Tegmark’s Vast Math

I recently had a surprise chance to meet Max Tegmark, and so I first quickly read his enjoyable new book The Mathematical Universe. It covers many foundations of physics topics that he correctly says are unfairly neglected. Since I’ve collected many opinions on foundation of physics over decades, I can’t resist mentioning the many ways I agree and disagree with him.

Let me start with what Tegmark presents as his main point, which is that the total universe is BIG, almost as big as it could possibly be. There’s a vast universe out there that we can’t see, and will never see. That is, not only does space extent far beyond our cosmological horizon, but out there are places where physics sits in very different equilibria of fundamental physics (e.g., has a different number of useful dimensions), and nearby are the different “many worlds” of quantum mechanics.

Furthermore, and this is Tegmark’s most unique point, there are whole different places “out there” completely causally (and spatially) disconnected from our universe, which follow completely different fundamental physics. In fact, all such mathematically describable places really exist, in the sense that any self-aware creatures there actually feel. Tegmark seems to stop short, however, of David Lewis, who said that all self-consistent possible worlds really exist.

Tegmark’s strongest argument for his distinctive claim, I think, is that we might find that the basic math of our physics is rare in allowing for intelligent life. In that case, the fact of our existence should make us suspect that many places with physics based on other maths are out there somewhere:

Why do the five arrows in the bottom panel conspire to allow any habitable range of Higgs properties? This could well be a fluke: five random arrows would allow some range with 19% probability, so we need only invoke a small amount of luck. … However, it’s perfectly plausible that further physics research could uncover more striking fine-tuning of this discrete type with, say, ten or more arrows conspiring to allow a habitable range for some physical parameter or parameters. And if this happens, then we can argue … that this is evidence for the existence … of other universes where the laws of physics are different, giving quite different requirements for life!

I accept that this consideration would modestly push us in this direction, just as the existence of physics parameters fine-tuned for life modestly pushes us to believe that other spatial regions out there have different values of those parameters. I accept these because I accept the self-indication principle in indexical inference, which says that the info that you exist should increase your belief in possible worlds where you might have existed, in proportion to the number of slots where you could have existed.

However, it isn’t clear that Tegmark accepts the self-indication principle, since he also says that you should believe in many worlds quantum mechanics if you survive many rounds of quantum suicide, which is where you randomly suicide depending on a quantum event. The self-indication principle does not say that seeing this should increase your belief in many worlds, as there aren’t substantially more slots for you to occupy in scenarios where quantum suicide attempts fail. Since both many worlds and stochastic quantum mechanics predict exactly the same conditional probabilities of observations given quantum suicide attempts, such data can’t distinguish them. So I wouldn’t see surviving quantum suicide as further evidence for many worlds.

Tegmark also argues that we should reject a description of reality that has math plus extra “baggage.” Apparently, if there are many possible math descriptions of universes, then it would be extra “baggage” for the universe to point to only one math and say “that’s the math that really exists.” This is pretty close to the argument of David Lewis. But Tegmark also seems to accept that there is some “measure” over all these maths, and that the reason to take some actions over others is to favorably change that measure. Yet this measure also seems to also be an extra “baggage” over and beyond the math descriptions themselves. The main difference I see between adding a “this exists” pointer and adding a measure over the space of maths is that the later approach has more slots for me to exist, and so is favored by self-indication.

Tegmark does say that we struggle to find reasonable measures for many physics problems, and as a result we don’t actually know what inflation implies. He suspects this is due to our over-eager embrace of the concept of infinity. But Tegmark also follows Everett in using infinity to derive the Born probability rule in quantum mechanics. Everett noted that the relative measure of Born-rule-deviant worlds approached zero after an infinity of measurements, and thus the final state is “as if” those worlds didn’t exist. To get the same effect, Tegmark instead considers a superposition of an infinity of spatial regions out there containing the same exact observer. But this trick wouldn’t work with a finite set of such regions, and I have my doubts that it is fair to treat such regions is if they hadn’t decohered. (My approach to deriving the Born rule in many worlds stays with strictly finite everything.)

