Let <* be any well-ordering of the real numbers. Run a countable infinity of independent random processes each of which picks out a random real number in exactly the same way and with the same continuous (or just atom-free) distribution. For instance, maybe in each of an infinite number of universes you toss a dart at a target in exactly the same way and measure the x-coordinate. Number the processes 1,2,3,....
Let Xn be the real number picked out by the nth process. Almost surely, all the numbers X1,X2,... are different: the probability of a repetition in a countable number of trials given a continuous distribution is zero. Thus, almost surely, there is an N such that XN<*Xn for all n≠N. This N counts as the choice in our lottery. All the processes being on par, it seems that we now have an infinite fair lottery outcome N, where the lottery tickets are 1,2,3,....
This process isn't guaranteed to work, since sometimes we will get a repetition. But most of the time the process will succeed.
Now, mathematically speaking, N is not going to be measurable on our product probability space.
But don't think about this mathematically. Think about it physically: Given an infinite multiverse, this could actually happen!
Maybe the lesson to be learned is that an infinite number of independent random trials is impossible and hence an infinite multiverse is impossible?
Am I missing something? Surely, with probability 1, there is no least* outcome? Suppose for example that the “target” range is (0, 1), the distribution is uniform and the ordering is the natural one. Then, with probability one, min(X1, ... , Xn) → 0 as n → ∞.
ReplyDelete... Given an infinite multiverse, this could actually happen... To make this work, you would have to be able to communicate between universes. Isn’t it a basic requirement of a multiverse that such communication is either impossible or restricted? (Otherwise, the “universes” would in effect be different parts of a single universe.) Also, finding the minimum would be a supertask, at least if you had to do it in finite time. It is hard to see this as a good argument against a multiverse, even if the probability theory worked.
Remember that the ordering is a well-ordering, and every non-empty subset of a well-ordered set has a least element.
ReplyDeleteWell, it's true that I couldn't do it, but God could. :-)
(One can also do the whole thing with a countable infinity of independent fair coin tosses arranged into a two-dimensional array. Well-order all countably infinite sequences of heads/tails results. With probability 1, no two columns match. Each column defines a countably infinite sequence of heads/tails. Choose the column whose sequence is least according to the well-ordering. The column number is the winning number in the lottery.)
For instance, maybe in each of an infinite number of universes you toss a dart at a target in exactly the same way and measure the x-coordinate. Number the processes 1,2,3,....
ReplyDeleteThis is really interesting. If the dart is thrown in "exactly the same way", you should surely expect repetition, assuming the worlds are similar in other ways. It is not difficult to throw a fair coin in exactly the same way and have it fall heads consistently, for instance. Or to roll dice exactly the same way and have the same numbers come up. And that is true even assuming our world is indeterministic. It would not be difficult to devise a machine that tosses fair coins tails consistently (I'm pretty sure such a device has already been constructed).
So why would it be unlikely that a dart tossed in exactly the same way, under the same conditions, would land on the same real? Doubts would arise only if there are doubts about tossing the dart in exactly the same way or about the conditions being exactly the same.
Surely the whole point of indeterminism is that if you do an exactly similar experiment you may well get a different result.
ReplyDeleteI've just given you a counterexample. Our world is indeterministic, but there are devices that toss fair coins that fall heads every time. But I don't need a machine for this, since it would not be difficult to become such a coin-tosser. You could train yourself to toss heads consistently.
ReplyDeleteBut setting that aside, I'm simply urging that you should expect lots of repetition, and that is consistent with your view about the point of indeterminism.
When a device is set up that can repeatedly toss a coin the same way, that's not a fair coin toss. (When one talks of fair coins, one is really talking of fair coin tosses.)
ReplyDeleteBut you're right that you need more than just indeterminism. You need that the fundamental objective chances of heads and tails be, respectively, 1/2. In the case where the device is set up to always toss heads, the fundamental objective chance of heads is close to 1.
When a device is set up that can repeatedly toss a coin the same way, that's not a fair coin toss. (When one talks of fair coins, one is really talking of fair coin tosses.)
ReplyDeleteThat's a really interesting thing to say since there is nothing unfair about the tosses. For instance, no one is in a position to complain that the toss is not fair in the case described. The device flips the coin at a certain velocity and with a certain rotation, but, no matter how the device tosses the coin, the toss are perfectly consistent, in indeterministic worlds, with the coin falling heads or tails. That's definitive of indeterministic worlds. And there is no toss that is consistent with there being a 50/50 chance of the coin falling heads. There is no such toss. So, I'm not sure what the argument is that the toss of the device is unfair.
