Re: White Rabbit vs. Tegmark
Dear Russell and Hal: thanks for the compassionate speedy replies. I would be happy to cpomply with Hal's advice, alas, I have no browser working yet(?), only the mailbox. Cluttered with garbage. I didn't believe a friend who had similar troubles when installing Symantec, I lost my cyberways by the 2005 ed. And Excuse me for keeping that irrelevant subject line. It is quite unfair to the rabbit. I have no difficulty with 'plain' HTML, OE6 can handle it - or my Word-viewer. It was the frightening 270 posts among the accumulated 13 - 1500 spam what I collected first out, into an 'everything' folder for simplicity's sake G (I did the same with the other 7 listposts as well.) * Could anybody still remember for me WHAT was originally what we could just simply accept... - in that former (other) (stupid) Subject line? I always try to read only but sometime I cannot keep my mouse shut. Will catch up hopefully soon. Best wishes John M - Original Message - From: Hal Finney [EMAIL PROTECTED] To: everything-list@eskimo.com Sent: Tuesday, June 14, 2005 7:41 PM Subject: Re: White Rabbit vs. Tegmark
Re: White Rabbit vs. Tegmark
Dear Russell and list: this is a personal problem due to my extremely feeble skills in computering. I had (optimistically in past tense) problems with my internet e-mail connection and could not get/send e-mail since the date of this post. Then 2 times I was lucky and got hundreds of email at a time, (the browser is still closed from my usage). To separate chaff from seed, I extracted the list-post before deleting the slumscumspam. From May 24 to June 13 I accumulated 270 listposts -absolutely impossible to scan for topical and content interest. Those posts were accessible (for me) that started with a statement of the writer and not a lot of copies with some reply-lines interjected. I know (and like to use) to copy the phrases to reply to but even in a 2-week archiving it turns sour. After the first 30-40 post-reviews I got dizzy. This pertains to the regular listpost. Your posts are one degree worse: You had 32 attachment-convoluted posts in the 270 of the 20 day everything list. The procedure to glance at your text (and I like to read what you wrote) I have to select the attachment, then call it up, then select open, then read it, and - If I want to save it: copy or cut it to file in 3-5 more clicks. A regular e-mail requires ONE click to read and ONE to save (in my Outlook Express). I don't want to reformulate this list, just tried to communicate a point that may make MY life easier. Apologies for whining John Mikes PS: I lost the 'original' post of the Subject: RE:What do you lose if you simply accept... and reading the posts I could not find out WHAT to accept? the posts were all over the Library of Congress, nothing to the original subject to explain. Just curious. JM == - Original Message - From: Russell Standish [EMAIL PROTECTED] To: Hal Finney [EMAIL PROTECTED] Cc: everything-list@eskimo.com Sent: Tuesday, May 24, 2005 2:45 AM Subject: Re: White Rabbit vs. Tegmark
Re: White Rabbit vs. Tegmark
John Mikes wrote: ... Those posts were accessible (for me) that started with a statement of the writer and not a lot of copies with some reply-lines interjected. I know (and like to use) to copy the phrases to reply to but even in a 2-week archiving it turns sour. After the first 30-40 post-reviews I got dizzy. This pertains to the regular listpost. And with regard to Russell's posts: Your posts are one degree worse: You had 32 attachment-convoluted posts in the 270 of the 20 day everything list. The procedure to glance at your text (and I like to read what you wrote) I have to select the attachment, then call it up, then select open, then read it, and - If I want to save it: copy or cut it to file in 3-5 more clicks. The list has certainly been active lately! John, you might want to read the list via the archive at http://www.escribe.com/science/theory/ . It's very readable and it does not choke on Russell's posts, which are actually perfectly legal email formats and are legible on many mail readers. Plus you won't be bothered by email interruptions. You can also use the thread index there which will help you to follow conversations. However I would like to echo John's complaint about excessive quoting. There is no need to quote the complete content of another message before replying. Especially lately, people are conversing quickly and the previous message was typically sent only a few hours or a day earlier. We can all remember what was said, or go back in the thread and see it. Just a brief quote to set the stage should be enough. If you want to reply point by point, fine, in that case it does make sense to quote each point before replying. But quoting the whole message is almost never necessary. Hal Finney
RE: White Rabbit vs. Tegmark
Jonathan Colvin writes: That's rather the million-dollar question, isn't it? But isn't the multiverse limited in what axioms or predicates can be assumed? For instance, can't we assume that in no universe in Platonia can (P AND ~P) be an axiom or predicate? No, I'd say that you could indeed have a mathematical object which had P AND ~P as one of its axioms. The problem is that from a contradiction, you can deduce any proposition. Therefore this mathematical system can prove all well formed strings as theorems. As Russell mentioned the other day, everything is just the other side of the coin from nothing. A system that proves everything is essentially equivalent to a system that proves nothing. So any system based on P AND ~P has essentially no internal structure and is essentially equivalent to the empty set. One point is that we need to distinguish the tools we use to analyze and understand the structure of a Tegmarkian multiverse from the mathematical objects which are said to make up the multiverse itself. We use logic and other tools to understand the nature of mathematics; but mathematical objects themselves can be based on nonstandard logics and have exotic structures. There are an infinite number of formal systems that have nothing to do with logic at all. Formal systems are just string-manipulation engines and only a few of them have the axioms of logic as a basis. Yet they can all be considered mathematical objects and some of them might be said to be universes containing observers. Hal Finney
Re: White Rabbit vs. Tegmark
- Original Message - From: Hal Finney [EMAIL PROTECTED] To: everything-list@eskimo.com Sent: 27 May 2005 19:19 Subject: RE: White Rabbit vs. Tegmark . . To summarize, logic is not a property of universes. It is a tool that our minds use to understand the world, including possible universes. We may fail to think clearly or consistently or logically about what can and cannot exist, but that doesn't change the world out there. Rather than expressing the AUH as the theory that all logically possible universes exist, I would just say that all universes exist. And of course as we try to understand the nature of such a multiverse, we will attempt to be logically consistent in our reasoning. That's where logic comes in. I basically agree with this (and with Brent's similar comment). In philosophy we tend to use 'logically possible' to distinguish from 'physically possible', the first usually means 'not violating deductive logic', the second 'not violating physical laws'. So no 'round squares' in the case of 'logically possible' - but, as you imply, that should really be thought of as a feature of descriptions, not universes themselves. (I make that very point in my paper.) Alastair
Re: White Rabbit vs. Tegmark
Le 27-mai-05, à 20:19, Hal Finney a écrit : Brent Meeker writes: I doubt that the concept of logically possible has any absolute meaning. It is relative to which axioms and predicates are assumed. I agree but that is the reason why if we want to talk *about* or to find measure *on* logical possibilities, we should make clear our axioms and inference rules. (And this is as in think compatible with Hal Finney's comment). BTW Hal, sorry for having misinterpret your use of UD. (Universal distribution versus Universal Dovetailer). Bruno http://iridia.ulb.ac.be/~marchal/
RE: White Rabbit vs. Tegmark
Hal: To summarize, logic is not a property of universes. It is a tool that our minds use to understand the world, including possible universes. We may fail to think clearly or consistently or logically about what can and cannot exist, but that doesn't change the world out there. Rather than expressing the AUH as the theory that all logically possible universes exist, I would just say that all universes exist. And of course as we try to understand the nature of such a multiverse, we will attempt to be logically consistent in our reasoning. That's where logic comes in. But all universes exist is merely a tautology. To say anything meaningful, one is then faced with attempting to define what one means by universe, or exist. The concept of logically possible seems to me to be useful in this latter endeavor. Jonathan Colvin
RE: White Rabbit vs. Tegmark
Brent: I doubt that the concept of logically possible has any absolute meaning. It is relative to which axioms and predicates are assumed. That's rather the million-dollar question, isn't it? But isn't the multiverse limited in what axioms or predicates can be assumed? For instance, can't we assume that in no universe in Platonia can (P AND ~P) be an axiom or predicate? Not long ago the quantum weirdness of Bell's theorem, or special relativity would have been declared logically impossible. That declaration would simply have been mistaken. Is it logically possible that Hamlet doesn't kill Polonius? Certainly. I'm sure there are people named Hamlet who have not killed a person named Polonius. Is it logically possible that a surface be both red and green? If you are asking whether it is logically possibly that a surface that can reflect *only* light at a wavelength of 680 nm can reflect a wavelength of 510 nm, the answer would seem to be no. Jonathan Colvin
Re: White Rabbit vs. Tegmark
Hi Jonathan, Should we not expect Platonia to be Complete? Stephen - Original Message - From: Jonathan Colvin [EMAIL PROTECTED] To: 'Everything-List' everything-list@eskimo.com Sent: Saturday, May 28, 2005 1:30 PM Subject: RE: White Rabbit vs. Tegmark Brent: I doubt that the concept of logically possible has any absolute meaning. It is relative to which axioms and predicates are assumed. That's rather the million-dollar question, isn't it? But isn't the multiverse limited in what axioms or predicates can be assumed? For instance, can't we assume that in no universe in Platonia can (P AND ~P) be an axiom or predicate? Not long ago the quantum weirdness of Bell's theorem, or special relativity would have been declared logically impossible. That declaration would simply have been mistaken. Is it logically possible that Hamlet doesn't kill Polonius? Certainly. I'm sure there are people named Hamlet who have not killed a person named Polonius. Is it logically possible that a surface be both red and green? If you are asking whether it is logically possibly that a surface that can reflect *only* light at a wavelength of 680 nm can reflect a wavelength of 510 nm, the answer would seem to be no. Jonathan Colvin
RE: White Rabbit vs. Tegmark
Stephen: Should we not expect Platonia to be Complete? I'd like to think that it should not be (by Godel?); or that it is not completely self-computable in finite meta-time. Or some such. But that's more of a faith than a theory. Jonathan Colvin Brent: I doubt that the concept of logically possible has any absolute meaning. It is relative to which axioms and predicates are assumed. That's rather the million-dollar question, isn't it? But isn't the multiverse limited in what axioms or predicates can be assumed? For instance, can't we assume that in no universe in Platonia can (P AND ~P) be an axiom or predicate? Not long ago the quantum weirdness of Bell's theorem, or special relativity would have been declared logically impossible. That declaration would simply have been mistaken. Is it logically possible that Hamlet doesn't kill Polonius? Certainly. I'm sure there are people named Hamlet who have not killed a person named Polonius. Is it logically possible that a surface be both red and green? If you are asking whether it is logically possibly that a surface that can reflect *only* light at a wavelength of 680 nm can reflect a wavelength of 510 nm, the answer would seem to be no. Jonathan Colvin
