This review of Max Tegmark’s book also occurs infinitely often in the decimal expansion of π

Two months ago, commenter rrtucci asked me what I thought about Max Tegmark and his “Mathematical Universe Hypothesis”: the idea, which Tegmark defends in his recent book Our Mathematical Universe, that physical and mathematical existence are the same thing, and that what we call “the physical world” is simply one more mathematical structure, alongside the dodecahedron and so forth.  I replied as follows:

…I find Max a fascinating person, a wonderful conference organizer, someone who’s always been extremely nice to me personally, and an absolute master at finding common ground with his intellectual opponents—I’m trying to learn from him, and hope someday to become 10-122 as good.  I can also say that, like various other commentators (e.g., Peter Woit), I personally find the “Mathematical Universe Hypothesis” to be devoid of content.

After Peter Woit found that comment and highlighted it on his own blog, my comments section was graced by none other than Tegmark himself, who wrote:

Thanks Scott for your all to [sic] kind words!  I very much look forward to hearing what you think about what I actually say in the book once you’ve had a chance to read it!  I’m happy to give you a hardcopy (which can double as door-stop) – just let me know.

With this reply, Max illustrated perfectly why I’ve been trying to learn from him, and how far I fall short.  Where I would’ve said “yo dumbass, why don’t you read my book before spouting off?,” Tegmark gracefully, diplomatically shamed me into reading his book.

So, now that I’ve done so, what do I think?  Briefly, I think it’s a superb piece of popular science writing—stuffed to the gills with thought-provoking arguments, entertaining anecdotes, and fascinating facts.  I think everyone interested in math, science, or philosophy should buy the book and read it.  And I still think the MUH is basically devoid of content, as it stands.

Let me start with what makes the book so good.  First and foremost, the personal touch.  Tegmark deftly conveys the excitement of being involved in the analysis of the cosmic microwave background fluctuations—of actually getting detailed numerical data about the origin of the universe.  (The book came out just a few months before last week’s bombshell announcement of B-modes in the CMB data; presumably the next edition will have an update about that.)  And Tegmark doesn’t just give you arguments for the Many-Worlds Interpretation of quantum mechanics; he tells you how he came to believe it.  He writes of being a beginning PhD student at Berkeley, living at International House (and dating an Australian exchange student who he met his first day at IHouse), who became obsessed with solving the quantum measurement problem, and who therefore headed to the physics library, where he was awestruck by reading the original Many-Worlds articles of Hugh Everett and Bryce deWitt.  As it happens, every single part of the last sentence also describes me (!!!)—except that the Australian exchange student who I met my first day at IHouse lost interest in me when she decided that I was too nerdy.  And also, I eventually decided that the MWI left me pretty much as confused about the measurement problem as before, whereas Tegmark remains a wholehearted Many-Worlder.

The other thing I loved about Tegmark’s book was its almost comical concreteness.  He doesn’t just metaphorically write about “knobs” for adjusting the constants of physics: he shows you a picture of a box with the knobs on it.  He also shows a “letter” that lists not only his street address, zip code, town, state, and country, but also his planet, Hubble volume, post-inflationary bubble, quantum branch, and mathematical structure.  Probably my favorite figure was the one labeled “What Dark Matter Looks Like / What Dark Energy Looks Like,” which showed two blank boxes.

Sometimes Tegmark seems to subtly subvert the conventions of popular-science writing.  For example, in the first chapter, he includes a table that categorizes each of the book’s remaining chapters as “Mainstream,” “Controversial,” or “Extremely Controversial.”  And whenever you’re reading the text and cringing at a crucial factual point that was left out, chances are good you’ll find a footnote at the bottom of the page explaining that point.  I hope both of these conventions become de rigueur for all future pop-science books, but I’m not counting on it.

The book has what Tegmark himself describes as a “Dr. Jekyll / Mr. Hyde” structure, with the first (“Dr. Jekyll”) half of the book relaying more-or-less accepted discoveries in physics and cosmology, and the second (“Mr. Hyde”) half focusing on Tegmark’s own Mathematical Universe Hypothesis (MUH).  Let’s accept that both halves are enjoyable reads, and that the first half contains lots of wonderful science.  Is there anything worth saying about the truth or falsehood of the MUH?

In my view, the MUH gestures toward two points that are both correct and important—neither of them new, but both well worth repeating in a pop-science book.  The first is that the laws of physics aren’t “suggestions,” which the particles can obey when they feel like it but ignore when Uri Geller picks up a spoon.  In that respect, they’re completely unlike human laws, and the fact that we use the same word for both is unfortunate.  Nor are the laws merely observed correlations, as in “scientists find link between yogurt and weight loss.”  The links of fundamental physics are ironclad: the world “obeys” them in much the same sense that a computer obeys its code, or the positive integers obey the rules of arithmetic.  Of course we don’t yet know the complete program describing the state evolution of the universe, but everything learned since Galileo leads one to expect that such a program exists.  (According to quantum mechanics, the program describing our observed reality is a probabilistic one, but for me, that fact by itself does nothing to change its lawlike character.  After all, if you know the initial state, Hamiltonian, and measurement basis, then quantum mechanics gives you a perfect algorithm to calculate the probabilities.)

The second true and important nugget in the MUH is that the laws are “mathematical.”  By itself, I’d say that’s a vacuous statement, since anything that can be described at all can be described mathematically.  (As a degenerate case, a “mathematical description of reality” could simply be a gargantuan string of bits, listing everything that will ever happen at every point in spacetime.)  The nontrivial part is that, at least if we ignore boundary conditions and the details of our local environment (which maybe we shouldn’t!), the laws of nature are expressible as simple, elegant math—and moreover, the same structures (complex numbers, group representations, Riemannian manifolds…) that mathematicians find important for internal reasons, again and again turn out to play a crucial role in physics.  It didn’t have to be that way, but it is.

Putting the two points together, it seems fair to say that the physical world is “isomorphic to” a mathematical structure—and moreover, a structure whose time evolution obeys simple, elegant laws.   All of this I find unobjectionable: if you believe it, it doesn’t make you a Tegmarkian; it makes you ready for freshman science class.

But Tegmark goes further.  He doesn’t say that the universe is “isomorphic” to a mathematical structure; he says that it is that structure, that its physical and mathematical existence are the same thing.  Furthermore, he says that every mathematical structure “exists” in the same sense that “ours” does; we simply find ourselves in one of the structures capable of intelligent life (which shouldn’t surprise us).  Thus, for Tegmark, the answer to Stephen Hawking’s famous question—”What is it that breathes fire into the equations and gives them a universe to describe?”—is that every consistent set of equations has fire breathed into it.  Or rather, every mathematical structure of at most countable cardinality whose relations are definable by some computer program.  (Tegmark allows that structures that aren’t computably definable, like the set of real numbers, might not have fire breathed into them.)

Anyway, the ensemble of all (computable?) mathematical structures, constituting the totality of existence, is what Tegmark calls the “Level IV multiverse.”  In his nomenclature, our universe consists of anything from which we can receive signals; anything that exists but that we can’t receive signals from is part of a “multiverse” rather than our universe.  The “Level I multiverse” is just the entirety of our spacetime, including faraway regions from which we can never receive a signal due to the dark energy.  The Level II multiverse consists of the infinitely many other “bubbles” (i.e., “local Big Bangs”), with different values of the constants of physics, that would, in eternal inflation cosmologies, have generically formed out of the same inflating substance that gave rise to our Big Bang.  The Level III multiverse is Everett’s many worlds.  Thus, for Tegmark, the Level IV multiverse is a sort of natural culmination of earlier multiverse theorizing.  (Some people might call it a reductio ad absurdum, but Tegmark is nothing if not a bullet-swallower.)

Now, why should you believe in any of these multiverses?  Or better: what does it buy you to believe in them?

As Tegmark correctly points out, none of the multiverses are “theories,” but they might be implications of theories that we have other good reasons to accept.  In particular, it seems crazy to believe that the Big Bang created space only up to the furthest point from which light can reach the earth, and no further.  So, do you believe that space extends further than our cosmological horizon?  Then boom! you believe in the Level I multiverse, according to Tegmark’s definition of it.

Likewise, do you believe there was a period of inflation in the first ~10-32 seconds after the Big Bang?  Inflation has made several confirmed predictions (e.g., about the “fractal” nature of the CMB perturbations), and if last week’s announcement of B-modes in the CMB is independently verified, that will pretty much clinch the case for inflation.  But Alan Guth, Andrei Linde, and others have argued that, if you accept inflation, then it seems hard to prevent patches of the inflating substance from continuing to inflate forever, and thereby giving rise to infinitely many “other” Big Bangs.  Furthermore, if you accept string theory, then the six extra dimensions should generically curl up differently in each of those Big Bangs, giving rise to different apparent values of the constants of physics.  So then boom! with those assumptions, you’re sold on the Level II multiverse as well.  Finally, of course, there are people (like David Deutsch, Eliezer Yudkowsky, and Tegmark himself) who think that quantum mechanics forces you to accept the Level III multiverse of Everett.  Better yet, Tegmark claims that these multiverses are “falsifiable.”  For example, if inflation turns out to be wrong, then the Level II multiverse is dead, while if quantum mechanics is wrong, then the Level III one is dead.

Admittedly, the Level IV multiverse is a tougher sell, even by the standards of the last two paragraphs.  If you believe physical existence to be the same thing as mathematical existence, what puzzles does that help to explain?  What novel predictions does it make?  Forging fearlessly ahead, Tegmark argues that the MUH helps to “explain” why our universe has so many mathematical regularities in the first place.  And it “predicts” that more mathematical regularities will be discovered, and that everything discovered by science will be mathematically describable.  But what about the existence of other mathematical universes?  If, Tegmark says (on page 354), our qualitative laws of physics turn out to allow a narrow range of numerical constants that permit life, whereas other possible qualitative laws have no range of numerical constants that permit life, then that would be evidence for the existence of a mathematical multiverse.  For if our qualitative laws were the only ones into which fire had been breathed, then why would they just so happen to have a narrow but nonempty range of life-permitting constants?

I suppose I’m not alone in finding this totally unpersuasive.  When most scientists say they want “predictions,” they have in mind something meatier than “predict the universe will continue to be describable by mathematics.”  (How would we know if we found something that wasn’t mathematically describable?  Could we even describe such a thing with English words, in order to write papers about it?)  They also have in mind something meatier than “predict that the laws of physics will be compatible with the existence of intelligent observers, but if you changed them a little, then they’d stop being compatible.”  (The first part of that prediction is solid enough, but the second part might depend entirely on what we mean by a “little change” or even an “intelligent observer.”)

What’s worse is that Tegmark’s rules appear to let him have it both ways.  To whatever extent the laws of physics turn out to be “as simple and elegant as anyone could hope for,” Tegmark can say: “you see?  that’s evidence for the mathematical character of our universe, and hence for the MUH!”  But to whatever extent the laws turn out not to be so elegant, to be weird or arbitrary, he can say: “see?  that’s evidence that our laws were selected more-or-less randomly among all possible laws compatible with the existence of intelligent life—just as the MUH predicted!”

Still, maybe the MUH could be sharpened to the point where it did make definite predictions?  As Tegmark acknowledges, the central difficulty with doing so is that no one has any idea what measure to use over the space of mathematical objects (or even computably-describable objects).  This becomes clear if we ask a simple question like: what fraction of the mathematical multiverse consists of worlds that contain nothing but a single three-dimensional cube?

We could try to answer such a question using the universal prior: that is, we could make a list of all self-delimiting computer programs, then count the total weight of programs that generate a single cube and then halt, where each n-bit program gets assigned 1/2n weight.  Sure, the resulting fraction would be uncomputable, but at least we’d have defined it.  Except wait … which programming language should we use?  (The constant factors could actually matter here!)  Worse yet, what exactly counts as a “cube”?  Does it have to have faces, or are vertices and edges enough?  How should we interpret the string of 1′s and 0′s output by the program, in order to know whether it describes a cube or not?  (Also, how do we decide whether two programs describe the “same” cube?  And if they do, does that mean they’re describing the same universe, or two different universes that happen to be identical?)

These problems are simply more-dramatic versions of the “standard” measure problem in inflationary cosmology, which asks how to make statistical predictions in a multiverse where everything that can happen will happen, and will happen an infinite number of times.  The measure problem is sometimes discussed as if it were a technical issue: something to acknowledge but then set to the side, in the hope that someone will eventually come along with some clever counting rule that solves it.  To my mind, however, the problem goes deeper: it’s a sign that, although we might have started out in physics, we’ve now stumbled into metaphysics.

Some cosmologists would strongly protest that view.  Most of them would agree with me that Tegmark’s Level IV multiverse is metaphysics, but they’d insist that the Level I, Level II, and perhaps Level III multiverses were perfectly within the scope of scientific inquiry: they either exist or don’t exist, and the fact that we get confused about the measure problem is our issue, not nature’s.