But I think the notion of a fair toss is a really interesting one.
Professor Pruss: Yes, I had missed the point of well-ordering. That is an ingenious construction. I’m impressed. (But after Banach-Tarski, should anything surprise me...?)
ReplyDeleteYou can also do it without a well-ordering, using Choice directly. By Choice, there is a function f that assigns to every set S of countably many reals a member f(S) of S. Then let N be such that X_N = f({X_1,X_2,...}), assuming they are all distinct.
ReplyDeleteYes, that’s even niftier.
ReplyDeleteThe standard textbook example of a non-measurable set can be used in a similar way. (This is the one that uses Choice to partition the circumference of a circle into a countable number of sets all congruent under rational*pi rotations.) Enumerate the sets. Select a random point on the circle. The number of the set that contains the selected point wins the lottery. Note this requires only a single dart throw, which is equivalent to an infinite sequence of coin tosses. Maybe this was the inspiration for your example?
I still have trouble seeing that this says anything the real world. At most is adds to my feeling of discomfort with Choice, supertasks and Gods that can look across universes.
The problem with the standard textbook example is that it presupposes that we can generate a rotationally invariant distribution that remains rotationally invariant at finer level of examination, like those of the Vitali nonmeasurable sets. Unfortunately, I think the upshot of a lot of my thinking about this stuff is that at finer levels of examination rotational invariance will have to disappear.
ReplyDeleteOn the other hand, we are quite comfortable in talking of infinite sequences of independent trials, and really that's all we need here.
As for Choice, surely it would be possible to have an uncountable infinity of universes, in each of which there is an inscription of a list of countably many reals, with every set of countably many reals represented in exactly one universe. And if that were possible, surely it would also be possible that one of the reals in each list was underlined. And that's all we need for Choice.
Or, not for Choice, but for the version of Choice needed for my story.
ReplyDeleteI’m not sure I understand your first paragraph.
ReplyDeleteFirst, just to avoid misunderstanding, the construction I have in mind is described for example in the Wikipedia article Non Measurable Set. It is the same idea as Vitali, but done on a circle (or, if you prefer, on [0,1) with addition mod 1). My “random point” has to be selected from a uniform distribution. This could be done by an infinite sequence of coin tosses spelling out (H = 1, T = 0) the binary expansion of a real number in [0, 1].
The sets will not be rotationally invariant at any scale, nor do they have to be – symmetry between the lottery outcomes is ensured by the uniform distribution of the random point and the rotational congruence of the sets. But no doubt I’m missing your point?
Why am I uncomfortable? Like you, I feel that a fair infinite lottery makes no sense. But you seem to have constructed one. Your construction (the second version, with “coin tosses”) seems to use respectable purely mathematical concepts. So where is the catch? It is true, as you say, that the lottery outcomes are non-measurable events, so probability theory is not broken. But I can’t help feeling the symmetry.
I take it the idea is this. You partition the circle into countably many Vitali sets. Any two Vitali sets are rotationally equivalent. Then you choose a real number in a "uniform" way, and the Vitali set the number falls in is the winning lottery ticket.
ReplyDeleteThe problem is that this assumes that all the Vitali sets are equiprobable (otherwise the lottery isn't *fair*). This is supposed to follow from the fact that the choice of random number is "uniform". The technical meaning of "uniform" is probably: rotationally invariant. So, we derive equiprobability of lottery tickets from the rotational invariance of the Vitali sets.
But that's risky. We know there are sets which are rotationally equivalent with their own proper subsets. But intuitively (and the notion of "equiprobable" is meant to be intuitive) a set is not equiprobable with a proper subset.
Yes, that’s the idea. (Provided I read rotational equivalence (not invariance) in para 2, last sentence.) Does the problem you raise apply to Vitali sets? By construction, each rotates (trivially) to itself or else to another Vitali set. So no problem? Or have I missed something else?
ReplyDeleteFor any pair of Vitali sets, there is a (rational) rotation (say R) that takes the first to the second. So precisely when the random number selects the first set, the same random number rotated by R would select the second. But if the random number process is rotationally symmetric, rotating it by R should not change our judgments. So the two sets seem “equiprobable”. Just as in your example, this is not formally valid – the sets are not measurable, and mere correspondence won’t do. But it seems intuitively reasonable.
