Max, Massimo and Julia explore the arguments for such a theory, how it could be tested, and what it even means.
Max's pick: "Surely You're Joking, Mr. Feynman! Adventures of a Curious Character."
Rationally Speaking: an archived blog about science & philosophy, by Massimo Pigliucci
Monday, February 10, 2014
Rationally Speaking podcast: Max Tegmark and the Mathematical Universe Hypothesis
Max, Massimo and Julia explore the arguments for such a theory, how it could be tested, and what it even means.
Max's pick: "Surely You're Joking, Mr. Feynman! Adventures of a Curious Character."
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Have downloaded podcast and will listen tomorrow. I read his original paper yesterday and was interested that his argument for MUH bears a striking resemblance to Berkeley's argument for Idealism.
ReplyDeleteAnd as I said before, if MUH is falsified but his argument for it is sound then it more or less implies Idealism is the case, and therefore scientific acceptance of this hypothesis is implicitly a scientific acceptance of the possibility of Idealism.
Hi Robin,
Delete>if MUH is falsified but his argument for it is sound then it more or less implies Idealism is the case, and therefore scientific acceptance of this hypothesis is implicitly a scientific acceptance of the possibility of Idealism.<
I don't see that the MUH being false implies idealism.
Since it's more on-topic here, maybe you could present your clearest formulation of your argument against the MUH which predicts that we should see miracles everywhere. I still don't understand it. I get that the MUH predicts that there will be infinitely many "miraculous" universes created by mathematical demigods, but I genuinely don't understand why you think that these would outnumber the "natural" universes.
I have explained it more than once and don't get what you don't get.
DeleteUnder MUH there is an uncountable infinity of uncountable infinities of infinitely many mathematically distinct digital demigods dedicated solely to the task of producing universes just as we are observing now and which this particular exchange between you and I is taking place and to be producing arbitrarily many more demigods dedicated to the same purpose.
Do you agree with that?
If so then how many "natural" ways are there to get to a universe just as we are observing and in which this precise exchange between you and I is taking place?
We are talking about how many mathematically distinct objects there will be that will lead, with a probability of 1 this universe as we observe it and this particular exchange between you and I.
You have not shown me anywhere that my reasoning is wrong.
DeleteHi, Disagreeable Me,
DeleteAs for the Idealism thing, I think you will find that the failure of the MUH would lead very explicitly to Idealism by Tegmark's own argument. I don't have the link to his original paper but someone has it below.
His argument for MUH is that it is implied by the External Reality Hypothesis (ERH)
Thus if his reasoning is sound then the falsification of MUH would also imply the falsification of ERH - in other words Idealism.
I should stress of course that each and every one of these mathematically distinct was of getting to this particular situation we are in is the case with a probability of 1 = the same probability as those simple rules which you say can produce - not this particular situation right now - but just some kind of natural universe.
DeleteSo what you have got to look at, since the probability of each distinct entity is 1, the number of mathematically distinct methods of arriving at this precise situation in which we are in right now.
DeleteAnd we have got to count the number of mathematically distinct demigods who are dedicated to producing just this situation in which we find ourselves right now.
Hi Robin,
DeleteI think I see where I disagree. You think there are an "uncountable infinity of uncountable infinities of infinitely many" ways for the universe to have been created by a God, whereas there are not so many ways for it to be natural.
I disagree with this.
I understand that there are a hierarchy of infinities, with some infinities greater than others. I disagree that you have demonstrated that there is a greater infinity of God-created universes than natural universes.
I also think you are perhaps confusing a few different concepts about infinity. If there are infinite demigods, and for each of those there are infinite demigods, and for each of those there are infinite demigods, and so on, I'm not sure that we end up with more demigods than if we had just said we have infinite demigods. Infinity is tricky that way, as I tried to explain to you with my analogy to mutliples of a million. For every multiple of a million, there are an infinite multiple of that number, and also infinite multiples of each of those (or if you want a different set, each one of those plus one), and so on. That doesn't mean that there are more of these numbers than there are simply integers.
In fact the number of multiples of a million is the same as the number of integers, but even so, a typical integer only has a chance of one in a million of being a multiple of a million, because from our perspective multiples of a million are in some sense less arbitrary than other numbers (if those are the numbers we're interested in)..
Your demigod argument is wrong because the mathematical structure needed to define the mind of a demigod is evidently much more complex than the laws of physics in this universe, and so less arbitrary because this is the kind of universe you are interested in. Any arbitrarily chosen universe is overwhelmingly more likely to be Godless than Godly.
In answer to your question, I'm not sure that there are uncountably many demigods producing universes like this. It may be countable. Whatever the number is, I am confident, because of the relative complexity, that there are more ways for a natural universe to exist. I in fact believe there are an infinite number of universes in which we are having this exact conversation. All the air molecules of this earth could be in slightly different positions right now, for instance, and we could still be having the same conversation with no awareness that anything is different.
Finally, just to remind you that I regard all mathematical structures which are isomorphic as the same structure. So, if it is true that a demigod has created a universe with this conversation in it, but if that demigod does not make himself known with miracles, then that universe is the same as its natural analogue. The demigod has created nothing but has merely discovered the universe. The same is true if there are an infinite number of such demigods.
DM wrote: "Your demigod argument is wrong because the mathematical structure needed to define the mind of a demigod is evidently much more complex than the laws of physics in this universe, and so less arbitrary because this is the kind of universe you are interested in."
DeleteIt does not matter. Each mathemtically distinct instance of the demigod that is dedicated to producing just this universe and us having a discussion in it exists with a probability of 1.
And yes, under MUH it would be an uncountable infinity of uncountable infinties as I showed before because it could be defined on a continuous system and the number of different trigger ratios would be uncountable and within each trigger ratio the variations would be uncountable.
What you say about the number of ways this universe could have naturally occurred I already covered above. The best you could get from that is that there is no meaningful pairing to be made and therefore that the question of the probability of being in one type of a universe or the other is undecidable.
But for each natural state of this universe that could result in this conversation a digital demigod could replicate it plus arbitrarily many variations, for example teapots orbiting Ganymede etc.
And each of the stuctures I have defined are definitely not isomorphic with each other and would therefore be distinct.
And, even though it doesn't matter any more, you still have not supported your assumption about the relative complexity of universes as opposed to some sort of algorithmic intelligence.
DeleteHi Robin,,
Delete>But for each natural state of this universe that could result in this conversation a digital demigod could replicate it plus arbitrarily many variations, for example teapots orbiting Ganymede etc.<
And for each such conversation in a digital demigod universe, a natural universe could replicate it plus arbitrarily many variations, for example an electron orbiting Ganymede, etc.
You can pair each natural universe with infinite demigod universes and vice versa. That alone establishes nothing. This is just like pairing every integer with an infinite number of multiples of a million or pairing each multiple of a million with infinite integers.
It really does all come down to the relative complexity of intelligent mathematical structures versus structures more like the laws of physics. I agree I have not supported the assumption that demigods are more complex because I think it is obvious. If you want to debate that I would be happy to oblige.
It's not that controversial, just Pythagorean. In fact, Morris Kline the great historian of mathematics has written that
ReplyDelete_'Modern science adheres to the Pythagorean emphasis on number, though as we shall see, the modern doctrines are much more sophisticated form of Pythagoreanism.'_ -- Mathematics: The Loss of Certainty
Pythagoreanism may be old but that doesn't mean it isn't controversial. But the MUH is not the same as Pythagoreanism anyway. Pythagoreanism includes a lot of mystical nonsense and certainly did not imagine that there may be an infinite multitude of universes.
DeleteHi Massimo,
ReplyDeleteGreat podcast, and in particular I enjoyed your and Julia's questions.
Unfortunately, I feel Tegmark evaded your best questions. I think you caught him out on a couple of issues (perhaps not coincidentally where I disagree with him).
Julia's question about what it would look like for Tegmark's thesis to be falsified by the discovery of a non-mathematical property of nature was very astute, and he simply couldn't answer it. It is inconceivable that such a property could be found, because if we really could discern no predictive mathematical pattern to it we could still describe it in terms of its frequency or probability. Such a property might end up looking much like quantum mechanics. Alternatively, if such a property really does exist, it might have been under our noses all along and we simply haven't recognised it or that it cannot be described mathematically since it yields no empirical predictions. The property of being physically real could be such a property, as could consciousness.
Of course that doesn't mean that I doubt the MUH, I only doubt that it is falsifiable or that it makes predictions in the way that Tegmark claims. As such I do not regard it as science.
The second issue where I think Tegmark dodged the question was on Massimo's point about whether he believes in a greatest integer. I doubt that Mathematical Platonism is compatible with being a skeptic of infinity, so it would have been interesting to hear Tegmark's answer here. However, I suspect his position may be that he is a skeptic of a physical infinity, that is he recognises that infinity plays a part in some mathematical structures but is skeptical that it does in our physical reality or in other universes that contain observers.
I feel Tegmark didn't really explain very well what he means by the claim that electrons have only mathematical properties. This is true if and only if the MUH is correct. If the MUH is not correct then they quite demonstrably have the property of "physical instantiation", which distinguishes them from other mathematical objects which are only abstract. He would have done well to make this more explicit. It seems to me that he is only willing to accept as legitimate properties which are amenable to mathematical analysis, so of course no non-mathematical properties will ever be found.
I really wish Tegmark would stop sidestepping the Godel issue by proposing the likes of the Computable Universe Hypothesis and the Finite Universe Hypothesis. There is no Godel issue. Godel's incompleteness theorems do not present any reason to doubt the MUH whatsoever, though perhaps they give the superficial appearance of an issue if no attempt is made to formulate an actual argument. Godel shows that we cannot prove that any powerful mathematical system is consistent. So what? This doesn't mean that the system is not consistent. As long as it is possible for consistent mathematical structures to exist, and it surely is, then there is no problem from Godel whatsoever.
Massimo, do you plan to read Our Mathematical Universe? If so, will you be sharing your thoughts on it?
I haven't listened to the interview yet, but based on the last time this subject came up, I agree that the Godel results are completely orthogonal to this discussion. Unless Tegmark's thesis is somehow committed to the claim that ALL the truths of the universe must be codifable into a single, recursively-enumerable axiomatic system, Godel is simply irrelevant to the conversation.
DeleteThat being said, I have one small correct for DM:
>>Godel shows that we cannot prove that any powerful mathematical system is consistent.
That's not quite correct. What the theorems show is that a powerful mathematical system cannot prove *itself* to be consistent. But *we* can prove mathematical systems to be consistent, simply by using more powerful systems (e.g., we can use set theory to prove the consistency of Peano Arithmetic). Like you said, though, this should prompt an even more emphatic, "So what?"
Thanks for the correction, C.
DeleteGodel's theorems are often misinterpreted and misapplied, and I'm chagrined to have contributed to this lamentable tradition. I took a shortcut by taking on face value what Tegmark said about the inability to show that there was no theorem to show that 0=1.
However, as I understood it, Peano arithmetic has only been proven consistent by appealing to a system which presupposes the existence of infinity. Some mathematicians are skeptical as to whether this is a legitimate move (including perhaps Tegmark himself). I am not myself such a skeptic.
So in other words it wasn't a great podcast. He avoided every pertinent question and went on to ramble about how we need to avoid the apocalypse.
DeleteBeat me to it about Godel.
DeleteHowever I wonder if the binary expansion of a Chaitin Constant exists in the mathematical universe?
Also I wonder if a Metamathematical Universe Hypothesis is required :)
By the way, axiomatic systems are not ontologies. No system presupposes the existence of infinity. Infinity can be an axiom or a consequence but its existence is another department altogether.
Delete- Robin
DeleteTo answer your question about Chaitin's constant, yes you could say that it does exist in the universe of mathematics (or the "mathematical universe") as its really the summation of outputs from all programs that compute something and eventually halt. If all answers already exist then this is just another part of the edifice of mathematical existence.
After listening to about half the interview, I noticed that Tegmark thinks that you can program a computer to "spit out" all the mathematical structures. There's a sense in which this could be taken to conflict with Godel's theorem. For example, if he thinks a computer can be programmed to give a FULL and consistent specification of the structure of the natural numbers, then that directly conflicts with Godel. But the bit about the computer isn't really essential to his core thesis, so he really shouldn't have even said it. It would be perfectly consistent with his broader position to say that mathematical structures are so rich that they cannot be fully described by any algorithm.
Delete>> As I understood it, Peano arithmetic has only been proven consistent by appealing to a system which presupposes the existence of infinity. Some mathematicians are skeptical as to whether this is a legitimate move (including perhaps Tegmark himself). I am not myself such a skeptic.
Yes, that's right. But such an infinity skeptic probably wouldn't want to believe the axioms of Peano Arithmetic anyway.
Hi Disagreeable Me,
DeleteThank you for making these excellent points about Gödel's theorem! I discuss these issues at length in Chapter 12 of the book, and I'm sorry I wasn't able to do more better justice to them during the brief time allotted to the topic during the podcast.
I completely agree with your statement that "Gödel's incompleteness theorems do not present any reason to doubt the MUH whatsoever, though perhaps they give the superficial appearance of an issue if no attempt is made to formulate an actual argument."
Given any sufficiently powerful mathematical system, Gödel indeed showed that we cannot use it to prove its own consistency - but so what? This doesn't mean that it's inconsistent or that we have a problem. Indeed, our cosmos doesn't show any signs of being inconsistent, despite showing hints that it may be a mathematical structure. Moreover, if a mathematical system *could* be used to prove its own consistency, we'd remain unconvinced that it actually was consistent, since an inconsistent system can prove anything. We'd only be convinced if a *simpler* system that we have better reason to trust the consistency of could prove the consistency of a more powerful system - and Gödel precludes this. I've never heard a mathematician suggest that the mathematical structures that dominate modern physics (pseudo-Riemannian manifolds, Calabi-Yau manifolds, Hilbert spaces, etc.) are actually inconsistent or ill-defined.
Also, since you're being so rigorous, please note that I'm using "mathematical system" to refer to "formal system", as distinct from "mathematical structure": the former can describe the latter and the latter can be a set-theoretic model of the former. The MUH that our external physical reality is a mathematical structure thus precludes fundamental randomness as in your first paragraph.
I certainly don't think there's a greatest integer in Peano arithmetic - please see my infinity comment above.
Apologies if it sounded like I was trying to dodge this. I hope to reply to your interesting "physical instantiation" issue later.
On a personal note, a key reason that I spent 3 years writing this 400-page book (http://mathematicaluniverse.org) was because I wanted to finally explain my MUH ideas thoroughly and properly, to be followed by many interesting discussions about these ideas, in the spirit of those here on this site. So I must confess that I'll feel a bit disappointed if the discussions forever remain limited merely to what I've said in brief podcasts and blog posts!