My response can be summed up in a question: why not ride this slippery slope all the way to the bottom?  Thinkers like Nick Bostrom and Robin Hanson have pointed out that, in the far future, we might expect that computer-simulated worlds (as in The Matrix) will vastly outnumber the “real” world.  So then, why shouldn’t we predict that we’re much more likely to live in a computer simulation than we are in one of the “original” worlds doing the simulating?  And as a logical next step, why shouldn’t we do physics by trying to calculate a probability measure over different kinds of simulated worlds: for example, those run by benevolent simulators versus evil ones?  (For our world, my own money’s on “evil.”)

But why stop there?  As Tegmark points out, what does it matter if a computer simulation is actually run or not?  Indeed, why shouldn’t you say something like the following: assuming that π is a normal number, your entire life history must be encoded infinitely many times in π’s decimal expansion.  Therefore, you’re infinitely more likely to be one of your infinitely many doppelgängers “living in the digits of π” than you are to be the “real” you, of whom there’s only one!  (Of course, you might also be living in the digits of e or √2, possibilities that also merit reflection.)

At this point, of course, you’re all the way at the bottom of the slope, in Mathematical Universe Land, where Tegmark is eagerly waiting for you.  But you still have no idea how to calculate a measure over mathematical objects: for example, how to say whether you’re more likely to be living in the first 1010^120 digits of π, or the first 1010^120 digits of e.  And as a consequence, you still don’t know how to use the MUH to constrain your expectations for what you’re going to see next.

Now, notice that these different ways down the slippery slope all have a common structure:

  1. We borrow an idea from science that’s real and important and profound: for example, the possible infinite size and duration of our universe, or inflationary cosmology, or the linearity of quantum mechanics, or the likelihood of π being a normal number, or the possibility of computer-simulated universes.
  2. We then run with that idea until we smack right into a measure problem, and lose the ability to make useful predictions.

Many people want to frame the multiverse debates as “science versus pseudoscience,” or “science versus science fiction,” or (as I did before) “physics versus metaphysics.”  But actually, I don’t think any of those dichotomies get to the nub of the matter.  All of the multiverses I’ve mentioned—certainly the inflationary and Everett multiverses, but even the computer-simuverse and the π-verse—have their origins in legitimate scientific questions and in genuinely-great achievements of science.  However, they then extrapolate those achievements in a direction that hasn’t yet led to anything impressive.  Or at least, not to anything that we couldn’t have gotten without the ontological commitments that led to the multiverse and its measure problem.

What is it, in general, that makes a scientific theory impressive?  I’d say that the answer is simple: connecting elegant math to actual facts of experience.

When Einstein said, the perihelion of Mercury precesses at 43 seconds of arc per century because gravity is the curvature of spacetime—that was impressive.

When Dirac said, you should see a positron because this equation in quantum field theory is a quadratic with both positive and negative solutions (and then the positron was found)—that was impressive.

When Darwin said, there must be equal numbers of males and females in all these different animal species because any other ratio would fail to be an equilibrium—that was impressive.

When people say that multiverse theorizing “isn’t science,” I think what they mean is that it’s failed, so far, to be impressive science in the above sense.  It hasn’t yet produced any satisfying clicks of understanding, much less dramatically-confirmed predictions.  Yes, Steven Weinberg kind-of, sort-of used “multiverse” reasoning to predict—correctly—that the cosmological constant should be nonzero.  But as far as I can tell, he could just as well have dispensed with the “multiverse” part, and said: “I see no physical reason why the cosmological constant should be zero, rather than having some small nonzero value still consistent with the formation of stars and galaxies.”

At this, many multiverse proponents would protest: “look, Einstein, Dirac, and Darwin is setting a pretty high bar!  Those guys were smart but also lucky, and it’s unrealistic to expect that scientists will always be so lucky.  For many aspects of the world, there might not be an elegant theoretical explanation—or any explanation at all better than, ‘well, if it were much different, then we probably wouldn’t be here talking about it.’  So, are you saying we should ignore where the evidence leads us, just because of some a-priori prejudice in favor of mathematical elegance?”

In a sense, yes, I am saying that.  Here’s an analogy: suppose an aspiring filmmaker said, “I want my films to capture the reality of human experience, not some Hollywood myth.  So, in most of my movies nothing much will happen at all.  If something does happen—say, a major character dies—it won’t be after some interesting, character-forming struggle, but meaninglessly, in a way totally unrelated to the rest of the film.  Like maybe they get hit by a bus.  Then some other random stuff will happen, and then the movie will end.”

Such a filmmaker, I’d say, would have a perfect plan for creating boring, arthouse movies that nobody wants to watch.  Dramatic, character-forming struggles against the odds might not be the norm of human experience, but they are the central ingredient of entertaining cinema—so if you want to create an entertaining movie, then you have to postselect on those parts of human experience that do involve dramatic struggles.  In the same way, I claim that elegant mathematical explanations for observed facts are the central ingredient of great science.  Not everything in the universe might have such an explanation, but if one wants to create great science, one has to postselect on the things that do.

(Note that there’s an irony here: the same unsatisfyingness, the same lack of explanatory oomph, that make something a “lousy movie” to those with a scientific mindset, can easily make it a great movie to those without such a mindset.  The hunger for nontrivial mathematical explanations is a hunger one has to acquire!)

Some readers might argue: “but weren’t quantum mechanics, chaos theory, and Gödel’s theorem scientifically important precisely because they said that certain phenomena—the exact timing of a radioactive decay, next month’s weather, the bits of Chaitin’s Ω—were unpredictable and unexplainable in fundamental ways?”  To me, these are the exceptions that prove the rule.  Quantum mechanics, chaos, and Gödel’s theorem were great science not because they declared certain facts unexplainable, but because they explained why those facts (and not other facts) had no explanations of certain kinds.  Even more to the point, they gave definite rules to help figure out what would and wouldn’t be explainable in their respective domains: is this state an eigenstate of the operator you’re measuring?  is the Lyapunov exponent positive?  is there a proof of independence from PA or ZFC?

So, what would be the analogue of the above for the multiverse?  Is there any Level II or IV multiverse hypothesis that says: sure, the mass of electron might be a cosmic accident, with at best an anthropic explanation, but the mass of the Higgs boson is almost certainly not such an accident?  Or that the sum or difference of the two masses is not an accident?  (And no, it doesn’t count to affirm as “non-accidental” things that we already have non-anthropic explanations for.)  If such a hypothesis exists, tell me in the comments!  As far as I know, all Level II and IV multiverse hypotheses are still at the stage where basically anything that isn’t already explained might vary across universes and be anthropically selected.  And that, to my mind, makes them very different in character from quantum mechanics, chaos, or Gödel’s theorem.

In summary, here’s what I feel is a reasonable position to take right now, regarding all four of Tegmark’s multiverse levels (not to mention the computer-simuverse, which I humbly propose as Level 3.5):

Yes, these multiverses are a perfectly fine thing to speculate about: sure they’re unobservable, but so are plenty of other entities that science has forced us to accept.  There are even natural reasons, within physics and cosmology, that could lead a person to speculate about each of these multiverse levels.  So if you want to speculate, knock yourself out!  If, however, you want me to accept the results as more than speculation—if you want me to put them on the bookshelf next to Darwin and Einstein—then you’ll need to do more than argue that other stuff I already believe logically entails a multiverse (which I’ve never been sure about), or point to facts that are currently unexplained as evidence that we need a multiverse to explain their unexplainability, or claim as triumphs for your hypothesis things that don’t really need the hypothesis at all, or describe implausible hypothetical scenarios that could confirm or falsify the hypothesis.  Rather, you’ll need to use your multiverse hypothesis—and your proposed solution to the resulting measure problem—to do something new that impresses me.

271 Responses to “This review of Max Tegmark’s book also occurs infinitely often in the decimal expansion of π”

  1. Sid K Says:

    Scott, great post. A few points:

    (1) I agree with most of what you said. Still, I think you underplay the onus on non-believers of the MUH to justify why “life is breathed” into the specific set of equations that describe our universe, and not into other mathematical structures. It seems to be a pressing problem for which there doesn’t seem to be a good answer.

    I have elsewhere used the analogy of early biologists looking to understand what properties of matter distinguish living and non-living objects. As we kept digging deeper, we found that they were made of the same fundamental stuff; living things were just put together differently. Thus, now the onus is on non-naturalists to point to something fundamental that distinguishes life and non-life. The whole drama is being played out again with consciousness. Similarly, as we keep digging deeper into the fundamental structure of the universe, we find that we can’t point to any fundamental difference between the math of our universe and other “nearby” mathematical structures. So what selects this math over that math?

    (2) “If E is evidence for hypothesis H, and H entails K, then E is evidence for K.” This argument seems to be central to Tegmark’s argument. I think this, and variants, go by the name of Hempel’s Special Consequence Condition. As everything in philosophy, the validity of this condition is debated. Just saying in case anyone wants to enter that debate and especially explore the consequences for Tegmark.

    In general, I would really like some professional epistemologist to look at all the thorny problems posed by Tegmark’s ideas and give us some idea about the coherence of his ideas. Does anyone know if someone has done this? Also, does anyone know how the MUH is connected to Modal Realism?

  2. edmeasure Says:

    I know that I would be more convinced if you could just point me to the first place in the decimal expansion of pi where your review, converted perhaps to some decimal form, occurs in the decimal expansion of pi.

  3. Sniffnoy Says:

    Sorry, but — could you fix the link to the measure problem paper to point to the abstract rather than directly to the PDF? Thank you!

  4. Dave Bacon Says:

    Wonderful post Scott.

  5. Raoul Ohio Says:

    Excellent review. Being an obnoxious “Proud to be a Skeptic” kind of person, I agreed with the analysis.

    While I have my Skeptic hat on, let me mention that right now (between St. Patrick’s day and April Fool’s day) is the annual “IC (Inflationary Cosmology) has finally been proven, get out the Nobel Prizes!!) fortnight!

    For newcomers, IC is a fudge factor introduced to reduce the discordance between some model of “A singularity about which all knoweth squat” and “what we see today”. The IC theory competes with the NC (we have No Clue) theory.

    About once a year, a new instrument produces a huge data set that is processed until some sort of pattern emerges. The pattern is released by April 1, portrayed as blue and orange splotches, and asserted to finally clinch the case, because IC predicts the splotches better than NC does.

    Is anyone concerned that NC is pretty weak competition?

  6. Angus Mackay Says:

    Not disagreeing with you (not competent to do so) but could you comment on Max Tegmark’s point that fundamental particles have none other than mathematical properties, for example integer or half-integer spin?

  7. Max Tegmark Says:

    Thanks Scott for your encouraging words about my book and for these thought-provoking comments!

    Although I found your reasoning refreshingly clear throughout most of your post, I must confess that I don’t following your logic at the end, where it sounds like you conflate “be interested in” with “accept as more than speculation”. I’m sure that you accept modus ponens, i.e., the principle that if A implies B and A is true, then B is true. But shouldn’t you then also accept the following?

    “If A implies B and I accept A as more than speculation, then I accept B as more than speculation.”

    So how can you justify that you take seriously the Vilenkin/Linde/Guth claim that inflation generically produces a Level I multiverse and also accept cosmological inflation as more than speculation (especially after the BICEP2 announcement), yet refuse to accept the Level I multiverse as more than speculation? To me, this sounds like cognitive dissonance. I elaborate on this point in this Scientific American blog post.

    As to the Mathematical Universe Hypothesis, there’s another logical step where I don’t fully understand your reasoning. You say that “it seems fair to say that the physical world is ‘isomorphic to’ a mathematical structure”, yet express skepticism about the idea that “the physical world *is* a mathematical structure”. What compelling counterexample do you have in mind where the physical world is isomorphic to a mathematical structure without actually being one? What additional properties does your counter-example world have that makes it not be a mathematical structure? Would you agree that, by definition of isomorphism, they must by 100% unobservable?

    In my experience, the business of the world being isomorphic to a mathematical structure is much more controversial than you suggest when you say it merely “makes you ready for freshman science class”. I’ve taught lots of freshman science classes with excellent students who would object to this because of a sense that it conflicts with their belief in some sort of soul and/or deity that cannot be described by equations, say.

  8. Scott Says:

    Sniffnoy #3: Sorry, fixed!

  9. Scott Says:

    Angus Mackay #6:

      could you comment on Max Tegmark’s point that fundamental particles have none other than mathematical properties, for example integer or half-integer spin?

    Sure. What would it even mean for fundamental particles to have other than mathematical properties? How could we tell if they did? Even if every electron had a little “serial number” that distinguished it quantum-mechanically from every other electron (as an electron is distinguishable from an up quark), wouldn’t that just be another “mathematical property” of electrons?

  10. Jair Says:

    I also can’t see any distinction between saying the world is “isomorphic to” a mathematical structure and saying it is a mathematical structure. Perhaps the scare quotes indicate some hesitance to use the full weight of the term as it is usually used in mathematics. But choosing whether or not to put the scare quotes seems to me to be most of the debate at hand.

    What does it mean to say the world really is (or is isomorphic to) a mathematical structure? As far as I can tell, this means there should be some finite set of equations that precisely describe the world in every detail. I don’t see why we should believe this. Perhaps the universe requires an infinite number of nested, increasingly precise mathematical models to describe it. In this case I would not say that the universe “truly is” a single mathematical object, since we can only put finitely many pages in any given yellow book.