Re: White Rabbit vs. Tegmark
- Original Message - From: Patrick Leahy [EMAIL PROTECTED] To: Brent Meeker [EMAIL PROTECTED] Cc: Everything-List everything-list@eskimo.com Sent: 26 May 2005 19:54 Subject: RE: White Rabbit vs. Tegmark . . . * But the arbitrariness of the measure itself becomes the main argument against the everything thesis, since the main claimed benefit of the thesis is that it removes arbitrary choices in defining reality. I don't think we can reject the thesis that all logically possible universes exist, just because its analysis is proving tricky (or even if it were to prove beyond us), and certainly not if we have reasonable candidates for a measure basis. I use fundamental philosophical principles effectively to infer the existence of all possible distinguishable entities (which would include both individual, and clusters of, physical universes), much if not all of which should be modellable by formal systems. In the context of SAS-containing universes, this seems to me a reasonable starting point for an in-principle measure basis, particularly when consideration of 'compressed' entities (not logically disbarred) leads to a prediction of the predominance of simpler universes. But I agree that going any further with particular predictions, particularly after factoring in Tegmark's other levels, is likely to prove, shall we say, at least a little way off ;) Alastair Philosophy paper at: http://www.physica.freeserve.co.uk/pa01.htm
Re: White Rabbit vs. Tegmark
Le 26-mai-05, à 18:03, Hal Finney a écrit : One problem with the UD is that the probability that an integer is even is not 1/2, and that it is prime is not zero. Probabilities in general will not equal those defined based on limits as in the earlier paragraph. It's not clear which is the correct one to use. It seems to me that the UDA showed that the (relative) measure on a computational state is determined by the (absolute?) measure on the infinite computational histories going through that states. There is a continuum of such histories, from the first person person point of view which can not be aware of any delay in the computations emulated by the DU (= all with Church's thesis), the first persons must bet on the infinite union of infinite histories. Bruno http://iridia.ulb.ac.be/~marchal/
Re: White Rabbit vs. Tegmark
Bruno Marchal writes: Le 26-mai-05, à 18:03, Hal Finney a écrit : One problem with the UD is that the probability that an integer is even is not 1/2, and that it is prime is not zero. Probabilities in general will not equal those defined based on limits as in the earlier paragraph. It's not clear which is the correct one to use. It seems to me that the UDA showed that the (relative) measure on a computational state is determined by the (absolute?) measure on the infinite computational histories going through that states. There is a continuum of such histories, from the first person person point of view which can not be aware of any delay in the computations emulated by the DU (= all with Church's thesis), the first persons must bet on the infinite union of infinite histories. Sorry, I was not clear: by UD I meant the Universal Distribution, aka the Universal Prior, not the Universal Dovetailer, which I think you are talking about. The UDA is the Universal Dovetailer Argument, your thesis about the nature of first and third person experience. I simply meant to use the Universal Distribution as an example probability measure over the integers, where, given a particular Universal Turing Machine, the measure for integer k equals the sum of 1/2^n where n is the length of each program that outputs integer k. Of course this is an uncomputable measure. A much simpler measure is 1/2^k for all positive integers k. I don't know whether there would be any probability measures over the integers such that the probability of every event E equals the limit as n approaches infinity of the probability of E for all integers less than n. Actually on further thought, it's clear that the answer is no. Consider the set Ex = all integers less than x. Clearly the probability of Ex being true for integers less than n goes to zero as n goes to infinity. But the only way a probability measure can give 0 for a set is to give 0 for every element of the set. That means that the measure for all elements less than x must be zero, for any x. And that implies that the measure must be 0 for all finite x, which rules out any meaningful measure. Hal Finney
RE: White Rabbit vs. Tegmark
-Original Message- From: Alastair Malcolm [mailto:[EMAIL PROTECTED] Sent: Friday, May 27, 2005 8:53 AM To: Patrick Leahy Cc: Everything-List Subject: Re: White Rabbit vs. Tegmark - Original Message - From: Patrick Leahy [EMAIL PROTECTED] To: Brent Meeker [EMAIL PROTECTED] Cc: Everything-List everything-list@eskimo.com Sent: 26 May 2005 19:54 Subject: RE: White Rabbit vs. Tegmark . . . * But the arbitrariness of the measure itself becomes the main argument against the everything thesis, since the main claimed benefit of the thesis is that it removes arbitrary choices in defining reality. I don't think we can reject the thesis that all logically possible universes exist, just because its analysis is proving tricky (or even if it were to prove beyond us), and certainly not if we have reasonable candidates for a measure basis. I doubt that the concept of logically possible has any absolute meaning. It is relative to which axioms and predicates are assumed. Not long ago the quantum weirdness of Bell's theorem, or special relativity would have been declared logically impossible. Is it logically possible that Hamlet doesn't kill Polonius? Is it logically possible that a surface be both red and green? Brent Meeker
RE: White Rabbit vs. Tegmark
Brent Meeker writes: I doubt that the concept of logically possible has any absolute meaning. It is relative to which axioms and predicates are assumed. Not long ago the quantum weirdness of Bell's theorem, or special relativity would have been declared logically impossible. Is it logically possible that Hamlet doesn't kill Polonius? Is it logically possible that a surface be both red and green? I agree. We went around on this logically possible stuff a few weeks ago. A universe is not constrained by logical possibility. Our understanding of what is or is not a possible universe is constrained by our mental abilities, which include logic as one of their components. If I say a universe exists where glirp glorp glurp, that is not meaningful. But it doesn't constrain or limit any universe, it is simply a non-meaningful description. It is a problem in my mind and my understanding, not a problem in the nature of the multiverse. If I say a universe exists where p and not p, that has similar problems. It it is not a meaningful description. Similarly if I say a universe exists where pi = 3. Saying this demonstrates an inconsistency in my mathematical logic. It doesn't limit any universes. More complex descriptions, like whether green can be red, come down to our definitions and what we mean. Maybe we are inconsistent in our minds and failing to describe a meaningful universe; maybe not. But again it does not limit what universes exist. To summarize, logic is not a property of universes. It is a tool that our minds use to understand the world, including possible universes. We may fail to think clearly or consistently or logically about what can and cannot exist, but that doesn't change the world out there. Rather than expressing the AUH as the theory that all logically possible universes exist, I would just say that all universes exist. And of course as we try to understand the nature of such a multiverse, we will attempt to be logically consistent in our reasoning. That's where logic comes in. Hal Finney
Re: White Rabbit vs. Tegmark
I don't see what practical difference there is between saying all universes exist and we need to think with logical consistency about the subject and all logically possible universes exist. Whatever the case, a related point might be worth considering: whether logic is the only such invariant involved. Level IV is supposed to involve variations of mathematical structure. But would there be various universes for various structures in deductive mathematical theories of optimization, probability, information, or logic? If probability theory supplied various different structures for different universes, it could certainly complicate the measure problem -- one could be contending with various theories of measure itself as sources of differences between various universes. If, on the other hand, all these mathematics (and they amount together to a great deal of mathematics, maths often regarded as applied, but also often regarded as mathematically deep) place no additional constraints, nor broaden the variety of universes, then they seem correlated not to Level IV, but perhaps to Level III, since Level III (if I understand and phrase it correctly) neither adds constraints nor broadens the variety of universes on top of what one already has at Level II, and also since they seem to be mathematics of the structures of the kinds of alternatives which MWI says are all actualized. Best, Ben Udell Hal Finney writes: Brent Meeker writes: I doubt that the concept of logically possible has any absolute meaning. It is relative to which axioms and predicates are assumed. Not long ago the quantum weirdness of Bell's theorem, or special relativity would have been declared logically impossible. Is it logically possible that Hamlet doesn't kill Polonius? Is it logically possible that a surface be both red and green? I agree. We went around on this logically possible stuff a few weeks ago. A universe is not constrained by logical possibility. Our understanding of what is or is not a possible universe is constrained by our mental abilities, which include logic as one of their components. If I say a universe exists where glirp glorp glurp, that is not meaningful. But it doesn't constrain or limit any universe, it is simply a non-meaningful description. It is a problem in my mind and my understanding, not a problem in the nature of the multiverse. If I say a universe exists where p and not p, that has similar problems. It it is not a meaningful description. Similarly if I say a universe exists where pi = 3. Saying this demonstrates an inconsistency in my mathematical logic. It doesn't limit any universes. More complex descriptions, like whether green can be red, come down to our definitions and what we mean. Maybe we are inconsistent in our minds and failing to describe a meaningful universe; maybe not. But again it does not limit what universes exist. To summarize, logic is not a property of universes. It is a tool that our minds use to understand the world, including possible universes. We may fail to think clearly or consistently or logically about what can and cannot exist, but that doesn't change the world out there. Rather than expressing the AUH as the theory that all logically possible universes exist, I would just say that all universes exist. And of course as we try to understand the nature of such a multiverse, we will attempt to be logically consistent in our reasoning. That's where logic comes in. Hal Finney
Re: White Rabbit vs. Tegmark
- Original Message - From: Patrick Leahy [EMAIL PROTECTED] To: Alastair Malcolm [EMAIL PROTECTED] Cc: EverythingList everything-list@eskimo.com Sent: 24 May 2005 22:10 Subject: Re: White Rabbit vs. Tegmark . [Patrick:] This is very reminiscent of Lewis' argument. Have you read his book? IIRC he claims that you can't actually put a measure (he probably said: you can't define probabilities) on a countably infinite set, precisely because of Cantor's pairing arguments. Which seems plausible to me. [Alastair:] It seems to depend on whether one can find an intrinsic ordering (or something similar), such that relative frequency comes into play (so prime numbers *would* be less likely to be hit). An example occurs which might be of help. Let us say that the physics of the universe is such that in the Milky Way galaxy, carbon-based SAS's outnumber silicon-based SAS's by a trillion to one. Wouldn't we say that the inhabitants of that galaxy are more likely to find themselves as carbon-based? Now extrapolate this to any large, finite number of galaxies. The same likelihood will pertain. Now surely all the statistics don't just go out of the window if the universe happens to be infinite rather than large and finite? Alastair
