tag:blogger.com,1999:blog-3891434218564545511.post2905225344934557509..comments2024-03-28T19:56:42.305-05:00Comments on Alexander Pruss's Blog: Probability on infinite sets and the Kalaam argumentAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger37125tag:blogger.com,1999:blog-3891434218564545511.post-35438894150731858862010-07-06T16:10:15.909-05:002010-07-06T16:10:15.909-05:00This comment has been removed by the author.Unknownhttps://www.blogger.com/profile/15527111653387665566noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-16711158147796081652010-07-06T16:09:06.452-05:002010-07-06T16:09:06.452-05:00Dr. Pruss,
I agree with hatsoff that your conclus...Dr. Pruss,<br /><br />I agree with hatsoff that your conclusion to this thought experiment is too hasty. Rather than pointing to problems with infinities, this more likely points to problems with assigning probabilities. Such problems are nothing new. Our intuitions about "natural" distributions can become confused as setups become more complicated. Faced with an apparent paradox involving assignments of probabilities, I would much sooner give up on probabilities than leap to such a sweeping metaphysical conclusion.<br /><br /><br />In this particular situation it is important to realize that in the second case we have additional information, and it is only natural to expect that additional information changes our probability assignments. At least, it is natural on the view that probability is a function of our ignorance and the measure of our uncertainty. I don’t actually see an inconsistency in assigning p=9/10 in the first case (where all we know is the state of affairs today) and p1=9/10, p2=1/10 in the second case (where we know about the teleportation). Our knowledge of the universe has changed – therefore, our probability assignments must change too.<br /><br />In fact, your conundrum is not specific to scenarios involving infinities. Consider this modified version of the thought experiment:<br /><br />(a) There are 10 paving stones labeled 1-10. , on each of which there is a blindfolded person. You are one of these persons. That's all you know. How likely is it you’re not on number 10? The obvious answer is: 9/10.<br /><br />(b) But now I give you a bit more information. Yesterday, all the same people were standing on the paving stones, but differently arranged. At midnight, all the people were teleported, in such a way that you end up on number 10. Should you change your estimate of the likelihood you're on a number 10?<br /><br />What makes your original and my modified scenarios similar is that additional information that you receive in (b) modifies your initial ignorance-based assignment of probabilities. In your case the effect is achieved by a non-uniform mapping of the initial distribution, which would not be possible in my case. But that detail does not seem significant. What makes the two experiments essentially similar is that our knowledge (or our ignorance) changes from (a) to (b), and our probability assessment follows.<br /><br /><br />In the two paragraphs starting with "Here is a suggestion..." and "This assumes, however, that there is a process...," you hint at the Principle of Indifference. The Principle of Indifference, or some generalized form of it, is pretty much the only game in town when it comes to objectively assigning prior probabilities. However, it has known challenges: apparent paradoxes not unlike the one you outlined here (see for instance the Bertrand paradox). These issues are not necessarily related to infinities.Unknownhttps://www.blogger.com/profile/15527111653387665566noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-18594822744445941802010-07-05T11:49:21.175-05:002010-07-05T11:49:21.175-05:00I wouldn't suggest we have to take what physic...I wouldn't suggest we have to take what physicists believe with no grains of salt, but: most of them think the spatial extent of our universe is infinite, due to GR considerations. IOW, it is neither just an expanding cloud of stuff with empty space beyond (an explosion in classical space) nor a closed hypersphere of finite volume.<br /><br />I do accept that the paradoxes discussed here are thought-provoking and give pause to glib acceptance of the reasonableness of infinite sets of real objects. However, if space is infinite it's hard to imagine it not being populated with the same kinds of objects "forever" into space (or else we'd have a special boundary, even if "pure space" could go on.) That means an Aleph_null of every card game, whatever. And as far as I'm concerned the probabilities are still what they're "supposed to be."<br /><br />If we put people on squares, I guess it depends on <i>how</i> we do that and not the sheer logic of set theory that matters most. Note also in cases like coin tossing there is an intrinsic "generator" of the probabilities, not just a hollow comparison between outcome sets. But that doesn't help my own argument about expectations for possible worlds as much as I'd like ...Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-23971380178564783212010-07-05T11:44:32.426-05:002010-07-05T11:44:32.426-05:00This comment has been removed by the author.Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-29365102488913842762010-07-05T11:33:02.738-05:002010-07-05T11:33:02.738-05:00Different locations in a non-Euclidean space can h...Different locations in a non-Euclidean space can have different geometrical properties. If so, then an infinite space on its own, assuming we're substantivalists about space (if we're relationalists, I don't know that the hypothesis of an infinite space without infinitely many objects or objects of infinite extent makes sense) can be enough to generate the sorts of problems involved here. Instead of people on different locations, just imagine different bits of space with different local geometrical properties.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-73243888819289983732010-07-05T08:53:53.478-05:002010-07-05T08:53:53.478-05:00Hi again, Alex. I notice that at the end of last m...Hi again, Alex. I notice that at the end of last month you wrote: "I came to the question of the Kalaam argument with very strong intuitions that an infinite universe is possible. I find the conclusions of arguments, like the present one, against the possibility of an infinite universe to be very counterintuitive. It seems almost obvious that there could be an infinite universe."<br /><br />I'd like to reiterate my observation that such arguments as this, even if they do work, do not show that an infinite universe is impossible. They would seem to show most directly that simple, countable infinities of physical objects are impossible. But mathematicians have long believed that the natural numbers (the intuitive ones, not the formal ones within ZF) might be indefinitely extensible. If so then we could only have finite numbers of ordinary objects in an infinite space, but we could have the infinite space. The reason why we could not use all that space to get a simple transfinity of objects would be the nonsensicality of such transfinities.<br /><br />I've thought about this observation quite deeply over several years, and I can see why most people (myself included, most of the time) do not make it. But it appears to be correct.Martin Cookehttps://www.blogger.com/profile/11425491938517935179noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-17918541303951414112010-07-01T14:27:50.982-05:002010-07-01T14:27:50.982-05:00This stuff is tough. Right now, my project is a m...This stuff is tough. Right now, my project is a modality book. Once that's out the door--it's due Sept. 15--I can get on to other publication projects, and this is one option. I am also thinking of running a conference on probability and infinity, with some good people (I have two really good people who in principle agreed to come). If I run the conference, I might just present this stuff at it, and then probably there'd be a book from the conference.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-67442414530039392022010-07-01T13:33:21.448-05:002010-07-01T13:33:21.448-05:00Yes of course Alexander, a regrouping attempt can ...Yes of course Alexander, a <i>regrouping</i> attempt can rearrange the people and make the expectation different than before. But if my original reasoning is sound, and since it contradicts what you consider the implications of regrouping - then we have a paradox. Maybe it just can't be resolved, that's the sort of problem it is.<br /><br />Perhaps you can write up the problem and with my attempted rebuttal, and see what splash it makes. (I'd love to get credit even as informal citation.) I will put something up at my blog meanwhile, but go ahead and write it up if you wish.Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-75416495688640520082010-07-01T09:52:26.853-05:002010-07-01T09:52:26.853-05:00Neil:
I expect problems like this have been produ...Neil:<br /><br />I expect problems like this have been produced independently by lots of people, but no, I didn't get my initial stones scenario from anybody else.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-7677838770883466112010-07-01T09:25:37.664-05:002010-07-01T09:25:37.664-05:00So, start with the groups 1-10, 11-20, etc. Maybe...So, start with the groups 1-10, 11-20, etc. Maybe each group is defined by the fact that they have the same color of shoes. (There are infinitely many colors, I suppose.)<br /><br />Question: How likely is it that your number is divisible by 10? You want to say: 1/10.