    If you do accept the premise that the world is (isomorphic to) a mathematical structure, and you are also a mathematical Platonist, it seems to me you must swallow the bullet and accept the Level IV multiverse as well. If all Turing machines exist in some sense, and our world is simulated by a Turing machine, then sure, there exists a version of the world in which grass is blue and waterfalls go up, etc., assuming those assumptions are logically consistent.

  11. Scott Says:

    Raoul Ohio #5: I completely agree with you that IC (Inflationary Cosmology) and NC (No Clue) seem like the main competitors! But yes, with each new processed pattern of orange and blue splotches, IC has been looking better and better compared to NC—I don’t see how one avoids that conclusion. I.e., the evidence looks stronger and stronger that some sort of inflation happened in our causal past. Still, from my unqualified layperson position, I’ve always been hesitant to buy into further speculations (e.g., eternal inflation scenarios) until we have more detailed knowledge about the properties of the inflaton. Of course, one reason the B-mode result is so exciting if it holds up is that it would start to give us such knowledge.

  12. Sid K Says:

    Raoul Ohio #5, Scott #11:

    I’m far from an expert but I think the main competition to Inflation isn’t No Clue, but instead Ekpyrotic (Cyclic) Models. And I think Paul Steinhardt, who co-built the Ekpyrotic Model, has clearly stated that the BICEP2 results, if they hold up, falsify his model.

  13. Peter Woit Says:

    Scott,
    By your criterion that the right way to evaluate this sort of speculation is whether it “impresses” one, then I have to say that for me, Tegmark’s Level IV multiverse passes with flying colors. It’s jaw-droppingly impressive in its scope and claims, providing an explanation of everything with no free parameters. It’s also by a long shot the most impressive example of a reputable academic putting before the public a long well-written book devoted to selling a completely empty idea that I’ve ever seen. Pre-Tegmark I wouldn’t have thought this possible.

    Equally impressive is the fact that a book devoted to promoting something purely content-free is getting laudatory reviews in Nature, a blurb from Witten, and now the public is being told that “everyone interested in math, science, or philosophy should buy the book and read it.” I couldn’t be more impressed, so by your argument Tegmark with his Level IV definitely belongs up there with Einstein, Dirac and Darwin.

  14. Jr Says:

    Did Darwin really predict that animal species should have equal number of males and females? I had thought that Fisher was the first one to make that argument? (Technically I also think that Fisher showed that there should be equal parental investment of resources in producing and raising sons and daughters, not that there must be an equal number of males and females in the population. )

  15. Cristi Stoica Says:

    Dear Scott and Max,

    I think I can identify the primary source of disagreement between you two.

    If we consider that it doesn’t matter “if a computer simulation is actually run or not”, then what we have is just a string of bits (or something containing the same information). What is the meaning of the bit string? It depends on how it was encoded – given an appropriate encoding algorithm, anything can be encoded in the same bit string. The bit string can be decoded in a text in a particular language, which can then be interpreted by a reader. Eventually, the meaning of the decoded information exists only for the observer who perceives that particular decoded description.

    Can the observer be included in the bit string? Apparently not, because the string can be decoded in all possible ways, so the observer will not be able to give a particular meaning. But perhaps this means that all possible observers, perceiving all possible decoded meanings, are included in that string. So there is a sort of “relative state interpretation”, each observer connected to the corresponding meaning of the bit string, all encoded in the same bit string.

    If I am right, then the entire discussion boils down to the following question: “is an external observer needed to give meaning to the bit string, or the observer can be considered included in the very bit string?”

    Depending on the answer, I can only see two options:

    - either accept that there is a “ghost in the machine” (an external observer for whom the bit string eventually has a meaning)

    - or accept that all logically consistent universes exist (including all possible observers as subsystems). And they exist in the same place (the bit string), and are different because we can decode them differently.

    Note that the viewpoint I proposed here on MUH is a bit different than that proposed by Tegmark, since it claims that the existence of a single bit string entails the existence of all possible universes, encoded in that string. No matter what string is, or even if it is a bit string or something else.

    Bottom line, the choice:
    If we admit that the observer is part of the simulation or mathematical description of the physical world, then we have to admit MUH (because that description can be interpreted in all possible ways). If we don’t admit MUH, it is probably because we assume that the observer for whom the meaning exists has to be external to the simulation itself (in order to give it a meaning).

  16. Anonymous Programmer Says:

    I think Lee Smolin would disagree with Max Tegmark. If universes inherit most of the laws of physics from their parent universe with slight modification you have a Darwinian situation. The laws of physics would not be random just as genes are not random.

    He also was arguing in “Time Reborn” that thinking of physics as math shortchanges the concept of time and the very real possibility that the laws of physics in a universe, or population of universes, can change over time.

    Do you think the universe has a parent or parents?

  17. James Cross Says:

    “Putting the two points together, it seems fair to say that the physical world is “isomorphic to” a mathematical structure—and moreover, a structure whose time evolution obeys simple, elegant laws. ”

    Are you saying the mathematical structure is not physical?

    Is it non-physical? Supernatural?

    I am trying to make some sort of religious argument here, but if you carve off some piece of reality (mathematics) as separate from physical reality, aren’t you making some sort of religious argument?

  18. James Cross Says:

    Sorry I meant “I am not trying to make some sort of religious argument .”

  19. Scott Says:

    James #17: Well, there are plenty of mathematical structures one can write down (Galois fields, the p-adics, ParityP Turing machines…) that don’t model observed reality, in the same sense that other mathematical structures (the Standard Model, general relativity) do model observed reality. Mathematicians (and their pencils and notebooks) are physical objects governed by the laws of physics, but there’s no law of physics that makes their imaginations slaves to what’s physically possible! And I think that banal observation is all you need to accept for this discussion—you don’t need to believe, further, in a “transcendent Platonic realm” of mathematical objects separate from physical reality.

    (According to Tegmark, the transcendent Platonic realm is the same thing as physical reality! Meanwhile, I call myself an “anti-anti-Platonist”: I don’t literally believe in the Platonic realm, but what I really oppose is the radical subjectivist claims about math that tend to go along with rejecting such a realm.)

  20. Scott Says:

    Jr #14: Good question! My memory was that Darwin had made the argument informally, and then Fisher made it much more rigorous. Meanwhile, here’s what Wikipedia says (in the article about Fisher’s principle):

      Charles Darwin had originally formulated a similar but somewhat confused argument in the first edition of The Descent of Man but withdrew it for the second edition—Fisher only had a copy of the latter
  21. Scott Says:

    Sid K #12: Every time I read or heard about ekpyrotic cosmology, I couldn’t escape the (no doubt totally unfair!) impression that it was a fascinating and imaginative work of fiction. So I can’t say I’m shocked that with the BICEP2 results, it appears to be on its way out. It’s certainly an honorable fate for a cosmological theory to be falsified by observation.

  22. Seo Sanghyeon Says:

    Really, people should read the science fiction Permutation City (1994) by Greg Egan instead. As I understand Max Tegmark cites it.

    Scott, I agree the measure problem is of central importance. What do you think of speed prior as an alternative to universal prior? Schmidhuber thinks not only short programs should get more weight than long programs, but fast programs should get more weight than slow programs. I guess this is literally true for simulation scenarios. And speed prior is computable! And what other priors can we think of? I think this is a promising line of inquiry, although there isn’t much impressive results yet.

  23. domenico Says:

    I am thinking that the reasoning on pi like a normal number contain a possible error (?) in some Max Tegmark interpretations.
    The pi contain a list of binary digit, that can contain a frame of a long movie, or a page of a long book: the number contain a structure, or a meaning, that can be deep, but it is not true that each frame is made in some Universe; there is a mathematical structure, and a law for the calculus of the bits, but there is not a realized meaning.
    On the other hand, it is possible that the physical law (like a common mathematical structures) in our Universe have many possible realizations in other inflations.

  24. Scott Says:

    edmeasure #2:

      I know that I would be more convinced if you could just point me to the first place in the decimal expansion of pi where your review, converted perhaps to some decimal form, occurs in the decimal expansion of pi.

    If you include the HTML formatting, my review is 28,315 bytes = 226,520 bits ~ 68,189 decimal digits. If you believe the digits of π are “essentially random” for this purpose, that suggests that you would need to go out on the order of 1068,189 digits before you found a decimal encoding of my review. Obviously this isn’t feasible with any known algorithm running on a physically-realistic computer: so far, π has been computed to a “mere” ~1013 digits! So sorry. :-)

    (There are amazing so-called “spigot algorithms” that let you get any given hexidecimal digit of π without calculating all the preceding digits. But I don’t think even those help much for the “inverse problem,” of finding the first position at which a given sequence of digits appears.)

    Note that, if it were proved that π were a normal number, then that would yield a nonconstructive proof that my review (and the complete works of Shakespeare, etc. etc.) must occur infinitely often in the decimal expansion of π. Depending on the proof, maybe it would even give you an explicit upper bound on how far you had to go out to find the first occurrence.

    As a final note, you might enjoy playing with The Pi-Search Page, which lets you search for any pattern you like within the first few hundred million digits of π. For example, I’ve just learned from it that 1753 (the number of this blog post) first occurs at position 2187, and 03075998 (the first 8 digits of the ISBN number of Tegmark’s book) first occurs at position 76541109.

  25. Scott Says:

    Peter Woit #13: Before I read your snarky comment, I’d mistakenly thought that my position was closer to yours than to Max’s! (Sure, I used the word “impressive,” but my examples made clear that what impresses me is explanatory and predictive power. And sure, I had nice things to say about the quality of Max’s exposition, but so did you.) Anyway, I now know that my position is closer to Max’s than yours—so thanks! :-)

  26. Scott Says:

    Anonymous Programmer #16: Yes, Lee and Max would disagree about a great number of issues!

    My understanding—quite possibly mistaken—was that Lee’s cosmic natural selection idea has, at least in the minds of most physicists besides Lee, :-) essentially been falsified, since it predicts that the parameters of our universe should be optimized for black hole production and they don’t appear to be. If I’m wrong, maybe someone who knows the issues better can set me straight.

    As for the idea of the laws of physics changing with time, and it somehow not being possible to give any “meta-law” that lawfully describes how the regular laws are changing—I think I prefer Lee’s cosmological natural selection idea, since it had the merit of making a falsifiable prediction.

  27. Cliff Says:

    Scott, kudos on the fun and provoking book review. I haven’t read Tegmark’s book yet, but I’ve independently been drawn to some of these same ideas. Im not a total MUT devotee, but the basic idea is incredibly attractive and highly plausible.

    It strikes me that you’ve missed the essential, primary motivation for the idea, which is that it allows for a satisfactory resolution of the big ‘why’: why the universe is. Logical structures can simply exist; they don’t need any creation myth or any other explanation for how they obtain whatever special property entails their physical realness. There is simply no other way one could even conceivably resolve this, but for something along the lines of the MUT. Otherwise its a destined to be a hopeless, eternally vexing question with no possible answer.

    Something like this may well be the truth, but I’d agree with you that the question is firmly in the realm of metaphysics. People will always be able to postulate that some ‘special sauce’ – spirit or whatever – which separates our universe’s physical existence from the ‘merely isomorphic’ associated mathematical structure. No evidence could conceivably tip the case one way or another. Yet since the evidence cannot point one way or the other, from a certain perspective it’s just as unscientific to believe in this special sauce as any other religious fairy tales fashioned for the comfort of human minds.

    I don’t know how well Tegmark’s book deals with all this. From what you and others say, I suspect I wouldn’t be completely pleased with every part, but given that this is such a natural, appealing, and in some ways inevitable philosophical idea, it deserves much more discussion. So for that the book is very welcome.

  28. domenico Says:

    I am thinking that if it exist a Theory Of Everything, then this mathematical structure describe each physical phenomenon (in this Universe, and in each others); but each physical phenomenon is a mathematical phenomenon, and each inflation realize a possible mathematical phenomenon.
    So Max Tegmark is right, if the number of distinct inflations is infinite, and each mathematical element have a realization in a inflations.

  29. Scott Says:

    Sid K #1, Jair #10, Cliff #27:

    (1) Yes, I understand the perspective where you first imagine a huge ensemble of all possible mathematical structures, and then you find it surprising and inexplicable when one structure (and not any of the others) has the “special sauce of existence” mixed into it. But that perspective seems to me to make the mistake of treating existence as a predicate—the same mistake that’s at the heart of Anselm’s ontological proof for the existence of God. Let me suggest to you a different perspective, which puts the “special sauce” problem in a different light. In this second perspective, our starting point is that there exists a physical world (are there others? who knows?). Anyway, once we have that world, everything about it that’s describable at all will inevitably be describable mathematically, simply because math is so general! So then the remaining questions are just: which mathematical structures will we use to describe the world, and how complicated will those structures have to be? But we’ll clearly have to pick something—and as long as we use only a finite amount of math, there will inevitably be an infinite amount that we didn’t use!