Hi again Despicable Me,
DeleteThanks for raising these interesting issues about purely mathematical properties and “physical instantiation”. I think it’s important to remember the broader context of this discussion. Many people I know insist that our cosmos has non-mathematical (and indeed non-physical) properties related to souls, spirits, deities, etc. Some very respected physicists also object to describing everything in terms of particles etc. - for example, Niels Bohr famously put humans center stage with his dictum “no reality without observation”. I don’t feel that the can dismiss the MUH trivially true when so many people disagree with it.
The argument that I give for the MUH in chapter 10 of my book doesn’t involve any assumptions about “physical instantiation”, and if you buy it and the corrollaries I give, then the conclusion is that “physical instantiation” means nothing: that mathematical existence and physical existence are one and the same.
Hi Max,
DeleteGreat to hear back from you. I've been following your development of the argument for some time, and if I'm critical it's only because I very strongly agree with you and am occasionally frustrated by what I perceive as somewhat suboptimal ways of tackling questions thrown your way (particularly Godel).
Looking forward very much to reading your book. I have been anticipating a book on the MUH for some time. I agree with you that physical instantiation is meaningless, but I think that it would be good to address this point when asked to explain your position that electrons have only mathematical properties. Perhaps you feel you cannot justify the claim that it is meaningless in a short podcast or talk?
I think the most serious criticism of the MUH is the worry that the universe seems to be too simple. If the laws of physics can be written on a t-shirt, then that seems to be unlikely if all possible mathematical structures exist. Surely a typical mathematical universe would be extremely complicated, as there is no obvious upper bound for complicatedness but there is a lower bound.
I have my own ideas on how to resolve this apparent paradox but would be interested to hear yours. Perhaps it's in the book!
I don't think I've heard yet how Godel's theorem's are not a problem for Tegmark's multiverse-MUH. So far all I've seen in this thread at least is the statement that they are not a problem and orthogonal to the issue. Statements, not arguments.
DeleteHowever, I think the question of whether Godel's theorem is a problem or not can only be assessed after we know which formalism(s) define Professor Tegmark's multiverse-MUH.
We'd also need to know if Tegmark's multiverse-MUH is populated by mathematical structures that can not be proved by the formalism(s) that define the multiverse-MUH. If these structures are not permitted, then it would seem that Tegmark's statement that all mathematical structures are realized in his multiverse-MUH, is incorrect. If they are permitted, then one wonders how a mathematical structure can physically exist in a multiverse governed by a formalism that can not prove said mathematical structure.
No, I think in order to conclusively figure out what Godel's theorems imply for Tegmark's multiverse-MUH we will need to hear Professor Tegmark layout the rules for this multiverse-MUH and the axiom's it depends upon.
Hi manyoso,
DeleteAs answered elsewhere, it does not rest on any particular formalism. All formalisms are valid as long as they are consistent.
I agree that there has not been a clear argument for why Godel is not a problem. The reason for this is that there is no clear argument for why it should be.
Let me put the argument in your terms. Let's consider those universes that can be defined according to one specific formalism, which form a subset of the mathematical multiverse ensemble.
Only those universes which are consistent exist. That's essentially what "exists" means when we are discussing mathematical objects. Any structures which are not consistent (e.g. square circles, a negative number greater than zero, a pythagorean triple for an exponent greater than two) do not exist as mathematical objects and so do not correspond to universes.
>We'd also need to know if Tegmark's multiverse-MUH is populated by mathematical structures that can not be proved by the formalism(s) that define the multiverse-MUH.<
What does it mean to "prove" a structure? How would you prove the integers, or the circle? Only statements about structures can be proven, not the structures themselves.
Hi Disagreeable Me,
DeleteOK, so you propose a thought experiment involving a reduced case of what you claim is Tegmark's multiverse-MUH?
"Let's consider those universes that can be defined according to one specific formalism..."
Stop! Need more details about this specific formalism. Is it comprised of a finite set of axioms or are the axioms at least computable?
Only those universes which are consistent exist.
So each Universe in this thought experiment is a "mathematical structure" that is consistent, right? And the multiverse for these purposes is the set of all such universes, right?
"What does it mean to "prove" a structure?"
First, please define what you mean by "structure" here and then we can talk about it more. By structure are you talking about a model of the axiomatic system in question? Are you talking about the axioms themselves? Are you talking about statements about the axioms? What?
Let's be precise as possible :)
Hi manyoso,
Delete>Need more details about this specific formalism. Is it comprised of a finite set of axioms or are the axioms at least computable?<
I would think it would need a finite set of axioms, otherwise it is probably not well defined. But are we talking about the formalism used to express certain mathematical structures or are you treating the formalism itself as a structure? I would think the formalism itself can be reasonably small, most formalisms are. I've never heard of any mathematician trying to work with a formalism with an arbitrarily large number of axioms. A formalism might be something like Peano arithmetic or ZFC or Euclidean geometry.
>So each Universe in this thought experiment is a "mathematical structure" that is consistent, right? And the multiverse for these purposes is the set of all such universes, right?<
Well, kind of. Fundamentally, the multiverse is the set of consistent mathematical structures. Some of these will be very simple and/or unlike our universe (e.g. a circle), so it feels unnatural to call these universes. It might be helpful to identify as universes only those structures which are capable of supporting conscious observers, although really this is just a convenience of language than a major ontological distinction.
>please define what you mean by "structure" here and then we can talk about it more.<
Any well-defined mathematical object that can be analysed, simulated etc. A set. A circle. An equation. The integers. A system of axioms. The real numbers. Euclidean space. Conway's Game of Life. The laws of physics of our universe. An algorithm.
The ones that are most interesting are the ones that are plausibly capable of supporting consciousness. These would necessarily need something analogous to time. Cellular automata are interesting, circles are not.
Disagreeable Me,
DeleteOK, although not precise in terms of definition of 'mathematical structure' maybe we have enough to play out this thought experiment and see if Godel's theorem spells trouble for the multiverse-MUH or not.
Consider a formal system such as ZFC and add your axiom, "Any mathematical structure that is consistent must exist."
Let's assume this formal system is the one that defines this multiverse-MUH that contains us.
If our universe is a model of this formal system and we consider the statement that the universe exists we arrive at a contradiction. If the universe exists, then by your axiom above we know it must be consistent. If it is consistent, then the formalism that defines it must be consistent. However, we know by Godel's Second Incompleteness Theorem that no formal system can prove its own consistency. Therefore our universe can not exist.
How is this not a problem for multiverse-MUH?
Hi manyoso,
DeleteThat is an intriguing and clever argument. Well done!
Let's see if I can pick it apart.
Firstly (and this is nitpicking), I don't recognise your additional axiom as valid, because in my view consistency is just the same thing as existence, therefore this axiom is a tautology and so not really an axiom. To make it an axiom you would need to have a robust distinction between existence and consistency, and I don't think there is such a robust distinction.
But this doesn't much affect your argument.
My first response would be that perhaps observing the universe to exist does not constitute a proof in the sense that Godel's theorem discusses. You are not making a formal set of deductions within the scope of a formal system, but merely observing that the universe exists, shortcutting the argument.
Another way of answering the problem would be that when you observe that the universe exists, you know only that some formal system exists, not what precisely that formal system is. It may be impossible for us to know precisely what the laws of physics are, therefore our observation of the existence of the universe does not constitute a proof of the consistency of the laws of physics because we can remain uncertain about whether what we think are the laws of physics are in fact the actual laws of physics.
Another answer, and I'm not too certain about this one, is that Godel's incompleteness theorem only applies to certain mathematical structures, not to all. It may be that our universe is not such a structure. While this is at first attractive to me, it seems dubious on reflection because we can at least work with Godel-type systems in this universe, so one must think it must be possible to derive Godel-type systems from the laws of physics.
In any case, I'm not swayed, but congratulations on providing the first interesting argument for why Godel might be a problem that I have so far come across. It bears some thinking about, certainly.
I've been thinking about this a bit more and I think there are some more problems with the argument.
DeleteFirstly, just to clear this up, ZFC cannot be the formal system that describes this universe, because the universe is not an axiomatisation of mathematics. The mathematical structure that is the universe is comprised of the laws of physics - whatever the Grand Unified Theory of everything happens to be, uniting QM with GR. It is essentially the algorithm you would use in a computer to simulate the universe.
I think on reflection that my third intuition is perhaps correct after all, because axiomatisations of arithmetic can be codified in such a way that computer programs can understand them, e.g. for the purposes of automatic checking of mathematical proofs. Now, though these axiomatisations cannot be proven to be consistent, the computer runtime that is working with them can. Computers are logically very simple and it is very clear what to do at every stage of a computation - it is not possible that the rules of a simple Turing machine could contradict each other and determine two different states at any given step. Therefore even though computers can work with ZFC, it is not possible that ZFC is actually derived from the basic rules of computing.
In the same way, even though humans are material objects existing in the universe and we work with ZFC, I don't think that means ZFC is derived from the laws of physics. As such, it is very doubtful that the laws of physics have the properties needed for Godel's theorem to apply. They do not constitute a "formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability".
Disagreeable Me,
DeleteBut this doesn't much affect your argument."
Correct.
My first response would be that perhaps observing the universe to exist does not constitute a proof in the sense that Godel's theorem discusses.
Given any axiom at least as powerful as the one you have suggested regarding the equivalency of consistency and existence I believe the argument can be made formal and rigorous and the contradiction observed. The problem is that you have imposed a new axiom that essentially asserts its own consistency.
Another way of answering the problem would be that when you observe that the universe exists, you know only that some formal system exists, not what precisely that formal system is.
Ahh, but this way lies dragons. At best this is an admission that if we are just a mathematical structure, then we can never understand ourselves or even define ourselves. At worst, we can say that the mathematical multiverse itself is not a mathematical structure that we can rigorously define. Moreover, no being in any of the universes that make up the supposed multiverse can define it. The hypothesis is thus empty.
Another answer, and I'm not too certain about this one, is that Godel's incompleteness theorem only applies to certain mathematical structures, not to all.
Actually, it applies to all mathematical formalisms that are consistent and computable... ie, they either have a finite number of axioms or an infinite number of axioms that can be generated in a deterministic manner. This theorem has been proved since the early thirties and I don't think there are any living notable mathematicians who doubt its veracity.
Firstly, just to clear this up, ZFC cannot be the formal system that describes this universe, because the universe is not an axiomatisation of mathematics.
You said that we could assume ZFC. But it doesn't matter, you said we could assume *some* formal system like ZFC. Are you now backtracking and saying that our universe can not be defined by any formal system? How then can it be a mathematical structure?
Hi manyoso,
Delete>Given any axiom at least as powerful as the one you have suggested regarding the equivalency of consistency and existence<
Again, I don't see this as an axiom. Consistency is the *only* account of existence of mathematical objects that makes sense to me. Physical existence seems to me to be incoherent, for reasons outlined on my blog.
>Ahh, but this way lies dragons.<
You could say the same of the Godel theorems themselves. We can't prove mathematics consistent. That doesn't mean we give up mathematics. We can't prove we have ever finished the job of physics. That doesn't mean we abandon physics entirely. Whether or not the universe or multiverse can be accurately defined doesn't mean that there is no definition. Furthermore, I did not say that it could not be accurately defined, I only speculated that it may be impossible to be certain that our definition is correct, in the same way that it is impossible to be certain that our mathematics is consistent.
>Actually, it applies to all mathematical formalisms that are consistent and computable... ie, they either have a finite number of axioms or an infinite number of axioms that can be generated in a deterministic manner.<
Erm, no. It does not for example apply to Euclidean geometry. The system must among other things be able to define the natural numbers, which Euclidean geometry cannot. Read up on it in Wikipedia if in doubt. I speculate that the laws of physics are more like Euclidean geometry than like ZFC.
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Limitations_of_G.C3.B6del.27s_theorems
>This theorem has been proved since the early thirties and I don't think there are any living notable mathematicians who doubt its veracity.<
I'm not doubting Godel, I'm claiming that your interpretation is incorrect.
>You said that we could assume ZFC.<
Yes, but it seems you misunderstood either my earlier agreement to assume ZFC or this more recent statement. Let me clarify.
You wanted to know what formalism Tegmark was adopting. I suggested we could assume whatever formalism you like. If we assume ZFC, that does not mean that the universe *is* ZFC, only that whatever the laws of physics are, they can be expressed *in* ZFC (I have no idea if this is true, by the way). The universe itself is the mathematical object corresponding to the laws of physics. Though it can be expressed in ZFC (and presumably other formal systems also), the universe itself is not ZFC and so probably not Godelisable.
In any case, though I feel rather confident that your argument does not succeeed, it is very interesting, and I would be interested in hearing Tegmark's thoughts.
Disagreeable Me,
DeleteYou seem to be changing the game and backtracking on the premise of this thought experiment to my appearances...
"Consistency is the *only* account of existence of mathematical objects that makes sense to me."
Fine, call it an axiom or a theorem, you have chosen to equate existence with mathematical consistency. You hypothesize that you are a part of a mathematical structure that is defined by one or more formal systems. If that formal system is sufficiently strong to meet the criterion of Godel's theorems, then you have a problem. If you can prove to yourself that you exist - hopefully you can - then you are proving that the formal system that you are based upon exists. Since exists == consistent, then that formal system has a "structure" asserting its own consistency and we know that this leads to contradiction when Godel's Second Incompleteness Theorem is applied. Given these premises you must accept either that you don't exist, or that you are not part of a mathematical structure defined by a Godelian formal system.
Erm, no. It does not for example apply to Euclidean geometry.
I'm sorry, but the whole point of this exercise was that we would assume a formal system sufficiently strong where Godel's theorems apply. That was the premise as I understood it.
Are you now modifying your claim that "all mathematical structures that are consistent must exist" and your claim about not having any free parameters? Because if you are restricting your multiverse-MUH to only those formalisms that are not sufficiently strong enough to where Godel's theorems apply, then you are certainly imposing more rules than you claimed. I would also say that the multiverse-MUH just got a lot less rich.
On the other hand, this backtracking does have positive implications for falsifiability. All we'd need to do is show a physical system in this universe that can not be described by a weak formal system in the Godelian sense. Thus the non-godelian-multiverse-MUH would be ruled out.
If we assume ZFC, that does not mean that the universe *is* ZFC, only that whatever the laws of physics are, they can be expressed *in* ZFC (I have no idea if this is true, by the way).
I never said it was and my argument does not rely upon that. I don't even know what it would mean to say the universe is a formal system of axioms... I don't really know what you mean when you say that the universe is a "mathematical structure" either, but it seems to make more sense than saying the universe is a mathematical formalism.
What I'm trying to point out to you is to talk about mathematical structures as existing makes *no sense* without also talking about the set of axiom's that you are assuming. We can't properly define a mathematical structure without a set of axioms. Math without axioms is vacuous.
And so even to talk about this multiverse-MUH requires first to establish the rules for the conversation... we need to know what formalism is the arena for this game.