Re: White Rabbit vs. Tegmark
On Thu, 26 May 2005, Alastair Malcolm wrote: An example occurs which might be of help. Let us say that the physics of the universe is such that in the Milky Way galaxy, carbon-based SAS's outnumber silicon-based SAS's by a trillion to one. Wouldn't we say that the inhabitants of that galaxy are more likely to find themselves as carbon-based? Now extrapolate this to any large, finite number of galaxies. The same likelihood will pertain. Now surely all the statistics don't just go out of the window if the universe happens to be infinite rather than large and finite? Alastair Well, it just does, for countable sets. This is what Cantor showed, and Lewis explains in his book. Cantor defines same size as a 1-to-1 pairing. Hence as there are infinite primes and infinite non-primes there are the same number (cardinality) of them: (1,3), (2,4), (3,6), (5,8), (7,9), (11,12), (13,14), (17,15), (19,16) etc and so ad infinitum You might say there are obviously more non-primes. This means that if you list the numbers in numerical sequence, you get fewer primes than non-primes in any finite sequence except a few very short ones. But in another sequence the answer is different: (1,2,4) (3,5,6) (7,11,8) (13,17,9) etc ad infinitum. In this infinite sequence, each triple has two primes and only one non-prime. Hence there seem to be more primes than non-primes! For the continuum you can restore order by specifying a measure which just *defines* what fraction of real numbers between 0 1 you consider to lie in any interval. For instance the obvious uniform measure is that there are the same number between 0.1 and 0.2 as between 0.8 and 0.9 etc. Why pick any other measure? Well, suppose y = x^2. Then y is also between 0 and 1. But if we pick a uniform measure for x, the measure on y is non-uniform (y is more likely to be less than 0.5). If you pick a uniform measure on y, then x = sqrt(y) also has a non-uniform measure (more likely to be 0.5). A measure like this works for the continuum but not for the naturals because you can map the continuum onto a finite segment of the real line. In m6511 Russell Standish describes how a measure can be applied to the naturals which can't be converted into a probability. I must say, I'm not completely sure what that would be good for. Paddy Leahy
Re: White Rabbit vs. Tegmark
On Thu, May 26, 2005 at 11:20:35AM +0100, Patrick Leahy wrote: A measure like this works for the continuum but not for the naturals because you can map the continuum onto a finite segment of the real line. In m6511 Russell Standish describes how a measure can be applied to the naturals which can't be converted into a probability. I must say, I'm not completely sure what that would be good for. Paddy Leahy Umm, doomsday argument reasoning when the total number of people who ever will have lived is infinite (but countable). Explain more later, if interested. -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgpShn8j0UGVW.pgp Description: PGP signature
Re: White Rabbit vs. Tegmark
Paddy Leahy writes: For the continuum you can restore order by specifying a measure which just *defines* what fraction of real numbers between 0 1 you consider to lie in any interval. For instance the obvious uniform measure is that there are the same number between 0.1 and 0.2 as between 0.8 and 0.9 etc. Why pick any other measure? Well, suppose y = x^2. Then y is also between 0 and 1. But if we pick a uniform measure for x, the measure on y is non-uniform (y is more likely to be less than 0.5). If you pick a uniform measure on y, then x = sqrt(y) also has a non-uniform measure (more likely to be 0.5). A measure like this works for the continuum but not for the naturals because you can map the continuum onto a finite segment of the real line. In m6511 Russell Standish describes how a measure can be applied to the naturals which can't be converted into a probability. I must say, I'm not completely sure what that would be good for. I think it still makes sense to take limits over the integers. The fraction of integers less than n that is prime has a limit of 0 as n goes to infinity. The fraction that are even has a limit of 1/2. And so on. When you apply a measure to the whole real line, it has to be non-uniform and has to go asymptotically to zero as you go out to infinity. This happens implicitly when you map it to (0,1) even before you put a measure on that segment. The same thing can be done to integers. The universal distribution assigns probability to every integer such that they all add to 1. The probability of an integer is based on the length of the smallest program in a given Universal Turing Machine which outputs that integer. Specifically it equals the sum of 1/2^l where l is the length of each program that outputs the integer in question. Generally this will give higher measure to smaller numbers, although a few big numbers will have relatively high measure if they have small programs (i.e. if they are simple). Of course this measure is non-uniform and goes asymptotically to zero, as any probability measure must. One problem with the UD is that the probability that an integer is even is not 1/2, and that it is prime is not zero. Probabilities in general will not equal those defined based on limits as in the earlier paragraph. It's not clear which is the correct one to use. Going back to Alistair's example, suppose we lived in a spatially infinite universe, Tegmark's level 1 multiverse. Of course our entire Hubble bubble is replicated an infinite number of times, to any desired degree of precision. Hence we have an infinite number of counterparts. Do you see a problem in drawing probabilistic conclusions from this? Would it make a difference if physics were ultimately discrete and all spatial positions could be described as integers, versus ultimately continuous, requiring real numbers to describe positions? Note that in this case we can't really use the UD or a line-segment measure because there is no natural starting point which distinguishes the origin of space. We can't have a non-uniform measure in a homogeneous space, unless we just pick an origin arbitrarily. So in this case the probability-limit concept seems most appropriate. Hal Finney
RE: White Rabbit vs. Tegmark
On Thu, 26 May 2005, Brent Meeker wrote: I agree with all you say. But note that the case of finite sets is not really any different. You still have to define a measure. It may seem that there is one, compelling, natural measure - but that's just Laplace's principle of indifference applied to integers. The is no more justification for it in finite sets than infinite ones. That there are fewer primes than non-primes in set of natural numbers less than 100 doesn't make the probability of a prime smaller *unless* you assign the same measure to each number. Brent Meeker I'll answer both Brent and Hal (m6556) here. Yup, I hadn't thought through the measure issue properly. Several conclusions from this discussion: * As Brent says, you always have to assume a measure. Sometimes a measure seems so natural you forget you're doing it, as above. Another example is in an infinite homogeneous universe, where equally a uniform measure seems natural, and also limiting frequencies over large volumes are well defined, as per Hal's message. * As Hal points out, it *is* possible to assign probability measures to countably-infinite sets. * Alternatively, you can assign a non-normalizable measure (presumably uniform) and take limiting frequencies. But then as per Cantor the answer does depend on your ordering, which is something extra you are adding to the definition of the set (even for numerical sequences). * Different lines of argument can easily lead to different natural measures for the same set, e.g. Hal's Universal Distribution vs. Laplacian indifference for the integers. * For me, the only way to connect a measure with a probability in the context of an everything theory is for the measure to represent the density of universes (or observers or observer-moments if you factor that in as well). * Since the White Rabbit^** argument implicitly assumes a measure, as it stands it can't be definitive. * But the arbitrariness of the measure itself becomes the main argument against the everything thesis, since the main claimed benefit of the thesis is that it removes arbitrary choices in defining reality. Paddy Leahy ^** This is a song about Alice, remember? --- Arlo Guthrie
Re: White Rabbit vs. Tegmark
- Original Message - From: Patrick Leahy [EMAIL PROTECTED] To: Alastair Malcolm [EMAIL PROTECTED] Cc: EverythingList everything-list@eskimo.com Sent: 26 May 2005 11:20 Subject: Re: White Rabbit vs. Tegmark On Thu, 26 May 2005, Alastair Malcolm wrote: An example occurs which might be of help. Let us say that the physics of the universe is such that in the Milky Way galaxy, carbon-based SAS's outnumber silicon-based SAS's by a trillion to one. Wouldn't we say that the inhabitants of that galaxy are more likely to find themselves as carbon-based? Now extrapolate this to any large, finite number of galaxies. The same likelihood will pertain. Now surely all the statistics don't just go out of the window if the universe happens to be infinite rather than large and finite? Alastair Well, it just does, for countable sets. This is what Cantor showed, and Lewis explains in his book. Cantor defines same size as a 1-to-1 pairing. Hence as there are infinite primes and infinite non-primes there are the same number (cardinality) of them: (1,3), (2,4), (3,6), (5,8), (7,9), (11,12), (13,14), (17,15), (19,16) etc and so ad infinitum You might say there are obviously more non-primes. This means that if you list the numbers in numerical sequence, you get fewer primes than non-primes in any finite sequence except a few very short ones. But in another sequence the answer is different: (1,2,4) (3,5,6) (7,11,8) (13,17,9) etc ad infinitum. In this infinite sequence, each triple has two primes and only one non-prime. Hence there seem to be more primes than non-primes! I've got no problem with this, as far as it goes. The point I was trying to make - talking in terms of prime numbers if you prefer - is that in the circumstance I am referring to we equivalently *are* in the situation of a particular pre-defined sequence - and so relative frequency becomes relevant. Alastair
Re: White Rabbit vs. Tegmark
On Thu, May 26, 2005 at 07:54:03PM +0100, Patrick Leahy wrote:
* Since the White Rabbit^** argument implicitly assumes a measure, as it
stands it can't be definitive.
* But the arbitrariness of the measure itself becomes the main argument
against the everything thesis, since the main claimed benefit of the
thesis is that it removes arbitrary choices in defining reality.
Paddy Leahy
^** This is a song about Alice, remember? --- Arlo Guthrie
This measure is not arbitrary, but defined by the observer
itself. Every such observer based measure satisfies an Occam's razor
theorem {\em in the framework of the observer}.
I discuss why the arbitrariness of the choice of UTM does not matter
in my Why Occam's Razor paper. What I didn't show in that paper (I
wrote it nearly 5 years ago!) was that any mapping of descriptions to
meaning suffices (ie any observer, computational or not) - Turing
completeness is not needed. Turing completeness only gives a guarantee
that the complexities seen by different observers differ by at most a
constant independent of the description.
Cheers
--
*PS: A number of people ask me about the attachment to my email, which
is of type application/pgp-signature. Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.
