the natural numbers would have to be indefinitely extensible. That is because if instead they formed a set then aleph-null particles (named 1, 2 and so on) in some infinite space would plausibly be metaphysically possible, and each might move from one place to another. Particles might move (or be moved) one at a time from some region A (containing parts An for all natural numbers n) to region B (containing Bn similarly)—first particle 1 from A1 to B1, then particle 2 from A2 to B2, and so on—in such a way that each does so twice as quickly as the previous particle (e.g. because B2–A2 is half of B1–A1, and so on). But it is no less plausible that those movements could be done in reverse order—last of all A1 to B1, before that A2 to B2, and so forth—and that is to move aleph-null particles, one at a time, from A to B in such a way that the numbers of particles in B before successive movements are 0 followed by a reverse omega-sequence (a sequence like …, 3, 2, 1) of aleph-nulls.

enigMan said...
This comment has been removed by the author.
enigMan said...

Sorry for that deleted comment (I had made completely the wrong point about the Cosmological argument). I take it that you want to show that the natural numbers do not form an actual infinite set because that would help with the Cosmological argument? However, a problem with that strategy is that that argument requires that past time is not infinite, and time might be infinite in the past even if the natural numbers are indefinitely extensible (rather than a combinatorial set).

A line full of points could contain n points for every natural number n precisely because such natural numbered points would not collectively be a part of the line, but would be as indefinitely extensible as those numbers (and of course, it could also do that because they would be part of the line, were the natural numbers to form a set). Such a line would contain some infinite number of points, but not a transfinite number (e.g. it might be hash, an actual infinite cardinal number such that 0 times hash is an indeterminate form including 1, so that basically hash = 1/0, being the number of zero-length points in an arbitrary unit length of the line of points).

So past time could be infinite, with no beginning, even if the natural numbers are indefinitely extensible. Infinite past time means that there is a time n years ago for every n, but it does not need to contain all those years. If the natural numbers are indefinitely extensible then there are not all those numbers, for there to be all those years.

Alexander R Pruss said...

1. The trumping principle talks about what epistemic probability I should assign given the information about the distribution.

2. With the exception of the grim reaper paradox and these probabilistic paradoxes, I am quite unimpressed by paradoxes of infinity.

3. If there was an infinite past time, then it would also have been possible at each past day an immortal soul to come into existence. And if that happened, there would now be an actual infinity of souls. (This isn't my argument--it's al Ghazali's.) Which, if the arguments work, is absurd.

James Bejon said...

Regarding an earlier comment, Is it consistent for something to be possible yet have probability 0? Shouldn't something's probability amount to something like possible worlds where it is the case divided by possible worlds where it isn't the case? If so, doesn't something having a 0 probability imply that there are no possible worlds where it is the case?

enigMan said...

The trumping principle talks about epistemic probabilities, but I was wondering if you needed to talk about other kinds as well. But my coin-tossing objection might apply even in a deterministic world, in which the probabilities were Laplacean, were epistemic.

Suppose that a friend and I are gambling, that he tosses a coin and I guess whether it is heads or not. My epistemic probability of it being heads is 1/2, because I think it unlikely that he has guessed what I am going to guess and so tossed the coin as to get that result. But now I recall that in our previous games there were 60 heads and 40 tails. Should I change my probability to 3/5? (There are 101 games including this one, but that has little affect on the ratio.) I would say it might, but I might have some evidence that my friend generally tosses fairly, and so that that ratio was just a fluke. And now suppose that an angel tells me that in the whole universe at present the heads to tails ratio is 4.9 to 5.1. Should I revise my 1/2 slightly downwards? But those games might have involved biases that this game does not. And so forth.

enigMan said...

I was wondering if you needed to talk about other kinds of probabilities because the trumping principle might (for all I know) apply to some of them. Your p cannot be an epistemic probability, because you conform to it at the start of your scenario. Is it a propensity? But then I don't see how an observed frequency could trump it (for all that if the propensity were unknown the frequency would be evidence for its value). Is it based on a larger population? But again, I'm not sure that the smaller population that contains you should trump the larger population (although perhaps it does; I was just wondering (if that is the case) why it should)...

enigMan said...

(Sorry about the split-up posts, but my network connection is flakey at present:) Regarding the other paradoxes of infinity, I quite agree (with no exceptions), although I wonder if they amount to some evidence against the standard view.

Regarding 3, your premise is false in the way the cosmological argument needs it to be true, for all that it is true in another way. The way in which it is true is the epistemic possibility way. The problem with that is that it is also logically possible (and hence it ought to be epistemically possible) that such could not have happened:

Suppose that infinite past time was like the continuum I mentioned (which has not yet been shown to be an incoherent conception). Then we could count past days from now without end, but if the natural numbers were indefinitely extensible then that counting would be without end in an indefinitely extensible kind of way. That would not be because the time was not actually infinite into the past, but because the counting was indefinitely extensible, because the natural numbers were.