In any case, though I feel rather confident that your argument does not succeeed
Alright, I really don't know why, but perhaps you can at least admit that Godel's theorems *are* a problem worthy of investigation here at least assuming you aren't describing down Tegmark's multiverse-MUH to non-godelian-multiverse-MUH.
Cheers,
Adam
Hi Adam,
DeleteFirstly, sorry if you are getting a bit frustrated with me. I don't think I am backtracking, but I may be using some terms loosely or incorrectly. I am not a mathematician or a physicist but a computer scientist so I approach this from a different angle and may not be communicating what I mean effectively.
This conversation is very valuable to me because you are raising some of the most cogent objections to the MUH that I have seen, and doing so thoughtfully, politely and insightfully. I believe the MUH must necessarily be true, but I also believe it is unfalsifiable, so the only way to test my belief is by interaction with someone such as you.
Firstly, I want to clarify what we mean by formal system, as I'm not too sure. There is the idea of a mathematical structure, such as the Mandelbrot set, and the idea of a formal system of axioms, such as ZFC. I think I am part of something like the former and not something like the latter. I'm not sure which kind of thing you mean by "formal system".
The definition of the Mandelbrot set can be expressed in the context of ZFC but I would not say that the Mandelbrot set is itself defined *by* ZFC. In the same way I would not say that I am part of a mathematical structure that is defined *by* any formal axiomatisation, but that the structure I am a part of could be expressed in the context of many axiomatisations, e.g. ZFC or ZC or what have you. If any one of those axiomatisations turned out to be inconsistent, it would not have any bearing on the consistency of the mathematical structure of which I am a part, in the same way that a proof of the inconsistency of ZFC would not prevent the Mandelbrot set from existing as long as there were some other axiomatisation that could express the same idea. Conversely, a proof that the Mandelbrot set is well-defined and free of contradictions does not establish that ZFC itself is consistent.
So, though you appear to be arguing from the perspective that every formal axiomatisation of set theory would have its own distinct version of structures such as the Mandelbrot set, my position is that there is only one Mandelbrot set, on the grounds that all mathematical structures which are isomorphic to each other are essentially the same structure.
With that clarified, I think much of your argument is moot, but please bring it up again if you disagree.
Dear all you who are interested in discussing "formal systems" in mathematics and physics. While it is interesting to see such an interest so publically, at some point if would be helpful if the discussion were informed more by mathematical insight, less by what seem to be vague second-hand information about the foundations of mathematics and the role that for instance ZFC plays in there.
DeleteJust to mention a few keywords which would seem to be relevant for having a discussion as envisioned here be fruitful:
Mathematicians have a precise concept of theory (in logic), meaning a formal language in which one states axioms. This is usually done INSIDE something that is regarded as the bare minimum substrate of mathematics, which is traditionally taken to be first order logic. If you have never seen how this works, take a look for instance at this exposition by Todd Trimble which spells out in full detail how set-theoretic foundations are formalized inside first order logic. (This is about the structural variant of ZFC called ETCS, a variant which should be of interest to anyone worrying about constructiveness (infinity...) in mathematics and physics, more on that below). Foundational theories other than ZFC that mathematicians are fond of and have formalized inside first order logic include elementary function arithmetic or second order arithmetic (since Harvey Friedman was mentioned elsewhere...)
Moreover, those participants of this discussion worrying about concepts such as "infinity" not being _practically_ "realizable" should know that this is an issue that mathematicans have already thought long and hard about. The flavor of foundational mathematics which means to consider only those cocepts and truth which may genuinely be _constructed_ in practice is called constructive mathematics. Once upon a time just the philosophy of a handful of individuals, this has long become a genuine and rich branch of mathematics. In constructive mathematics the foundations are laid in what is called intuitionistic type theory, something a bit richer than first order logic, but more fundamental than for instance ZFC. This is something that plays a major role in computer science, and for just those reasons for which one would imagine participants of this discussion here should be interested in it: type theory provides a foundations of mathematics which is such that anything proven in it is _computably_ proven, is algorithmic, is, hence _physical_.
Indeed, just recently a little revolution happened at the foundations of mathematics. Constructivists (intuitionistic type theorists) had long had trouble with what seems like a little technical detail in the concept of constructive equality: if we only accept that we know that two things (terms in type theory) are equal if we may construct a proof of their equality (no "infinities", no oracle axiom of choice, just a finite string of constructive steps), then it does not follow that two proof of the same equality may be assume to be equal, instead one needs a constructive proof that the two constructive proofs are equal.
DeleteThis yields to a regression which had remained mysterious... untile a few years ago when Fields medalist Vladimir Voevodsky realized that this apparent embarrassement of constructive mathematics is instead a major virtue: he realized that this means that constructive type theory _autmomatically_ knows all about modern homotopy theory (which, incidentally, is ultimately the mathematics behind gauge theory, BV-BRST quantization and other mathematical aspects of modern QFT).
Ever since this little revolution, essentially what was previously just a niche approach to foundations called (intensional) type theory, is now known as Univalent Foundations in Homotopy Type Theory. The Institute of Advanced Study just hosted a 1-year special program devoted entirely to this new constructive foundations of mathematics.
Anyone interested in formally founding physics in mathematics must take note of this, or else you are playing soccer with a pingpong ball.
Once we are actually sure what we mean when we speak about formal mathematical structures in some given foundations of mathematics, then it is indeed interesting to see how in turn physical theory may arise in these foundations.
There is striking surprise here. In the foundations of homotopy type theory many concepts of modern physics that are usually regarded as sophisticated come out naturally from the axioms. A good bit of local Lagrangian gauge field theory flows almost as if by itself from the modern foundations of mathematics in homotopy type theory.
For some notes of mine with more pointers on what this refers to see here) and here.
Now whether you embrace my notes here or not, the point I'd like to make is that there is a potentially interesting question of how physics is formally founded in mathematics, but that in order to discuss this fruitfully, it might help to have a bit more of a formal toolbox. In the 21st century foundations of mathematics and physics are not in the realm of philosophy and hence must not be investigating by exchange of essays expressing opinions and vague thoughts. Foundations are an exact mathematical science with mathematical tools where if you want to make an interesting contribution, you first phrase a proposition in a precise sense and then you mathematically prove it.
I can imagine that there is are some interesting such precise and provable (or dis-provable) statements that one could look into in a discussion such as this one here. But the way the discussion has been proceeding it seems to be too uninformed of its very topic to be likely to yield much progress.
Hi 108081058828040288656,
DeleteYes, I'm uninformed, and have only a vague understanding of the foundations of mathematics, so any corrections would be welcome, however I don't necessarily agree that a deep understanding of mathematics or physics is necessary for the point I'm making. All I need is to know that the laws of physics are a mathematical structure that can be expressed in one or more formal mathematical theories. No more is required, because my argument is metaphysical rather than physical or mathematical. That's not to say that it wouldn't be great to understand this stuff better, but I have only so much time.
So, here are some propositions that you may be able to correct me on:
1) There is no one true formal theory of mathematics. There are various competing theories each with their own axioms.
2) Mathematical structures are formally expressed in some particular formal theory of mathematics.
3) Mathematical structures are not necessarily tied to any particular formal theory of mathematics - for most structures I imagine many of the competing theories will work to describe them.
4) The consistency of any particular mathematical structure is a separate concern from the consistency of a formal theory used to describe it.
5) On mathematical Platonism, all consistent mathematical structures (expressed in any formal system) can be said to exist - (plenitudinous Platonism)
I think that's all I need to make my point, but if you think I'm missing something please let me know.
Hello Disagreeable Me,
DeleteFirstly, sorry if you are getting a bit frustrated with me. I don't think I am backtracking, but I may be using some terms loosely or incorrectly.
I am not frustrated with you per se as I am frustrated at the lack of precision of the discussion in general. I think this is your fault only insofar as you are a proponent of MUH and MUH is no where defined in any kind of rigorous fashion. It is hard to take the idea seriously without defining terms which is what I was hoping you would do a bit more.
Firstly, I want to clarify what we mean by formal system, as I'm not too sure.
A set of axioms expressed in a formal language sufficient to form a model of the natural numbers using a system of logic. These axioms are taken as a priori as a starting point to reason from.
There is the idea of a mathematical structure, such as the Mandelbrot set...
Here we go. You do not define "mathematical structure" anywhere that I have seen. You've given examples, but it is not defined and I suspect this is a big problem with our conversation.
The definition of the Mandelbrot set can be expressed in the context of ZFC but I would not say that the Mandelbrot set is itself defined *by* ZFC.
This seems nonsensical to me. Read the sentence again and tell me if you can see where I have a problem with it. Let me paraphrase a bit.
"The definition of the Mandelbrot set can be be expressed in the context of a , but I would not say that the Mandelbrot set is ... defined by a .
Mathematicians do their work within the confines of axiomatic formal systems and logic systems. These are the rules of the game. Without the rules we have no game.
Depending upon how you define "mathematical structure" ... depending upon how you define MUH ... depending upon how you define multiverse ... you might have a problem with Godel's theorems. Definitely, the way you've been speaking about "all mathematical objects exist in the multiverse" and "consistency equals existence" leads me to believe this is the case.
Hello 108081058828040288656,
... at some point if would be helpful if the discussion were informed more by mathematical insight ...
Absolutely. I suspect that until MUH proponents can produce a rigorous definition of what they are talking about the grown ups - read: mathematicians - will not get involved.
Now whether you embrace my notes here or not, the point I'd like to make is that there is a potentially interesting question of how physics is formally founded in mathematics, but that in order to discuss this fruitfully, it might help to have a bit more of a formal toolbox.
Could not agree more. For MUH to be taken seriously as a physical theory will require falsifiable predictions or at least some genuine insight into the problems bedeviling modern particle physics or cosmology. For MUH to be taken seriously as a mathematical idea will require a rigorous defintion of the idea.
Cheers,
Adam
Disagreeable Me,
Delete4) The consistency of any particular mathematical structure is a separate concern from the consistency of a formal theory used to describe it.
I say this is false although I do not know your definition of "mathematical structure." For a structure to be consistent necessarily depends upon the formal system(s) that successfully define it to be consistent. In other words, asserting that the structure is consistent necessarily asserts that the formal system(s) it is embedded within are consistent. Perhaps the crux of our disagreement.
Cheers,
Adam
Hi Adam,
DeleteThanks for bearing with me a while longer. I will attempt to clarify what I'm saying further.
>A set of axioms expressed in a formal language sufficient to form a model of the natural numbers using a system of logic. These axioms are taken as a priori as a starting point to reason from.<
OK, so I am not claiming that the universe is a formal system then. I am claiming that it is a mathematical structure.
>You do not define "mathematical structure" anywhere that I have seen.<
OK, here is my attempt to define what I mean:
Definition: a mathematical structure is an abstract object which can be defined entirely unambiguously.
Explanation: If an abstract structure is defined unambiguously, then there can be no disagreement about its properties or what they entail. I think mathematics is the study of such objects. The lack of ambiguity makes it possible to e.g. prove theorems. To propose such a definition may in some cases require a formal system to be established first, but not necessarily as long as everyone can understand clearly what is meant. Formal systems themselves are mathematical structures on this definition, as is first order logic, which as the most basic formal system is not itself grounded in any deeper formal system.
>This seems nonsensical to me. Read the sentence again and tell me if you can see where I have a problem with it.<
That's why I emphasised the word "by". The idea "My car is grey" is not defined *by* English, it is defined *with* English. By this I mean that the existence of the English language does not entail that my car is grey or vice versa. I can also express the same idea with other languages. That is what I mean when I say the Mandelbrot Set is not defined *by* ZFC but *with* ZFC (or other formal systems).
Finally I would agree with you that the MUH is neither a scientific nor a mathematical idea. It is a metaphysical idea, but one which I think must necessarily be true if we accept:
1) naturalism
2) mathematical platonism
3) the computational theory of mind
I explain why on my blog.
Regards,
DM
Hi Adam,
DeleteHope you'll get back to me on this. Does my definition of mathematics make sense to you? Do you now see what I mean by the independence of a mathematical structure from any particular formal system?
DM
This comment has been removed by the author.
DeleteDisagreeable Me,
DeleteDoes my definition of mathematics make sense to you?
I assume you meant your definition of mathematical structure... In which case, no I am afraid it is still too ambiguous. When I say I want you to define what you mean I think the best way to do this is in the language of mathematics... ie, pick a formal system and language and define it vigorously.
Do you now see what I mean by the independence of a mathematical structure from any particular formal system?
No, not really. I get that you are saying that some of these structures you have in mind can be defined by more than one formalism and in this sense they are independent from any *one* formalism. However, they have to be defined by *at least one* formalism. No mathematical structure can be truly independent of *any* formalism. And I think that is what you would need to defeat my argument. Do you see why?
Formal systems of mathematics are usually found in non disjoint sets. Some are weaker and some are stronger. I guess in truth each mathematical structure has at least one minimally weakest formalism that is still powerful enough to define it. In this sense, it will never be independent.
And please see above where I take issue with your point #4. I think this is the crucial question and we'll never be able to resolve it unless/until you can define "mathematical structure" in a rigorous way I think or give an example of one of these structures which can be proven consistent and still can be defined by an inconsistent formalism. I don't think you'll find any such examples, but feel free to look.
Cheers,
Adam
Hi Adam,
Delete>I am afraid it is still too ambiguous<
Perhaps to do mathematics with, but I don't see why if we're having a metaphysical discussion. Can you challenge the definition with some border cases?
>pick a formal system and language and define it vigorously<
It is not possible to express my position in these terms because it assumes that there is a privileged formal system, which I don't believe. Instead, I could define a mathematical structure as anything that can be defined in any formal system.
>However, they have to be defined by *at least one* formalism.<
Mostly agreed, but that would probably exclude first order logic, as that is the most basic formal system and so not defined in any more basic system. I would count that as a mathematical structure too.
>I think that is what you would need to defeat my argument. Do you see why?<
No. I think as long as it is not tied to any one specific formalism, your argument is defeated, because proving the consistency of a structure does not prove the consistency of any specific formalism - it only proves that there exists *some* formalism that can handle it. Proving the inconsistency of a formalism similarly does not prove the inconsistency of a structure defined with it - the structure may still be consistent if some other formalism can handle it.
The Mandelbrot set as visualised by computers has to be consistent, because there are computer programs that can draw it out. Whatever they are doing, it is an operation defined by the algorithms they are running. It's not possible to implement an inconsistent or ambiguous (single-threaded) algorithm. Anything that is algorithmic has to be consistent.
Let's say the Mandelbrot set is typically defined in ZFC. We know the Mandelbrot set (as visualised by computers) is consistent, but that in itself does not prove that ZFC is consistent. The analogy to our universe is that knowing that the laws of physics are consistent does not prove that ZFC is consistent.
Now, suppose for a second that there is actually something inconsistent in the definition of the Mandelbrot set. To me, this means that the computer programs are not actually implementing the definition correctly, because this is impossible. So, we don't necessarily know that the Mandelbrot set is consistent, we only know that its visualisation algorithm is.