A/Prof Russell Standish Phone 8308 3119 (mobile)
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RE: White Rabbit vs. Tegmark
On Wed, 25 May 2005, Stathis Papaioannou wrote: SNIP Consider these two parallel arguments using a version of the anthropic principle: (a) In the multiverse, those worlds which have physical laws and constants very different to what we are used to may greatly predominate. However, it is no surprise that we live in the world we do. For in those other worlds, conditions are such that stars and planets could never form, and so observers who are even remotely like us would never have evolved. The mere fact that we are having this discussion therefore necessitates that we live in a world where the physical laws and constants are very close to their present values, however unlikely such a world may at first seem. This is the anthropic principle at work. (b) In the multiverse, those worlds in which it is a frequent occurence that the laws of physics are temporarily suspended so that, for example, talking white rabbits materialise out of thin air, may greatly predominate. However, it is no surprise that we live in the orderly world that we do. For in those other worlds, although observers very much like us may evolve, they will certainly not spend their time puzzling over the curious absence of white rabbit type phenomena. The mere fact that we are having this discussion therefore necessitates that we live in a world where physical laws are never violated, however unlikely such a world may at first seem. This is the *extreme* anthropic principle at work. If there is something wrong with (b), why isn't there also something wrong with (a)? --Stathis Papaioannou Good point, this is a fundamental weakness of the AP. If you take it to extremes, we should not be surprised by *anything* because the entire history of our past light-cone to date, down to specific microscopic quantum events, is required in order to account for the fact that you and I are having this particular exchange. To give the AP force, you have to work on the most general possible level (hence it was a big mistake for Barrow Tipler to restrict it to carbon-based life forms in their book, certainly not in line with Brandon Carter's original thought). Paddy Leahy
Re: White Rabbit vs. Tegmark
- Original Message - From: Patrick Leahy [EMAIL PROTECTED] To: Alastair Malcolm [EMAIL PROTECTED] Cc: EverythingList everything-list@eskimo.com Sent: 24 May 2005 22:10 Subject: Re: White Rabbit vs. Tegmark . . This is very reminiscent of Lewis' argument. Have you read his book? IIRC he claims that you can't actually put a measure (he probably said: you can't define probabilities) on a countably infinite set, precisely because of Cantor's pairing arguments. Which seems plausible to me. It seems to depend on whether one can find an intrinsic ordering (or something similar), such that relative frequency comes into play (so prime numbers *would* be less likely to be hit). As implied by my paper this would suggest a solution to the WR problem, but even if no ordering is possible or is irrelevant - the simple Cantorian situation - then there would be no WR problem anyway. (I have read hopefully the relevant passages in 'On the Plurality of Worlds' - I would think you are mainly referring to section 2.5; he doesn't actually mention either 'measure' or 'probability' here as far as I can see - more like 'outnumber', 'abundance' etc.) Lewis also distinguishes between inductive failure and rubbish universes as two different objections to his model. I notice that in your articles both you and Russell Standish more or less run these together. Lewis' approach to the inductive failure objection is slightly different, with the result that he can deploy a separate argument against it. Where he says Why should the reason everyone has to distrust induction seem more formidable when the risk of error is understood my way: as the existence of other worlds wherein our counterparts are deceived? It should not. [p117] ... he is basically saying that from a deductive-logic point of view we have some degree of mistrust of induction anyway, and this will not be affected whether we consider the possible worlds (where induction fails) to be real or imaginary. However, it is the (for me) straightforward 'induction failure' objection - that the world should in all likelihood become unpredictable from the next moment on - that I address in my paper, (which in many ways more closely links to Lewis's 'rubbish universe' objection); my mentioning of 'rubbish' is in the different context of *invisible* universes, which is in the appendix argument concerning predomination of simpler universes. Alastair
RE: White Rabbit vs. Tegmark
Stathis: I don't know if you can make a sharp distinction between the really weird universes where observers never evolve and the slightly weird ones where talking white rabbits appear now and then. Consider these two parallel arguments using a version of the anthropic principle: (a) In the multiverse, those worlds which have physical laws and constants very different to what we are used to may greatly predominate. However, it is no surprise that we live in the world we do. For in those other worlds, conditions are such that stars and planets could never form, and so observers who are even remotely like us would never have evolved. The mere fact that we are having this discussion therefore necessitates that we live in a world where the physical laws and constants are very close to their present values, however unlikely such a world may at first seem. This is the anthropic principle at work. (b) In the multiverse, those worlds in which it is a frequent occurence that the laws of physics are temporarily suspended so that, for example, talking white rabbits materialise out of thin air, may greatly predominate. However, it is no surprise that we live in the orderly world that we do. For in those other worlds, although observers very much like us may evolve, they will certainly not spend their time puzzling over the curious absence of white rabbit type phenomena. The mere fact that we are having this discussion therefore necessitates that we live in a world where physical laws are never violated, however unlikely such a world may at first seem. This is the *extreme* anthropic principle at work. If there is something wrong with (b), why isn't there also something wrong with (a)? This is the problem of determining the appropriate class of observer we should count ourselves as being a random selection on. There might indeed be something wrong with (a); replace The mere fact that we are having *this* discussion with, The mere fact that we are having *a* discussion to obtain a dramatically different observer class. Your formulation of (a) (*this* discussion) essentially restricts us to being a random selection on the class of observers with access to internet and email, discoursing on the everything list. Replacing this with a broadens the class to include any intelligent entity capable of (and having) a discussion. The problem of determining the appropriate class seems a rather intractable one. Choosing too broad a class can lead to unpleasant consequences such as the doomsday argument; too narrow a class leads to (b). Mondays, wednesdays and fridays, I believe that my appropriate reference class can be only one; Jonathan Colvin in this particular branch of the MW, since I could not have been anyone else. Weekends, tuesdays and thursdays I believe I'm a random observer on the class of observers. Jonathan Colvin
RE: White Rabbit vs. Tegmark
Paddy writes Stathis Papaioannou wrote: (b) In the multiverse, those worlds in which it is a frequent occurrence that the laws of physics are temporarily suspended so that, for example, talking white rabbits materialise out of thin air, may greatly predominate. However, it is no surprise that we live in the orderly world that we do. For in those other worlds, although observers very much like us may evolve, they will certainly not spend their time puzzling over the curious absence of white rabbit type phenomena. The mere fact that we are having this discussion therefore necessitates that we live in a world where physical laws are never violated, however unlikely such a world may at first seem. This is the *extreme* anthropic principle at work. Might it not also seem more probable (upon some of the hypotheses being entertained here) that observers *just* like us evolved, and then just today white rabbits began to appear out of nowhere? Good point, this is a fundamental weakness of the AP. If you take it to extremes, we should not be surprised by *anything* because the entire history of our past light-cone to date, down to specific microscopic quantum events, is required in order to account for the fact that you and I are having this particular exchange. But is the entire history... down to quantum events really necessary to account for this exchange? I say this because the set of all the observer moments *this* seems to require covers so many possibilities, including schizophrenia, etc. But... this being so tricky, how may I have misunderstood? Lee To give the AP force, you have to work on the most general possible level (hence it was a big mistake for Barrow Tipler to restrict it to carbon-based life forms in their book, certainly not in line with Brandon Carter's original thought). Paddy Leahy
Re: White Rabbit vs. Tegmark
On Mon, May 23, 2005 at 06:03:32PM -0700, Hal Finney wrote: Paddy Leahy writes: Oops, mea culpa. I said that wrong. What I meant was, what is the cardinality of the data needed to specify *one* continuous function of the continuum. E.g. for constant functions it is blatantly aleph-null. Similarly for any function expressible as a finite-length formula in which some terms stand for reals. I think it's somewhat nonstandard to ask for the cardinality of the data needed to specify an object. Usually we ask for the cardinality of some set of objects. The cardinality of the reals is c. But the cardinality of the data needed to specify a particular real is no more than aleph-null (and possibly quite a bit less!). In the same way, the cardinality of the set of continuous functions is c. But the cardinality of the data to specify a particular continuous function is no more than aleph null. At least for infinitely differentiable ones, you can do as Russell suggests and represent it as a Taylor series, which is a countable set of real numbers and can be expressed via a countable number of bits. I'm not sure how to extend this result to continuous but non-differentiable functions but I'm pretty sure the same thing applies. Hal Finney You've got me digging out my copy of Kreyszig Intro to Functional Analysis. It turns out that the set of continuous functions on an interval C[a,b] form a vector space. By application of Zorn's lemma (or equivalently the axiom of choice), every vector space has what is called a Hamel basis, namely a linearly independent countable set B such that every element in the vector space can be expressed as a finite linear combination of elements drawn from the Hamel basis: ie \forall x\in V, \exists n\in N, b_i\in B, a_i\in F, i=1, ... n : x = \sum_i^n a_ib_i where F is the field (eg real numbers), V the vector space (eg C[a,b]) and B the Hamel basis. Only a finite number of reals is needed to specify an arbitrary continuous function! Actually the theory of Fourier series will tell you how to generate any Lebesgue integral function almost everywhere from a countable series of cosine functions. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgpQXY4RMJ7BQ.pgp Description: PGP signature
Re: White Rabbit vs. Tegmark
Remember that Wolfram assumes a 1-1 correspondence between consciousness and physical activity, which, as you, I have refuted (or I pretend I have refuted, if you prefer). the comp hyp predicts physical laws must be as complex as the solution of the measure problem. In that sense, the apparent simplicity of the currently known physical laws is mysterious, and need to be explained (except that QM does predict too some non computational observation like the spin up of particles in superposition states up + down. Bruno Le 23-mai-05, à 23:59, Hal Finney a écrit : Besides, it's not all that clear that our own universe is as simple as it should be. CA systems like Conway's Life allow for computation and might even allow for the evolution of intelligence, but our universe's rules are apparently far more complex. Wolfram studied a variety of simple computational systems and estimated that from 1/100 to 1/10 of them were able to maintain stable structures with interesting behavior (like Life). These tentative results suggest that it shouldn't take all that much law to create life, not as much as we see in this universe. I take from this a prediction of the all-universe hypothesis to be that it will turn out either that our universe is a lot simpler than we think, or else that these very simple universes actually won't allow the creation of stable, living beings. That's not vacuous, although it's not clear how long it will be before we are in a position to refute it. I've overlooked until now the fact that mathematical physics restricts itself to (almost-everywhere) differentiable functions of the continuum. What is the cardinality of the set of such functions? I rather suspect that they are denumerable, hence exactly representable by UTM programs. Perhaps this is what Russell Standish meant. The cardinality of such functions is c, the same as the continuum. The existence of the constant functions alone shows that it is at least c, and my understanding is that continuous, let alone differentiable, functions have cardinality no more than c. I must insist though, that there exist mathematical objects in platonia which require c bits to describe (and some which require more), and hence can't be represented either by a UTM program or by the output of a UTM. Hence Tegmark's original everything is bigger than Schmidhuber's. But these structures are so arbitrary it is hard to imagine SAS in them, so maybe it makes no anthropic difference. Whether Tegmark had those structures in mind or not, we can certainly consider such an ensemble - the name is not important. I posted last Monday a summary of a paper by Frank Tipler which proposed that in fact our universe's laws do require c bits to describe themm, and a lot of other crazy ideas as well, http://www.iop.org/EJ/abstract/0034-4885/68/4/R04 . I don't think it was particularly convincing, but it did offer a way of thinking about infinitely complicated natural laws. One simple example would be the fine structure constant, which might turn out to be an uncomputable number. That wouldn't be inconsistent with our existence, but it is hard to see how our being here could depend on such a property. http://iridia.ulb.ac.be/~marchal/