Tegmark also seems to think that a similar approach explains the second law of thermodynamics, but that doesn’t at all make sense to me. The main thing to explain there is why our best measure for predicting the past is pretty much the inverse of our best measure for predicting the future. I didn’t see Tegmark addressing that at all. (Btw, Sean Carroll recently mostly rejected the solution he wrote a book elaborating.)

My last disagreement with Tegmark is that he sees the simulation argument as self-destructing because since we know nothing about the universe that might simulate us, it could also be simulated by another universe, which is a “a reductio ad absurdum.” But I think it quite reasonable to put substantial weight there being real costs of computation in the base universe, which means they would tend to limit the ability of universes they simulate to spawn yet further universes. Which makes it less likely that we are a second level simulation, relative to a first level one.

Let me conclude with some of the many ways I agree with the book:

  • We reject or accept total theories, not places. If parallel universes are implied by our best theories, we should believe in them.
  • It is vitally important to figure out good physics measures, and good physics explanations for them.
  • I also prefer to reason about theories without infinities, when nearly as plausible on other grounds. Take a limit to infinity as the very last theory step, if possible.
  • Our brains aren’t doing much quantum computing.
  • Quantum mechanics seems random because we observers are splitting; the universe is deterministic.
  • Foundations of physics gets unfairly neglected. People who do it have to hide it to save their careers.
  • The world is hurting itself by not coordinating to reduce existential risk. We have a vast great potential future if we don’t kill ourselves.
  • We don’t need a new understanding of consciousness to solve foundations of physics problems. “Consciousness is the way information feels when being processed in certain complex ways.” Saying that our math “exists” is pretty much the same as saying that the stuff that fits our math is the stuff that “feels” when arranged in certain ways.
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  • lump1

    I find the argument for parallel universes based on anthropic reasoning sort of weird. The way it’s usually presented is to say that given the familiar laws, it seems that the observed constants can only have a narrow range of values in order for there to be enough structure to support consciousness. It would be surprising if the universe hit that bullseye just by accident on the only shot it took, so we assume that there were many shots taken, with many misses that play out unobserved.

    This is not an a priori argument – the observation supporting it is the surprising narrowness of fundamental parameters that support life. This datum is explained by other universes. What puzzles me is whether that’s the right kind of evidence. For example, what if we discovered the opposite – that the parameters were not at all narrow, that basically any old combination of values for the fundamental constants would give rise to an interesting and potentially habitable universe. Would this be any less surprising? Wouldn’t cosmologists wonder at how it is that the universe seems positively tailored to make life an inevitability? And in groping around for an explanation for this, wouldn’t they hit on the story that this isn’t the only universe, and that its unusual property only makes sense if we imagine lots of “blander” (less robustly life-supporting) universes also existing?

    What I’m saying is that though this initially has the form of an empirical argument, the argument would work equally well no matter what specific data we actually observed. This makes it not an empirical argument at all. It basically amounts to saying “The universe is just-so, and its just-so-ness seems remarkable somehow. (We can too easily imagine not-so universes, and it a priori seems so much easier for a universe to be not-so rather than just-so!) Thus in order to explain the just-so-ness of the universe, we postulate the existence of other universes.”

    • IMASBA

      “For example, what if we discovered the opposite – that the parameters were not at all narrow, that basically any old combination of values for the fundamental constants would give rise to an interesting and potentially habitable universe. Would this be any less surprising? Wouldn’t cosmologists wonder at how it is that the universe seems positively tailored to make life an inevitability?”

      No, that would just mean “life” is very robust compared to the many ways the universe can logically exist.

      The many-worlds theory is also inspired by quantum mechanics, if it was just about there being life then a series of variations of this universe existing “after” each other would be just as good of an explanation as the many-worlds theory.

    • http://juridicalcoherence.blogspot.com/ Stephen Diamond

      Thus in order to explain the just-so-ness of the universe, we postulate the existence of other universes.

      What I’m not clear about in the anthropic argument for multiple universes is what’s so special or “interesting” about life? Life is just one kind of complexity, as judged by one metric. (Our choice of this metric, our metric, isn’t happenstance.)