Now let me bring this back to the universe. We know we are in a universe which is consistent, but perhaps we have not correctly identified the true laws of physics. The laws of physics we have found could be inconsistent, like the Mandelbrot set itself, but our universe is not actually following those laws, like the visualisation algorithm. So the existence of the universe does not prove that any particular mathematical structure is consistent, only that something like it is.
So the existence of our universe can neither prove the consistency of any particular formalism or of any particular mathematical structure, therefore there is no problem from Godel.
Hi Disagreeable Me, it's been fun, but I don't think we are getting anywhere. Cheers!
DeleteI think Max Tegmark may be approximately right, but the first thing I did was to look at his formal definition of 'mathematical structure' (Appendix A of http://arxiv.org/abs/0704.0646). I think a better (constructive) approach is to base the definition of structures on type theory – that is, a type-theoretic approach instead of a set-theoretic approach. I think this approach has greater 'physicality'. It also is the basis for exact analysis (via objects called completion monads) which I think could make Max happy related to his we don't actually need the infinite to do physics stance.
ReplyDeletePhilip Thrift mentioned mathematical foundations not in set theory, but in type theory.
DeleteIndeed, there is a good argument to be made -- and since a few years now prominently being made by Fields medalist Vladimir Voevosky -- that modern mathematics has preferable foundations in homotopy type theory. This is in some sense more fundamental than set theory, and in any case set theory may be recovered inside it (see here).
Homotopy type theory as a foundational framework stands out in its monolithc conceptual elegance and conceptual simplicity (the "homotopy"-aspect comes about not by adding but by removing a classical axiom!) combined with the naturality with which it expresses the profound features of modern mathematics. For instance if you are to prove some profound theorem of modern homotopy theory such as the Blakers-Massey theorem then in set theory even to set up what this means takes you on a large detour via point set topology or else simplicial set theory, while in homotopy type theory it is proven "synthetically" from the bare minimum of the theory.
In a rather similar way, it turns out, there is a good bit of modern fundamental physics -- of local Lagrangian gauge field theory -- naturally arising in homotopy type theory, not via the usual detour of setting up all the ingredients (differential geoemtry, operator algebra, distribution theory, etc. pp.) item-by-item via set theory, but naturally ("synthetically") form an utmost minimum of axioms. Some expository notes of mine on this are available here: "Synthetic QFT"
"Gauge field theory in Cohesive homotopy type theory"
"Homotopy-type semantics for quantization".
In this "synthetic" formalization of modern fundamental physics in the modern foundations of mathematics given by homotopy type theory, it is interesting to investigate which physical theories (which QFTs) arise more naturally from the axioms than others. It turns out, maybe surprisingly in any case provably, that what the axiomatization here produces naturally are higher Chern-Simons type TQFTs in some dimension d whose quantum anomaly cancellation is controled by aTQFT in dimension (d+1) and which on their (d-1)-dimensional boundaries have non-topological boundary field theories in a way directly analogous to how the 2d WZW model sits on the boundary of ordinary 3d Chern-Simons field theory (an mathematically precise instance of, as it were, the "holographic principle").
See for instance Daniel Freed's notes with the telling title 4-3-2 8-7-6 for more on the relevance of this in fundamental physics.
So there is something substantial to be further explored here.
Disappointing episode. It was weird to see a physicist at MIT be so vague and evasive with his pronouncements. There was virtually no substance presented in his argument. He seemed to just say that you can look at the universe mathematically and it's pretty cool. Ok, that's it? Valiant effort by you two to direct him toward giving some substance but he was just too balky.
ReplyDeleteI thought it was a great podcast, oddly enough for much the same reasons. I don't need convincing that the MUH is true, I have believed in it for a few years now. What was great to hear was the challenges to the idea posed by Massimo and Julia. They did a great job. I agree Tegmark could have been better.
DeleteHi Disagreeable Me,
DeleteCan you tell me on what basis you believe the MUH? Is it on Mr Tegmark's reasoning?
Hi Robin,
DeleteNo, not quite on Tegmark's reasoning although I think there are bound to be similarities. Tegmark views the MUH as a hypothesis - for me it is almost philosophical certainty based on some premises I am disinclined to seriously doubt, at least from an atheist naturalist perspective.
A full account of my reasoning is here:
http://disagreeableme.blogspot.co.uk/2013/12/the-universe-is-made-of-mathematics.html
My immediate reaction is that the digital demigod problem has another implication. Even if we could identify something that was mathematically impossible then there would exist, at a probability of 1, an algorithmic intelligence who could create a world that would make it seem that way..
ReplyDeleteSo the hypothesis is not falsifiable by that method.
I'm just a lowly physicist ... I haven't done serious physics in ages ... but this is very strange. If I understand Tegmark - but correct me if I'm wrong - there's a mathematical universe in which the parallel axiom is true, and then there's another in which it's not true.
ReplyDeleteBut .. how is it possible, then, that I can find a space without the parallel axiom in a Euclidian space with the parallel axiom? I'm talking about the Beltrami-Klein model of course. When exactly am I switching between these two universes - that seem remarkably similar, to be honest? When I draw a circle? When I define the points on the circle in a specific way? There seems to a universe - the universe in which I'm living - that easily can accommodate different axiomatic systems that lead to completely different theorems. The universe in which I'm living easily accommodates all those different axiomatic systems that make different universes necessary, per Tegmark.
How come?
How come I can work with exactly two truth values one moment, and with a continuum of truth values the next moment? Am I changing universes? And if I'm not changing universes, then what are these universes useful for in the first place?
I suspect it's a stupid question, but I'm curious.
Hi Patrick,
DeleteYou are not changing universes. All mathematical structures can in principle be analysed from the perspective of a given universe. The different universes Tegmark is talking about have different laws of physics, that's all. Some universes may have Euclidean space, for example, while others have completely different kinds of space corresponding to different axioms.
Thanks Patrick G for this interesting question about the parallel axiom!
DeleteConsider our physical space: here the parallel axiom is generally false, since matter curves space and can cause initially parallel laser beams to bend and cross each other. If it weren’t for quantum mechanics and other complications, this cosmos would correspond to a perfectly well-defined mathematical structure described by General Relativity: a 3+1-dimensional pseudo-Riemannian manifold with some tensor fields etc (for electromagnetism and other matter) obeying certain partial differential equations. It’s simply a more complicated mathematical structure than a Euclidean space satisfying the parallel axiom or a homogeneous non-Euclidean space such as those discovered by Lobachevsky, Bolyai et al.
Unfortunately, I have to disagree. In our physical space the parallel axiom is neither false nor correct. It all depends on your definition of "parallel", as the Beltrami-Klein model shows.
DeleteHi Patrick,
DeleteBut regardless, my point stands, does it not? The fact that we can imagine different geometries tells us nothing about our universe, and does not entail that we are switching between universes.
Disagreeable,
DeleteThe Beltrami-Klein model is embedded in a Euclidean plane. If I remember correctly, any differentiable manifold can be embedded in a "flat" space of sufficiently high dimension. A spere can be embedded in a 3 dim. Euclidian space. You can find every geometrical property on the surface of a sphere by doing mathematics in the 3 dim. Euclidean space.
Now, on a sphere two geodesics always meet. In a 3 dim Euclidean space not.
In other words: a differentiable manifold on which the parallel axiom isn't true, can be embedded in a space in which the parallel axiom is true.
Now tell me: which laws of physics are valid in this situation? Laws that somehow correspond to the mathematical structure on the manifold, or laws that correspond to the mathematical structure of the higher dimensional and "flat" space?
Let's say that photons move along geodesics. On a sphere, a geodesic is a circle in the embedding 3 dim. space. Something you can perfectly prove by doing mathematics in the 3 dim. space, although a geodesic in the 3 dim. space is a straight line (note how important definitions are ...).
Lots of physics in done on differentiable manifolds. These can be embedded in flat spaces, and the mathematics on the manifold stay the same. I'm really curious to know how Tegmark decides with which laws of physics this situation corresponds. Laws that demand that space is curved? Law that demand that space isn't curved at all? With some clever manipulation of mathematical definitions you can have both.
Hi Patrick,
DeleteSorry I didn't answer this sooner. I was distracted and forgot.
>In other words: a differentiable manifold on which the parallel axiom isn't true, can be embedded in a space in which the parallel axiom is true.<
The answer to this is that the parallel axiom is true in Euclidean space, your example not withstanding. The geodesics on the surface of the sphere are simply not geodesics from the point of view of Euclidean space. If you consider them to be geodesics then you are changing the definition of what geodesic means. If you draw a triangle on a sphere with three right angles, you have not contradicted Euclidean geometry because from the point of view of Euclidean geometry this is simply not a triangle.
>which laws of physics are valid in this situation?<
The laws of physics are those that affect the actual geometry of space-time, which is approximately Euclidean (at least on human scales, and notwithstanding the effects of gravity). The properties of the sphere which appear to contradict Euclidean geometry can be rephrased in terms which accord with it.
>Let's say that photons move along geodesics. On a sphere, a geodesic is a circle in the embedding 3 dim. space.<
OK, but what is a geodesic from the point of view of the geometry of the surface of a sphere is not generally a geodesic of spacetime (unless the photons are in a circular orbit around some mass). Photons always move in geodesics according to the geometry of spacetime itself, not geodesics of arbitrary manifolds we can represent within spacetime.
Does that answer your question?
I wish you had asked Professor Tegmark *which* mathematical formalism describes our universe.
ReplyDeleteDoes he believe that our universe is described by Zermelo-Fraenkel set theory WITH the axiom of choice?
or...
Does he believe that our universe is described by Zermelo-Fraenkel set theory WITHOUT the axiom of choice?
Because if you believe his theory that our universe is just a mathematical object in a sea of parallel mathematical universes... it behooves one to state the formalism he is using. Moreover, the choice has severe implications for the predictions we can expect of our universe.
For instance, ZFC implies that the Banach-Tarski paradox is a real property of our universe and we could expect to get to work on turning this property into real technological breakthrough. No more shortages of anything!
On the other hand, if he chooses ZF, then this implies that our universe *really* contains non-empty sets that when multiplied produce an empty set...
I wish some mathematicians would weigh in on Tegmark's "theory." They should be his prime constituents, but I suspect they would be too busy laughing for Tegmark to describe his "theory" sufficiently.
I am not sure that, even under MUH, ZF predicts that the Banach-Tarski paradox is a real property of *this* universe.
DeleteBut I think the point you are making is good. Mathematical Universe Hypothesis would need to rely on a continuing series of Meta-Mathematical Universe Hypotheses.
DeleteHi manyoso,
DeleteOur universe is not a mathematical formalism at all. The only axioms are the laws of physics. Think about it like a computer simulation of the laws of physics but without a computer to run it.
Our universe is not a mathematical formalism at all.
DeleteTegmark's MUH is a hypothesis to the contrary. More specifically, he is saying our universe is a mathematical structure defined by a mathematical formalism. Moreover, that a multitude of parallel universes each identified by their own mathematical structure exists.
I'm saying that for MUH to stand on any kind of footing it is imperative for Tegmark to explain which formalism he is using OR if he believes that all formalisms are represented in the pool of parallel universes.
Our universe is not a mathematical formalism at all.
DeleteTegmark's MUH is a hypothesis to the contrary. More specifically, he is saying our universe is a mathematical structure defined by a mathematical formalism. Moreover, that a multitude of parallel universes each identified by their own mathematical structure exists.
I'm saying that for MUH to stand on any kind of footing it is imperative for Tegmark to explain which formalism he is using OR if he believes that all formalisms are represented in the pool of parallel universes.
Tegmark disagrees. His MUH is a statement that our universe is a mathematical structure defined by some mathematical formalism. Which formalism he has not said and this is a big problem in trying to evaluate what his MUH even says.
DeleteI don't think the Axiom of Choice (or the Continuum Hypothesis, if we wish to go there as well) has any consequences in the physical world. If it did, then we should be able to devise physical tests that would show AC (or CH) to be true or false. It's not even a given that the physical universe is infinite. ZFC is a mental construct that, conveniently, can be used to produce number theory, analysis, and so on -- useful tools for studying our universe (or alternative universes, if we choose), but not direct constructs of our universe.
DeleteI would also take issue with the statement that "ZFC implies that the Banach-Tarski paradox is a real property of our universe and we could expect to get to work on turning this property into real technological breakthrough." Not if matter is quantized! I don't think a material object can be decomposed to produce non-measurable sets that are also material, even if you do accept the axiom of choice.
DeleteHi Manyoso,
Delete>Tegmark's MUH is a hypothesis to the contrary. More specifically, he is saying our universe is a mathematical structure defined by a mathematical formalism. <
I think this is wrong. He claims it is a mathematical structure, not a structure defined by any particular formalism. Which formalism you use does not matter as long as it can express that structure. Some of the axioms we use in mathematical formalisms are likely to be irrelevant to certain mathematical structures, e.g. I don't think you need to take a position on the axiom of choice in order to define the integers.
You ought to have the same concerns for other physicists. You should for example want to ask Einstein which formalism he is using when he claims that e=mc^2.
If the laws of physics are ever understood as a coherent mathematical structure, then Tegmark's view is that that structure is identical to our universe. Your question about formalism does not apply.
I think this is wrong. He claims it is a mathematical structure, not a structure defined by any particular formalism. Which formalism you use does not matter as long as it can express that structure.
DeleteAsking for a specific mathematical formalism might not matter for the idea that our universe is a mathematical structure, but it most certainly does matter for the idea that our universe is one of a multiverse of mathematical structures. To even define such a multiverse requires being specific about the mathematical formalism.
The MUH as I understand it is not just a hypothesis about our universe. Rather, it is a hypothesis about a multiverse of mathematical structures. To even be able to talk semi-intelligently about such an idea Professor Tegmark needs to say what formalism he believes is the basis for this multiverse.
Look, this MUH of his has no equations. It has no physical predictions that anyone can discern. It doesn't tell us which mathematical structure our universe is equivalent to. Can it not at least tell us which formalism it subscribes to? Is even *that* too much to ask for this seemingly vacuous theory?
Hi manyoso,
DeleteI think I see the problem.
Tegmark and I both subscribe to plenitudinous platonism.
http://plato.stanford.edu/entries/philosophy-mathematics/#PlePla
Basically, all possible formalisms are equally valid and there will be universes that are better described by some formalisms than others.
The fundamental idea that is so important about the MUH, that makes it so strong, so simple and so necessarily true, is that there are no free parameters. Everything that can exist does. Preferring some formalisms to others would violate this idea.
Richard,
DeleteI don't think the Axiom of Choice (or the Continuum Hypothesis, if we wish to go there as well) has any consequences in the physical world.
...