Re: White Rabbit vs. Tegmark
Le 24-mai-05, à 01:10, Patrick Leahy a écrit : On Mon, 23 May 2005, Hal Finney wrote: I've overlooked until now the fact that mathematical physics restricts itself to (almost-everywhere) differentiable functions of the continuum. What is the cardinality of the set of such functions? I rather suspect that they are denumerable, hence exactly representable by UTM programs. Perhaps this is what Russell Standish meant. The cardinality of such functions is c, the same as the continuum. The existence of the constant functions alone shows that it is at least c, and my understanding is that continuous, let alone differentiable, functions have cardinality no more than c. Oops, mea culpa. I said that wrong. What I meant was, what is the cardinality of the data needed to specify *one* continuous function of the continuum. E.g. for constant functions it is blatantly aleph-null. Similarly for any function expressible as a finite-length formula in which some terms stand for reals. You reassure me a little bit ;) PS I will answer your other post asap. bruno http://iridia.ulb.ac.be/~marchal/
Re: White Rabbit vs. Tegmark
Le 24-mai-05, à 00:17, Patrick Leahy a écrit : On Mon, 23 May 2005, Bruno Marchal wrote: SNIP> Concerning the white rabbits, I don't see how Tegmark could even address the problem given that it is a measure problem with respect to the many computational histories. I don't even remember if Tegmark is aware of any measure relating the 1-person and 3-person points of view. Not sure why you say *computational* wrt Tegmark's theory. Nor do I understand exactly what you mean by a measure relating 1-person 3-person. This is not easy to sum up, and is related to my PhD thesis, which is summarized in english in the following papers: http://iridia.ulb.ac.be/~marchal/publications/CCQ.pdf http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf or in links to this list. You can find them in my webpage (URL below). Tegmark is certainly aware of the need for a measure to allow statements about the probability of finding oneself (1-person pov, OK?) in a universe with certain properties. This is listed in astro-ph/0302131 as a horrendous problem to which he tentatively offers what looks suspiciously like Schmidhuber's (or whoever's) Universal Prior as a solution. Could be promising heuristic, but is deeply wrong. I mean I am myself very suspicious that Universal Prior can be used as an explanation per se. (Of course, this means he tacitly accepts the restriction to computable functions). You cannot really be tacit about this. If only because you can gives them basic role in more than one way. Tegmark is unclear, at least.. So I don't agree that the problem can't be addressed by Tegmark, although it hasn't been. Unless by addressed you mean solved, in which case I agree! To adress the problem you need to be ontologically clear. Let's suppose with Wei Dai that a measure can be applied to Tegmark's everything. It certainly can to the set of UTM programs as per Schmidhuber and related proposals. Most such proposals are done by people not aware of the 1-3 distinction. In the approach I have developed that difference is crucial. Obviously it is possible to assign a measure which solves the White Rabbit problem, such as the UP. But to me this procedure is very suspicious. I agree. You can serach my discussion with Schmidhuber on this list. (search on the name marchal, not bruno marchal: it is an old discussion we did have some years ago). We can get whatever answer we like by picking the right measure. I mainly agree. While the UP and similar are presented by their proponents as natural, my strong suspicion is that if we lived in a universe that was obviously algorithmically very complex, we would see papers arguing for natural measures that reward algorithmic complexity. In fact the White Rabbit argument is basically an assertion that such measures *are* natural. Why one measure rather than another? By the logic of Tegmark's original thesis, we should consider the set of all possible measures over everything. But then we need a measure on the measures, and so ad infinitum. I mainly agree. One self-consistent approach is Lewis', i.e. to abandon all talk of measure, all anthropic predictions, and just to speak of possibilities rather than probabilities. This suited Lewis fine, but greatly undermines the attractiveness of the everything thesis for physicists. With comp the measure *is* on the *possibilities*, themselves captured by the maximal consistent extensions in the sense of the logicians. I have not the time to give detail, but in july or augustus, I can give you all the details in case you are interested SNIP> more or less recently in the scientific american. I'm sure Tegmark's approach, which a priori does not presuppose the comp hyp, would benefit from category theory: this one put structure on the possible sets of mathematical structures. Lawvere rediscovered the Grothendieck toposes by trying (without success) to get the category of all categories. Toposes (or Topoi) are categories formalizing first person universes of mathematical structures. There is a North-holland book on Topoi by Goldblatt which is an excellent introduction to toposes for ... logicians (mhhh ...). Hope that helps, Bruno Not really. I know category theory is a potential route into this, but I havn't seen any definitive statements and from what I've read on this list I don't expect to any time soon. I'm certainly not going to learn category theory myself! At least you don't need them for reading my work. I have suppressed all need to it because it is a difficult theory for those who have not a sufficiently algebraic mind. In the long run I believe they will be inescapable though. If only to learn knot theory, which I have reason to believe as being very fundamental for extracting geometry from the UTM introspection (as comp forces us to believe unless my thesis is wrong somewhere ...). You overlooked a couple of direct queries to you in my posting: * You still havn't explained why
RE: White Rabbit vs. Tegmark
Russell writes You've got me digging out my copy of Kreyszig Intro to Functional Analysis. It turns out that the set of continuous functions on an interval C[a,b] form a vector space. By application of Zorn's lemma (or equivalently the axiom of choice), every vector space has what is called a Hamel basis, namely a linearly independent countable set B such that every element in the vector space can be expressed as a finite linear combination of elements drawn from the Hamel basis: ie \forall x\in V, \exists n\in N, b_i\in B, a_i\in F, i=1, ... n : x = \sum_i^n a_ib_i where F is the field (eg real numbers), V the vector space (eg C[a,b]) and B the Hamel basis. Only a finite number of reals is needed to specify an arbitrary continuous function! I can't follow your math, but are you saying the following in effect? Any continuous function on R or C, as we know, can be specified by countably many reals R1, R2, R3, ... But by a certain mapping trick, I think that I can see how this could be reduced to *one* real. It depends for its functioning---as I think your result above depends--- on the fact that each real encodes infinite information. Suppose that I have a continuous function f that I wish to encode using one real. I use the trick that shows that countably many infinite sets are countable (you know the one: by running back and forth along the diagonals). Take the digits of R1, and place them in positions 1, 3, 6, 10, 15, 21, ... of the MasterReal, and R2 in positions 2, 4, 7, 11, 16, 22, ... of the MasterReal, R3's digits at 5, 8, 12, 17, 23, ... of the MasterReal, and so on, using the first free integer position of the gaps that are left after specification of the positions of the real R(N-1). So it seems that countably many reals have been packed into just one. (A slightly more involved example could be produced for the Complex field.) Lee Actually the theory of Fourier series will tell you how to generate any Lebesgue integral function almost everywhere from a countable series of cosine functions. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics 0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02
RE: White Rabbit vs. Tegmark
Lee Corbin writes: Russell writes You've got me digging out my copy of Kreyszig Intro to Functional Analysis. It turns out that the set of continuous functions on an interval C[a,b] form a vector space. By application of Zorn's lemma (or equivalently the axiom of choice), every vector space has what is called a Hamel basis, namely a linearly independent countable set B such that every element in the vector space can be expressed as a finite linear combination of elements drawn from the Hamel basis I can't follow your math, but are you saying the following in effect? Any continuous function on R or C, as we know, can be specified by countably many reals R1, R2, R3, ... But by a certain mapping trick, I think that I can see how this could be reduced to *one* real. It depends for its functioning---as I think your result above depends--- on the fact that each real encodes infinite information. I don't think that is exactly how the result Russell describes works, but certainly Lee's construction makes his result somewhat less paradoxical. Indeed, a real number can include the information from any countable set of reals. Nevertheless I'd be curious to see an example of this Hamel basis construction. Let's consider a simple Euclidean space. A two dimensional space is just the Euclidean plane, where every point corresponds to a pair of real numbers (x, y). We can generalize this to any number of dimensions, including a countably infinite number of dimensions. In that form each point can be expressed as (x0, x1, x2, x3, ...). The standard orthonormal basis for this vector space is b0=(1,0,0,0...), b1=(0,1,0,0...), b2=(0,0,1,0...), With such a basis the point I showed can be expressed as x0*b0+x1*b1+ I gather from Russell's result that we can create a different, countable basis such that an arbitrary point can be expressed as only a finite number of terms. That is pretty surprising. I have searched online for such a construction without any luck. The Wikipedia article, http://en.wikipedia.org/wiki/Hamel_basis has an example of using a Fourier basis to span functions, which requires an infinite combination of basis vectors and is therefore not a Hamel basis. They then remark, Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. That would seem to imply, contrary to what Russell writes above, that the Hamel basis is uncountably infinite in size. In that case the Hamel basis for the infinite dimensional Euclidean space can simply be the set of all points in the space, so then each point can be represented as 1 * the appropriate basis vector. That would be a disappointingly trivial result. And it would not shed light on the original question of proving that an arbitrary continuous function can be represented by a countably infinite number of bits. Hal
Re: White Rabbit vs. Tegmark
Perhaps I can throw in a few thoughts here, partly in the hope I may learn something from possible replies (or lack thereof!). - Original Message - From: Patrick Leahy [EMAIL PROTECTED] Sent: 23 May 2005 00:03 . . A very similar argument (rubbish universes) was put forward long ago against David Lewis's modal realism, and is discussed in his On the plurality of worlds. As I understand it, Lewis's defence was that there is no measure in his concept of possible worlds, so it is not meaningful to make statements about which kinds of universe are more likely (given that there is an infinity of both lawful and law-like worlds). This is not a defense which Tegmark can make, since he does require a measure (to give his thesis some anthropic content). I don't understand this last sentence - why couldn't he use the 'Lewisian defence' if he wanted - it is the Anthropic Principle (or just logic) that necessitates SAS's (in a many worlds context): our existence in a world that is suitable for us is independent of the uncountability or otherwise of the sets of suitable and unsuitable worlds, it seems to me. (Granted he does use the 'm' word in talking about level 4 (and other level) universes, but I am asking why he needs it to provide 'anthropic content'.) There are hints that it may be worth exploring fundamentally different approaches to the White Rabbit problem when we consider that for Cantor the set of all integers is the same 'size' as that of all the evens (not too good on its own for deciding whether a randomly selected integer is likely to come out odd or even); similarly for comparing the set of all reals between 0 and 1000, and between 0 and 1. The standard response to this is that one *cannot* select a real (or integer) in such circumstances - but in the case of many worlds we *do* have a selection (the one we are in now), so maybe there is more to be said than that of applying the Cantor approach to real worlds, and also on random selection. I use the simple 'limit to infinity' approach to provide a potential solution to the WR problem (see appendix of http://www.physica.freeserve.co.uk/pa01.htm) - Russell's paper is not too dissimilar in this area, I think. This approach seems to cover at least the 'countable' region (in Cantorian terms), and also addresses the above problems (ie odd/even type questions etc). The key point in my philosophy paper is that it is mathematics (and/or information theory) that is more likely to map the objective distribution of types of worlds, compared to the particular anthropic intuition that is implied by the WR challenge. A final musing on finite formal systems: I have always considered formal systems to be a provisional 'best guess' (or *maybe* 2nd best after the informational approach) for exploring the plenitude - but it occurs to me that non-finitary formal systems (which could inter alia encompass the reals) may match (say SAS-relevant) finite formal systems in simplicity terms, if the (infinite-length) axioms themselves could be algorithmically generated. This would lead to a kind of 'meta-formal-system' approach. Just a passing thought... Alastair