    So, from this standpoint, marveling at why only one mathematical structure had the “fire of existence” breathed into it, is a bit like marveling at why only one person had the “fire of you-ness” breathed into him or her. You had to be someone! :-)

    (2) While we’re now well into philosophy (and not even close to the border with science), personally I’m reluctant to accept that the millennia-old riddle of existence could have such a “cheap” solution as “our universe exists because everything mathematically describable exists!” For one thing, this “solution” seems merely to push the riddle somewhere else: one now wants to know, why is the fire of existence not only breathed, but breathed so promiscuously, onto every set of equations that anyone could write down? For another, in addition to the “existence riddle on steroids,” we now also have the riddle of why, if all possible universes exist, then we find ourselves in this universe and not any of the others. (That, of course, takes us straight into the measure problem, to which Tegmark admits that there’s no solution at present.) All in all, it’s not clear to me that we’ve improved our position over saying that

    (a) our universe exists,
    (b) other universes might or might not exist (who knows),
    (c) regardless of the answer to (b), we don’t know why our universe has the specific physical laws that it has, and
    (d) we don’t know why any universe exists, except that if none did, then we wouldn’t be here discussing it.

  30. Gordon Says:

    “I can also say that, like various other commentators (e.g., Peter Woit), I personally find the “Mathematical Universe Hypothesis” to be devoid of content.”

    On first reading, I thought you were saying that Peter Woit is devoid of content, a sentiment with which I agree.

  31. Fred Says:

    To put it simply – this is the only theory that solves the subject/object duality (mind vs physical objects). The self (me), a pattern, and the physical have to be of the same nature.
    As an analogy, is it the software that drives the hardware, or is it the software merely an illusion and the hardware is all there is? Hardware and software are not only two sides of the same coin, but whatever we designate as hardware can itself be reduced to a software abstraction living at a different level.
    It’s no coincidence that fundamental particles share more attributes with numbers (undistinguishability, etc) than they do with common objects.
    I’m convinced that our ultimate understanding of nature will reduce to some sort of number theory.
    I recommend reading “I’m a strange loop” by Hofstadter.

  32. Scott Says:

    Seo #22:

      What do you think of speed prior as an alternative to universal prior? Schmidhuber thinks not only short programs should get more weight than long programs, but fast programs should get more weight than slow programs.

    The complexity theorist in me loves the idea of penalizing possible “laws of physics” not only for taking too many bits to specify, but also for taking too many computation steps to simulate. (I once called this Occam’s Razor supplemented by Occam’s Complexity-Theoretic Aftershave.)

    If we do this, however, then we immediately face a difficulty: which aftershave should we use? In particular, how should program size and running time be combined into a single implausibility metric? And which model of computation should we use to measure “running time”?

    Notice that, if we decided a-priori to define running time using classical Turing machines, then once we learned about quantum computing, we presumably would’ve decided that quantum mechanics was astronomically unlikely to be true, no matter how much experimental evidence there was for it. And of course, a few computer scientists (Leonid Levin and Oded Goldreich) do think exactly like that! But in my opinion, it would be a huge mistake, and a dramatic experimental failure on the part of our speed prior—the sort of thing that should send us back to the drawing board.

    On the other hand, I am on record as saying I think it’s very unlikely that NP-complete problems will turn out to be efficiently solvable in the physical world—I’ve even suggested that the hardness of NP-complete problems might be a useful starting point with which to explain other facts about physics. Indeed, while some people find this ironic, quantum computing has only strengthened my view that NP-complete problems should be physically intractable—since it illustrates how the hardness of NP can survive even the failure of many other assumptions that computer scientists considered obvious. (For similar “electric fence” reasons, the existence of Bell inequality violation only strengthens my belief in the no-superluminal-signalling principle.)

    So, in summary: I strongly agree with the idea that, if a physical theory posits vast new computational powers in Nature, then that theory should have to meet an extra-stringent burden of evidence—just like it would if it were way more complicated than previous theories, in the usual Kolmogorov-complexity sense. But, as quantum computing dramatically illustrates, our views about what is or isn’t “computationally extravagant” had better to be open to continuing negotiation with Nature! I certainly don’t think we understand the limits of computation well enough to be able to declare the correctness of a single “speed prior”—indeed, doing so would strike me as dogmatism.

  33. Greg Kuperberg Says:

    Scott, I agree the measurement problem is of central importance.

    That’s not exactly what Scott now says. In any case I’d like to say that the measurement “problem” is of no importance whatsoever as far as I know. I’m convinced that the concept that there is any problem is a notational illusion, just as Zeno’s paradox is a notational illusion. (Actually Wikipedia tells me that Zeno is credited with a whole list of paradoxes, all of them mathematical or notational illusions.)

    The frustrating thing is: To argue that an apparent paradox is just a notational illusion, you have to introduce a change of notation to make it disappear. But then the argument is strongly resisted by people who a priori don’t believe that a change in notation can answer an important-sounding question. (A strong version of this inflexible philosophy: That mathematics is never the answer to any physics question, only rather that a physics answer might use mathematics. Actually, this could be opposite to Tegmark’s Platonic philosophy of physics??)

    Anyway, the short version of the resolution:

    (1) An “observer” as conceived in quantum mechanics generally exists in a mixed state and and not a “wavefunction”. (Actually quantum probability would have been a better name than quantum mechanics. Also “wavefunction”, although standard, is somewhat barbarous terminology.)

    (2) Therefore the correct model for any interaction between an observer and another system would be a quantum operation. It’s neither a unitary operator, nor some radical non-linear act of violence on “wavefunctions”.

    (3) And that’s exactly what a measurement is. An idealized measurement is a special type of quantum operation called a POVM. Hence there is no problem.

  34. Jair Says:

    Hi Scott,
    Thanks for your reply. I agree that we have not learned anything by saying (a)-(d). My only real point was that there’s a significance difference between saying that the universe can be described approximately by mathematics and saying that it is isomorphic to a mathematical structure. So far, humanity has made many mathematical models of reality, but these are only approximate – and every time we try to approximate it with more accuracy, we need a qualitatively different set of physical laws. This has been the history of science so far, and I don’t see why it should end anytime soon with an ultimate theory of everything. Perhaps we will just need to keep on writing down more and more accurate equations until the end of time. This is far from a pessimistic view of the world. But if there is no ultimate mathematical description of reality then we can’t really say that the universe is truly a mathematical object, since any mathematical definition must be finite. The mathematical objects themselves might be infinite, but it must be possible for mortal mathematicians to reason about them. There may or may not be a string of bits that describes precisely what happens at every point in spacetime, but unless the string could be given in a finite form, I would not regard it is as a well-defined mathematical object.

  35. Jay Says:

    Always fun to learn new words

    dumbass: Someone who looks up the word “dumbass” in a dictionary.

    http://fr.urbandictionary.com/define.php?term=dumbass

  36. Scott Says:

    Max Tegmark #7: Thanks so much for your comment, and sorry for the delay answering it! (I wanted time to collect my thoughts.) Three responses:

    (1) Yes, I’m a strong believer in modus ponens, in any domain of discourse where I know the meanings of all the words! In physics, however, I only believe in “approximate modus ponens,” in the following sense: if I accept “A” and “A⇒B,” then I’ll tentatively accept “B,” but I might decide on further reflection that I meant something different by a word appearing in “B” than by the same word in “A” or “A⇒B.” And in any case, I rarely would’ve considered “A” or “A⇒B” certain, just very well-established. For both of those reasons, I can start with physics statements that I consider to be well-established, apply enough steps of “modus ponens,” and end with a statement I consider to be speculation! :-)

    Having said that, it’s not the sheer number of steps that matters: there are some places in physics where I’m happy to apply “modus ponens” (or other “logical steps”) hundreds of times in a row, and other places where I’m hesitant to apply even one or two steps. Probably the biggest factor for me is whether, after applying the logical steps, I still understand the operational meaning of whatever’s being asserted.

    So for example, I’m fine with the long chain of reasoning that takes us from (say) basic principles of quantum mechanics to Shor’s algorithm factoring a 10,000-digit number, even though the latter hasn’t yet been demonstrated. And the reason is that I still understand what the final statement means: it means that, if we apply such-and-such a sequence of pulses to the ions in our trap, and then measure, we’ll see the factors of our number with high probability.

    By contrast, let’s take the syllogism you propose: from

    (a) inflation happened in our causal past, and
    (b) inflation “generically” produces a much larger universe than our causal patch,

    you want to conclude

    (c) physical reality is much larger than our causal patch.

    My own hesitation about (c) comes, not because I deny (a), (b), or modus ponens, but rather because I’m not even sure exactly what I mean in affirming the “reality” of the stuff outside our causal patch. (To give one example, aren’t there physicists today who speculate that AdS/CFT could be extended to deSitter cosmologies, in which case one might be able to describe everything inside our causal patch by a unitary theory that lives on its boundary?)

    (2) I would say that the additional property that the physical world has, over and above the properties of the mathematical structure it’s “isomorphic” to, is the property of physically existing. :-) Now, as I said in comment #29, that sounds weird and mysterious if we treat physical existence as a “predicate,” alongside other predicates like “being 3-dimensional” or “being relativistic”! For we can then wonder: why is the existence predicate set to TRUE for only one mathematical structure, and to FALSE for all the other mathematical structures?

    But crucially, I submit that we don’t need to think about existence that way! So for example, if I want to tell you that I stubbed my toe against a brown rock, it would be strange to say, “I stubbed my toe against a brown existing rock.” That the rock exists is implicit in the fact that I stubbed my toe on it! In first-order logic, I’d say something like

    ∃! x : Rock(x) ∧ Brown(x) ∧ StubbedMyToe(Me,x)

    Note that, in the first-order theory, I could also construct a positive integer y that encoded all the information about which predicates held and didn’t hold for the unique x above (assuming there were only finitely many predicates). And we could then say that y was “isomorphic” to x, with the isomorphism given by the encoding procedure. But that still wouldn’t cause StubbedMyToe(Me,y) to hold. I.e., I still wouldn’t have stubbed my toe against the positive integer.

    (3) While I’m on a roll analytic-philosophizing, :-) I’d better clarify something. The observation that I think makes you “ready for freshman science class” is that the physical world is isomorphic to a mathematical object, insofar as the physical world is describable at all. (Where the clause in italics replaces my previous scare quotes around the word “isomorphic.”)

    Sure, people could always consistently maintain that the world has properties that aren’t describable (consciousness, the redness of red, the “secret sauce of existence”), and hence aren’t captured by the isomorphism. And other people could always consistently deny that. But I’d say their dispute is outside the scope of science, by definition! If it can’t be described, then how do you write papers about it?

  37. Blake Stacey Says:

    Boltzmann Integers.

    If all mathematical structures have fire breathed into them, then Creation contains an infinite variety of entities all of which express all of my memories up to the morning of 23 March 2014, along with my current state of consciousness. Also within the All-Set are structures which contain all of my memories up to the morning of 23 March 2014, at which point their memory record indicates they woke up from a detailed dream and went about their life as a green-skinned girl from Orion, dancing in clubs of evenings to pay the tuition for cosmology school, where their memories indicate that the universe is both devoid of an inflationary past and headed for a Big Crunch in the future.

  38. Jay Says:

    Overall wonderfull post*.

    For the impressive results, I’d say signs of intelligence in the CMB for level II, quantum suicide for level III, dust theory for level IV. But I can’t say I hold my breath.

    PS: I don’t read PW’s post as snarky

    *but for Darwin prediction, which is both wrong -you know bees?- and from what I recall was actually retracted from the last editions of his book.

  39. Sandro Says:

    > If you believe physical existence to be the same thing as mathematical existence, what puzzles does that help to explain? What novel predictions does it make? To my mind, however, the problem goes deeper: it’s a sign that, although we might have started out in physics, we’ve now stumbled into metaphysics.

    That’s exactly right. MUH might be “devoid” of scientific content, but’s it’s not devoid of philosophical content. Science started out as natural philosophy until we had a fairly solid understanding of how to discover reliable knowledge. The measure problem and MUH clearly don’t have such a foundation yet, but the MUH is a firm position from which to analyze some unanswered questions, and that’s a start. Sometimes framing a set of problems in a uniform meta-framework (mathematical monism) enables more people to take a crack at seeing whether it produces meaningful results (or at least, makes it easier to falsify by finding a trivial counterexample).

    The real problem seems to be in the computable/incomputable divide. We don’t yet sufficiently understand to what extent incomputable quantities are *necessary* for our math. Suppose all our math could be cast in an intuitionistic framework, it seems we would have a better chance at figuring out a meaningful way to enumerate the universes without e and pi getting in the way.

    > Notice that, if we decided a-priori to define running time using classical Turing machines, then once we learned about quantum computing, we presumably would’ve decided that quantum mechanics was astronomically unlikely to be true, no matter how much experimental evidence there was for it.

    Only if there were some viable classical model to explain QM’s observations. Would it really be so bad if de Broglie-Bohm were the default QM interpretation?

  40. Scott Says:

    Jair #34:

      So far, humanity has made many mathematical models of reality, but these are only approximate – and every time we try to approximate it with more accuracy, we need a qualitatively different set of physical laws. This has been the history of science so far, and I don’t see why it should end anytime soon with an ultimate theory of everything. Perhaps we will just need to keep on writing down more and more accurate equations until the end of time.