I would also take issue with the statement that "ZFC implies that the Banach-Tarski paradox is a real property of our universe and we could expect to get to work on turning this property into real technological breakthrough." Not if matter is quantized!
My apologies for being imprecise. I was speaking not of our universe per se, but of Tegmark's multiverse-MUH. Seemingly, if his multiverse is defined by one of these formalisms, then we expect some of the universes so defined to have real physical aspects governed by the Axiom of Choice. That is what I meant.
Perhaps, 'we' shouldn't get to work, but the inhabitants of the universe where Banach-Tarski paradox is physically manifest should, no?
To be clear, I would be very surprised if our universe is such a universe, but then I don't believe Tegmark's multiverse-MUH makes much sense.
I'm not sure what form "real physical aspects governed by the Axiom of Choice" even means, and that was the point of my counter. If all the universes produced under the MUH are all finite, then AC really means finite choice, which is a theorem in ZF, not a new axiom. If they are infinite (or at least some of them are), it does not follow that you could construct an experiment in *any* universe that would tell you which type (AC or non-AC) you were in.
DeleteRegarding BT, my objection was simply that BT requires some sets in the decomposition to be non-measurable. To say that a non-measurable set nevertheless could have some material reality is a very large stretch, in my opinion. Thus I doubt that BT can be physically manifest in any universe. I may be wrong -- a computer with continuum-sized memory might be able to simulate a BT decomposition, and that simulation would be "physically manifest". But is a universe that allows such a computer to exist even possible? I think it is an open question, and answers are only speculative at best.
Like you I'm skeptical of MUH, but mainly because I don't think it produces any testable predictions.
Disagreeable Me,
DeleteBasically, all possible formalisms are equally valid and there will be universes that are better described by some formalisms than others."
Are you positing the set of all formal systems? Do you allow for formalisms with an infinite set of axioms or non-computable set of axioms?
The fundamental idea that is so important about the MUH, that makes it so strong, so simple and so necessarily true, is that there are no free parameters. Everything that can exist does. Preferring some formalisms to others would violate this idea.
Where you see so powerful I see utterly vacuous, but perhaps I am misunderstanding. Still, I wonder about your answer to the above. Do we allow formalisms with non-computable axioms? If not, does this "violate this idea" as you say?
Richard,
To say that a non-measurable set nevertheless could have some material reality is a very large stretch, in my opinion. Thus I doubt that BT can be physically manifest in any universe.
Talking about any of this is hard because we don't have any real concrete proposal yet from Tegmark et al. Just loose language. But I take him at his word that he really means that the universe *is* a mathematical structure and that other universes in his multiverse-MUH *are* mathematical structures. If that is so, then the physical reality *must* manifest *in some way* all the abstract features of the math it is supposed to *be*.
Agreed that this is all highly speculative bikeshedding until/unless it gets more... formal.
Richard,
DeleteIf I'm not mistaken you need the axiom of choice to prove that every vector space has a basis. (I even think both statements are equivalent.)
Are you certain this doesn't have consequences in the physical world?
Hi Manyoso,
Delete>Are you positing the set of all formal systems? Do you allow for formalisms with an infinite set of axioms or non-computable set of axioms?<
I guess I am positing the set of all formal systems, but only because formal systems are mathematical objects and I think all mathematical objects exist. I don't think any formal system with infinite axioms exists because it is inherently impossible to define - if you can't define it then it is meaningless to say whether it exists or not, because it itself is meaningless. But the axioms could be arbitrarily many.
>If that is so, then the physical reality *must* manifest *in some way* all the abstract features of the math it is supposed to *be*.<
I think one of the things you learn from really grokking the MUH is that the concept of "physical reality" is meaningless. There is no difference between the abstract and the physical. We perceive certain objects to be physical only because they are part of the same mathematical structure as we are. Fundamentally, tables and chairs are abstract, but then so are we.
Patrick, I wasn't familiar with that theorem. But after a brief Google search (and I haven't had time for a careful read), a few things stand out in my mind.
Delete1. Since we already have Finite Choice as a *theorem* in ZF, AC isn't relevant for finite universes. As the wikipedia artilce on Axiom of Choice states, "Without the axiom of choice, these theorems may not hold for mathematical objects *of large cardinality*" (my emphasis).
2. "Large cardinality" may refer to cardinalities greater than any physical reality of the universe in question; for instance, perhaps an inaccessible or even a measurable cardinal is required, and these may lie far beyond what is required to do physics in a particular infinite universe. In other words, this would mean that the vector spaces without bases are non-physical, abstract ones.
3. If the theorem applies to physical vector spaces in *our* universe, AC would be settled, at least as far as our universe is concerned. Yet we still allow that it is an unsettled question.
To put my response to Patrick another way, if V = L (the axiom of constructibility) is assumed, then AC follows. So every constructible vector space must have a basis. I will now put forth as a reasonable conjecture that any mathematical object that has physical relevance is constructible. If so, then denying AC does not break the physics of any world; it only introduces purely abstract objects. And accepting AC permits other things, such as the paradoxical decomposition of a ball, that do not correspond to anything physical (in this case, because some of the sets involved are non-measurable -- here I am implicitly introducing another conjecture).
DeleteSo given two conjectures which I believe to be reasonable (non-constructible sets do not describe physical entities, and non-measurable sets do not describe physical entities), it follows that AC is irrelevant to physics -- not just of our world but of any world that might arise if MUH happens to be true. Therefore, for a given physical world, you could have at least two mathematical descriptions, one with AC and one with ~AC, which have identical consequences for the physics of that world.
"Any mathematical object that has physical relevance is constructible" and "non-measurable sets do not describe physical entities" ...
DeleteWhy do you say that these conjectures are reasonable? This would severely restrict the composition of the multiverse-MUH and introduce the 'free parameters' that Disagreeable Me says are so important to the value of multiverse-MUH.
It would also rule out other classes of axiom's that are incompatible with AC and V=L. This multiverse-MUH is starting to look pretty artificial and baked-in I have to say.
Hi Manyoso,
Delete>This multiverse-MUH is starting to look pretty artificial and baked-in I have to say.<
I don't think that follows. There are a number of ways of resolving this.
1) Richard's conjectures could be wrong.
2) Richard's conjectures could be right because their negation leads to inconsistencies.
3) Richard may be using the word "physical" in a specific sense which does not reflect on the mathematical ensemble as a whole. On the MUH, there is ultimately no difference between physical and abstract, so the world "physical" becomes ambiguous and problematic. It can be taken to refer to this universe specifically, or universes similar to our universe, or universes with conscious observers.
Manyoso,
DeleteIt seems to me that a physical entity must have some kind of meaningful measure in order for its interactions with other physical entities to be quantified. Even dark matter is measurable. And a physics that is forced to deal with non-constructible entities is a physics that would be unable to make predictions.
And no, other axioms would not be ruled out... They just wouldn't have physical consequences. I'm thinking of things like :existence of a measurable cardinal, continuum hypothesis, determinacy, etc.
DM, I'm not sure what your confusion about my use of "physical" stems from, but a general idea is that "physical" means "capable of interacting with other objects in the same universe." Or if we are talking mathematical objects, they are physically relevant if they are essential to a complete description of the interactions between physical objects.
Richard,
DeleteAnd no, other axioms would not be ruled out...
The Axiom of Constructibility is in contradiction to some large cardinal axioms involving zero sharp I believe.
Imposing the Axiom of Constructability on multiverse-MUH would therefore leave out some formalisms from enriching the multiverse. But this kind of conjecture just serves to show how meaningless this whole exercise is. Here we are speculating about math and physicality with absolutely no way to test anything or verify anything. For you physicality might mean computability - I guess on "intuition", but others might have a different belief. How will we ever know?
Hi Richard,
Delete>physical" means "capable of interacting with other objects in the same universe.<
OK, but then we need to understand what you mean by 'object', 'interaction' and 'universe'. A central idea of the MUH is that there is no real ontological distinction between our universe and any other mathematical structure, including ones as simple as circles or the set of natural numbers. We would not call these simple structures universes, but perhaps only because they are so dissimilar to our idea of what a universe is.
So, to illustrate the point, is Conway's Game of Life a universe? Are gliders objects in that universe? Do they interact with each other? Are they physical?
Or how about the Fibonacci numbers? Does each pair of numbers in the sequence interact with each other to produce the next? Does that mean that these numbers are physical objects?
I don't know, because to me those terms are relatively meaningless when we're considering entities outside our own universe. I would be inclined to reserve those words to describe either structures within our universe or substructures of mathematical structures similar to our universe.
So if your point is that these axioms cannot be relevant in mathematical structures that are in any way similar to our universe, then you might be right. Otherwise I'm not sure what you mean.
> The Axiom of Constructibility is in contradiction to some large cardinal axioms involving zero sharp I believe.
DeleteIndeed it is, as well as anything stronger than 0#, but if you accept my conjectures then you don't need those LCAs to get a complete physics (anywhere in the multiverse). But my conjectures don't preclude rejecting V=L, they simply remove the need for deciding V=L. So you can still have all the LCAs you want. That was what I meant by "and not be ruled out": That (in my expectation) V=L, AC, CH, AD, and LCAs that require rejection of V=L -- or the negations of these axioms -- won't have physical consequences (anywhere in the multiverse).
(As an aside, this doesn't preclude the possibility that *some* additional axioms required; ZF is almost surely not sufficient to axiomatize physics. I just don't think those axioms will require non-measurable entities, nor non-constructible ones.)
You are right, the final answer is unknowable, which is why I emphasized the underlying conjectures. But I do believe my conjectures are entirely reasonable for anything interesting enough and sufficiently constrained to be called "physics". As they are only conjectures, you are free to disagree.
DM: In our universe, objects can be particles, waves, fields, or more complex composites of these things. Or their equivalents in mathematical symbolism, if we are going to accept the kind of equivalency underlying MUH. I don't think a general definition is going to help, as each possible "type" of universe will have its own fundamental objects, and I'm not sure we can anticipate what those will look like in terms of mathematical structures.
Delete"Interaction" is easier. By that, I mean that each object has a state. A universe should have rules that define when and how an object's state may change; basically, a set of operators. Think of Feynman diagrams as an example. And if a particular state change is only allowed to take place in the presence of a second object (or of an additional N objects), then the operation is an interaction between the objects in that set. Note that "interactions" can also create and destroy particles, as for example in the case of electron-positron annihilation.
A really good definition of "universe" (in the physical rather than mathematical sense) might be elusive, but I can think of some minimum requirements. A universe should be closed. That excludes the Fibonacci numbers. Fibonacci numbers are produced by addition, but the set of Fibonacci numbers is not closed under addition (3 and 8 are Fibonacci numbers, but 3+8 is not). A universe needs a set of operators as discussed above, such that all state changes can be defined in terms of only those operators (perhaps with the introduction of concepts such as probability, for universes that aren't deterministic). So far, Game of Life qualifies as a universe. It may not be as rich as our physical universe, but I haven't eliminated it. True, neither of these conditions can distinguish purely mathematical universes from physical universes. I think my assumptions about measurability and constructibility might remove some universes from being considered "physical", although it might not be sufficient (and apparently in the minds of some here, not necessary). Maybe some specific types of rules must also be included (e.g., conservation laws). Maybe a physical universe additionally needs to be sufficiently complex to support emergence. Maybe (as you seem to believe) there is no distinction, but perhaps there is and we haven't identified it yet. These are just some ideas off the top of my head, and certainly everything here is in need of more formalized definition.
Hi Richard,
DeleteAll of that is probably fine for one particular definition of those terms. My only point really is that no particular definition should be assumed, and in particular for the terms "physical" and "universe".
It seems that interaction needs some analogue of time, and perhaps also some notion of events happening in parallel - you can't just compute the future of one particular object by extrapolating into the future without also doing the same for other objects. Perhaps a universe can be defined as a mathematical construct in which interactions can take place.
I'm not sure that your criteria of closedness really works. In the Fibonacci "universe", it's simply not allowed to add 3 to 8. You can only add successive numbers, so the operation you are describing is against the "laws of physics". It's perhaps not dissimilar from the way two events cannot interact in our universe if they are outside each other's light cones. We could rule out the Fibonacci series as a universe on the grounds I mentioned above because there is no way to construe it as a structure in which stuff happens in parallel.
>True, neither of these conditions can distinguish purely mathematical universes from physical universes.<
>Maybe (as you seem to believe) there is no distinction<
Yes, that is my view. And really the only view if we take the MUH seriously. Tegmark isn't saying that some mathematical structures have life breathed into them and become physical, he is saying that the concept of physical as opposed to mathematical is incoherent. If you think there is a fundamental distinction between what is physical and what is mathematically consistent, that may be a defensible view, but it seems to me that it amounts to a rejection of the MUH.
DM,
DeleteI was referring to this definition of closure: http://en.wikipedia.org/wiki/Closure_(mathematics)
Ordering of the elements does not come into play in that definition. So, the Fibonacci numbers are most definitely not closed under addition.
Regarding your last paragraph, I don't reject MUH outright. I might have been a firm believer once upon a time, but now I approach it with skepticism.
Hi Richard,
DeleteI understood what you meant. I know that the Fibonacci numbers are not closed under arithmetic. I'm just not sure that closure under arithmetic or any other operation is a good criterion for ruling out certain structures as universes. In the context of the Fibonacci numbers there are certain operations you can't perform while still restricting yourself to the Fibonacci numbers. To me, this is not obviously different from the fact that there are certain ways for the matrix of Conway's Game of Life to evolve which are forbidden by the rules. But perhaps it is different.
Might I ask what lead you from being a firm believer to being a skeptic? I'm obviously a firm believer myself, but I want to test my conviction to see if it holds up to various counter-arguments. Were there any in particular that impressed you?
DM,
Delete"To me, this is not obviously different from the fact that there are certain ways for the matrix of Conway's Game of Life to evolve which are forbidden by the rules."
I see a significant difference. Within the context of the natural numbers, the operation of addition has an unrestricted domain. But unrestricted addition in the domain of the Fibonacci numbers generates numbers outside of that "universe". In GoL, there is an operator that defines how the matrix evolves, so: state[i+1] = G( state[i] ). For any state of the matrix at stage i of the game, the operation will generate a valid state for stage i+1. The GoL universe is closed under this operation, with no restriction of the operator's domain within the universe of GoL. That certain paths of evolution are forbidden is therefore beside the point, since the universe consists of cells and states. It does not consist of an unrestricted set of "paths".
As to your question, there wasn't really any specific counterargument that changed my mind. Rather, it was disenchantment with mathematical platonism in general, triggered in part by Woodin's Omega logic proposal, which seemed to me (to the limited extent that I could interpret it) to be overreaching.
Hi Richard,
DeleteI understand what you are saying but I do think that I could make a case for the GoL not being closed under certain operations. But anyway, it doesn't matter at all so we can drop it.