Re: White Rabbit vs. Tegmark
On Tue, 24 May 2005, Alastair Malcolm wrote: Perhaps I can throw in a few thoughts here, partly in the hope I may learn something from possible replies (or lack thereof!). - Original Message - From: Patrick Leahy [EMAIL PROTECTED] Sent: 23 May 2005 00:03 . SNIP This is not a defense which Tegmark can make, since he does require a measure (to give his thesis some anthropic content). I don't understand this last sentence - why couldn't he use the 'Lewisian defence' if he wanted - it is the Anthropic Principle (or just logic) that necessitates SAS's (in a many worlds context): our existence in a world that is suitable for us is independent of the uncountability or otherwise of the sets of suitable and unsuitable worlds, it seems to me. (Granted he does use the 'm' word in talking about level 4 (and other level) universes, but I am asking why he needs it to provide 'anthropic content'.) You have to ask what motivates a physicist like Tegmark to propose this concept. OK, there are deep metaphysical reasons which favour it, but the they arn't going to get your paper published in a physics journal. The main motive is the Anthropic Principle explanation for alleged fine tuning of the fundamental parameters. As Brandon Carter remarks in the original AP paper, this implies the existence of an ensemble. Meaning that fine tuning only ceases to be a surprise if there are lots of universes, at least some of which are congenial/cognizable. But this bare statement is not enough to do physics with. But suppose you can estimate the fraction of cognizable worlds with, say the cosmological constant Lambda less than its current value. If Lambda is an arbitrary real variable, there are continuously many such worlds, so you need a measure to do this. This allows a real test of the hypothesis: if Lambda is very much lower than it has to be anthropically, there is probably some non-anthropic reason for its low value. (Actually Lambda does seem to be unnecessarily low, but only by one or two orders of magnitude). The point is, without a measure there is no way to make such predictions and the AP loses its precarious claim to be scientific. There are hints that it may be worth exploring fundamentally different approaches to the White Rabbit problem when we consider that for Cantor the set of all integers is the same 'size' as that of all the evens (not too good on its own for deciding whether a randomly selected integer is likely to come out odd or even); similarly for comparing the set of all reals between 0 and 1000, and between 0 and 1. The standard response to this is that one *cannot* select a real (or integer) in such circumstances - but in the case of many worlds we *do* have a selection (the one we are in now), so maybe there is more to be said than that of applying the Cantor approach to real worlds, and also on random selection. This is very reminiscent of Lewis' argument. Have you read his book? IIRC he claims that you can't actually put a measure (he probably said: you can't define probabilities) on a countably infinite set, precisely because of Cantor's pairing arguments. Which seems plausible to me. Lewis also distinguishes between inductive failure and rubbish universes as two different objections to his model. I notice that in your articles both you and Russell Standish more or less run these together. SNIP A final musing on finite formal systems: I have always considered formal systems to be a provisional 'best guess' (or *maybe* 2nd best after the informational approach) for exploring the plenitude - but it occurs to me that non-finitary formal systems (which could inter alia encompass the reals) may match (say SAS-relevant) finite formal systems in simplicity terms, if the (infinite-length) axioms themselves could be algorithmically generated. This would lead to a kind of 'meta-formal-system' approach. Just a passing thought... I think this is the kind of trouble you get into with the mathematical structure = formal system approach. If you just take the structure as mathematical objects, you are in much better shape. For instance, although there are aleph-null theorems in integer arithmetic, and a higher order of unprovable statements, you can just generate the integers with a program a few bits long. And the integers are the complete set of objects in the field of integer arithmetic. Similarly for the real numbers: if you just want to generate them all, draw a line (or postulate the complete set of infinite-length bitstrings). No need to worry about whether individual ones are computable or not. Paddy Leahy
Re: White Rabbit vs. Tegmark
Le 23-mai-05, à 06:09, Russell Standish a écrit : On Mon, May 23, 2005 at 04:00:39AM +0100, Patrick Leahy wrote: Hmm, my lack of a pure maths background may be getting me into trouble here. What about real numbers? Do you need an infinite axiomatic system to handle them? Because it seems to me that your ensemble of digital strings is too small (wrong cardinality?) to handle the set of functions of real variables over the continuum. Certainly this is explicit in Schmidhuber's 1998 paper. Not that I would insist that our universe really does involve real numbers, but I'm pretty sure that Tegmark would not be happy to exclude them from his all of mathematics. A finite set of axioms describing the reals does not completely specify the real numbers, unless they are inconsistent. I'm sure you've heard it before. I guess you mean natural numbers. By a theorem by Tarski there is a sense to say that the real numbers are far much simpler than the natural numbers (for example it has taken 300 years to prove Fermat formula when the variables refer to natural numbers, but the same formula is an easy exercice when the variable refers to real number). The system the axioms do describe can be modelled by a countable system as well. Sure. You could say that it describes describable functions over describable numbers. Then you get only the constructive real numbers. This is equivalent to the total (defined everywhere) computable function from N to N. This is a non recursively enumerable set. People can read the diagonalization posts to the everything-list in my url, to understand better what happens here. http://www.escribe.com/science/theory/m3079.html http://www.escribe.com/science/theory/m3344.html It may even be the case that you only have computable functions over computable numbers, and that describable, uncomputable things cannot be captured by finite axiomatic systems, but I'm not sure. Juergen Schmidhuber knows more about such things. It is a bit too ambiguous. What I would argue is what use are undescribable things in the plenitude? I don't think we can escape them; provably so once the comp hyp is assumed. Hence my interpretation of Tegmark's assertion is of finite axiomatic systems, not all mathematic things. I don't think Tegmark would agree. I agree with you that the whole math is much too big (inconsistent). The bigger problem with Tegmark is that he associates first person with their third person description in a 1-1 way (like most aristotelian). But then he should postulate non-comp, and explain the nature of that association with a suitable theory of mind (which he does not really discuss). It is mainly from a logician point of view that Tegmark can hardly be convincing. As I said often, physical reality cannot be a mathematical reality *among other*. The relation is more subtle both with or without the comp hyp. I have discussed it at length a long time ago in this list. Category theory and logic provides tools for defining big structure, but not the whole. The David Lewis problem mentionned recently is not even expressible in Tegmark framework. Schmidhuber takes the right ontology, but then messed up completely the mind-body problem by complete abstraction from the 1/3 distinction. Tegmark do a sort of 1/3 distinction (the frog/bird views) but does not take it sufficiently seriously. Bruno http://iridia.ulb.ac.be/~marchal/
Re: White Rabbit vs. Tegmark
On Sun, 22 May 2005, Hal Finney wrote: Regarding the nature of Tegmark's mathematical objects, I found some old discussion on the list, a debate between me and Russell Standish, in which Russell argued that Tegmark's objects should be understood as formal systems, while I claimed that they should be seen more as pure Platonic objects which can only be approximated via axiomatization. SNIP I see that Russell is right, and that Tegmark does identify mathematical structures with formal systems. His chart at the first link above shows Formal Systems as the foundation for all mathematical structures. And the discussion in his paper is entirely in terms of formal systems and their properties. He does not seem to consider the implications if any of Godel's theorem. Actually he does both. Most of the time he implies that universes are formal systems, e.g. formal systems can be self-consistent but objects just exist: only statements about objects (e.g. theorems) have to be consistent. But he also says that *our* universe corresponds to the solution of a set of equations, i.e. a mathematical object in Platonia. This is definitely muddled thinking. Specifying a universe by a set of statements about it seems to be a highly redundant way to go about things, whether you go for all theorems or the much larger class of true statements. (Of course, many statements, including nearly all the ones interesting to mathematicians, apply to whole classes of objects). Hence despite all the guff about formal systems, I think any attempt to make sense of Tegmark's thesis has to go with the object = universe. Paddy Leahy
Re: White Rabbit vs. Tegmark
Hi Patrick, Sorry for having been short, especially on those notions for which some background of logic is needed. Unfortunately I have not really the time to explain with all the nuances needed. Nevertheless the fact that reals are simpler to axiomatize than natural numbers should be a natural idea in the everything list, given that the everything basic idea is that taking all objects is more simple than taking some subpart of it. Now, concerning the [natural numbers versus real numbers] this has been somehow formally capture by a beautiful theorem by Tarski, which I guess should be on the net, let me see, googlle: tarski reals, ok it gives Wolfram dictionnary: so look here for the theorem I was alluding to: http://mathworld.wolfram.com/TarskisTheorem.html It is not so astonishing. reals have been invented for making math more easier. Concerning the white rabbits, I don't see how Tegmark could even address the problem given that it is a measure problem with respect to the many computational histories. I don't even remember if Tegmark is aware of any measure relating the 1-person and 3-person points of view. Of course I like very much Tegmark's idea that physicalness is a special case of mathematicalness, but on the later he is a little naive, like physicist often are when they talk about math. Even Einstein, and that's normal. More normal and frequent, but more annoying also, is that he seems unaware of the mind-body problem. John Archibald Wheeler law without law is quite good too. My favorite paper by Tegmark is the one he wrote with Wheeler on Everett more or less recently in the scientific american. I'm sure Tegmark's approach, which a priori does not presuppose the comp hyp, would benefit from category theory: this one put structure on the possible sets of mathematical structures. Lawvere rediscovered the Grothendieck toposes by trying (without success) to get the category of all categories. Toposes (or Topoi) are categories formalizing first person universes of mathematical structures. There is a North-holland book on Topoi by Goldblatt which is an excellent introduction to toposes for ... logicians (mhhh ...). Hope that helps, Bruno Le 23-mai-05, à 12:51, Patrick Leahy a écrit : Now I'm really confused! I took Russell to mean that real numbers are excluded from his system because they require an infinite number of axioms. In which case his system is really quite different from Tegmark's. But if Bruno is correct and reals only need a finite number of axioms, then surely Russell is wrong to imply that real-number universes are covered by his system. Sure, they can be modelled to any finite degree of precision, but that is not the same thing as actually being included (which requires infinite precision). For instance, Duhem pointed out that you can devise a Newtonian dynamical system where a particle will go to infinity if its starting point is an irrational number, but execute closed orbits if its starting point is rational. On Mon, 23 May 2005, Bruno Marchal wrote (among other things): Le 23-mai-05, à 06:09, Russell Standish a écrit : Hence my interpretation of Tegmark's assertion is of finite axiomatic systems, not all mathematic things. I don't think Tegmark would agree. I agree with you that the whole math is much too big (inconsistent). Since Tegmark defines mathematical structures as existing if self-consistent (following Hilbert), how can his concept be inconsistent? But there may be an inconsistency in (i) asserting the identity of isomorphic systems and (ii) claiming that a measure exists, especially if you try both at once. It is mainly from a logician point of view that Tegmark can hardly be convincing. As I said often, physical reality cannot be a mathematical reality *among other*. The relation is more subtle both with or without the comp hyp. I have discussed it at length a long time ago in this list. Category theory and logic provides tools for defining big structure, but not the whole. As I understand it, this is because the whole is unquantifiably big, i.e. outside even the heirarchy of cardinals. Correct? The David Lewis problem mentionned recently is not even expressible in Tegmark framework. It might be illuminating if you could explain why not. On the face of it, it fits in perfectly well, viz: for any given lawful universe, there are infinitely many others in which as well as the observable phenomena there exist non-observable epiphenomenal rubbish. The only difference from the White Rabbit problem is the specification that the rubbish be strictly non-observable. As a physicist, my reaction is that it is then irrelevant so who cares? But this can be fixed by making the rubbish perceptible but mostly harmless, i.e. White Rabbits. Paddy Leahy http://iridia.ulb.ac.be/~marchal/