      Is there some absolute metric for complexity? I don’t think so: degree of complexity seems language-relative.

  • efalken

    OK, dumb questions, perhaps you have this in some FAQ you can link to:
    1) I know when the universe split, but don’t understand how. What force copies and pulls the universe in two?
    2) when probabilities aren’t 50%, how does this split happen?

    • http://overcomingbias.com RobinHanson

      “Decoherence” is the physical process of splitting. With non 50/50 it happens in exactly the same way, but vectors have different lengths in the different sides of the split.

    • trent

      1) It’s misleading to think of the universe as splitting. Rather, think of both (or all) universes as always have existed. Pre “split” the different copies of the universe interact with eachother, but afterwards they cannot.

      2) The same way as with a 50/50 split, only the “mass” of the wave function won’t be evenly split between the two.

      Most of my knowledge of many words comes from LW’s quantum physics sequence, maybe you’ll find it interesting too:
      http://lesswrong.com/lw/r5/the_quantum_physics_sequence/

      • efalken

        So, there’s an infinite number of universes, with a google universes decohering every second. I am constantly getting split apart when a wave function collapse happens somewhere in the universe, but I don’t notice because I’m just one of the infinite versions of me. I think I can understand that idea, though I must admit some skepticism.

      • IMASBA

        “1) It’s misleading to think of the universe as splitting. Rather, think of both (or all) universes as always have existed. Pre “split” the different copies of the universe interact with eachother, but afterwards they cannot.”

        That’s not the whole story: in order to mask a deterministic multiverse as a probabilistic universe you really need splitting, or you need versions of “you” jumping between universes all the time. I’m pretty sure having a bunch of deterministic universes (all possible permutations) existing side by side since the beginning of time would not be compatible with observed quantum mechanical effects.

      • http://juridicalcoherence.blogspot.com/ Stephen Diamond

        or you need versions of “you” jumping between universes all the time

        It seems to this diletante that MW’s “determinism” is (like the “collapse of the wave function”) a sleight of hand: there’s not only no way to predict which “you” will be the observer in the experiment, but not even a hypothetical mechanism causing the allocation of the versions of “you.” No way, in principle, to distinguish any version of “you” from another.

        Let me also enter an unrelated dilettantish comment. Robin deplores the low status accorded explorers of qm’s foundations. But is it sensible to try to provide foundations for a theory you know must be fundamentally false?

        [In fact, both relativity and qm must be false, since they each make wrong predictions at different physical extremes, a fitting rejoinder to literally interpreting our “acceptance of theories” as belief in their truth, even provisionally.]

      • IMASBA

        Well, if you could collect the “yous” in one universe they would have different memories so they are distinguishable, it’s just that you have no way of collecting them. Yet, through QM nature has given us some reason to believe the other “yous” do exist (it is of course also possible the universe really is probabilistic).

        “But is it sensible to try to provide foundations for a theory you know must be fundamentally false?”

        Well, QM might be entirely true with the fault lying with relativity, or both might have relatively minor flaws or both are true within a certain domain (like classical mechanics), being underpinned by a deeper layer of physics that connects them. It is almost without question that some of the “weird” results from QM will hold even if a deeper theory is eventually found and those results need explaining, in the same way the doppler effect’s explanation still stands even though we now know classical mechanics is not the whole story.

      • http://juridicalcoherence.blogspot.com/ Stephen Diamond

        Is there really any question that qm is false in its assumption of absolute space and time?

        Here’s an historical parallel. Newton declined to philosophize about foundations, but Leibnitz accepted the challenge and arrived at a kind of relativistic conception. Newton’s absolutist assumptions were false, and laying foundations would have been misleading. (Absolute time and space was [it is said] a necessary assumption at that stage in the development of physical science.)

        The basic falsity of this assumption was not a bar to making correct predictions within acceptable tolerance in many practical circumstances, but it was nonetheless a false clue about reality.

      • IMASBA

        QM does not assume absolute time and space, the subfield of QED (quantum electrodynamics) incorporates relativistic principles.