Feel free to say no more of my example – I don’t want to waste your time. Or else to continue – I do want to correct my thinking. I’m more interested to know what you make of your example. In your previous post you called the fair infinite lottery “incoherent”. This seems right – any winning number would seem too small. But you seem to have devised one. So what has gone wrong? Surely maths is not broken. So the problem must be somewhere in the gap between formal mathematics and intuition. But where exactly?
How about this? We know from the Hausdorff Paradox that on the (surface of) a sphere rotational invariance for a probability measure will have to break down for some nonmeasurable sets. And we know that there are sets that rotate into proper subsets on the circle. Thus in general we should be suspicious of rotational invariance as soon as we move away from the classical setting, either by working with nonmeasurable sets or by more finely comparing probabilities that classically are equal (say, because they're classically zero).
ReplyDeleteAn infinite fair lottery would lead to paradoxes about rationality. We would have cases of reasoning to a foregone conclusion. For instance, I toss a coin. If it's heads, I then do an ordinary lottery on the naturals with P(n) = 2^-n. If it's tails, I do an infinite fair lottery on the naturals. I announce to you the number I got. Let's say it's 10. Then P(10 is announced | heads) = 2^-10, but P(10 is announced | tails) = 0, so by standard Bayesian reasoning P(heads | 10 is announced) = 1. But the same is true no matter what I announce. So I can manipulate you into having conclusive evidence of heads, without any dishonesty on my part and no matter how things turn out. We would also have cases where it is rational to pay not to get information. (If you're going to be gambling on the coin toss in my story, it will typically be rational for you to pay me not to hear the result of the lottery. For if you hear it, you will be almost certain that it's heads, and you know that will lead to accepting stupid gambles.)
So what's going on? Well, there is no contradiction in the mathematics. The lottery that I've constructed isn't mathematically-speaking an infinite fair lottery, since mathematics does not assign any probability to the outcomes of this lottery. Just as in your case, this is a case of a nonmeasurable set.
The problem is a metaphysics or epistemology problem. One option is to deny the metaphysical presuppositions needed for it. For instance, one of the presuppositions is the metaphysical possibility of an infinite number of independent coin tosses where we've got the appropriate permutation symmetries. On the metaphysics side, a strict finitism will allow that presupposition to be denied. Another thing that might disrupt the assumptions on the metaphysics side is some versions of theism. If God is in some sort of control of indeterministic outcomes (as is the case in Molinism and Thomism), then maybe God will see distinctions between the coin tosses that destroy the independence or symmetry.
On the epistemology side, we might deny the implicitly applied principle of indifference that says that because our evidence is exactly balanced between all the outcomes of the lottery, therefore we should assign equal probabilities to the outcomes. Perhaps in cases where the evidence is exactly balanced between infinitely many outcomes, we should not assign any probabilities. (On this move, we haven't constructed an infinite fair lottery, I guess.)
There are problems with all of these solutions.
What I would most like would be a moderate finitism of some sort. For instance, to get out of the Grim Reaper Paradox, I posit the finitist thesis that one cannot have infinitely many things causally prior to a single thing. That's weaker than strict finitism. My main worry about strict finitism is that it either leads to intuitively absurd and theologically problematic theses about the future (say as that there can only be finitely many future events, or at least stochastic ones) or it require an open future metaphysics, which has its own problems, mainly theological ones, though also some with science and induction and other things.
But it seems that a strict finitism is what is needed to block the present argument, and even a future infinity of coin tosses will cause the problems.
I think *something* needs a deep rethinking. But I don't know whether it's on the metaphysics or the epistemology side.
Regarding the note in my first paragraph, consider this. One way to generate a classically rotationally invariant point on the circle is to do an infinite sequence of fair ten-sided die tosses, use that to represent a decimal number in [0,1), multiply by 2 pi, and get an angle. This will satisfy the classical probabilistic conditions for a rotationally invariant choice. But intuitively this isn't rotationally invariant. Indeed, intuitively some points are twice as likely as others. The initial procedure for selecting a number in [0,1) has only one way of getting us the number 1/3 (namely, an infinite sequence of 3s), but two ways of getting us 1/4 (namely, .250000... and .249999...). So rotational invariance at a cruder level (the level of classical probabilities) disappears at a finer level of examination.
ReplyDeleteThat’s a fair bit to think about.I will refrain from cheap comment, at least for a while. Thank you for your patience and tolerance.
ReplyDelete’m back. You said earlier that one can construct a set that rotates to a proper subset of itself, and that this undermines the intuitive idea of “equiprobablility”. But permutation symmetry has a similar problem.