Re: White Rabbit vs. Tegmark
On Mon, 23 May 2005, Bruno Marchal wrote: SNIP Concerning the white rabbits, I don't see how Tegmark could even address the problem given that it is a measure problem with respect to the many computational histories. I don't even remember if Tegmark is aware of any measure relating the 1-person and 3-person points of view. Not sure why you say *computational* wrt Tegmark's theory. Nor do I understand exactly what you mean by a measure relating 1-person 3-person. Tegmark is certainly aware of the need for a measure to allow statements about the probability of finding oneself (1-person pov, OK?) in a universe with certain properties. This is listed in astro-ph/0302131 as a horrendous problem to which he tentatively offers what looks suspiciously like Schmidhuber's (or whoever's) Universal Prior as a solution. (Of course, this means he tacitly accepts the restriction to computable functions). So I don't agree that the problem can't be addressed by Tegmark, although it hasn't been. Unless by addressed you mean solved, in which case I agree! Let's suppose with Wei Dai that a measure can be applied to Tegmark's everything. It certainly can to the set of UTM programs as per Schmidhuber and related proposals. Obviously it is possible to assign a measure which solves the White Rabbit problem, such as the UP. But to me this procedure is very suspicious. We can get whatever answer we like by picking the right measure. While the UP and similar are presented by their proponents as natural, my strong suspicion is that if we lived in a universe that was obviously algorithmically very complex, we would see papers arguing for natural measures that reward algorithmic complexity. In fact the White Rabbit argument is basically an assertion that such measures *are* natural. Why one measure rather than another? By the logic of Tegmark's original thesis, we should consider the set of all possible measures over everything. But then we need a measure on the measures, and so ad infinitum. One self-consistent approach is Lewis', i.e. to abandon all talk of measure, all anthropic predictions, and just to speak of possibilities rather than probabilities. This suited Lewis fine, but greatly undermines the attractiveness of the everything thesis for physicists. SNIP more or less recently in the scientific american. I'm sure Tegmark's approach, which a priori does not presuppose the comp hyp, would benefit from category theory: this one put structure on the possible sets of mathematical structures. Lawvere rediscovered the Grothendieck toposes by trying (without success) to get the category of all categories. Toposes (or Topoi) are categories formalizing first person universes of mathematical structures. There is a North-holland book on Topoi by Goldblatt which is an excellent introduction to toposes for ... logicians (mhhh ...). Hope that helps, Bruno Not really. I know category theory is a potential route into this, but I havn't seen any definitive statements and from what I've read on this list I don't expect to any time soon. I'm certainly not going to learn category theory myself! You overlooked a couple of direct queries to you in my posting: * You still havn't explained why you say his system is too big (inconsistent). Especially the inconsistent bit. I'm sure a level of explanation is possible which doesn't take the whole of category theory for granted. Also, if you have a proof that his system is inconsistent, you should publish it. * Is it correct to say that category theory cannot define the whole because it is outside the heirarchy of the cardinals? And another mathematical query for you or anyone on the list: I've overlooked until now the fact that mathematical physics restricts itself to (almost-everywhere) differentiable functions of the continuum. What is the cardinality of the set of such functions? I rather suspect that they are denumerable, hence exactly representable by UTM programs. Perhaps this is what Russell Standish meant. I must insist though, that there exist mathematical objects in platonia which require c bits to describe (and some which require more), and hence can't be represented either by a UTM program or by the output of a UTM. Hence Tegmark's original everything is bigger than Schmidhuber's. But these structures are so arbitrary it is hard to imagine SAS in them, so maybe it makes no anthropic difference. Paddy Leahy
Re: White Rabbit vs. Tegmark
Paddy Leahy writes: Let's suppose with Wei Dai that a measure can be applied to Tegmark's everything. It certainly can to the set of UTM programs as per Schmidhuber and related proposals. Obviously it is possible to assign a measure which solves the White Rabbit problem, such as the UP. But to me this procedure is very suspicious. We can get whatever answer we like by picking the right measure. While the UP and similar are presented by their proponents as natural, my strong suspicion is that if we lived in a universe that was obviously algorithmically very complex, we would see papers arguing for natural measures that reward algorithmic complexity. In fact the White Rabbit argument is basically an assertion that such measures *are* natural. Why one measure rather than another? By the logic of Tegmark's original thesis, we should consider the set of all possible measures over everything. But then we need a measure on the measures, and so ad infinitum. I agree that this is a potential problem and an area where more work is needed. We do know that the universal distribution has certain nice properties that make it stand out, that algorithmic complexity is asymptotically unique up to a constant, and similar results which suggest that we are not totally off base in granting these measures the power to determine which physical realities we are likely to experience. But certainly the argument is far from iron-clad and it's not clear how well the whole thing is grounded. I hope that in the future we will have a better understanding of these issues. I don't agree however that we are attracted to simplicity-favoring measures merely by virtue of our particular circumstances. The universal distribution was invented decades ago as a mathematical object of study, and Chaitin's work on algorithmic complexity is likewise an example of pure math. These results can be used (loosely) to explain and justify the success of Occam's Razor, and with more difficulty to explain why the universe is as we see it, but that's not where they came from. Besides, it's not all that clear that our own universe is as simple as it should be. CA systems like Conway's Life allow for computation and might even allow for the evolution of intelligence, but our universe's rules are apparently far more complex. Wolfram studied a variety of simple computational systems and estimated that from 1/100 to 1/10 of them were able to maintain stable structures with interesting behavior (like Life). These tentative results suggest that it shouldn't take all that much law to create life, not as much as we see in this universe. I take from this a prediction of the all-universe hypothesis to be that it will turn out either that our universe is a lot simpler than we think, or else that these very simple universes actually won't allow the creation of stable, living beings. That's not vacuous, although it's not clear how long it will be before we are in a position to refute it. I've overlooked until now the fact that mathematical physics restricts itself to (almost-everywhere) differentiable functions of the continuum. What is the cardinality of the set of such functions? I rather suspect that they are denumerable, hence exactly representable by UTM programs. Perhaps this is what Russell Standish meant. The cardinality of such functions is c, the same as the continuum. The existence of the constant functions alone shows that it is at least c, and my understanding is that continuous, let alone differentiable, functions have cardinality no more than c. I must insist though, that there exist mathematical objects in platonia which require c bits to describe (and some which require more), and hence can't be represented either by a UTM program or by the output of a UTM. Hence Tegmark's original everything is bigger than Schmidhuber's. But these structures are so arbitrary it is hard to imagine SAS in them, so maybe it makes no anthropic difference. Whether Tegmark had those structures in mind or not, we can certainly consider such an ensemble - the name is not important. I posted last Monday a summary of a paper by Frank Tipler which proposed that in fact our universe's laws do require c bits to describe themm, and a lot of other crazy ideas as well, http://www.iop.org/EJ/abstract/0034-4885/68/4/R04 . I don't think it was particularly convincing, but it did offer a way of thinking about infinitely complicated natural laws. One simple example would be the fine structure constant, which might turn out to be an uncomputable number. That wouldn't be inconsistent with our existence, but it is hard to see how our being here could depend on such a property. Hal Finney
Re: White Rabbit vs. Tegmark
On Mon, May 23, 2005 at 11:17:04PM +0100, Patrick Leahy wrote: And another mathematical query for you or anyone on the list: I've overlooked until now the fact that mathematical physics restricts itself to (almost-everywhere) differentiable functions of the continuum. What is the cardinality of the set of such functions? I rather suspect that they are denumerable, hence exactly representable by UTM programs. Perhaps this is what Russell Standish meant. I noticed a couple of people have already responded to this with the obvious, but you can consider the set of analytic functions (which tend to be popular with physicists), which are representable as a Taylor series, so can be represented by a countable set of real coefficients. If you further restrict the coefficients to be computable (namely the limit of some well defined series) then the cardinality of your set is countable. Not sure if this observation translates to the bigger set of almost everywhere differentiable functions. Also, can one create a consistent theory of differential calculus based on a field of computable numbers (it must be a field, because it is bigger than the set of rationals). The reason I persist in this notion is that the set of all descriptions has the interesting property of having zero information. This set does have cardinality c, however. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgpg90KdANHqY.pgp Description: PGP signature
Re: White Rabbit vs. Tegmark
Paddy Leahy writes: Oops, mea culpa. I said that wrong. What I meant was, what is the cardinality of the data needed to specify *one* continuous function of the continuum. E.g. for constant functions it is blatantly aleph-null. Similarly for any function expressible as a finite-length formula in which some terms stand for reals. I think it's somewhat nonstandard to ask for the cardinality of the data needed to specify an object. Usually we ask for the cardinality of some set of objects. The cardinality of the reals is c. But the cardinality of the data needed to specify a particular real is no more than aleph-null (and possibly quite a bit less!). In the same way, the cardinality of the set of continuous functions is c. But the cardinality of the data to specify a particular continuous function is no more than aleph null. At least for infinitely differentiable ones, you can do as Russell suggests and represent it as a Taylor series, which is a countable set of real numbers and can be expressed via a countable number of bits. I'm not sure how to extend this result to continuous but non-differentiable functions but I'm pretty sure the same thing applies. Hal Finney
White Rabbit vs. Tegmark
I looked into this mailing list because I thought I'd come up with a fairly cogent objection to Max Tegmark's version of the everything thesis, i.e. that there is no distinction between physical and mathematical reality... our multiverse is one particular solution to a set of differential equations, not privileged in any way over other solutions to the same equations, solutions to other equations, and indeed any other mathemetical construct whatsoever (e.g. outputs of UTMs). Sure enough, you came up with my objection years ago, in the form of the White Rabbit paradox. Since usage is a bit vague, I'll briefly re-state it here. The problem is that worlds which are law-like, that is which behave roughly as if there are physical laws but not exactly, seem to vastly outnumber worlds which are strictly lawful. Hence we would expect to see numerous departures from laws of nature of a non-life-threating kind. This is a different objection to the prediction of a complete failure of induction... it's true that stochastic universes with no laws at all (or where laws abruptly cease to function) should be vastly more common still, but they are not observed due to anthropic selection. A very similar argument (rubbish universes) was put forward long ago against David Lewis's modal realism, and is discussed in his On the plurality of worlds. As I understand it, Lewis's defence was that there is no measure in his concept of possible worlds, so it is not meaningful to make statements about which kinds of universe are more likely (given that there is an infinity of both lawful and law-like worlds). This is not a defense which Tegmark can make, since he does require a measure (to give his thesis some anthropic content). It seems to me that discussion on this list back in 1999 more or less concluded that this was a fatal objection to Tegmark's version of the thesis, although not to some alternatives based exclusively on UTM programs (e.g. Russell Standish's Occam's Razor paper). Is this a fair summary, or is anyone here prepared to defend Tegmark's thesis? Paddy Leahy == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618