So, there could be an infinite past time, and the natural numbers could also be indefinitely extensible, such that a countable infinity of souls was impossible, and also any transfinite number of souls was impossible; and it might also be the case that any actual infinity of souls was impossible, because it might be that souls could not exist like points in a continuum, but would have to exist in a more discrete manner.

So the arguments do not show that there could not be an infinite past time (my continuum not having been shown to be incoherent:)

enigMan said...

Regarding trumping, your final argument is best construed as an argument against that principle. Your final contradiction is derived from taking the true probability to be the one given by the process that bestows the Rs and Qs. So you have got your final contradiction from denying the trumping principle.

Perhaps your final argument was misconstrued; but even so, even if the trumping principle is not false (in its applications here), another problem with your main argument is that infinity divided by infinity is most plausibly construed as an indefinite form, in general. A nice argument for that is based on your previous post.

Suppose you know only that there are natural numbered stones scattered about (one per natural number), and that you are on one of them (and blindfolded). If you think of the stones as numbered 1, 2, 3, ..., 9, 10, 11, ..., then you may well think of an epistemic probability of 1/10 that you are on a round numbered stone. These being epistemic probabilities, there is not much wrong with that.

But if you think of the stones as numbered 1, 10, 2, 20, 3, 30, ..., 9, 90, 11, 100, 12, 110, ..., 19, 180, 21, 190, 22, 200, ..., then you may well think of an epistemic probability of 1/2. Again, why not? And if you thought of the first and then the second, and realised that they were two true descriptions of your situation, then you might reasonably conclude that there is no objective or logical probability; or rather more precisely, that the probability is infinity divided by infinity, and that that is an indeterminate form.

If you knew more, say about how you had been put onto your stone, then you might have a reason to prefer one of the possible values of such an indeterminate form. In the absence of any such information, you simply have no good reason to prefer one to any other. In a situation where you should rationally have a definite epistemic probability, then you might rationally pick any of them. That is not a contradiction that could prove that your situation was not real (was perhaps a dream), rather than that there were many different beliefs that were equally rational. But arguably, it would be wisest to know that there was no best epistemic probability, and to have as a result no epistemic probability.

Such cases are perhaps quite ordinary, even in finite situations. E.g. I am thinking of an object; what is the probability that it is round? You might think it is 1/2 because you have no idea whether it is round or not. Or you might think that objects can be round, cubic, furry or otherwise, giving you an epistemic probability of 1/4. You might even try to guess about the sort of object that I might be thinking of, and try to deduce a probability from that. Or a probability might just spring into your consciousness and feel like the right sort of number. And maybe none of those is particularly irrational.

So, suppose that infinity divided by infinity is in general an indeterminate form. Then when you learn about the distribution of Q in the population, that it is infinity divided by infinity, there is no contradiction with your already knowing that it is p. And indeterminate form is often called 'undefined', but that does not contradict any definite value being assigned by some other process of reasoning. The value is undefined by one process, but defined by another.

Now, I agree that the true value is p; but again, that seems to indicate merely that the trumping principle is not always true. The indeterminate value found from the actual distribution has been trumped by the particular value given by the process bestowing the property in each case (a single-case propensity?).

hatsoff said...

Dr. Pruss,

This is an even more interesting paradox than the last. However, I see a hidden problem: When the population is denumerable, what does it mean to say that a property Q is assigned to its members with a probability p? I suggest that such an idea either lacks content or relevance.

We might wish to say that p refers to the ratio of members out of a subset S of the population who guess correctly whether or not they have Q. But how do we choose S? As we know, it is possible to frame our choice to satisfy any rational value of p. So such an interpretation of p is not unique, that is, not well-defined.

Alternatively, we might wish to imagine a lengthy process by which the population is assigned, one by one, Q or no-Q. So, if q is the number of people with Q after n assignments, then we could interpret p as the limit of q/n as n goes to infinity. However, without an end to this process, we will never have all members of the denumerable population granted an assignment at any one time.

In order to give p content, we might wish to apply the above limit-driven interpretation, and also invoke a Xeno-like game, such that the speed of the process in question steadily increases and an infinity of assignments are completed in a finite time interval. In that case, p loses relevance to our assessment of the situation at or after the moment of completion.

Now, this is hardly exhaustive of possible interpretations, but the point is, one must be accepted before we draw our conclusions. For myself, I am simply unable to make sense of the claim that the probability of each member having Q is p, if the population of interest is infinite. As before, I regard this as a limitation of probability.

However, I must thank you for another intuition-bending paradox. They're a lot of fun to tackle!