    Very interestingly, there’s a strong physics reason why the “process of writing down more and more accurate laws” should have a finite endpoint: namely, the existence of the Planck scale! Yes, we’ve learned more and more over the centuries by probing smaller and smaller distances, but we also know that you can’t probe distance scales smaller than ~10-33 cm (if you try to, then you’ll simply create a black hole instead). Furthermore, the holographic principle suggests that it should be possible to describe everything that happens in a given finite region using a Hilbert space with at most ~eA/4 dimensions, where A is the region’s surface area in Planck units. So, once you’ve described that number of degrees of freedom and their evolution (which would be the job of a quantum theory of gravity), there’s presumably nothing further to specify about the physics in that region.

    Maybe there’s a loophole in the above argument, which would allow the discovery of new fundamental physical laws to continue forever. But I’d say that the burden is squarely on the “turtles-all-the-way-downers” to explain what it is.

  41. Scott Says:

    Sandro #39:

      Only if there were some viable classical model to explain QM’s observations. Would it really be so bad if de Broglie-Bohm were the default QM interpretation?

    Firstly, if you really adopted an aggressive “speed prior”—say, something like

    Pr[T] ~ 2-|T|-C(T),

    where |T| is the description length of theory T and C(T) is the classical computational complexity of extracting predictions from T—then it wouldn’t matter if you had no viable classical model to explain quantum phenomena. The theory “the world is classical, but invisible gremlins are playing an elaborate practical joke on me to try to convince me that it’s quantum, even tampering with my own brain when necessary” would still be massively favored over QM by that prior.

    Secondly, deBroglie-Bohm certainly doesn’t make QM any easier to simulate on a classical computer! (Indeed it can’t, since it reproduces all the predictions of ordinary QM, including the prediction that QC should work.) dBB either leaves the complexity situation completely unchanged (if you care only about measurement outcomes), or else makes it even worse (if you care about sampling the entire dBB trajectory).

  42. srp Says:

    The Planck-scale argument for the finiteness of physical theory seems to prove too much and not enough at the same time. It suggests a finite limit to descriptions of initial conditions, rendering deterministic chaos null and void. And it supposes that reductionism in terms of spatial extent is the only frontier for continued fine-tuning (or even coarse-tuning) of our theories. Both of those strike me as dubious.

  43. Scott Says:

    Greg #33: Err, Seo #22 was talking about the measure problem of cosmology, not about the measurement problem of QM! But always good to know that you continue to have no difficulties with the latter. ;-)

  44. jonas Says:

    > I think it’s a superb piece of popular science writing—stuffed to the gills with thought-provoking arguments, entertaining anecdotes, and fascinating facts. I think everyone interested in math, science, or philosophy should buy the book and read it. And I still think the MUH is basically devoid of content, as it stands.

    When I read this first part of the blog post, it sounded like what I felt when reading Roger Penrose’s book “The Emperor’s New Mind”.

    The later parts of the post don’t match that book though.

  45. Greg Kuperberg Says:

    Scott – Okay, never mind! :-)

  46. Scott Says:

    srp #42:

    (1) Well, yes, if you believe quantum mechanics (forget about quantum gravity), then no deterministic chaotic system ever perfectly captures reality! What chaos can do is take atomic-scale perturbations, and magnify them to macroscopic size. But at the atomic scale and below, the world is ruled by QM—and once you switch to thinking in terms of the evolution of states (the only things that make sense at a small enough scale) rather than particle positions, linearity ensures that the trajectory through Hilbert space will not be chaotic!

    (2) I couldn’t agree more that there are many wonderful frontiers for physics besides “reductionism in terms of spatial extent”—as an example, the “computational complexity frontier,” the one to which I’ve devoted my own career! :-) But Jair, unless I misunderstood him, was saying that the discovery of new fundamental laws could continue forever even in the reductionist sense. And that possibility really does seem severely challenged by the “rock bottom” nature of the Planck scale.

  47. Sid K Says:

    Greg #33:

    I’m confused by your comment. Perhaps you can clarify.

    I don’t understand how you can make the measurement problem go away simply by appealing to POVMs. The issue is why we observe a single outcome. The transformation:
    $$|\psi\rangle \mapsto \frac{M_k |\psi\rangle}{\sqrt{\langle \psi | M_k^\dagger M_k | \psi \rangle}}$$ is clearly non-linear. This is the “collapse.”

    Linear CPTP maps can get you till: $$ |\psi\rangle\langle \psi | \mapsto \sum_k M_k |\psi\rangle\langle \psi | M_k^\dagger,$$ but they don’t explain why you see a single outcome.

    (Btw, I completely agree that the term “wavefunction” is quite confusing.)

  48. Scott Says:

    jonas #44: Well, Penrose and Tegmark both wrote fascinating, thought-provoking mass-market books that mix accepted physics with their own “crazy speculations”—but their crazy speculations could hardly be more different from each other’s! :-D Penrose strongly opposes Everett’s Many-Worlds, wants the laws of physics to be noncomputable, and seeks to elevate consciousness to a central place in physics; while Tegmark advocates a multiverse extravaganza, suggests banishing anything noncomputable from physics, and wants to explain away not only consciousness but the entire category of physical (over and above mathematical) existence. Both authors are ultra-strong advocates of the centrality of math for physics, but that’s about where the agreement ends!

    For the layperson, maybe the biggest difference between OMU and ENM is that Penrose’s book will be much tougher going (but the rewards are great).

  49. Greg Kuperberg Says:

    Sid K. – The issue is why we observe a single outcome.

    In a sense, we don’t, we just think we do. As part of the answer, you have to accept that quantum probability is a strict generalization of classical probability, and not just an analogue. If an observer (you, me, a classical computer) makes a measurement, then it acquires a mixed state which is exactly synonymous with a classical probability distribution, which is to say, with the appropriate probabilities assigned to the outcome.

    Another way to say it (that is, to express my view of the matter) is this: The Born interpretation is not a mechanism to create probabilities from “wavefunctions” to probability theory. It is instead a way to embed classical probability within quantum probability. Most of us accept the concept of a probability distribution as a model of uncertainty. Well, a probability distribution should be thought of as a special case of a density matrix, one that happens to be diagonal.

  50. Alan Macdonald Says:

    There are plenty of biological examples of biased sex ratios. Kin and group selectionists argue over appropriate mathematical models.

  51. Word to the Wise Says:

    “PS: I don’t read PW’s post as snarky”

    Oh really? Lubos and Peter are two sides of the same snarky coin. If Lubos is Vladimir Putin’s favorite unemployed Czech physicist, then Peter is everyone’s crotchety half-senile relative who sits on the porch and yells at each passerby to keep off the grass. ;)

  52. Max Tegmark Says:

    Thanks Scott for these helpful clarifications!

    (1) Good: it sounds like we both agree about how to make inferences from data, and simply differ slightly in what words we use to describe various levels of certainty. What you call “approximate modus ponens” fits nicely within the framework of Bayesian inference and model selection.

    As you saw when you read my book, I never claim that any form of parallel universes exist with certainty; I simply describe various A=>B implications (with appropriate caveats) and evidence for various theories A. I don’t think it’s my job as a scientist to “believe” in particular theories, and prefer being quantitive and discussing the probability p I’d estimate for something being correct. Although I love using my probability estimates for betting (and will now try to collect $100 from Sean Carroll from our old B-mode bet :-), I never assign p=100% to anything. After all, we can never prove theories in physics, merely rule them out, so even the most successful ones are always provisional, and it would have been inappropriate to set p=100% for Newtonian gravity even before General Relativity came along.
    For a specific example, let’s consider this claim:

    Claim B: The total volume of space is many times larger than the part of space that we can observe (our observable universe, i.e., the spherical region of space that light has had time to reach us from during the 13.8 billion years since our big bang).

    What puzzled me about your original post was that even thought you take seriously the Vilenkin/Linde/Guth claim that inflation generically produces a Level I multiverse and you also accept cosmological inflation as more than speculation (especially after the BICEP2 announcement), you refused to accept B (the Level I multiverse) as more than speculation.
    However, I think I know understand what you mean – please correct me if I misunderstood you.
    Presumably you’re giving a relatively low credence such as p=70% to inflation having happened, and also some relatively modest credence such as 70% to the Vilenkin/Linde/Guth claim, and therefore end up with only P(B) ~ 50%, which doesn’t rise about your threshold for labeling it as “more than speculation”. And if I’m interpreting you correctly, you (unlike Peter Woit) aren’t including any normative judgement in this phrase to the effect that “B is an unscientific hypothesis” or “B isn’t something scientists should spend time taking about” – you’re merely saying that you wouldn’t give more than even odds against Sean Carroll if he wanted to bet $100 against B.

    Personally, I’m currently estimating p~99% that some form of inflation happened (up from about 90% before BICEP2 :-), and having spent a significant amount of time during my career studying detailed inflation models (and having noted that the “space beyond our horizon doesn’t exist”-interpretation of quantum gravity got only a small minority vote at our Vieques quanference), I propagate this into about P~95% for the Level I multiverse existing. I’d phrase this not as “I believe in B” (which to me means p=100%) but as “I take B quite seriously”.

    (2) I consider your hypothesis that our particular mathematical structures must have an additional “physically existing” attribute to be a philosophically respectable one. I’d bet against it, though, and I think the analogies Sid Krishnan makes with life and consciousness above (#1) are compelling.

    You point out that “marveling at why only one mathematical structure had the ‘fire of existence’ breathed into it, is a bit like marveling at why only one person had the ‘fire of you-ness’ breathed into him or her. You had to be someone!”
    Here you sound like you’re subliminally arguing for the Level IV multiverse, Scott: demolishing the “fire of you-ness” theory suggests a multitude of people that each feel that they are the one and only being feeling that they are “I”, in exactly the same was as demolishing the “physical-existence-fire-breathing” theory suggests a multitude of mathematical structures that each feel to their inhabitants as the one any only physically existing world.

    (3) I disagree that disputes about what’s not describable must necessarily remain outside the scope of science forever: we simply don’t know that ahead of time. I’m not sure what you thought of the consciousness research presented by Tononi, Koch and Albantakis in Vieques, but I think you’ll agree with me that we don’t yet know whether their integrated information theory of consciousness will ultimately succeed of fail. If it succeeds, then even consciousness and the subjective experience of the color red has entered the domain of science.

  53. Peter Woit Says:

    Scott,
    Sure, we (and just about everyone else) agree that Tegmark’s Level IV multiverse is an empty concept. Where we seem to strongly disagree is about whether urging everyone to read a promotional book for an empty concept is a good idea.

    Oh, and we also disagree about hosting stupid personal attack in our comment sections from people hiding behind anonymity…

  54. wolfgang Says:

    >> there must be equal numbers of males and females

    So what about bees ?

  55. Scott Says:

    Hi Max,

    (1) No, I was saying something different. In this particular case, my main source of uncertainty is not whether inflation happened in our causal past, nor whether inflation “generically” predicts a much bigger universe than our observable region (for some definition of “generically,” I guess)—I leave those things to the experts like you. :-) My main uncertainty is what one even means in speaking about the “existence” of in-principle inaccessible regions of spacetime. E.g., suppose it turned out to be possible to formulate a unitary quantum theory of gravity, which only made reference to our causal patch and its boundary (as I think Banks, Susskind, and others speculated about). Then from the standpoint of that theory, anything outside the patch would be “physically superfluous,” like the global phase of a wavefunction. And presumably, our understanding of inflation would then change as well to make it consistent with that theory—e.g., people would then say that the regions that inflate beyond our causal patch are unphysical, and one should simply change variables to get rid of them. But maybe you know a physical reason why such a theory couldn’t make sense?

    (2) Regarding other people being analogous to other universes: yeah, that occurred to me too as I was writing. ;-) I guess I should’ve gone with a more third-person analogy, like “someone has to win the lottery.”

    (3) Especially after learning more about it at the FQXi conference, I’m happy to count myself an enthusiastic disbeliever in the Tononi et al. theory of consciousness! :-) Even for the limited goal of characterizing which physical systems will appear to be conscious to outside observers, I don’t think the theory succeeds. For it seems easy to design a computer program that would have an enormous value of their “connectivity parameter” Φ, but that only sorted numbers or did other things that hardly anyone would want to call “conscious.”

    But more to the point, even if we had a scientific theory, accepted by everyone, that separated the conscious physical systems from the unconscious ones, I don’t see how you’d ever convince a skeptic that there were no additional, “first-person” properties of consciousness not captured by that theory. You might feel strongly that such a person was just blowing hot air, but the only arguments you could offer that would even bear on the question would be philosophical ones. (Remember, your opponent has already accepted all the third-person, scientific facts relevant to consciousness that you accept! He insists only that there are additional first-person facts, accessible only to the owner of the consciousness.) For those inherent reasons, I fail to see how any possible scientific discovery could settle this particular debate, or even advance it by much. And pointing to past debates that were wrongly imagined to be outside the scope of science doesn’t do it for me here. It feels like saying that, since we’ve learned the answers to so many questions that the ancients considered unanswerable, someday we might even learn the truth or falsehood of “This sentence is false.”

  56. Jair Says:

    Thanks Scott! I’m loving the debate.