Re: White Rabbit vs. Tegmark
Without getting into a long hurrang, I think that Tegmark is correct, at least in part. Briefly, there has to be a reason why these alternate worlds exist. I'm referring to the Everett-Wheeler hypothesis and not just wishful thinking. Granted, if I remember correctly, Tegmark does deal with the whole issue in terms of mathematical models, which I don't really care for, however, I can tolerate it. As for the issue of measurement, all possible worlds can be viewed as still existing, without measurement, in a superpositional state, until they are observed (or measured). So you still get a plurality of worlds. My question is how does anyone know whether law-like worlds vastly outnumber lawful ones? If I have to, I can go look up Tegmark's Scientific American article on it all, to get a refresher on it. I'm just hoping I don't have to. I met Tegmark about a year ago or so. Nice guy. I stumbled into his office, since the door was open, and introduced myself, mentioning that I was also an Everett man. Everett-man? he replied puzzled. Yeah, you know - Hugh Everett, Everett/Wheeler hypothesis? at which point he excitedly jumped out of his seat and shook my hand, laughing. - Original Message - From: Patrick Leahy To: EverythingList Subject: White Rabbit vs. Tegmark Date: Mon, 23 May 2005 00:03:55 +0100 (BST) I looked into this mailing list because I thought I'd come up with a fairly cogent objection to Max Tegmark's version of the everything thesis, i.e. that there is no distinction between physical and mathematical reality... our multiverse is one particular solution to a set of differential equations, not privileged in any way over other solutions to the same equations, solutions to other equations, and indeed any other mathemetical construct whatsoever (e.g. outputs of UTMs). Sure enough, you came up with my objection years ago, in the form of the White Rabbit paradox. Since usage is a bit vague, I'll briefly re-state it here. The problem is that worlds which are law-like, that is which behave roughly as if there are physical laws but not exactly, seem to vastly outnumber worlds which are strictly lawful. Hence we would expect to see numerous departures from laws of nature of a non-life-threating kind. This is a different objection to the prediction of a complete failure of induction... it's true that stochastic universes with no laws at all (or where laws abruptly cease to function) should be vastly more common still, but they are not observed due to anthropic selection. A very similar argument (rubbish universes) was put forward long ago against David Lewis's modal realism, and is discussed in his On the plurality of worlds. As I understand it, Lewis's defence was that there is no measure in his concept of possible worlds, so it is not meaningful to make statements about which kinds of universe are more likely (given that there is an infinity of both lawful and law-like worlds). This is not a defense which Tegmark can make, since he does require a measure (to give his thesis some anthropic content). It seems to me that discussion on this list back in 1999 more or less concluded that this was a fatal objection to Tegmark's version of the thesis, although not to some alternatives based exclusively on UTM programs (e.g. Russell Standish's Occam's Razor paper). Is this a fair summary, or is anyone here prepared to defend Tegmark's thesis? Paddy Leahy == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618 -- ___ Sign-up for Ads Free at Mail.com http://promo.mail.com/adsfreejump.htm
Re: White Rabbit vs. Tegmark
On Mon, May 23, 2005 at 12:03:55AM +0100, Patrick Leahy wrote: ... A very similar argument (rubbish universes) was put forward long ago against David Lewis's modal realism, and is discussed in his On the plurality of worlds. As I understand it, Lewis's defence was that there is no measure in his concept of possible worlds, so it is not meaningful to make statements about which kinds of universe are more likely (given that there is an infinity of both lawful and law-like worlds). This is not a defense which Tegmark can make, since he does require a measure (to give his thesis some anthropic content). It seems to me that discussion on this list back in 1999 more or less concluded that this was a fatal objection to Tegmark's version of the thesis, although not to some alternatives based exclusively on UTM programs (e.g. Russell Standish's Occam's Razor paper). Is this a fair summary, or is anyone here prepared to defend Tegmark's thesis? Paddy Leahy I think most of us concluded that Tegmark's thesis is somewhat ambiguous. One interpretation of it that both myself and Bruno tend to make is that it is the set of finite axiomatic systems (finite sets of axioms, and recusively enumerated theorems). Thus, for example, the system where the continuum hypothesis is true is a distinct mathematical system from one where it is false. Such a collection can be shown to be a subset of the set of descriptions (what I call the Schmidhuber ensemble in my paper), and has some fairly natural measures associated with it. As such, the arguments I make in Why Occam's razor paper apply just as much to Tegmark's ensemble as Schmidhuber's. Conversely, if you wish to stand on the phrase all of mathematics exists then you will have trouble defining exactly what that means, let alone defining a measure. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgpwakb1EX2cL.pgp Description: PGP signature
Re: White Rabbit vs. Tegmark
I would agree with Russell, here. That's what I meant when I said that I didn't like Tegmark's mathematical model but I could tolerate it. In the end, it gives me what I need in that it supports parallel universes and doesn't threaten E/W, etc. At the same time, I don't have a dog in every fight, so until I see something about his theory that is simply untenable, I'll let it slide. - Original Message - From: Russell Standish To: Patrick Leahy Subject: Re: White Rabbit vs. Tegmark Date: Mon, 23 May 2005 09:47:22 +1000 On Mon, May 23, 2005 at 12:03:55AM +0100, Patrick Leahy wrote: ... A very similar argument (rubbish universes) was put forward long ago against David Lewis's modal realism, and is discussed in his On the plurality of worlds. As I understand it, Lewis's defence was that there is no measure in his concept of possible worlds, so it is not meaningful to make statements about which kinds of universe are more likely (given that there is an infinity of both lawful and law-like worlds). This is not a defense which Tegmark can make, since he does require a measure (to give his thesis some anthropic content). It seems to me that discussion on this list back in 1999 more or less concluded that this was a fatal objection to Tegmark's version of the thesis, although not to some alternatives based exclusively on UTM programs (e.g. Russell Standish's Occam's Razor paper). Is this a fair summary, or is anyone here prepared to defend Tegmark's thesis? Paddy Leahy I think most of us concluded that Tegmark's thesis is somewhat ambiguous. One interpretation of it that both myself and Bruno tend to make is that it is the set of finite axiomatic systems (finite sets of axioms, and recusively enumerated theorems). Thus, for example, the system where the continuum hypothesis is true is a distinct mathematical system from one where it is false. Such a collection can be shown to be a subset of the set of descriptions (what I call the Schmidhuber ensemble in my paper), and has some fairly natural measures associated with it. As such, the arguments I make in Why Occam's razor paper apply just as much to Tegmark's ensemble as Schmidhuber's. Conversely, if you wish to stand on the phrase all of mathematics exists then you will have trouble defining exactly what that means, let alone defining a measure. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics 0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 2.dat -- ___ Sign-up for Ads Free at Mail.com http://promo.mail.com/adsfreejump.htm
Re: White Rabbit vs. Tegmark
On Mon, 23 May 2005, Russell Standish wrote: I think most of us concluded that Tegmark's thesis is somewhat ambiguous. One interpretation of it that both myself and Bruno tend to make is that it is the set of finite axiomatic systems (finite sets of axioms, and recusively enumerated theorems). Thus, for example, the system where the continuum hypothesis is true is a distinct mathematical system from one where it is false. Such a collection can be shown to be a subset of the set of descriptions (what I call the Schmidhuber ensemble in my paper), and has some fairly natural measures associated with it. As such, the arguments I make in Why Occam's razor paper apply just as much to Tegmark's ensemble as Schmidhuber's. Hmm, my lack of a pure maths background may be getting me into trouble here. What about real numbers? Do you need an infinite axiomatic system to handle them? Because it seems to me that your ensemble of digital strings is too small (wrong cardinality?) to handle the set of functions of real variables over the continuum. Certainly this is explicit in Schmidhuber's 1998 paper. Not that I would insist that our universe really does involve real numbers, but I'm pretty sure that Tegmark would not be happy to exclude them from his all of mathematics. Conversely, if you wish to stand on the phrase all of mathematics exists then you will have trouble defining exactly what that means, let alone defining a measure. I don't wish to, but this concept has been repeated by Tegmark in several well publicised articles (e.g. the Scientific American one). Again, lack of mathematical background forbids me from making definitive claims, but I suspect that it could be proved impossible even to define a measure over *all* self-consistent mathematical concepts. In which case Lewis was right and Tegmark's level 4 multiverse is essentially content-free, from the point of view of a physicist (as opposed to a logician). Paddy Leahy
Re: White Rabbit vs. Tegmark
Patrick Leahy writes: Sure enough, you came up with my objection years ago, in the form of the White Rabbit paradox. Since usage is a bit vague, I'll briefly re-state it here. The problem is that worlds which are law-like, that is which behave roughly as if there are physical laws but not exactly, seem to vastly outnumber worlds which are strictly lawful. Hence we would expect to see numerous departures from laws of nature of a non-life-threating kind. I think the question is whether we can assume the existence of a measure over the mathematical objects which compose Tegmark's ensemble. If so then we can use the same argument as we do for Schmidhuber, namely that the simpler objects would have greater measure. Hence we would predict that our laws of nature would be among the simplest possible in order to allow for life to exist. If we assume that a mathematical object (never clear what that meant) corresponds to a formal axiomatic system, then we could use a measure based on the size of the axiomatic description. I don't remember now whether Tegmark considered his mathematical objects to be the same as formal systems or not. Hal Finney
Re: White Rabbit vs. Tegmark
Regarding the nature of Tegmark's mathematical objects, I found some old discussion on the list, a debate between me and Russell Standish, in which Russell argued that Tegmark's objects should be understood as formal systems, while I claimed that they should be seen more as pure Platonic objects which can only be approximated via axiomatization. The discussions can generally be found at http://www.escribe.com/science/theory/index.html?by=Daten=25 under the title Tegmark's TOE Cantor's Absolute Infinity. In particular http://www.escribe.com/science/theory/m4034.html http://www.escribe.com/science/theory/m4045.html http://www.escribe.com/science/theory/m4048.html In this last message I write: I have gone back to Tegmark's paper, which is discussed informally at http://www.hep.upenn.edu/~max/toe.html and linked from http://arXiv.org/abs/gr-qc/9704009. I see that Russell is right, and that Tegmark does identify mathematical structures with formal systems. His chart at the first link above shows Formal Systems as the foundation for all mathematical structures. And the discussion in his paper is entirely in terms of formal systems and their properties. He does not seem to consider the implications if any of Godel's theorem. Note, Tegmark's paper has moved to http://space.mit.edu/home/tegmark/toe_frames.html . See also http://www.escribe.com/science/theory/m4038.html where Wei Dai argues that even unaxiomatizable mathematical objects, even infinite objects that are too big to be sets, can have a measure meaningfully applied to them. However I do not know enough math to fully understand his proposal. Hal Finney
Re: White Rabbit vs. Tegmark
On Mon, May 23, 2005 at 04:00:39AM +0100, Patrick Leahy wrote: Hmm, my lack of a pure maths background may be getting me into trouble here. What about real numbers? Do you need an infinite axiomatic system to handle them? Because it seems to me that your ensemble of digital strings is too small (wrong cardinality?) to handle the set of functions of real variables over the continuum. Certainly this is explicit in Schmidhuber's 1998 paper. Not that I would insist that our universe really does involve real numbers, but I'm pretty sure that Tegmark would not be happy to exclude them from his all of mathematics. A finite set of axioms describing the reals does not completely specify the real numbers, unless they are inconsistent. I'm sure you've heard it before. The system the axioms do describe can be modelled by a countable system as well. You could say that it describes describable functions over describable numbers. It may even be the case that you only have computable functions over computable numbers, and that describable, uncomputable things cannot be captured by finite axiomatic systems, but I'm not sure. Juergen Schmidhuber knows more about such things. What I would argue is what use are undescribable things in the plenitude? Hence my interpretation of Tegmark's assertion is of finite axiomatic systems, not all mathematic things. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgpwI90o4J8im.pgp Description: PGP signature