I find it curious how one particular proposal could undermine platonism as a whole. Perhaps you were not a full-blooded/plenitudinous Platonist? This view of Platonism, my own, holds that all consistent axiomatic systems are equally valid, so there is no one true mathematics. On this view, if any one axiomatic system is flawed (i.e. inconsistent), then the others are still fine.
So here is my objection:
ReplyDeleteUnder MUH there is, for every observable state of the Universe, an uncountable infinity of uncountable infinities of infintely many infinities of mathematically distinct universes created by algorithmic intelligences which would be observably identical and at most, infinitely many non-algorithmic paths
So it is overwhelmingly likely that I live in a universe created by an algorithmic intelligence.
An algorithmic intelligent could make a universe in which there is sometimes an appearance of mathematical consistency but change that at will - in other words to produce miracles.
There are, by many orders of magnitude, more ways the next five minutes could pan out if the laws of physics might be altered than if they held.
So the probability that I will experience a noticeable miracle in the next 5 minutes is arbitrarily close to 1
But I live my entire life without expriencing any noticeable miracles and so the MUH is false.
There is a similar argument for the CUH.
One possible out is if consciousness is not computable. That would rule out CUH in any case. But even under MUH it would still be possible for there to be demigods using hypercomputing or whatever mathematical principle our consciousness operates under.
Another way out is that, in QM there may be an uncountable infinity of uncountable infinities of ways that this universe I am experiencing could arise.
That would be neat - it would explain why we are in such a universe.
But then you would have to demonstrate that it was mathematically impossible for each of the digital demigods to appear in a similar way in the first place.
And miraculous universes would still be vastly more numerous than non-miraculous ones.
But the best then you could say is that it, under MUH, it is maximally undecidable whether or not you will experience a miracle in the next second.
And that is not a compressible theory as he suggests.
Tegmark does not believe in infinity. I guess this means he believes the number of finite mathematical structures is itself finite hence the number of parallel universes is also finite.
DeleteHe uses this dismissal of infinity to hand wave away the very real problems that Godel's Incompleteness Theorem presents for his "theory."
Of course, he never bothers to tell us under what mathematical formalism his "theory" rests so it is impossible to say much of anything.
If he has in mind the standard ZF axioms combined presumably with something like, "There exists no set whose cardinality is infinite" he can get rid of the real numbers. And the irrational numbers too for that matter.
Yeah, that will be fun... Try describing the real world without pi that most famous of irrational numbers... Ask him at what decimal do we stop since pi must be finite in his "theory?"
BTW, what set theorists "believe" is of very little import to this discussion IMO.
DeleteWhat they can *prove* on the other hand...
Hi Robin,
Delete>There are, by many orders of magnitude, more ways the next five minutes could pan out if the laws of physics might be altered than if they held.<
I don't think this is true. On quantum mechanics, more or less anything is possible. In fact, there are fewer ways for miraculous events to happen because miraculous events are essentially an infinitesimal subset of what can happen on quantum mechanics.
There is some finite but tiny probability that any particular particle will suddenly jump a meter to the left. There is therefore an infinitesimal probability that all the particles that make you up will suddenly jump a meter to the left - a miraculous teleportation of a person. But there are many many more ways that quantum events to pan out that we would regard as unremarkable, and that's what almost always happens. We don't see miracles because they are impossible but because there are vastly more non-miraculous ways for the universe to be.
It's like the idea of gas molecules in a room. In principle, they could all happen to bunch up in one tiny corner of the room. There's no fundamental law of physics that rules this out. They don't because there are vastly more ways for them to be spread out then there are for them to be in a corner. The mundane is so because it is common, and it is common because there are so many ways for it to be.
Your idea that there are vastly more ways for miracles to happen is exactly backwards.
Incidentally I have a simpler, (and I think more accurate) description of mathematics. Two in fact but I think they are analogous - one procedural the other ontological.
ReplyDeleteDef 1 Mathematics: "Manipulating some set of symbols you have made up, according to some set of rules you have made up"
Def 2 Mathematics: "Any information that is new to you which can be derived by the method described in Def 1"
And I think that it is a neat little paradox that we have discovered necessary truths by manipulating a bunch of symbols we have made up according to a set of rules we have made up.
DeleteAnd let me clarify def2, "... derived just by using the method described in Def 1"
ReplyDeleteRobin -
ReplyDeleteI think you're a bit confused about the parallel axiom. There are multiple different geometries where that axiom fails, but every single one can be instantiated without a problem in our physical universe. You're not creating a new universe when you draw a circle as opposed to a line. Its a different type of system with different characteristics, but its all mathematics. Its simply that the parallel postulate is not a "foundational" axiom that MUST be true, as so many from Euclid on had assumed (or doubted).
This only means that those axioms aren't necessary as primitive truths that any "foundational" axiomatic theory is built on (think Set Theory, as every area of mathematics should in principle be reducible to sets and their relations).
As for your objection based on algorthmic intelligences, you make some claims that have no possibility of ever being proven and some heavy assumptions as well (why are these intelligences trying to make math inconsistent? Why are we even talking about weird demigods that probably aren't even metaphysically coherent?).
Manyoso -
As far as your similar concerns, they at least relate to the foundations of mathematics rather than some particular geometric system. That being said, many set theorists (Hugh Woodin, Peter Koellner, etc) argue that there is only one mathematical universe and that we will be able to utilize techniques of set theory and large cardinal axioms to discover once and for all the ultimate foundation of mathematical existence (which would put a rest to whether or not the CH and other statements are true or false). Others are pluralists about set theory, believing that any consistent axiomatic system is admissible, while also arguing that each type is as real as any other (prominent example being Joel David Hamkins). Thus Platonism (or the MUH) has very little to fear from that idea.
I am afraid it does not get that far. Tegmark does not believe in infinity as used to describe the physical world.
DeletePresumably this means he thinks there are a finite number of finite math structures.
Perhaps if he would define his formal system this claim could be evaluated, but without his formalism we are left to guess at what he means.
Too much hand waving to really evaluate whether his theory is even consistent let alone whether if it makes any predictions or describes the real world.
Pete wrote: "(why are these intelligences trying to make math inconsistent?)"
DeleteIt is mathematically possible that those intelligences doing anything can exist and under MUH anything mathematically possible *does* exist and therefore they exist.
No assumption there whatsoever- it is a very definite consequence of MUH.
And why are you saying that algorithms are metaphysically incoherent? I don't get that. The digital demigods are a mathematically well defined entity and therefore they exist under MUH.
And what part of what I said is contradicted by what you said about the parallell axioms?
Pete,
DeleteAlso, what claim do you say I make that has no possiblility of being proven?
Pete - and finally the digital demigods are not trying to make maths inconsistent, they are merely trying to make perceived physical reality mathematically inconsistent to the observer.
DeleteRobin -
DeleteThat's not a consequence of the MUH at all. To use your exact phrasing:
"An algorithmic intelligent could make a universe in which there is sometimes an appearance of mathematical consistency but change that at will - in other words to produce miracles."
What does that even mean? The MUH makes a claim about abstract mathematical entities existing and forming the basis of reality. Why are you talking about miracles and changes in consistency? That is what's incoherent about your argument, not any definition of "algorithm."
As far as parallel axioms, apologies for not splitting that up for "Patrick G" and his question on the parallel axioms (his name was right next to yours. Rookie mistake).
Now getting back to these super duper demigods, they have absolutely no way of being empirically proven and they don't really add anything to the discussion. I could start believing that a couple of demons in my car engine are what cause the pistons to turn, but there's probably a reason no one seriously considers the possibility. Why would they even go through the trouble of making the universe seem mathematically inconsistent?
Like they would implant the idea that 1+1=3 or something? I'm at a loss. I don't even think a rational brain that knows 1+1=2 could believe that without all of that syntax meaning something very different.
Thanks to all of you for all the great questions above, and for providing a dramatically higher level of discussion than I've seen on some other blogs! Let me briefly address the infinity question, and get back to the others later tonight after some urgent chores.
ReplyDeleteI was not taking issue with infinity in *mathematics*, but in *physics*, arguing that we lack experimental evidence that there's anything truly infinite in the physical world - for example, an infinite number of particles, an infinite magnitude of something (energy, density, curvature, etc.) or a true spatial continuum with an infinite number of points. Of course I assume the contrary in every single physics class I teach at MIT, by introducing 3+1-dimensional pseudo-Riemannian manifolds to model spacetime, by introducing real-valued quantities such as the electric field components in this morning's electromagnetism class, etc. However, whereas a generic real number requires infinitely many decimals to specify, we've never measured anything to more than 17 significant digits in physics, and have good reason to think that too many significant digits don't even make sense: for example, that the continuum model of space breaks down below the Planck scale of about 10^(-34) meters.
Moreover, many examples of where I use the continuum in my teaching are *known* to be fundamentally incorrect: whenever I use the concepts of temperature, density and pressure as continuous functions of position to derive the equations of fluid dynamics, the properties of sound waves, etc., I know that the continuum is merely convenient approximation, since real substances are made of individual atoms for which it doesn't make sense to speak of a temperature, say.
To me, this raises an embarrassing question for us physicists: if infinity is merely a convenient approximation in *some* circumstances, how do we know that it isn't merely a convenient approximation in *all* circumstances?
As I describe in chapter 11 of the book, it is precisely the hypothesis that space is a true continuum that can expand by arbitrarily large amounts that is responsible for one of the most embarrassing problems facing physics today: the cosmological measure problem that prevents us from making reliable predictions. I therefore feel that the infinity assumption in physics must not be exempt from the list of assumptions that we question. I feel that this point deserves to be discussed independently of and separately from any issues to do with the MUH.
/Max
Hi Max - thanks for joining in. I would be interested to know if you agreed that if your reasoning about the MUH is true but the MUH is falsified then it would mean that the External Reality Hypothesis was also falsified.
DeleteHello Max,
DeleteFirst, kudos to you for answering the points raised. And I apologize if my scare quotes around "theory" is dismissive, but you confused me in the podcast by saying MUH isn't a theory, but a prediction... maybe you can clarify what you meant.
Now, on to the question of your skepticism of infinity conflated with your MUH. I am not sure they can be separated so easily.
For MUH, if I understand correctly, you are positing the existence of a collection of universes each with a mathematical structure defining it. Well, does the formalism that defines these structures include infinite sets or not?
If so, then by definition you believe that infinity is necessary to describe the mathematical/physical world.
If not, then your MUH is based on an entirely unknown set of axioms com
err... If not, then your MUH is based on an entirely unknown set of axioms forming a mathematical system completely unrelated to the "math" that physicists take for granted. In that case, I would say that your MUH is ruled out by the extraordinary success of the standard mathematical axioms which permit the existence of infinite sets.
DeleteWhat's more, you are the one that brought up infinities and your skepticism of them in the podcast in relation to a question about MUH. If you wish to separate the two discussions, then please reformulate your answer to Massimo's question with out regard to your skepticism of infinities. In particular, I think he was asking about the implications for MUH of Godel's Incompleteness Theorem...
Thanks Robin & manyoso for these interesting questions.
DeleteRobin: if you agree with my argument that ERH implies MUH (which you're certainly not obliged to), then if MUH is falsified, you'd indeed need to reject ERH - unless you reject modus tollens, which I suspect you don't!
manyoso: My point was that it's important to question the assumption of infinity in physics - and that this is important regardless of whether one has read my book or not and regardless of any views that one has about the Mathematical Universe Hypothesis.
The Mathematical Universe Hypothesis is simply the hypothesis that our external physical reality is a mathematical structure, so the hypothesis itself doesn't say anything about the Level IV multiverse. I give an argument in the book that MUH implies the Level IV multiverse, but this is again an argument you may choose to object to.
My point was that it's important to question the assumption of infinity in physics - and that this is important regardless of whether one has read my book or not and regardless of any views that one has about the Mathematical Universe Hypothesis.
DeletePoint taken. But my point is that these two ideas: that our universe does not have any infinite examples, and the multiverse-MUH seem mutually incompatible.
I will look forward to reading your book when I can, but if your MUH implies a multiverse of mathematical structures, then under what formalism?
By what rules is this mathematical multiverse defined? Does it allow for infinity or not?
Just to clarify - a "digital demigod" or "algorithmic intelligence" can mean some Turing Machine + some strings. So there is nothing incoherent about it.
ReplyDeleteAnother issue - if infinities were impossible, on what basis would they be impossible? After all, even in the computable universe infinities and continuous values are mathematical possibilities.
ReplyDeleteSo ruling out some mathematical possibilities would imply some ontology which would really just reduce the CU to a physical realm.
The argument is not that infinities are logically impossible, but that we ought to be skeptical of whether they happen to exist in this universe.
DeleteShould we be equally skeptical if they happen in other universes in Tegmark's multiverse-MUH?
DeleteIf I understand Tegmark correctly, he regards the infinite as a consistent mathematical idea and so there must be mathematical structures that incorporate the idea of infinity. This universe could be one of them, we don't know. Whether you call these structures universes is up to you.
DeleteRobin -
ReplyDeleteThat's not a consequence of the MUH at all. To use your exact phrasing:
"An algorithmic intelligent could make a universe in which there is sometimes an appearance of mathematical consistency but change that at will - in other words to produce miracles."
What does that even mean? The MUH makes a claim about abstract mathematical entities existing and forming the basis of reality. Why are you talking about miracles and changes in consistency? That is what's incoherent about your argument, not any definition of "algorithm."
As far as parallel axioms, apologies for not splitting that up for "Patrick G" and his question on the parallel axioms (his name was right next to yours. Rookie mistake).
Now getting back to these super duper demigods, they have absolutely no way of being empirically proven and they don't really add anything to the discussion. I could start believing that a couple of demons in my car engine are what cause the pistons to turn, but there's probably a reason no one seriously considers the possibility. Why would they even go through the trouble of making the universe seem mathematically inconsistent?
Like they would implant the idea that 1+1=3 or something? I'm at a loss. I don't even think a rational brain that knows 1+1=2 could believe that without all of that syntax meaning something very different.
No need for them to be empitically proven, they.are algorithms running programs. Are you saying the mathemafical universe can't have algorithms running programs. Imaginw Godzilla turns up tomorrow in hot pants and a top hat dancinf the macarena in Central Park. That sort of thing. The only way it might be impossible is if consciousness is not computanle. But if that is ghe case then the whole MUH is in trouble.
DeleteSo they are well defined mathematical objects, so of course MUH predicts they will exist with a probability of 1.
Hi pete,
DeleteWe're both on board with the MUH, but I don't think Robin's suggestion is as incoherent as you think. If the Computational Theory of Mind is true, then minds are mathematical structures. There must exist a mathematical structure which corresponds to some hugely intelligent mind. This mind can essentially imagine a universe just like ours, thereby creating it. So far, I think he is right. Robin's argument is that the probability of any observer finding themselves inside such a universe is greater than finding themselves in a physics-driven universe, and so we should be seeing miracles all over the place. The fact that we do not is supposedly evidence against the MUH.