        Newton’s assumption about absolute time and space was wrong but even today his laws of motion still stands (although you have to tweak the math of the second law to make it compatible with relativity). In the same way QM undoubtedly contains principles that will hold no matter what deeper physics will be found and we have some indications of which principles those will be. It is worthwhile to do foundational research into the principles even if a few of them will later be proven false (or more likely, get a correcting term added to them), that is at the current levels of spending (which are low compared to global GDP or even the science budgets of the world’s nations), not in the least because this kind of research tends to create useful mathematical spin-off.

      • IMASBA

        On the low status of these researchers. I think that has to do with laymen thinking their will never applications of the research and other scientists suspecting the fundamental researchers of hiding away in an ivory tower and not being productive (even for another physicist it’s hard to judge whether a theoretical physicist is working hard or just doodling and drinking coffee most of the day, pretending to work).

      • Peter David Jones

        If the many worlds interpretation could offer predictability as well as in-principle determinism , it would be a theory not an interpretation.

      • http://juridicalcoherence.blogspot.com/ Stephen Diamond

        The problem seems to me to be that MW doesn’t, in fact, provide in-principle determinism: there is no deterministic mechanism (or any mechanism) hypothesized assigning any particular version of “you” to a “world.”

      • Peter David Jones

        @Stephen

        There is no need for MWI to assign different versions of you to different worlds.
        Nothing labels your counterparts other than their having had different histories as a result of having lived in different worlds, which is to say, their identities and the identites of he worlds they live in match up automatically without ant special mechanism.

        Consider the MWI response to Schrodingers Cat.

    • Peter David Jones

      The splitting comes from the behaviour of the Schrodinger Wave Equation absent collapse. It is not an additional post of MWI.

  • IMASBA

    “In that case, the fact of our existence should make us suspect that many places with physics based on other maths are out there somewhere:”

    You mean with physics based on other laws, right? So not that 1+1=15 in an other universe (which would be inconceivable to the human mind and also invalidate any calculations, probabilistic or otherwise one can let loose on the multiverse).

    “Quantum mechanics seems random because we observers are splitting; the universe is deterministic.

    The world is hurting itself by not coordinating to reduce existential risk. We have a vast great potential future if we don’t kill ourselves.”

    These statements seem contradictory: the concept of “we” becomes very fuzzy in an eversplitting multiverse and in such a multiverse some version of us always survives because being responsible stewards of the world is a possible outcome and therefore has to happen in some of the universes and “we” have zero control over how much time “we” get to spend in these universes before splitting off into more bleak universes. It’s mind boggling really.

    • Dave Lindbergh

      “But Tegmark also seems to accept that there is some “measure” over all these maths, and that the reason to take some actions over others is to favorably change that measure.”

      If you buy that (I do), it’s not contradictory. Our actions influence the proportions of the different outcomes.

      • IMASBA

        Ok, that seems reasonable, with the addition that in a deterministic multiverse the proportions were really already fixed before we were born. That Robin Hanson and Max Tegmark would propose these things was already fixed and the way I would react to it was fixed as well, as was the way the world will react to the proposals.

  • Tyrrell_McAllister

    Typo here: “Saying that our math “exists” is pretty much the same as saying stuff
    our stuff the fits that math “feels” when arranged certain ways.”

  • Robert Koslover

    I confess to being a little (OK, a lot) rusty on my QM, but re: “Quantum mechanics seems random because we observers are splitting; the universe is deterministic.” — If the universe is truly deterministic, isn’t that the same as saying it obeys a “hidden variables” theory? And if so, haven’t the many EPR experiments (tests of Bell’s theorem, etc.) already proven that to be wrong?

    • http://overcomingbias.com RobinHanson

      No, those are different concepts. Many worlds doesn’t predict violations of the usual QM predictions in tests of Bell’s theorem.

      • Robert Koslover

        Thank you, but I didn’t mean to suggest that the many-worlds view, all by itself, implied violation of quantum-mechanics. Rather, my contention was that, many-worlds or not, the experimental evidence says that the universe is NOT deterministic, because hidden-variables have been proven inconsistent with experiments. But… you called the universe deterministic! Why? Can it still be deterministic if there are no hidden variables? (I apologize in advance if answering this question properly is simply too involved for this forum.) Thanks.