ReplyDeleteHere is an example. Start with your construction using an array of coin tosses. Define a permutation P of the columns as a product of cycles as follows: P = (1)(2, 3)(4, 5, 6)(7, 8 9, 10)..... (Each cycle is 1 unit longer that the previous). Define a special sequence of outcomes S = H, H, T, H, T, T, H, T, T, T.... (i.e all heads in the first column of each cycle, all tails elsewhere). Then the sequences S, PS, PPS, PPPS ... are all distinct (because P contains arbitrarily long cycles). Define the event E = {S, PS, PPS, PPPS, ...} . Then PE is a proper subset of E (because it does not contain S). [This construction is similar to the usual one for rotation – the permutation P corresponds to the irrational rotation].
So it seems that permutation symmetry is inconsistent with intuitive equiprobablilty. Maybe one could restrict the symmetry to some subset of permutations? Or maybe we have to abandon equiprobability.
A further thought: a suitable subgroup of permutations may be hard to find. To swap two whole columns, we need a permutation with an infinite number of swaps of tosses. Suppose we admit all such permutations. (I have trouble seeing a principled way to admit some but not all.) The permutation (1, 2) (3, 4) (5, 6) ... is one such. (This is a permutation of individual tosses, not of whole columns as in the previous post.) The permutation (2, 3) (4, 5) (6, 7) ... is another. Apply them successively and get the infinite cycle (... 6, 4, 2, 1, 3, 5, 7...). This can be used in the same way as P in the previous post. So maybe permutation-based equiprobability is in trouble?
ReplyDeleteIf I’m thinking straight, we may lack a good reason to regard the coin-based lottery as “fair”. The dart-based version may avoid these problems. Suppose we take the throws as symmetric over finite permutations only (i.e. those that permute only a finite number of throws). This would be enough to make the lottery “fair” between any two throws. But no finite permutation will work in the construction in my previous post.
All very good points.
ReplyDeleteThe coin-based lottery is fair insofar as every toss is being treated equally. We could even suppose that each coin is in a separate exactly similar (at least in relevant respects) causally isolated universe. It is very hard to deny the possibility of causally isolated universes which are exactly similar in relevant respects. We can give a pretty much complete physical description of the setup. All you need is the collapse of an electron spin wavefunction like |up>+|down> upon spin measurement, say.
In the rotational case, it's harder to give a complete physical description. One might try to set up a uniform position wavefunction over the surface of a circle, and then measure the position. But one will have to measure the exact position, and I don't know that we can describe an experiment where that's done.
Plus it might turn out that necessarily all quantities are discrete.
Granted, we can imagine an infinite number of relevantly similar independent coin tosses. The problem is how to assign probabilities to events defined on the outcomes, or at least judge two events as equiprobable.
ReplyDeleteI guess I’m taking the epistemology / deny indifference / have not really constructed a fair infinite lottery route in your long post above. Equiprobability based on finite permutations makes sense. If we could apply it here, we would have a positive reason to think the lottery fair. But we can’t, so we seem to have only indifference based on ignorance.
For the rotational case (and equally for the dart-based lottery), I agree that it is hard to think of a plausible physical setup.
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ReplyDeleteIf it's not fair, whom does it favor or disfavor?
ReplyDeleteI guess the thought is that there are subtler ways of being unfair than by favoring or disfavoring someone. And I guess I already knew that, though I had forgotten it.
I know no reason to see any outcomes as more (or less) favoured. But the point of constructing a fair infinite lottery is that the fairness follows visibly from the construction. Indifference based on ignorance won’t do. (An angel tells me he will choose a natural number. He gives me no more information. Is this a fair infinite lottery?) At first sight, the coin-based lottery seems to work - permuting two columns takes a win for one to a win for the other, so the corresponding outcomes seem equiprobable. But the required permutation is infinite, and events related by an infinite permutation of coin tosses need not be equiprobable. So I have no reason to see the lottery as fair. Maybe there is a way to argue it is in fact fair. But I have not seen it.
ReplyDeleteFairness does follow visibly from the construction. Each toss is exactly like every other toss (except in outcome). Each sequence of tosses defining a real number is exactly like every other sequence of tosses defining a real number.
ReplyDeleteApologies, I had missed the basic point that the coin tosses are supposed to be irreducibly probabilistic (like quantum measurements, but unlike real coin tosses). My fault, you said it clearly. For such coin tosses, the lottery (if it could exist) would be visibly fair.
ReplyDeleteYeah, though I do worry about the following: Can one combine objective chances across causally isolated scenarios?
ReplyDelete