Robin's argument is wrong because I think he has the probability backwards, and also because I'm not sure that we should be seeing miracles all over the place even if we were inside the mind of a god.
But the idea of a creator god itself is not incoherent.
Hi Pete,
DeleteYou also ask the "why" question again. But as I have pointed out, according to the hypothesis mathematically possible objects exist. So long as these algorithms are mathematically possible (and of course they are) it is a consequence of the hypothesis that they exist.
You need just 3 things for there go be digital demigods:
ReplyDelete1. Consciousness is computable
2. The logic of a universal computer can be implemented
3. Sufficeintly long programs
Which do you say are ompossoble in ghe mathematical universe?
Hmm... "ompossoble"? Note to self - don't answer these on phone.
DeleteHi Max
ReplyDeleteThe problem is more to how a particle or any mass can exist infinitely temporally and retain spatial dimensionality.
In particular, whether mass can exist before a Big Bang and for how long before the event?
This saves the embarrassment of "something from nothing" and retains "matter cannot be created or destroyed". But can it be compressed for an infinite time awaiting an neutralizing expansion?
I assume it can have rest mass to convert to energy to expand (accepted by Einstein himself) but existing infinitely like that is the issue, as well as the trigger for expansion.
These are real physical states and events with real infinity of the state exists forever, and the things math describes. A real application of infinity - as a useful hypothesis.
According to what I heard, the fact that there are arbitrary symbols and names for integers, it means that we can ignore the things to which integers refer when we use them to building something? How we refer to integers is not the same as how integers refer to things - like concrete pouring. Without things to which they refer they are called abstracts.
ReplyDelete(like in building something)
On the "no infinities in physics" proposition: There are the hypothetical Malament–Hogarth spacetimes (which allow for the implementation of certain non-Turing computable tasks), but this may just be physics fiction. :)
ReplyDelete"I was not taking issue with infinity in *mathematics*, but in *physics*, arguing that we lack experimental evidence that there's anything truly infinite in the physical world..." Max
ReplyDeleteAll that is lacking in science or physics is certainty, the absolute. =
Questions: Is the Universe measurable or immeasurable, finite or infinite? If it is measurable, what is its measure, and if the Universe is the latter, immeasurable, what of its parts? Are not the parts that science measures and divides from the Universe equally immeasurable too? Is not the whole the sum of its parts? Isn’t the crux of physics today, the uncertainty of its measure? The sciences that measure and divide the Universe with probability or dice, have they counted all the measurable or is it immeasurable pieces yet? How many parts are there in the Universe, should they even try? And lastly to help Einstein resolve the unification problem of energies and mass, are they or are we, it is us, One equals One, divisible or indivisible; is not the Universe equal, infinite, self-evident and free?
=
Re the “digital demigod” problem, maybe I am wrong about the probabilities but I don’t think so.
ReplyDeleteI would like to go through it just once more as there has been some misunderstanding.
The “digital demigod” is some universal automaton plus bit strings as necessary.
So it is a perfectly well defined mathematical entity.
It can compute any observable state of affairs in the universe including the minds of the observers.
For each state of affairs there can be many distinct mathematical structures which can compute this.
There are infinitely many bit strings which can be equivalent algorithms.
For each of these the machine might be running on a non-discrete substrate and so there can be an uncountable infinity of variations in trigger ratio and within each ratio an uncountable infinity of local fluctuation.
Each automaton can model itself recursively and each can create arbitrarily many of these computations.
If any mathematically possible structure can exist then all of these distinct entities will exist for every observable state of affairs in the universe including this one at a probability of 1
And there can be infinitely many variations in the algorithm which we do not observe - teapots round Ganymede, Tellytubbies on Io etc.
And the number of ways the next 60 seconds could pan out, if there could be miracles, is many orders of magnitude higher than if the laws of physics hold.
So if I could be any mathematical entity which can account for my experiences right now then the probability that I will experience a miracle in the next 60 seconds is arbitrarily close to 1.
I think that this really does pose a serious problem to the Mathematical Universe Hypothesis.
And, even if you disagree about the probability, you are still committed to the proposition that there are universes (among others) where you consciously observe Godzilla coming over the horizon in hot pants and top hat, dancing the macarena.
I have to ask - do you *really* think that?
Hi Robin,
Delete>There are infinitely many bit strings which can be equivalent algorithms.<
OK, but in that case I would say there is only one algorithm. If the algorithm corresponds to the universe, then all those bit strings are describing the same universe.
>here can be an uncountable infinity of variations in trigger ratio<
This is kind of meaningless to me. For the reasons outlined above, I don't care about the machine but about the algorithm. Do these variations in trigger ratio have an affect on the algorithm? How so? Might these be perceived as quantum fluctuations, for example?
>Each automaton can model itself recursively and each can create arbitrarily many of these computations.<
Hmm, maybe? I don't know, this sounds like the kind of thing that might run into paradoxes like the self-applicability problem. I'm not sure if what you propose is consistent, but either way I don't think there's a problem so I'll accept it for now.
>If any mathematically possible structure can exist then all of these distinct entities will exist for every observable state of affairs in the universe including this one at a probability of 1<
So, for every mathematical structure there is an algorithm that can simulate it? Not exactly. For every computable mathematical structure, yeah. But there are uncomputable structures. But I think this universe is probably computable so that doesn't matter too much.
>And there can be infinitely many variations in the algorithm which we do not observe - teapots round Ganymede, Tellytubbies on Io etc.<
Agreed.
>And the number of ways the next 60 seconds could pan out, if there could be miracles, is many orders of magnitude higher than if the laws of physics hold.<
No. Precisely wrong.
Let's consider the example of a teapot orbiting Ganymede. Let's suppose that it's a tiny teapot made of 1000 carbon atoms. It would indeed be miraculous if we found an object orbiting Ganymede in the shape of a teapot.
But it would not be regarded as miraculous if those carbon atoms were a blob. And there are many more ways of being a formless blob than there are of being a teapot. Or of being ten formless blobs. Or of each carbon atom orbiting Ganymede separately.
It would be miraculous if a teapot were orbiting Ganymede only because that is such a specific and improbable organisation of matter, improbable because there are so very many ways for the same matter to be arranged that we would find unremarkable. As such, there are many orders of magnitude more ways for the universe to be non-miraculous than for it to be miraculous.
>you are still committed to the proposition that there are universes (among others) where you consciously observe Godzilla coming over the horizon in hot pants and top hat, dancing the macarena.<
Sure, but this is also possible given the perfectly mainstream understanding of the laws of quantum mechanics. A huge number of particles could spontaneously so organise themselves and make this event happen, but it is so infinitesimally improbable as to be effectively impossible. Forgetting the more controversial MUH, even the Many Worlds interpretation of QM says there is a universe where this happens, as strange as it sounds. But it only seems strange if you don't really grasp what infinitesimally improbable means, or the magnitude of the disparity between the number of universes where nothing so outlandish happens and those where it does.
You are missing the point about the teapot around Ganymede. It can be a teapot or a blob, the point I am making is that it is a variation.
DeleteAnd there is a difference between infinitessimal probability in MWI and a probability of 1 under MUH.
Actually it is only recently that MWI became an actual hypothesis rather than an interpretation.
Hi Robin,
Delete>You are missing the point about the teapot around Ganymede. It can be a teapot or a blob, the point I am making is that it is a variation.<
With respect, I think it is you who misses the point. The point is that there are an essentially infinite range of variations we would regard as unremarkable, and a vastly smaller range of variations we would regard as miraculous. There very well could be a blob of 1000 carbon atoms orbiting Ganymede. If this were discovered, it would surprise nobody. It would only be surprising if it were very definitely and perfectly in the shape of some recognisable artificial structure such as a teapot, and these configurations are a tiny subset of the possible configurations.
There may be many ways for miracles to happen, but there are many many more ways for nothing miraculous to happen. The air molecules in the room in which you sit alone could be in infinite different positions and velocities without your noticing any difference between them. But if all the air in your room happened to leave that room you would surely notice, and you would regard it as a (negatively?) miraculous event.
>And there is a difference between infinitessimal probability in MWI and a probability of 1 under MUH.<
No there isn't. On MWI, there necessarily exists, with a probability of one, a world where Godzilla manifests dancing the macarena. What is infinitesimally improbable from the point of view of an observer is that this event will ever be witnessed, because there are vastly more universes where this never happens, so that it can essentially be regarded as impossible even though it technically isn't. The same logic holds for the MUH.
Under MUH mathematically distinct digital demigods for each and every possible state of the multiverse are uncountably more probable than the entire multiverse and the same goes for each miraculous variation on it. You seem to be thinking of the MU as a sort of physical realm.
Delete>Under MUH mathematically distinct digital demigods for each and every possible state of the multiverse are uncountably more probable than the entire multiverse and the same goes for each miraculous variation on it.<
DeleteI don't see that, at all.
>You seem to be thinking of the MU as a sort of physical realm.<
Why do you say that?
Hitler's cosmology gig is up!
ReplyDeletehttp://youtu.be/U6_adWD5H-s
It seems obvious that we will never observe anything infinite in the physical
ReplyDeleteworld. If some quantity were infinite, how would we even measure it? We can
only ever get lower bounds on observables, or a finite number of measurements.
So the fact that there is a "lack of experimental evidence that there's
anything truly infinite in the physical world" doesn't actually tell us
anything about the existence of infinity in the real world. When a theory
predicts infinity, it's usually considered as a failure or limitation of that
theory (e.g. GR and singularities). For me the claim of lack of infinity in
physics is not falsifiable.
A fact however is that the extreme success of physics is due to mathematics
with the explicit use of infinity. Without the concept of infinity we would not
have gotten far. Try to obtain predictions under QM or GR without differential
or integral calculus. To claim that this is possible even "in principle" would
be pretty bold, and should be backed up with strong evidence.
So if Max wants to restrict himself to finite maths to avoid paradoxes and inconsistencies, he is biting off the hand that feed his hypothesis, namely that
physical reality is well described by maths.
Hi LLDE,
DeleteYou are right to be skeptical of a finite universe, but Tegmark is probably right to be skeptical of infinity. His point is that the only defensible position is infinite-agnosticism, and too often physicists assume that just because their simplified equations treat space and time as continuous that it really is so. GR predicts a singularity at the center of a black hole, but we don't actually know if this is true, and skepticism/agnosticism is appropriate until we know more.
If we ever do.
Hi manyoso and others, who wonder what the point is of questioning infinity/the continuum. The point is that we IMHO have a hugely embarrassing problem in modern cosmology, brought on by the Level II multiverse of eternal inflation: the measure problem, which (as I explain in detail in the book) ruins our ability to make predictions. This problem needs to be solved one way or another, and the assumption of a continuous space that can be eternally stretched by inflation is a direct cause of the problem.
ReplyDeleteYou reject infinity in order to define some inflation problem that ruins predictions? I thought that you disavowed infinity in order to evade some bogus Godel problem. You believe in many-worlds that ruins predictions anyway, as all possible predictions take place in some of the universes.
DeleteHi Roger: We're currently stuck with the inflationary measure problem (that we can't make predictions) whether we like it or not. It only occurs if space can be infinitely stretched, which is why I think it's a good idea to test that assumption rather than simply take for granted that it's true.
DeleteWe *don't* have good evidence that we need to get rid of infinity because of Gödel's incompleteness theorem, which I explore in detail in chapter 12 of the book (http://mathematicaluniverse.org).
Is the matematical universe hypothesis ruled out by Gödel's incompleteness theorem? No, not as far as we know. Given any sufficiently powerful formal system, Gödel showed that we cannot use it to prove its own consistency, but his doesn't mean that it is inconsistent or that we have a problem. Indeed, our cosmos doesn't show any signs of being inconsistent or ill-defined, despite showing hints that it may be a mathematical structure. Moreover, what were we hoping for? If a mathematical system could be used to prove its own consistency, we'd remain unconvinced that it actually was consistent, since an inconsistent system can prove anything. We'd only be somewhat convinced if a simpler system that we have better reason to trust the consistency of could prove the consistency of a more powerful system - unsurprisingly, that's impossible, as Gödel also proved. Of the many mathematicians with whom I'm friends, I've never heard anyone suggest that the mathematical structures that dominate modern physics (pseudo-Riemannian manifolds, Calabi-Yau manifolds, Hilbert spaces, etc.) are actually inconsistent or ill-defined.
Please note that I'm using "mathematical system" to refer to "formal system", as distinct from "mathematical structure": the former can describe the latter and the latter can be a set-theoretic model of the former.
PS: Roger: It's not correct that Many-Worlds "ruins predictions"; there's a perfectly well-defined prescription for calculating probabilities for what we'll observer. In the same way, the the appearance of probabilities in the Copenhagen Interpretation interpretation doesn't "ruin predictions": it's just that with quantum mechanics, some of our predictions become statistical.
DeleteExcellent episode.
ReplyDeleteI especially appreciate Julia’s question, “Is the Math a tool for physics? Or, is Math itself giving rise to physics laws?”
My opinion is that Math itself gives rise to physics laws. But I do not take the Multiverse view. I will show this by giving the ‘Ultimate Nature of Reality’ an operational definition first.
One, It must be ‘eternal’, that is, it is time-independent; not created in-time and cannot be destroyed at the end of time.
Two, It must give rise to ‘this’ universe; an itemized list of realities, such as,
1. gives rise to nature constants {Alpha, e (electric charge), c (light speed), ħ (Planck constant), etc.},
2. gives rise to the particle zoo of the Standard Model,
3. gives rise to baryongenesis,
4. gives rise to … everything in ‘this’ universe.
A failure of any one of the ‘gives rise to …’, it cannot be the Ultimate theory.
Now, I will introduce a mathematical structure. A system of two persons (I and a ghost) and one game (flipping an American quarter 10 times as a game).
By playing alone (me only), the probability of the outcome patterns (one head/9 tails; …, 9 heads/one tail, etc.) should all be the same after played a large number (such as, 10^500) of times. Now, the ghost (not visible by me) comes in and tries to mess up my play either randomly or with a planed-system (of sabotages).
Now, there is a “Ghost-rascal conjecture”:
“For a coin flipping game (head vs tail), T is the number times flip as one ‘game’, N is the number times that that ‘game’ is played. If T >= 10 and N >= 10^500, then no amount of sabotage from a Ghost can change the outcome of this game.”
Thus, the bigger the T is, the smaller the N is needed. When T = 1, the conjecture could fail unless the ‘N’ goes to infinite. When T >= 3, the power of immutable becomes strong. Thus, I will show a game with T = 3, as below.
Game 1: (tail, head, head)
Game 2: (head, tail, head)
Game 3: - (tail, tail, head)
Can these three games make contact to this physical universe? Let’s add three spices for this game.
One, hot-juice: the head carries 1/3 of electric charge, the tail with zero (0).