      • http://juridicalcoherence.blogspot.com/ Stephen Diamond

        Because the whole ensemble of worlds is deterministic. But although the multiverse is deterministic (according to many worlds), we can’t make any deterministic predictions, only probabilities.

        Why does the distribution of our experienced probabilities correspond to the distribution of worlds? (By what mechanism do we land in one world or another?) I don’t think this is explained, although it gives the initial appearance of being self-explanatory.

        [Note also that in versions of MW where the number of world is infinite, relative proportions are meaningless.]

      • Peter David Jones

        @Stephen

        MWI gives the correct probabilities for observed outcomes because it uses the standard QM apparatus, which works.

        You split along with the worlds, you don’t jump from one to the other.

        Relative proportions are perfectly meaningful with infinities. Eg even numbers and integers exist in the proportion 1:2.

        See The Fabric of Reality.

      • IMASBA

        Like Guest said before (sort of): Bell’s theorem and determinism are compatible is you accept that “locality” is false. Locality means that information cannot travel faster than the speed of light. In the many-worlds theory locality is true in every individual universe but it’s false/meaningless when you talk about the whole multiverse.

      • Peter David Jones

        ….and relativity says
        locality is true, which is how you get from either-nonlocality-or-indeterminism to indeterminism.

    • Guest

      EPR proves that either QM is nonlocal, or we’re importantly misdescribing QM’s experimental outcomes. Bohm’s interpretation grabs the first horn (it’s a nonlocal deterministic theory that seems to reproduce QM’s experimental results, though no one’s made a field theory for it yet). Everett’s ‘many worlds’ or ‘relative state’ interpretation grabs the second horn (we’re neglecting important parts of the experimental outcome).

  • truth_machine

    What a conceptually confused pile of drivel.

    “there are whole different places “out there” completely causally (and spatially) disconnected from our universe”

    If there are, we cannot possibly know it.

    • Dave Lindbergh

      “We reject or accept total theories, not places. If parallel universes are implied by our best theories, we should believe in them.”

      If our best theory succeeds in explaining the data we observe, then we take that theory seriously – we say it’s our best model of what reality is.

      If that same theory predicts facts that we can’t directly observe, we’re not free to reject those facts as a “pile of drivel” – if we accept the theory as (provisional) truth, then the things it predicts must also be true.

      • http://juridicalcoherence.blogspot.com/ Stephen Diamond

        The maxim begs the question, which is whether our best theories will ever imply parallel universes. Why would a good theory ever imply something radically untestable, when testability is central to what makes a theory good.

      • Dave Lindbergh

        Any theory is a statement that “the world works like X”. X may imply facts A, B, and C, of which only A and B are measurable.

        If you accept the theory because it correctly predicts testable facts A and B, then you have to accept that it’s right about C also.

        Or, come up with a better theory.

        But what you can’t do is accept the theory but not it’s consequences. You don’t get to pick and choose consequences – only theories, together with their baggage.

        [FWIW, my impression is that most physicists _already_ think our best theories imply parallel universes. Tegmark’s book is a good place to start…]

      • http://juridicalcoherence.blogspot.com/ Stephen Diamond

        You’re amalgamating facts that are untestable in practice with those untestable in principle.

      • arch1

        Stephen, you (and truth_machine) appear to be conflating “verifiable truth” and “truth.”
        It would indeed be vastly more satisfying to find/converge toward a *verifiably* (in the sense of passing increasingly stringent tests) true theory of the universe, and all else equal, such theories should be given search priority; but as far as I can see, reality is under no obligation to satisfy us by being verifiable.

      • http://juridicalcoherence.blogspot.com/ Stephen Diamond

        The question is whether it is possible to know a truth that is unverifiable. If reality doesn’t satisfy us by being verifiable, it may (will?) also fail to satisfy us in being knowable.

        (Perhaps the barriers to precision in qm are epistemic rather than ontic, but are absolute, so we never will know the foundations.)

        [I don’t want to press this point too hard because it requires drawing and motivating a distinction between contingent unverifiability and unverifiability in principle.]