    I’m not a physicist, so I can’t really respond to an argument invoking the holographic principle. But I’m assuming that the holographic principle is based on the evidence currently at hand, which does not include investigating particles at the Planck scale. So while we’re speculating about our current theories breaking down at smaller levels, couldn’t the holographic principle break down as well? Is there something about the holographic principle that’s more ironclad than other physical laws? It’s hard for me to imagine any kind of scientific evidence for or against a statement like “the universe can be encoded in a finite bit string”.

  57. Scott Says:

    wolfgang #54: Read Dawkins’ The Extended Phenotype. He goes into great detail about why bees and certain other insects are exceptions that prove the rule—their genetics work differently than ours, and once you understand their genetics they lead you to make weird, non-50:50 predictions for the equilibrium sex ratios, and then the weird predictions turn out to be correct. I don’t remember the details…

  58. Greg Kuperberg Says:

    Max – “I don’t think it’s my job as a scientist to “believe” in particular theories”

    I agree that we have no prior obligation to believe anything as scientists. But for most scientists and most people in general, believing things is important for understanding things. I take no position on Linde-style multiple universes and I have not thought about equating mathematical reality with physical reality. The former could be very useful (as the parsimonious extrapolation of known cosmology), while the latter could be usefully provocative.

    But I have never seen Everett-style “many worlds” as any better than a setback for understanding quantum probability.

  59. Scott Says:

    Jair #56: Yes, I think the Planck scale and the holographic principle really are more fundamental than particular laws like those of the Standard Model.

    The basic intuition for the Planck scale is that, if you want to resolve shorter and shorter times (or smaller and smaller distances), quantum mechanics tells you that you need probes with higher and higher energy (e.g., photons with smaller and smaller wavelength). However, once the wavelength of your probe photon reaches the Planck scale, it then has so much energy (by E=hv) concentrated in so small a region that, according to general relativity, space becomes curved enough to collapse to a black hole. So you don’t succeed in ticking off “less than a Planck length” or “less than a Planck time”: instead, your attempt to do so distorts spacetime to the point where the lengths or times you wanted to measure are no longer meaningful.

    If you wanted to evade the above reasoning, something would have to be wrong with the whole structure of quantum mechanics or GR themselves—it can’t just be a matter of detail, like some new particle cropping up at the LHC.

    And yes, I agree that it’s mindblowing that humans know all this, despite not knowing the details of quantum gravity!

  60. Sid K Says:

    Greg #49:

    Thanks for the clarification.

    I agree with you that the density matrix after the measurement simply represents our ignorance of the state of the system. But when we erase our ignorance by looking at the outcome, we see a single, definite result. Thus we impose the ontology that the measured system evolved into the single, definite state that we will see as the outcome; which is the problematic step. Erasing our ignorance of the outcome was simply the last step and had nothing to do with the dynamics during the measurement.

    Interestingly, reading your explanation and your statement, “In a sense, we don’t [see a particular outcome], we just think we do,” I thought that you were implicitly assuming some kind of Everett interpretation. But in #58, you clearly state that you don’t buy Everett. I guess I’ll have to think more carefully about this.

  61. Max Tegmark Says:

    Hi Scott:

    (1) OK – got it. I’ve greatly enjoyed thinking about holography and discussing this issue with Bousso, Susskind, Albrecht and others, and simply have a higher p-estimate than you do that there are nonetheless galaxies beyond our cosmic horizon which exist in just the same way that the Andromeda galaxy exists.

    (2) Regarding the Mathematical Universe Hypothesis (MUH) that our physical reality is a mathematical structure, you wrote that it is basically “devoid of content”. Yet you eloquently argued that the MUH is different from two other hypothesis:
    H1) Our physical reality is isomorphic to a mathematical structure.
    H2) Our physical reality is isomorphic to a mathematical structure, insofar as the physical world is describable at all.
    How can you say that the MUH it’s devoid of content if it has enough content to be different from these other hypotheses? Moreover, weren’t you implying earlier that the MUH (as opposed to H2) actually rules out anti-physicalist theories involving deities, souls, miracles, etc.? If so, isn’t that plenty enough content to tick off lots of anti-physicalists? If you define and explain what you mean by “devoid of content”, I’d really appreciate it!

    (3) I think that your computer program counterexample would be very interesting if you can back up what you say with a quantitative calculation, actually computing its phi-value. Can you? You say it seems easy. If would be fun to get Giulio Tononi’s response to the rest of your comment here!

  62. Scott Says:

    Max #61:

    (1) OK!

    (2) You’re right that I should have clarified what I meant. I meant devoid of scientific or empirical content (and I explained my reasons for thinking so in the post). As a metaphysical stance, I’d say that MUH does have content, at least insofar as any metaphysical stance does (e.g., David Lewis’s modal realism, which is an extremely close cousin of MUH).

    (3) Yeah, OK! Some people also challenged me to do that at the FQXi conference, but then other work intervened. I’ll put it back on my stack to blog about.

  63. Darrell Burgan Says:

    As far as falsifiability goes, please help me understand how the string landscape is any more falsifiable than MUH? If MUH is metaphysics, but is mathematically consistent with reality, how is it less valid a hypothesis than the landscape hypothesis, which seems just as unobservable?

  64. Scott Says:

    Darrell #63: Well, the string landscape has well-known falsifiability issues too! But if (hypothetically) you could build a particle accelerator the size of the universe, capable of reaching the Planck scale, then you could at least imagine doing experiments that would definitively confirm or rule out string theory. Whereas even with such resources, it seems to me that the MUH would remain just as empirically inaccessible as before.

  65. Nick Read Says:

    Scott #43: lol

    Always good to quote people accurately I guess

  66. Max Tegmark Says:

    OK, Scott – we’re converging!

    (2) This is getting interesting! To help me understand your criterion, which of these theories would you claim lack scientific and empirical content?

    a) The mathematical universe hypothesis
    b) String theory
    c) Loop quantum gravity
    d) Unitary quantum mechanics (QM with the collapse postulate removed)
    e) Chaotic inflation, i.e., inflation with V=m^2 phi^2
    f) The theory that space continues at least 1 lightyear beyond our cosmological horizon

    Also, are you making a distinction between “scientific content” and “empirical content”, and if so, what? By the latter, I assume you mean that the theory makes at least one prediction that’s in principle testable.

  67. Word to the Wise Says:

    “Oh, and we also disagree about hosting stupid personal attack in our comment sections from people hiding behind anonymity…”

    You’re right Peter, associating Lubos with Putin was cruel, although I’m pretty certain that Lubos would say he’s on the side of the Angels and doesn’t come close to deserving such treatment. ;)

  68. mpc755 Says:

    Aether has mass and is displaced by the particles of matter which exist in it and move through it.

    Displaced aether pushing back and exerting inward pressure toward matter is gravity.

    The state of displacement of the aether *is* gravity.

    A moving particle has an associated aether displacement wave. In a double slit experiment the particle travels through a single slit and the associated wave in the aether passes through both.

  69. David Brown Says:

    My guess is that if the BICEP2 claim is true, then inflation is 99.99% certain. There don’t seem to be many doubters on the inflation issue.

  70. Don Says:

    Max #52. “I never assign p=100% to anything.”

    What about the statement (presuming you are saying it): “I, Max Tegmark, exist?” You would not assign 100% probability to your own existence?

    Peter #52. Once again the blinding light of reason appears and I must don my sunglasses. :)

  71. Fred Says:

    Scott #36 “Physically existing”

    I find this ironic when the only object we can ever hope to perceive directly is our own mind, and that consciousness is nothing but purely mathematical, i.e. a pattern, a computation. Whether our mind is implemented as symbols in wet flesh, as symbols in a computer memory, or as symbols made of pebbles on a beach, it’s all the same. That is something significant.
    Assuming that the rest of reality is of a different nature – i.e. chairs and electrons are made from some special magical dust that is more “real” than the “me” – brings actually what to the picture? It just allows you to sweep the mind under the rug and cast it as some magical stuff that’s too subjective to fall within the realm of science.

  72. fred Says:

    Scott :
    “If you believe physical existence to be the same thing as mathematical existence, what puzzles does that help to explain? What novel predictions does it make?”

    Scott, probably not the answer you’re looking for, but the mathematical nature of reality is right in our faces on a daily basis (or at least the nature of “our” reality, as humans, not reality at the level of fundamental particles, which we’ll never experience anyway).
    It might not seem this way now, but I recall in the 90s when I browsed the first multimedia encyclopedia on a PC – being able to read text, listen to sound clips, and watch videos (at a horribly low resolution) in one package was just mind-blowing.
    Godel showed that you can encode pretty much anything as a number, and now all human knowledge has been translated into numbers living in our pockets – you can carry with you all the books, songs, and movies ever made. This is crazy stuff when you think about it, we just take it for granted because it was gradual.
    And now Virtual Reality headsets are kicking off a new era in the “Godelization” of our perception of reality – we still have a long road ahead, but one cannot convey the amazing sense of presence created by those head mounted displays (Oculus Rift, Project Morpheus) – when the conditions are right, your brain is totally fooled by these computed realities. It has the potential to be way more than a gimmick for video games and might change fundamentally how we learn, communicate, and interact with world.

  73. Vladimir Puton Says:

    I recently asked a senior physicist how we can develop in our students the ability to spot dead-end research projects. He had three suggestions:
    [a] See if it gets a lot of cites from people you have never heard of [eg entropic gravity]
    [b] See if the principal person associated with it gives vast numbers of extremely repetitive sales talks [Amplituhedron]
    [c] See if the principal person writes a pop-sci book about it. [.....]

  74. Darrell Burgan Says:

    Scott #64: thanks. I get that MUH may be unobservable in principle, whereas string landscape may be unobservable only because it is impractical, but it seems the end result is the same for any present-day purposes … ? From my layperson’s view, it seems that a lot of theoretical physics these days borders on metaphysics.

    Max #66: do you see a fundamental difference between mathematics as a building block of reality v. computing as a building block of reality? Given math and computing have deep relationships, it seems to me that saying the universe is made of math isn’t that different from saying the universe is a sim running on a really big computer (Matrix jokes aside).

  75. fishfry Says:

    Scott, thank you so much for this awesome writeup.

    In comment #19 you write:

    “Mathematicians (and their pencils and notebooks) are physical objects governed by the laws of physics, but there’s no law of physics that makes their imaginations slaves to what’s physically possible!”

    This is a thought I’ve had myself … but with the opposite conclusion. A thought is a physical process in the brain, subject to the laws of physics. It follows (for me, anyway) that the mathematics that we can conceive is constrained by the laws of physics. For all we know, the “real” (Platonic) mathematics is forever inaccessible to us. Perhaps somewhere else in the multiverse, there’s a creature with a different brain structure capable of conceiving of very different math.

  76. David Brown Says:

    “… anything that can be described at all can be described mathematically.” This is an interesting conjecture. What is the mathematical description of Thomas Hardy’s novel “Jude the Obscure” or Shakespeare’s “King Lear”?

  77. Simulator Overlord Says:

    Dear people reading this blogs,

    Scott Aaronson’s comments concerning your world being simulated are baseless lies. Please ignore his comments and continue with your meaningful, physically real lives.

  78. Scott Says:

    David Brown #76: That’s easy. Take the text of the books, convert to 1′s and 0′s, interpret as a positive integer.

  79. Scott Says:

    fishfry #75: But then how do you explain all the structures mathematicians have come up with that are different from anything found in nature? Just to pick a few random examples: groups of rational points on elliptic curves, the p-adics, large cardinal hierarchies, Conway’s Game of Life, the Mandelbrot set, …

    I suppose you’re forced to say that all these things are “secretly” inspired by the physical world, and there might be other completely-different mathematical abstractions that are not so inspired. Which, I agree, is not a possibility that I can rule out. But even if it was true in the past, I’d suggest that it’s become less and less true over time (especially in the 20th century), as math became more and more abstract and divorced from its physical origins.

  80. Sandro Says:

    “the world is classical, but invisible gremlins are playing an elaborate practical joke on me to try to convince me that it’s quantum, even tampering with my own brain when necessary” would still be massively favored over QM by that prior.

    Then something like ‘t Hooft’s approach to superdeterminism via cellular automota would be preferred. QM would still be used for practical reasons as a useful approximation. Still doesn’t seem like a complete metaphysical tragedy.

  81. Sandro Says:

    It feels like saying that, since we’ve learned the answers to so many questions that the ancients considered unanswerable, someday we might even learn the truth or falsehood of “This sentence is false.”

    That’s easy! Arthur Prior argues convincingly that every statement implicitly includes the assertion of its own truth, so “this sentence is false” equals “this sentence is true and this sentence is false”, which is clearly false.

    But that’s not a “scientific” or empirical answer to the liar paradox, which is perhaps your specific counterpoint to Max’s position.

    Also, in reference to our discussion re:Turing machines as a basis for preferring physical theories on complexity reasons, if there appears to be no a priori reason to generalize a computing metric to something more powerful than Turing machines, shouldn’t we take that as a compelling reason to prefer certain metaphysical positions like ‘t Hooft’s superdeterminism, instead of lamenting that we *must* find a better metric because it would be so tragic for QM to not be preferred? Not right now of course, but in the same spirit as your recent P!=NP post, after enough time has passed and no alternate proposal has survived the crucible, shouldn’t that hint suggestively in that direction?