Two, color: the first flip (or spin) is red, yellow the second and the blue the third. Then, every game carries a color-tag which is the color of the ‘single’, such as the Game 1 is red; Game 3 is blue.
Three, twister: flipped by left hand (sabotaged by the Ghost) is marked with a negative sign. Flipped by the right hand carries a positive sign for the game.
With these spices, this game can actually describe all the Standard Model particles (excluding the bosons) symbolically.
In summary, this game has the following attributes.
1. It is time-independent, not created in time and will not be destroyed at the end of time.
2. It is immutable; no amount of sabotage can change the outcome of this game.
3. The left-hand game (sabotaged by the Ghost, marked with a negative sign) is the ‘source’ of ‘orderliness’. That is, there is a right-hand-genesis which is similar to the baryongenesis.
This is a bit different from Tegmark's ‘Ultimate Nature of Reality’.
In recent months, three major issues were discussed.
Delete1. Why is there something rather than ‘Nothing’?
2. Is the Math a tool for physics? Or, is Math itself giving rise to physics laws?”
3. Is the ‘Ultimate Nature of Reality’ a reality (reachable in physics)?
Then, there are some sub-issues (extended from the above).
a. Falsifiability,
b. Multiverse.
In Tegmark’s works, it has touched the most of these issues. Although I am 100% in agreement with Tegmark in principle about that the Math is fundamental while the physics laws are the emergent, I disagree with him on all other issues. Of course, it will not be fair to him if I disagree only by using the tongue in cheek.
His view of number is just an ‘address’ or some kind of relations is of course not wrong but is useless in the arena of ‘Ultimate Reality’. The key is the first issue; there is something become that it is only the ‘emergent’ of “Nothing”. Thus, we must show that the ‘Nothing (zero)’ is a physical reality in addition to as a mathematics concept.
The Ghost-rascal game in my previous comment is very much ‘physical’ which can be played by a first grader. And, the probability of each game-pattern at any given number N (i) can be defined with a probability function Λ, such as,
Function Λ (i) = P (top, the largest probability of a game-pattern) – P (bottom, the smallest probability of a game-pattern) at any given N (i).
Yet, by the definition of the conjecture, the probability of each game-pattern should all be the same at a number N (z), that is Λ (z)= 0 (exactly) at N (z). For a number N (i) < N (z), Λ (i) > 0. When this N (i) is very large, this Λ (i) should be almost to be zero. And now, zero (0) is precisely defined operationally in a finite {N (z) is very large but finite} ‘physical’ game, in addition to be only as a math concept. Furthermore, we should find such a Λ function in ‘this’ universe. If we do, there is no need for a multiverse. Furthermore, there is a concrete substance {the Nothing (zero)} in ‘this’ universe as the ‘Ultimate Reality’.
"... we lack experimental evidence that there's anything truly infinite in the physical world ..." — Max Tegmark
ReplyDeleteAccording to Kroupa, Pawlowski, and Milgrom, "Understanding the deeper physical meaning of MOND remains a challenging aim. It involves the realistic likelihood that a major new insight into gravitation will emerge, which would have significant implications for our understanding of space, time and matter.”
http://arxiv.org/pdf/1301.3907v1.pdf “The failures of the standard model of cosmology require a new paradigm”, Jan. 2013
Assume nature is finite and digital + string vibrations are confined to the Leech lattice.
Google "pavel kroupa youtube" & "space roar dark energy".
MOND (MOdified Newtonian Dynamics) is really off-topic, and has nothing to do with the discussion about infinity in the physical world.
ReplyDeleteAnyways, MOND as an alternative for dark matter is pretty much on the downgrade, since explaining objects like the bullet galaxy cluster with MOND requires additional, non-baryonic dark matter (http://arxiv.org/abs/astro-ph/0609125). It seems like you're getting to live with dark matter, no matter what...
Some good commentary here so far, but I did want to mention that there are actually ways that we might be able to indirectly prove that infinity actually exists, or at least has definite implications for mathematics of a more finite and "natural" bent. If you're a Platonist or realist about mathematics, then this would definitely force one to take it as very interesting evidence in favor of infinity's reality (thought nominalists might also have to reassess things as well).
ReplyDeleteThe idea of concrete incompleteness is something mathematician Harvey Freidman has worked on extensively. It is the idea that certain patterns inherent in numbers (that can be easily observed) actually depend on certain large cardinal axioms. I know, its pretty mind blowing. If you want to take a look at some of this, start with this article that was published a few years back: http://www.telegraph.co.uk/science/8118823/Large-cardinals-maths-shaken-by-the-unprovable.html
It would be interesting to get some input from Max himself or others who are debating the nature of infinity in light of a pretty incredible development like this. The discoveries are pretty recent, but this could be a monumental development in the field of mathematics
Hi Pete: I agree with you that it would be fascinating if we could devise a physical experiment whose outcome depends on whether something truly infinite exists. It would be even more intriguing if we could perform an experiment (performed in a finite volume of space during a finite time) whose outcome depended on a Gödel-undecidable statement.
DeleteIf you can think of such an experiment, please let me know!
Anyway, good discussion and thanks to Max for joining in. Got to get all this stuff out of my mind for a while now, so see you all later. :)
ReplyDeletePete: "The idea of concrete incompleteness is something mathematician Harvey Freidman has worked on extensively. It is the idea that certain patterns inherent in numbers (that can be easily observed) actually depend on certain large cardinal axioms."
ReplyDeleteWe're getting a little off-topic, but that would be very unexpected if Friedman had shown such a thing. But what the linked article says is a bit different: "In particular, the existence of large cardinals is the condition needed to tame Friedman's unprovable theorems. If their existence is assumed as an additional axiom, then it can indeed be proven that his numerical patterns must always appear when they should. But without large cardinals, no such proof is possible."
To show that the patterns depend on LCAs, you need to show that assuming the LCA implies the pattern exists, and assuming the negation of the LCA implies the pattern does not exist. But this is actually "assuming the negation of the LCA implies that there can be no proof that the pattern exists." And this leaves the possibility that the pattern exists independent of the LCA. So, not really anything new...
I suspect that what Friedman has done is shown how a finite proof of such a statement can be written if we assume some particular LCA (still a remarkable accomplishment). Then the statement is (finitely) provable in (ZFC+LCA), but not in ZFC (assuming ZFC is consistent). That means the statement is true, provided that (ZFC+LCA) is consistent. But in the natural numbers, either a pattern holds out to infinity, or does not (decidable in a countable number of steps), so if (ZFC+LCA) is consistent, then the statement is true (but not provable) in ZFC, something we already expect to find, thanks to Godel.
Richard -
DeleteThat's an excellent point. I should have further elaborated on one of the main reason's why this is an important result for the philosophy of mathematics. There are many mathematicians, including examples such as Lakatos (who definitely subscribed to a strand of Platonism), that believe elements of empiricism should be incorporated into mathematics to discover new truths. Even though proof theory has been ingrained into mathematical practice (and very much deservedly so), these individuals believe that empirical evidence has an extremely important place in uncovering new mathematical truths and practicing mathematics. In fact, empiricism came well before mathematics in the development and evolution of the discipline, preceding the formal axiomatic systems by many centuries.
If these patterns are discovered in nature, and they hold for a large number of cases (which grows over time), then that should be taken as strong evidence in favor of it being a mathematical truth. If this result can then be proven only through large cardinal axioms, this would by extension lead one to believe that large cardinal are themselves real.
The philosopher James Robert Brown has an excellent book about the philosophy of mathematics (he strongly leans towards Platonism) that develops just this type of argument for empiricism. I would definitely encourage anyone interested in the subject to read it:
http://www.goodreads.com/book/show/469677.Philosophy_of_Mathematics
"I understand what you are saying but I do think that I could make a case for the GoL not being closed under certain operations." Certain operations -- but not the operation that defines GoL, right? Otherwise I'm not sure what that case would look like.
ReplyDeleteI share the view that any consistent set of axioms is as valid as any other -- at least, as far as mathematics is concerned. Nothing about Woodin's approach undermines Platonism as a whole, but studying it made me revisit the idea more thoroughly. Woodin is pushing (as a Platonist) for Omega consistency (rather than consistency) as the standard for validity. It is a stronger standard that would resolve CH in the negative, but it is unnecessary in my opinion. I would think formalism is less constrained over standards than Platonism, and holds more strictly to the view that "there is no one true mathematics". But I don't know if I can really associate myself with any of these "isms". Or maybe for me it is just "pragmatism". I'm not going to look for empirical reasons to accept or reject AC or CH, for instance, because I don't think they will be found, and I don't think they will ever have physical consequences. It is enough to leave them undecided.
Hi Richard,
ReplyDelete>Certain operations -- but not the operation that defines GoL, right?<
My thinking was not to define Fibonacci in terms of ordinary addition but in terms of a restricted kind of addition, say "faddition" which is only permitted on adjacent Fibonacci numbers. It would be as illegal to "fadd" two non-adjacent Fibonacci numbers as it is to divide by zero. Possibly this way of thinking about it does not really make sense, but that was my intuition. In a similar way, the operation that defines the GoL is a restricted form of the more general operation that says "set the state of the matrix to M).
I don't see why formalism is more true to the view that there is no one true mathematics than *plenitudinous* or *full-blooded* Platonism. That's basically the central idea.
I'm not sure if it is possible to define "3+5=8" without the implicit knowledge that there are natural numbers in between those numbers. In fact, the Fibonacci set, taken outside the context of the natural numbers, loses much if not all of its semantics. "13" is no longer "the successor to 12" as it is in Peano arithmetic or set theory.
DeleteIn my case, by Platonism, I was referring to traditional Platonism, which might not be related to MUH at all, but mainly, I don't really think any of these isms are going to produce profound truths about mathematics. They are really just views about mathematics. And I'm just more inclined (nowadays) to be skeptical of ideas that I might have once accepted because of alignment with some particular ism.
Hi Richard,
DeleteYou're probably right on Fibonacci.
In a way I think you're right about Platonism. I've made a similar point about this on my blog. Basically, there is no fact of the matter about whether mathematical objects exist or not. It depends what you mean by exists. The concept of existence becomes ambiguous when referring to objects outside our universe. I prefer to say that mathematical objects exist because I think this is an intuitive and useful way to think about them.
So you're right, it doesn't matter much either way. Until you come to the MUH at least. So what does no fact of the matter mean for the MUH? It just means that it doesn't make sense to ask whether this or any other universe exists. The concept of existence doesn't apply to universes any more than it does to mathematical objects. Either way, the existence of the universe is explained. Either its existence is an illusion or it does exist, but only in the way mathematical objects do.
To a formalist, "existence" is simply a predicate which has syntactic -- not semantic -- consequences for other objects (which may or may not "exist"). But few people would say that "existence" in the physical world is devoid of semantics. So formalism has its own consequences for MUH (except to those few), and those consequences are the opposite of Platonism. (I don't think either ism is more justified than the other.)
ReplyDeleteIf "existence" does have semantics in mathematics, it doesn't seem to be the same semantics that the word has in physics. The empty set exists, but you can't really do anything *to* it, although you can do things *with* it, like instantiate it in other sets. I don't even think that counts as doing something "with" it, because the containing set actually exists a priori, so all we can do is talk about the sets of a universe, not create them. On the other hand, a proton exists, and in the physical world, "existence" implies that you can do things *to* it (accelerate it through the LHC, for example).
Hi Richard,
DeleteAgain, I'd say there is no fact of the matter on whether mathematical objects exist. It depends on your perspective. From the formalist's perspective, they don't. But this is compatible with the MUH if we assume the universe doesn't actually exist. This is a strange point of view to adopt which is why I prefer Platonism.
>If "existence" does have semantics in mathematics, it doesn't seem to be the same semantics that the word has in physics.<
This is compatible with my view that physical existence is a relative predicate. The chair I am sitting on physically exists from my point of view. The chair Luke Skywalker is sitting on is real from his point of view. Neither of our chairs exist from the point of view of the other.
Let's take it back to math. I would note that if you need interaction to exist, we should consider only mathematical structures with something analogous to time, so the empty set is not the best example.
A glider exists from the point of view of another in Conway's Game of Life, because it can do stuff to it. A character in a computer simulation exists from the point of view of another, but not to an observer outside the simulation.
I therefore don't see any basis for confidence that there is any fundamental difference between abstract existence and physical existence.
Richard -
DeleteYou mention the existence of protons and us being able to "do" things to them, but do we honestly even have a handle on what they are? I remember bringing up this same point on Massimo's post from 12/2013:
"This problem is not one that is unique to the idea that the world is literally mathematics. Physicalism, a doctrine that virtually all scientists and a large portion of philosophers subscribe to, also runs into the same difficulties. I have always considered myself a physicalist as well, but after really contemplating what that means I start to revert to simply considering myself a naturalist, which contrary to many opinions can embrace abstract objects without a problem. When one really begins to delve into the nature of the "real" objects that are out there, their decidedly ephemeral nature is exposed. For one, objects in the universe are made almost entirely of empty space. The atoms that compose physical entities are something like 99.999% empty space, with a very tiny nucleaus and some extremely small electrons that occupy probability clouds around it. The nucleus itself is composed of even tinier quarks and gluons, the true nature of which (along with every other elementary particle in existence) is extremely hard to pin down. Maybe they're "vibrating strands of energy" as String Theory posits, or “knots” in the fabric of space-time as postulated by Loop Quantum Gravity. (I'll add to the original that the present description of fundamental particles is that they are in fact "point particles," which means they're 0-dimensional mathematical points).
The point is, what we think about as being physical is really no such thing at all. At the bottom, it really seems to become mathematical equations and relations. And at that point, a la Massimo's invocation of no miracles and the idea of indispensability to our understanding of the world, you have a strong case for mathematical realism. The ironic part of the opposite position, that of nominalism with regard to mathematics, where something must be physically instantiated in space-time if it’s to be considered real (something which makes mathematics nonexistent), is its inherent assumption of physicality that is largely misunderstood and nonexistent in and of itself."
Pete: Does it matter if we know what protons are? We have evidence of their existence, we can detect them, we can distinguish them from other particles, and as I said, we can manipulate them. Empirically, we know that a proton you detect today will have the exact properties as one I detected yesterday, and we have no reason to expect that will ever change. They are as physical as anything can be, and I say they exist, independent of our concept of "substance".
DeleteI can also say that mathematical objects exist. But in mathematics, I don't think the word has the same meaning. I can't take the number 5 and put it in a particle accelerator and collide it with the number 7.
DM, I agree, gliders exist in a real sense, from the point of view of inhabitants of the game of life. The game has fundamental particles (or the analogs of particles) which are the states of cells in the matrix. I'm not sure I agree about Luke Skywalker's chair, but that's because we can't establish that he and the chair inhabit a universe with a coherent set of rules governing its progress. What happens to Luke or the chair is what George Lucas tells us (or shows us), and narrative alone does not create physical worlds, no matter what some theologians may want to believe.