      • Peter David Jones

        @Stephen
        Positing that the barriers to precision in QM are epistemic rather than ontic is what leads to hidden variables, nonlocality, etc.

      • Joe Teicher

        >If that same theory predicts facts that we can’t directly observe, we’re not free to reject those facts as a “pile of drivel”

        Why not? It seems like we can do so without consequence.

      • Dave Lindbergh

        Because that’s equivalent to saying that you don’t accept the theory in the first place.

        You either do or don’t accept a theory as (provisionally) correct.

        If you do, then you have to accept all of it. Cherry-picking only the consequences you like is equivalent to proposing a modified version of the theory.

        (Which is fine, if you’re prepared to explain why your modified theory seems more likely to be correct. But you need a _reason_ if you hope to convince anyone else.)

      • http://juridicalcoherence.blogspot.com/ Stephen Diamond

        You either do or don’t accept a theory as (provisionally) correct.

        “Provisional acceptance” [I don’t think there’s such as thing as “provisional correctness”] is cashed out as treating scientific theories as if they were making truth claims, in order to apply a strict falsificationist acceptance standard.

        But when the theory is said to have implications without implication for perceived reality, these claims play no potential role in the theory’s falsification. (They’re “without consequence.”)

    • Peter David Jones

      Things are even worse that that. You don’t have direct evidence of *this* world.

  • http://juridicalcoherence.blogspot.com/ Stephen Diamond

    Hadn’t Robin rejected self-indication when he accepted Katja Grace’s rebuttal?

  • SanguineEmpiricist

    “I also prefer to reason about theories without infinities, when nearly as plausible on other grounds. Take a limit to infinity as the very last theory step, if possible.”

    The importance and relevance of this is highly under-emphasized in today’s mathematical education. With all the mandatory calculus teaching, one would think it would be mentioned, but I’ve seen it only in a few places.

    Knowing integrals and limits without knowing this seems like giving people flame-throwers without telling them the appropriate usage.

  • Peter David Jones

    “Quantum mechanics seems random because we observers are splitting; the universe is deterministic.”

    Presumably, this applies at level III. At level IV, you have every mathematically expressible variation on time, determinism and indeterminism, somewhere.

  • Peter David Jones

    “We don’t need a new understanding of consciousness to solve foundations of physics problems. ”

    We may need a new understanding of physics to solve consciousness problems. Consciousness may be how physics feels from the inside, but physics doesnt predict that anything should feel like anything from the inside.

    • IMASBA

      Yes, consciousness would be declared unscientific nonsense if it weren’t for the empirical evidence in favor of it. We don’t even know how to measure it in a mind that is not our own and that will have severe social consequences when AI develops to the complexity where it could have human level consciousness (of course we would not know it when it arrives, that is the point), we (will) live in fascinating times (at least those of us under the age of 30, like myself).

  • arch1

    “Foundations of physics gets unfairly neglected. People who do it have to hide it to save their careers.”
    Lee Smolin, in the last third of his superb book “The Trouble with Physics,” speaks eloquently on this general topic.
    Smolin [2006] believes that the reason the list of the five biggest outstanding problems in physics hasn’t changed at all in 30 years (a situation I believe he characterizes as unprecedented in recent centuries) is that
    a) unlike the 1930s & 1940s. when the challenge was to do calculations/extend models within an existing framework, the current key problems will require fundamental new insights for their solution (much as in the 1st 2 decades of the 20th century);
    b) the current sociology of the physics community actively and severely discourages revolutionary thinkers willing to question foundations and generate such insights, in favor of superb technicians who demonstrate the ability to push ahead on currently-favored lines of research.
    c) one major enabler of b) is that geezers such as himself have way too much power over the young people.
    Great book, and not just for the physics. If you read the last two pages you will be at risk of reading it backwards to the beginning, so grab a bite to eat first.

  • Philip Goetz

    “he sees the simulation argument as self-destructing because since we know nothing about the universe that might simulate us, it could also be simulated by another universe, which is a “a reductio ad absurdum.”

    There is nothing absurd about that.