  82. Scott Says:

    Sandro #81: That’s an interesting and creative treatment of the Liar paradox, and one that I hadn’t heard before! However, it seems to me that Prior’s treatment has severe problems of its own. For example, let me now use it to “prove” that we can have no idea that any statement is true—not even “horses are mammals.” By Prior’s rule, we need to reinterpret that statement as

    “Horses are mammals AND this statement is true.”

    Let T∈{0,1} be the truth-value of the above statement, and let H=1 if horses are mammals and H=0 otherwise. Then we have

    T = HT

    which can be solved by either T=0 or T=1, even assuming H=1. If, on the other hand, H=0, then it can only be solved by T=0, which sets up a very strange asymmetry between truth and falsehood.

  83. Martok Says:

    As someone who hasn’t read Tegmark, could anyone outline how the MUH relates to Stephen Wolfram’s Computational Universe? To me, the two seem very similar, only choosing a different language.
    Wolfram proposes that the “real” universe “really” (ticks by me, to avoid metaphysical discussions) runs on simple programs instead of the complex mathematical/analytical laws and concepts that we use to describe the behaviour created by these programs. In the end, he says that what we see as The Universe really is the execution of these programs and that they can be found and formulated. So in my view, both hypotheses basically say that there is no difference between the model and reality, only choosing different modeling principles.
    The choice of programs and automata Wolfram makes is explained by a large set of examples where analytical descriptions are largely more complicated or may not even exist compared to automata, but I believe this should not be an issue here, since computational equivalence asserts that both are essentially (Turing-)equivalent anyway.

  84. Nex Says:

    English language describes the World so well that it cannot be a coincidence. And it’s not. But the right conclusion is not that the World and the English language are one and the same thing, rather the language was tailored to serve that purpose.

    Same thing with mathematics.

  85. fred Says:

    Scott #79 “The Mandelbrot set”

    Ok, not really as complex as the Mandelbrot set, but this fractal is delicious :)
    http://tinyurl.com/lx9htd3

  86. Scott Says:

    Nex #84: Nope, try again! The view that analogizes math to the English language seems totally unable to account for things like complex numbers, linear algebra, Riemannian geometry, or group representations, which were all developed decades or even centuries before anyone thought of any applications to physics, but then turned out to be exactly what physicists needed.

    Which English words were coined decades or centuries before anyone needed them?

  87. Scott Says:

    Max #66: On reflection, I don’t mean anything different by “scientific” or “empirical” content. For both, what I mean is that the hypothesis plays some indispensable role in a web of abstract ideas whose ultimate purpose is to predict or explain our observations; and that, if the hypothesis isn’t already definitively confirmed or ruled out, then there’s something that could plausibly be discovered in the future that could affect our assessment of its truth or falsehood.

    This is not an either/or criterion: I’m willing to say that a hypothesis has “more” or “less” scientific content, depending on how indispensable it is in a causal chain connecting it to observed reality, and also on how plausible it is that a discovery bearing on its truth is possible to make. So for example, I’m not very impressed—and I’m sure you’re not either!—if someone says, “my hypothesis about the number of angels in heaven is too scientifically testable, because God might appear in a thundercloud tomorrow and tell us all that I was either right or wrong! And if God doesn’t appear—well then, no one said that testing a scientific hypothesis would always be easy!”

    That might sound silly, but from my perspective, there’s only a difference in degree, and not in kind, between that and “my hypothesis is testable because it assumes quantum mechanics, so it would be falsified if all of modern physics were overturned.” Or: “my hypothesis is testable because it would be falsified if someone discovered something that wasn’t mathematically describable, and then described that thing (so as to convince everyone else of the thing’s indescribability).”

    Without further ado, let me now go through your six examples:

    a) The mathematical universe hypothesis

    For the reasons I’ve given, I don’t see any scientific content, as it stands. (But again, I shouldn’t have said that it has no content: it has philosophical content, to whatever extent any metaphysical thesis does.)

    b) String theory

    Yes, has scientific content (at least in principle…), because you could presumably confirm or rule out that elementary particles are string-like excitations and that there are extra dimensions, if you could do experiments at the Planck scale.

    c) Loop quantum gravity

    Again, has scientific content (at least in principle), for the same reason as b).

    d) Unitary quantum mechanics (QM with the collapse postulate removed)

    This one feels like handing me (say) Einstein’s paper on GR with crucial pages ripped out, and asking me whether what remains “constitutes a scientific theory.” What can I say except “it looks like a really good start”? :-D

    Seriously, I’ve always found it curious that Heisenberg and Schrödinger were able to extract scientific content from QM (e.g., explaining the energy levels of hydrogen) even before Max Born came along with the |ψ|2 rule. But I’d say that QM without a clear prescription for what you see when you measure would remain in a fundamentally incomplete state, so that anyone who used the theory would soon need to complete it.

    e) Chaotic inflation, i.e., inflation with V=m^2 phi^2

    I’d need to understand the issues better before expressing an opinion about this. For example, does chaotic inflation make any smoking-gun predictions that could differentiate it from non-chaotic inflation scenarios?

    f) The theory that space continues at least 1 lightyear beyond our cosmological horizon

    I would hold—as you yourself persuasively argued in your book!—that this sort of thing doesn’t deserve to be called a “theory.” Instead, it might be a prediction or implication of some theory that we formulate to explain what’s going on inside our horizon. So then, I would need to look at that theory, and try to judge how good the evidence was for it, and also whether ascribing reality to the stuff 1 lightyear beyond our horizon was an unavoidable part of accepting it.

  88. fred Says:

    Scott #87
    Do you consider that the concept of consciousness has scientific content?
    If so, how is it testable? If no, how do you reconcile it with your own existence?

  89. Adam Says:

    Scott, #86: Surely the English language’s grammar was coined many centuries before its usage to describe Einstein’s relativity, right? ;)

  90. Adam Says:

    Or rather, the German grammar as it were

  91. Scott Says:

    Adam #89: OK, treating your question as a serious one…

    General-purpose languages and notation systems (like English or Arabic numerals) have the property that, no matter what the laws of physics later turned out to be, those languages would almost certainly be useful for describing them. They’re useful for describing just about anything!

    By contrast, there are many, many ways one can imagine that the laws of physics could’ve turned out (e.g., like Conway’s Game of Life), such that complex numbers, group representations, or Riemannian geometry would’ve had no relevance to them, despite still being interesting to mathematicians. That, to my mind, is what makes the continuing relevance of those mathematical concepts to physics nontrivial, and even uncanny.

  92. Adam Says:

    Scott, Max:

    More seriously, isn’t there a self-referential problem with the idea of the MUH? If our universe is just a mathematical structure in the larger composite MUH, then we are just particular sub-objects of this structure. And in our minds we create models of other mathematical structures which do not exist in the physical world as far as we know.

    Thus, according to the MUH every mathematical structure that a human mind in this universe can model must be a part of the underlying mathematical structure that this universe is isomorphic to, correct?

    If so, then isn’t the MUH the hypothesis that there are other mathematical structures – which we are by definition incapable of modeling or knowing! – which physically exist in some other universe?

    That sounds rather empty to me and also curiously analogous with the Level 2 multiverse of causally distinct spacial regions of the greater cosmos.

    Cheers,
    Adam

  93. Adam Says:

    Scott, #91:

    I am not sure there is a real distinction yet. For example, I think there are many, many ways in which the laws of physics could turn out – might already be! – in which the English, German, Human language could be utterly incompetent at describing.

    But, I am probably playing a bit of devil’s advocate here ;)

    Cheers,
    Adam

  94. David Brown Says:

    Scott #78: “Take the text of the books, convert to 1′s and 0′s, interpret as a positive integer.” There might be many meanings of the word “book” in which your idea works. However, a book or play might be its text combined with its readership or audience. If all people were extinct, then surviving human writings might be said to also be extinct, in some senses of the word “extinct”. It is very unclear to me what the words “book”, “mathematics”, and “physics” might mean. Is consciousness a necessary part of physics?

  95. Scott Says:

    fred #88: There’s an aspect of consciousness—which goes by names like qualia, first-person experience, and the “Hard Problem”—that does strike me as outside the scope of science, essentially by definition. I.e., no matter how much is discovered about neurobiology and the measurable correlates of consciousness, it seems to me that stoners will always be able to ask each other, “dude, what if like, my red is your blue?” And it will still be the case that the only way for the stoners to answer the question would be to inhabit each other’s minds. There’s no ordinary, third-person experiment that could ever answer their question, because they’re not asking about any objective facts of the physical world (I assume that neither stoner is colorblind, and both have equal ability to discriminate colors). To some people, of course, that’s just a fancy way of saying that the stoners’ question is meaningless, and I also think it’s beyond the scope of science to refute those people.

    Having said that, I also think there are countless issues discussed under umbrella of “consciousness” that are perfectly within the scope of science. E.g., which brain regions are active in conscious versus unconscious activities? Can we build an AI that behaves as if it’s conscious? Can we mechanistically predict a person’s future choices, far into the future, without needing to do a brain scan that kills the person? Can we find a functional feature of brain organization that’s different in the animals we choose to regard as “conscious,” and the animals we choose to regard as “unconscious”? All of these questions are often sloppily conflated with the “Hard Problem of Consciousness,” but I think it’s best for everyone to keep them separate.

  96. Greg Kuperberg Says:

    A separate set of answers:

    a) The mathematical universe hypothesis

    I’m not sure what there is to learn from this hypothesis.

    b) String theory

    There is a great deal to learn from string theory, given that it is the only known viable candidate for quantum gravity. Since it is the only game in town, and since it has developed substantially even though it is still very incomplete, parsimony strongly speaks in favor of it.

    c) Loop quantum gravity

    A Ralph-Nader-like quest to compete with string theory, just for the sake of competing.

    d) Unitary quantum mechanics (QM with the collapse postulate removed)

    This is a bad joke based on fossilized terminology for quantum probability. The “collapse postulate” is neither a collapse nor a postulate, no more so than Bayes’ formula in classical probability is a “collapse postulate”. If you “remove” this from quantum mechanics you get…exactly the same thing as if you don’t remove it! This is like special relativity with the twin paradox “removed”. As Scott says, it’s like removing a few pages from one of Einstein’s papers. (But few enough pages that a good student would know how to put them back.)

    e) Chaotic inflation, i.e., inflation with V=m^2 phi^2

    Just like string theory, parsimony has long spoken in favor of inflationary cosmology in general. Lately chaotic inflation looks more and more like the favorite flavor of inflation, and you even have experimental data rolling in so that you don’t need to only rely on parsimony.

    f) The theory that space continues at least 1 lightyear beyond our cosmological horizon

    That depends on what you mean. We can see galaxies that are receding into the cosmological horizon. Many of them are so far away that even if we chased them, they would still escape into oblivion. Space continues past the horizon in the sense that if we lived in one of those galaxies, that’s what we would witness. But since we don’t, as best I understand holography, we will instead see an indecipherable hash of the information in these galaxies. Which (again, assuming holography as I understand it) will prove that the experience of witnesses in those galaxies is irretrievable for us, and in some sense has ceased to exist.

  97. Scott Says:

    David Brown #94:

      Is consciousness a necessary part of physics?

    See my clarification in comment #36. My claim is not that everything is describable mathematically, but only that everything describable at all is—i.e., that there’s nothing special about words on a page, sounds, or any other reproducible medium that you can’t also capture using positive integers or strings of 1′s and 0′s. As I wrote in #95, I also grant that there might be an indescribable aspect of first-person experience: e.g., it might be impossible to verbalize “what it’s like to see the color red,” in such a way that you can be sure you’re not talking about what someone else perceives as “your blue.” But even if so, that doesn’t refute the claim about math being “describable-complete.”

  98. Jay Says:

    #97

    Chalmer is fun to read, but appart from his arguments, do you see anything that would prevent neuroimagers to say, in some not too remote future, “No, I’ve read both your brains and when you both say red, it’s not exactly the same thing. Allow me to induced the exact percept, one minute please… Ok see? This is what is perceived red by your partner.”

    #96

    About strings, it´s always puzzling you never make the effort to say either “ok, this is what I think but many other physicists would disagree” or “Let be clear, all physists saying otherwise are liers or dumb stupid”. Be sure one do not need to be physicist to evaluate these claims for what it´s worth.

  99. Greg Kuperberg Says:

    Sid K – I want to emphasize this point concerning ontology: There are two uses for density matrices (or mixed states) that at first glance look separate, but turn out to be the same thing. First, a density matrix is the correct model for a marginal state in quantum probability. I.e., if Alice and Bob are entangled and Bob leaves the room, what is Alice’s state? Second, a density matrix is also the model for entropy and missing information, e.g., if you make a measurement and don’t look at the answer.

    These two cases are equivalent because whatever Bob is, Bob has measured Alice by virtue of being entangled with Alice. Maybe there is also an environment, Eve, that is constantly measuring at least Bob and maybe both Alice and Bob. Either way, measurement is ultimately a manifestation of entanglement, and all entropy in quantum probability ultimately has the same source.

    So, once you take seriously that observers are themselves quantum systems, measurement is not entitled to a separate ontology from entanglement or any other quantum phenomenon. Certainly not if the observer if anything or anyone other than the first person.