<br /><br />Fine. But keep the very same people, but recolor the shoes. So now the people with numbers 1,2,3,4,5,6,7,10,20 have the same color of shoes; the people with numbers 8,9,11,12,13,14,15,16,30,40 form another group, with new shoe colors; so do the people with numbers 17,18,19,21,22,23,24,25,50,60, etc. You get the point. After recoloring the shoes, two out of ten people in each group have a number divisible by 10. So using the grouping method, we now conclude that the probability you have a number divisible by 10 is 2/10=1/5. <br /><br />But your probability of being on a number divisible by 10 should not change when someone repaints your shoes! <br /><br />Perhaps, though, you think there is something special about a case where the grouping is done spatially, rather than by color. I don't know exactly why. Though, I kind of feel a pull of that claim when the grouping is done temporally.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-11646384326678713562010-06-30T21:11:46.470-05:002010-06-30T21:11:46.470-05:00I have a further argument directly about the pavin...I have a further argument directly about the paving stones. It is very much like my previous arguments. Let's say we had groups of ten blindfolded people each, and indeed an infinite number (A-null) of those groups of ten. You're in one of them. Each group is assigned to a set of stones: 1-10, 11-20, ... 4361-4370, .... You don't know which set you end up on. So I expect that I and my nine comrades will end up arranged "randomly" on some set of stones. <br /><br />How could I not think it plausible that I have a "1/10 chance" of ending up on #10, or maybe # 23,780, etc? After all, each group doesn't have to give a hoot - <i>in advance of any further meddling</i> - whether there are other groups at all, or even other stones! Their existence is not "felt" by my group, how could it be?<br /><br />Now sure, if you rearrange people that changes the chances but almost by circular argument. You're taking the people who were on e.g. 1, 2, ... 9, 11, 12, ... 137, 138, 139, 141, ... and switching them with those on 10, 20, 30, ... I know you can do that, it's just another Hilbert Hotel. But you changed something from what it was before. If we accepted the original probability as accurate and random, then I should believe I was likely not on a decadal stone. Hence it would make sense to change, regardless of it being done it that manner.<br /><br />So I think it is the argument justifying the original probability that matters. Sure, after the shuffle the sets are "equivalent" but the second set wasn't "used" to establish the probability directly. I don't know how else to handle this. I admit it is perplexing and we have conflicts of interest here over two compelling ways to look at it: it's a paradox!<br /><br />I want to put this problem up at my own blog. Is the stones argument specifically yours Dr. Pruss, and in any case I'd like a way to cite. txNeil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-32143411516356206152010-06-30T20:52:38.225-05:002010-06-30T20:52:38.225-05:00OK ... I can't blame you for wanting to take s...OK ... I can't blame you for wanting to take seriously the implications of an argument that seems valid. I'm not sure your argument about the stones has the implication that probability for infinite sets (given the actual context) simply must be absurd in all cases. All I can say is, the counterargument e.g. that local context can't be invalidated by infinite boundary conditions also seems valid, so we have conflict over what to accept. The limit graining argument seems valid to me also.<br /><br />And again, consider the "0/0" is in itself absurd, yet dy/dx somehow makes sense. Have you asked colleagues about this argument? I wonder if "surreal numbers" and other ways of handling infinites and infinitesimals (as in non-standard analysis) could help out here.Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-34452482632717483302010-06-30T20:30:58.399-05:002010-06-30T20:30:58.399-05:00This comment has been removed by the author.Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-20715409358926068522010-06-30T15:28:45.579-05:002010-06-30T15:28:45.579-05:00Why should it matter if the universe is infinite? ...Why should it matter if the universe is infinite? Because of this argument (and the ones in my other posts on the issue). :-)<br /><br />I came to the question of the Kalaam argument with very strong intuitions that an infinite universe is possible. I find the conclusions of arguments, like the present one, against the possibility of an infinite universe to be very counterintuitive. It seems almost obvious that there could be an infinite universe. <br /><br />But sometimes arguments force one to accept one thing that is counterintuitive, in order to escape another that is even more counterintuitive. Even so, I am reluctant to accept the conclusions--that's just a psychological statement about me. But I don't know of a plausible refutation of my arguments that lead to the conclusion that it's impossible to have an infinite universe (or, more precisely, that it's impossible for there to be an infinite number of items in the past of any item).Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-33119893250547897522010-06-30T14:45:05.872-05:002010-06-30T14:45:05.872-05:00OK, what if we were in an infinite universe. Do yo...OK, what if we <i>were</i> in an infinite universe. Do you think probability would cease to be either meaningful or a tool for study and expressing expectations? Really? Why should it matter, whether or not there's all that "out there," to what happens here?Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-43330524818215792532010-06-30T09:50:41.539-05:002010-06-30T09:50:41.539-05:00"There are infinite cases of the tosses etc. ..."There are infinite cases of the tosses etc. yet still we expect (and really do find) the appropriate probabilities."<br /><br />The "really do find" depends on the claim that we are actually in an infinite universe, which we do not know. The "expect" could be explained away as an expectation unduly generalized from finite cases.<br /><br />As for frequentism, unless there are well-defined probabilities, there is no guarantee that there are any limiting frequencies. The limit might just not exist. (For instance, in the pattern 010011000011110000000011111111..., there is no limiting frequency of 1's.) Even if there are well-defined probabilities, the existence of the limit is not logically guaranteed--it is only guaranteed to exist with probability 1.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-73519839569884314292010-06-30T07:22:09.497-05:002010-06-30T07:22:09.497-05:00Alexander, Allen: I accept the intrinsic logical p...Alexander, Allen: I accept the intrinsic logical problems with infinite sets per se. Right, we can't compare infinite ratios like we can finite sets. But that's just taking them in the abstract and asking which is bigger and by how much - and finding we can't, "per se." As we've seen argued, that isn't all there is to it. We need some context. Whether that has to be the idea of structure that Allen poses or not is perhaps debatable.<br /><br />So I want people to peel away from the pure abstraction some more and consider again the counter-example of betting in an infinite universe. There are infinite cases of the tosses etc. yet still we expect (and really do find) the appropriate probabilities. That is supposed to show that the pure abstract argument is inadequate by <i>reductio</i>. We should not imagine, as Mr. Pruss seems to, that the counter-argument must be wrong because the abstraction is unassailable.<br /><br />I think that taking the limit of fine graining even if the "real case" goes to continuum is also valid, and there could be more.<br /><br />Here's a final challenge to taking problems with infinity too seriously, albeit it regards "potential infinity" and not the expressed set per se: probability theory itself is often couched in terms of, how the frequentist relative proportions converge to a distinct ratio as we approach an infinite number of trials (presumably Aleph null but could be others I suppose.) In a logically possible world I could keep tossing "forever" and mosts thinkers would say, I can expect whatever chances per ordinary probability theory. (Note: some thinkers find problems with extension into an infinite past, e.g. "I was always doing it and never started ...." One could argue it's just a time reversal of the former.) <br /><br />Few would think that approaching infinity as the idea of the definition in probability, invalidates the essential concept. (However, note that probabilistic claims are not strictly falsifiable in Popperian terms, since no particular run (and a stated "set of runs" is of course just a fragmented "run") can be dispositive! It's a judgment call, FWIW ...) So it seems to be a valid concept. The pure abstractions of set theory are inadequate as a critique.<br /><br />BTW, I recommend Rudy Rucker's <i>Infinity and the Mind</i> for mind-bending excursions into the transfinite world.Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-28741313657080145112010-06-30T00:00:01.057-05:002010-06-30T00:00:01.057-05:00How about this:
Lets assume we have an infinitely...How about this:<br /><br />Lets assume we have an infinitely long array of squares. And a fair 6-sided dice.<br /><br />We roll the dice an infinite number of times and write each roll's number into a square. <br /><br />When we finish, how many squares have a "1" written in them? An infinite number, right?<br /><br />How many squares have an even number written in them? Also an infinite number.<br /><br />How many squares have a number OTHER than "1" written in them? Again, an infinite number.<br /><br />Therefore, the squares with "1" can be put into a one-to-one correspondence with the "not-1" squares...correct?<br /><br />Now, while we have this one-to-one correspondence between "1" and "not-1" squares set up, let's put a sticker with an "A" on it in the "1" squares. And a sticker with a "B" on it in the "not-1" squares. We'll need the same number of "A" and "B" stickers, obviously. Aleph-null.<br /><br />So, if we throw a dart at a random location on the array of squares, what is the probability of hitting a square with a "1" in it?<br /><br />What is the probability of hitting a square with a "A" sticker?<br /><br />The two questions don't have a compatible answers, right? So, in this scenario, probability is useless. It just doesn't apply. You should have no expectations about either outcome.<br /><br />BUT. NOW. Let's erase the numbers and remove the stickers and start over.<br /><br />This time, let's just fill in the squares with a repeating sequence of 1,2,3,4,5,6,1,2,3,4,5,6,1,2,3,...<br /><br />And then, let's do our same trick about putting the "1" squares into a one-to-one mapping with the "not-1" squares, and putting an "A" sticker on the "1" squares, and a "B" sticker on the "not-1" squares.<br /><br />Now, let's throw a dart at a random location on the array of squares. What is the probability of hitting a square with a "1" in it?<br /><br />What is the probability of hitting a square with a "A" sticker on it?<br /><br />THIS time we have some extra information! There is a repeating pattern to the numbers and the stickers. No matter where the dart hits, we know the layout of the area. This is our "measure" that allows us to ignore the infinite aspect of the problem and apply probability.<br /><br />For any area the dart hits, there will always be an equal probability of hitting a 1, 2, 3, 4, 5, *or* 6. As you'd expect. So the probability of hitting a square with a "1" in it is ~16.67%.<br /><br />Any area where the dart hits will have a repeating pattern of one "A" sticker followed by five "B" stickers. So the probability of hitting an "A" sticker is ~16.67%. <br /><br />The answers are now compatible, thanks to the extra "structural" information that gave us a way to ignore the infinity.<br /><br />In other words, you can't apply probability to infinite sets, but you can apply it to the *structure* of an infinite set.<br /><br />If the infinite set has no structure, then you're out of luck. At best you can talk about the method used to generate the infinite set...but if this method involves randomness, it's not quite the same thing.Allenhttps://www.blogger.com/profile/10637109782433403780noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-6778045703502184402010-06-29T16:15:25.178-05:002010-06-29T16:15:25.178-05:00If there were a small number of sites, it would ma...If there were a small number of sites, it would make sense to assume that each site is equally likely, and then the number of flukish sites would be a small fraction of the total number of sites, so the probability of a fluke would be low. But (a) it doesn't seem to make sense to assume all sites are equally likely given an infinite number of sites, and (b) the number of flukish sites is the same as the number of non-flukish ones.<br /><br />Moreover, whether it makes sense to talk of the "local", if it includes other close-by universes, depends on what kind of a multiverse we have. A multiverse where the universes are not embedded in a metric space may not allow one to talk of which universes are local to us.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-6298594795159238162010-06-29T15:29:29.865-05:002010-06-29T15:29:29.865-05:00Thanks for a quick reply, Dr. Pruss! Well you have...Thanks for a quick reply, Dr. Pruss! Well you have a point but we could wonder about similar issues of "am I stuck in a statistical fluke" if the universe had "only" 10^1200 such tossings or card games going on. My point was: do you really think you can't expect a likely outcome, just because you're part of an infinite set of such activities v. a finite one, however large the latter? How could that be? Our universe likely is infinite and that means infinite copies of us and what we do - should I care? Like I said, the number being infinite should be of no consequence, only the local proportionality I observe at each successive scale. (That is, ignore the boundary condition.)Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-34720888852845914252010-06-29T15:00:36.630-05:002010-06-29T15:00:36.630-05:00Neil:
But in the infinite sequence of coin tosses...Neil:<br /><br />But in the infinite sequence of coin tosses, there are subsequences where it's just heads for a very, very long time, and there are subsequences where it's just tails for a very, very long time. So how do we know that we're in a place in the infinite sequence where heads and tails are roughly in equal proportion, rather than in one of those portions where it's just heads or just tails?<br /><br />Well, we might say: There are a lot more portions in the sequence where heads and tails are roughly in equal proportion. But what does "a lot more portions" means in respect of an infinite sequence?Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-29989563252360915082010-06-29T13:54:31.104-05:002010-06-29T13:54:31.104-05:00Your argument is interesting, and makes me think o...Your argument is interesting, and makes me think of general objections to probability involving infinite sets. We can't compare the size of say, the integers and the odd numbers since both can be Cantor-matched as equivalent cardinality. So you're tempted to say, there's no way to talk about probabilities if there are infinite sets involved.<br /><br />However, consider a coin-toss exercise. You expect 50/50 from a fair coin over time, and whatever appropriate results from doing other things like cards, dice. But suppose the universe is infinite (and it seems flat, so it supposedly really is.) Now there are Aleph-null people tossing coins, Aleph-null tossings, etc. all over this infinite universe. Aleph null head-landings and Aleph null tails ... Do you really want to say that this exotic boundary condition makes it impossible for you to "expect" normal outcomes?<br /><br />Somehow, the proportions for probability are based on intrinsic tendencies of the finite limit and not the idealized set properties. If you "fill it in" by making it infinite, that doesn't change the proportion (almost like taking dy/dx in reverse.) So suppose I looked at integers between one and ten and the chance of hitting two, we'd say"1/10". If it's numbers made into tenths like 1, 1.1, ... 9.9, 10 then the chance of hitting from say 1.6 through 2.5 is again 1/10. We can keep making it finer, and the ratio holds. Indeed, we can take a range "1.5-2.5" of the continuum (Aleph = ?!) and it's still intelligible despite there being infinitely many targets.<br /><br />That's how I look at the problem of constants and features of possible universes. Imagine it being grainy to some fine degree (like 0.001 increments to each spec.), and there's various chances of this or that. Then cut the grain to 1/10, and so on ... The proportionality should hold. That's what matters, not the limit infinities. Think of it more like the chance a dart would hit one colored region rather than another on a picture. So yes there are various "chances" we're likely to end up in various universes depending upon the other factors (ie, Bayesian reasoning.) It is not invalid.Neil Bateshttps://www.blogger.com/profile/04564859009749481136noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-23822979836600674932010-04-08T07:32:47.630-05:002010-04-08T07:32:47.630-05:00The post for the day after this one addresses the ...The post for the day after this one addresses the in-universe case. If that argument works, then you can't assign probabilities to the outcomes of random processes. And that would be pretty bad, since science needs to be able to do that.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-70103443281903827372010-04-08T07:20:23.549-05:002010-04-08T07:20:23.549-05:00Okay, I can see how that might be an interesting p...Okay, I can see how that might be an interesting point: An infinite multiverse poses problems for probability assignments insofar as it prevents us from assigning certain values which could potentially be helpful under a finite multiverse hypothesis. But it seems like you're taking that problem further than I'd be willing to do.<br /><br />I would make two claims regarding how to temper your view: First, probability assignments regarding physical systems inside our own universe remain demonstrably useful. I think this is pretty clear from an empirical perspective, and I don't see how we could demonstrate that an infinite multiverse leads us to any logical problems, so long as we stay within the confines of our own context---this universe. Second, I only find it to be problematic as a scientific hypothesis because it seems not to be falsifiable. So, if you have some other criticism in mind when you call it "self-defeating," I'd be curious to hear what that is.<br /><br />Anyway, I don't mean to overwhelm you with comments. I thank you for your responses so far.Ben Wallishttps://www.blogger.com/profile/00131358613835119782noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-19377204288613974762010-04-08T00:16:35.194-05:002010-04-08T00:16:35.194-05:00If we go for the abstention position, then we seem...If we go for the abstention position, then we seem to get this interesting result: In an infinite multiverse, we can't do probability. But science requires probability. Thus, insofar as the multiverse is a scientific hypothesis, it is self-defeating.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.com