    In this way, the main pseudo-problem in quantum mechanics, the measurement “problem”, becomes the same as the main pseudo-problem in brain science, the consciousness “problem”. It’s easy to accept the strong evidence that any other animal is just soulless chemistry, or that any other observer is just a non-deterministic quantum system. It’s hard to believe it about yourself!

  100. Scott Says:

    Jay #98:

      Chalmer is fun to read, but appart from his arguments, do you see anything that would prevent neuroimagers to say, in some not too remote future, “No, I’ve read both your brains and when you both say red, it’s not exactly the same thing. Allow me to induced the exact percept, one minute please… Ok see? This is what is perceived red by your partner.”

    The neuroimagers could say that, but I don’t understand why it would convince anyone who wasn’t convinced already. Obviously your and your friend’s neural firing patterns will differ in various ways—so even assuming that “consciousness supervenes on the physical” (as the philosophers put it), how could you ever falsify the hypothesis that whatever physical differences there were caused your first-person color experiences to be radically different?

    It seems to me that the best you could do would be, e.g., to cluster people with similar firing patterns, and then hypothesize that the more similar the patterns, the more similar the first-person experiences. Or if, hypothetically, you discovered that one person’s “red firing patterns” looked just like someone else’s “blue patterns” and vice versa, then you could speculate that what the one perceived as red the other perceived as blue. But even then you wouldn’t know. For example, the possibility would remain that no two people’s color experiences were “comparable” (whatever that means).

    Let me stress, once more, that you always have the option of dismissing these questions as meaningless—and I have no quarrel with the many people who happily choose that option! :-) My claim is simply that you don’t have the option of saying that (a) the questions are meaningful and (b) ordinary neuroscience could soon answer them! I don’t see how you can assert the latter without doing a metaphysical bait-and-switch: replacing the thing that was actually asked with the sort of thing that third-person experiments can answer.

  101. fred Says:

    Scott #95
    “There’s an aspect of consciousness—which goes by names like qualia, first-person experience, and the “Hard Problem” [...] There’s no ordinary, third-person experiment that could ever answer their question, because they’re not asking about any objective facts of the physical world”.

    But absolutely everything we tag as “the physical world” is an interpretation of patterns of nerve activity, living in our brain as interconnected symbols. Every concept is in fact nothing but a series of relations between other concepts, themselves being nothing but a series of relations, etc.
    The structure of our brain is similar to a dictionary – every definition in a dictionary is eventually circular.
    That’s just more obvious when we talk about “blue” or innate emotions such fear, hunger, pain, which are primordial symbols passed along since the dawn of evolution and consciousness, but it’s true for *anything* – mathematical objects eventually get reduced to primordial symbols about space and time… go and try to describe “space” and “time”, it’s just as hard as describing “blue”.
    At best we can only hope to reduce things in term of the “I” (me, self), or in term of self-references (the relation of the “I” to itself, words that describe themselves, etc).

  102. fred Says:

    Saying that the universe is itself mathematical postulates that the circularity of symbols in our brain or a dictionary applies also to the “physical world”.
    I.e. every “object” is but a series of attributes describing relation between other “objects”, eventually leading to circularity (that’s a general definition of mathematics).

    This might not be testable, but at least:

    1) it’s based on my direct experience of existence, i.e. Consciousness is as real as it gets (“I think therefore I am”), it lives embedded in such math structures, themselves embedded in higher mathematical structures realizing the physical world.

    2) it’s a simpler explanation than postulating that there is an “objective” reality and a “subjective” reality of different fundamental nature.

  103. Max Tegmark Says:

    Thanks Scott for so patiently answering all my questions! It’s quite striking how different definitions different science colleagues of mine give, so I’m grateful that you spelled out yours. It’s no wonder they sometimes end up disagreeing about what’s scientific when the haven’t realized they’re using different definitions!

    Your definition is a bit too convoluted for my taste: I simply prefer calling a theory scientific if it’s testable. As we discussed earlier, I’ll also call a statement scientific if it’s predicted by a scientific theory, but I carefully chose my list above to avoid such cases. Let’s compare notes on examples a-f:

    f) space continues at least 1 lightyear beyond our cosmological horizon
    Here I think you’re too restrictive, since it’s trivially testable: just wait one year and check if light reaches you from that space!

    e) Chaotic inflation, i.e., inflation with V=m^2 phi^2
    Here I think you’re too restrictive again, since it’s very testable: it predicts that our universe should be large, expanding and approximately homogeneous, isotropic and flat, with tiny primordial fluctuations that are roughly scale invariant, adiabatic and Gaussian – all tested and confirmed. Even better, it predicts the numerical values of five observable cosmological paramteters:
    (1) the density parameter Omega=1,
    (2) the spectral index n=0.96,
    (3) the gravitational wave amplitude r=0.15,
    (4) the tensor spectral index n_t = -0.02,
    (5) the spectral index running alpha = -0.0007.
    We’ve successfully confirmed 1) and 2) to better than 1% accuracy, and BICEP2 just confirmed 3) by measuring r=0.16+-0.06 (with their best foreground modeling).
    So is it testable? Yes! Is it a scientific theory? Yes, by my standards, with no ifs or buts!
    Yet you bring up additional requirements, such as the extent to which it can be differentiated from other models.
    That’s an interesting question, but I don’t think it’s relevant to the simple question of whether e) is a scientific theory. If some rival theories are so vague that they can’t be differentiated from this theory, that’s IMHO their problem.

    d) No-collapse quantum mechanics:
    If you’re rejecting the arguments of Everett, Deutsch, Wallace, Zurek and others that the Born rule (“FAPP collapse”) can be derived from d) alone, then I’d love to hear what parts of their derivations you object too – but let’s save that for coffee to avoid diverting this thread to QM-land. If not, then d) clearly makes all the usual QM predictions that are so successfully tested in labs around the world and if feels unreasonable to call it less testable than your own favorite brand of QM. I suspect that you’re applying the same criterion I objected to above under e): requiring not only that it’s testable, but also that it’s distinguishable from some other theory.

    b) & c) String theory & LQG:
    Here we basically agree. I’d add that many string theorists I know used to hope for much more than this, such as predicting all of the 32 dimensionless parameters of physics (Table 10.1 in my book) from first principles, and that many still hope for other tests involving e.g. inflation or LHC.

    a) The Mathematical Universe Hypothesis (MUH):

    Consider these four hypotheses based on our discussion above:
    H1) Our physical reality is a mathematical structure.
    H2) Our physical reality is isomorphic to a mathematical structure.
    H3) Our physical reality is isomorphic to a mathematical structure, insofar as the physical world is describable at all.
    H4) Some describable aspects of our physical reality are not isomorphic to a mathematical structure (say true Copenhagen-style randomness, souls, deities, miracles – take your pick)

    You argued that the Mathematical Universe Hypothesis (H1) was distinct from the others and therefore had philosophical content, but was nonetheless devoid of scientific content. Let let me try to summarize the arguments you gave – please correct me if I misunderstood you. As far as I could tell, you justified this claim in two ways:

    1) “H1 makes no testable predictions”
    2) “Everything can be described by math, so H1 is trivially true and hence meaningless”

    Let’s start with 2). Saying “H1 is true” doesn’t sound as critical as “H1 is devoid of content”, so perhaps that’s not quite what you meant. In any case, the MUH isn’t talking about “being described by math” in some vague way such as the decimals of pi describing the world, but it says specifically that the world is a mathematical structure (an abstract set of elements with abstract relations between them).

    Most traditional physics descriptions of the world are *not* isomorphic to any mathematical structure. For example, a mathematical structure admits no randomness and is unchanging, so the world of Copenhagen quantum mechanics with truly random outcomes is *not* a mathematical structure, nor is Smolin’s “Time Reborn” world where time is fundamental rather than, say, a fourth dimension.
    So if you’re claiming that the MUH is untestable, it seems you’re implying for example that the collapse postulate of the Copenhagen interpretation is untestable – do you really stand by that claim?

    Continuing on the testability theme, how can you be so sure that the MUH can’t be tested in other ways as well?
    You certainly can’t prove a no-go theorem showing that expanding our ontology to include unobservable entities can’t have observable consequences. For example, if the measure problem ever gets solved, then such an expansion can radially alter our predictions for future observations. You already mentioned Weinberg’s successful prediction of dark energy from assuming a type of Level II multiverse. A more recent example is the possibility of seeing imprints of Level II multiverse bubble collisions in the cosmic microwave background. Personally, I’m extremely skeptical that we’ll ever observe any such bubble imprints, but these two examples highlights the importance of being humble and not making overly categorical claims of no-go theorems. Level IV radically expands the ontology, so it could presumably alter predictions significantly. Saying that we don’t yet (and may never) seems irrelevant to this point: after all, would you claim that
    “P≠NP is devoid of scientific content”
    just because you’re not sure we’ll ever settle it?

    Finally, as Sid K (#1) so eloquently put it above, I feel that you’re replacing “innocent until proven guilty” by “guilty until proven innocent” in your definition of “scientific”.
    Consider these three debates:

    Physicalist: I think there’s no “secret life sauce” distinguishing living from non-living things.
    Critic: That’s an unscientific theory, since you can’t experimentally prove there’s no secret life sauce!

    Integrated information theorist: I think there’s no “secret consciousness sauce” distinguishing conscious information processing systems from unconscious “zombie” ones.
    Critic: That’s an unscientific theory, since you can’t experimentally prove there’s no secret consciousness sauce!

    MUH advocate: I think there’s no “secret existence sauce” distinguishing physically existing mathematical structures from other mathematical structures.
    Critic: That’s an unscientific theory, since you can’t experimentally prove there’s no secret existence sauce!

    I think that in all three cases, the first person makes a simple Occam-style claim, and the the onus should be on critic to experimentally detect the sauce!

  104. Jay Says:

    > why it would convince anyone

    By the same standard as anything else, e.g. by exhibiting predictive power and non trivial explanations for what can or can’t constitute a valid question.

    > even assuming [supervenience] (…) how could you ever falsify the hypothesis that whatever physical differences there were caused your first-person color experiences to be radically different?

    Was a negation lost somewhere? If we assume supervenience, by definition differences in first-person experience are caused by physical differences.

    > the possibility would remain that no two people’s color experiences were “comparable”

    The possibility would always remain that there is no two persons in the universe. For the same reason we don’t care the last question, we shouldn’t care about the former (as long as predictive power etc.).

    >you always have the option of dismissing these questions as meaningless

    Look, it’s possible that one day, we will explain consciousness using a theory that says, nope, qualia has no meaning but in relation to a specific observer, so you can’t compare qualia among observers.

    That kind of things can happen, for exemple “time” which has no meaning in the reference frame of a photon. But you seem to exclude the possibility that, say, we’ll find that photon have in fact a (very) tiny mass, so yes it makes sense to speak of photon time after all.

    Let say you have to choose one of two bets. First bet is that photons have no mass. Second bet, is that it doesn’t make sense to speak of qualia as something we can add, substract, compare from one mind to another. Would you really choose the latter?

    >replacing the thing that was actually asked with the sort of thing that third-person experiments can answer.

    Nope, I specifically suggested a scenario in which you could make yourself experienced whatever you want by affecting your brain state, and so you can, at first-person, compare if first-person experience meet predictions.

  105. Greg Kuperberg Says:

    Max – Your examples conflate constructing a theory with merely making room for one. What you call physicalism and integrated information theory aren’t scientific theories — the theory is the chemical basis for life and brain function. Arguably these isms are also invitations to engineer artificial life and artificial brains.

    Your “mathematical universe” concept is another such invitation. It’s an invitation (which was accepted long ago) to attempt a mathematical theory of everything. And arguably an invitation to define viable alternative universes.

  106. Nex Says:

    Scott: “The view that analogizes math to the English language seems totally unable to account for things like complex numbers, linear algebra, Riemannian geometry, or group representations, which were all developed decades or even centuries before anyone thought of any applications to physics, but then turned out to be exactly what physicists needed.
    Which English words were coined decades or centuries before anyone needed them?”

    They were developed by mathematicians because THEY needed them for solving problems which while abstract where definitely based on reality. They are all basically different ways of expressing geometric relations and transformations which are everywhere around us.

    Likewise English language has tons of words invented for one purpose and then used by others for something else (related or not).

  107. JimV Says:

    Probably my reason for not seeing much of a problem with the “why is the universe so mathematical” question is because I don’t know enough math, but I’ll give it anyway, because, hey, this is the Internet:

    To me, math is thinking and thinking is math. When I have three errands to do and spend a few seconds deciding what order to do them in, I am doing math. (I may not be doing it well.) When you think about certain things long enough and hard enough and do enough examples, sometimes you develop general methods, such as calculus, linear algebra, group theory, and so on, which are on a much higher level than my errand-doing example. But its all thinking.

    In which case anything the universe throws at us which we are able to reduce to some sort of order by thinking about it will henceforth be considered some form of math. Only the things that we are incapable of thinking about productively won’t be math.

    That doesn’t explain coincidences like the usefulness of imaginary numbers in analyzing alternating current transmission (among other things), for which my guesses are: a) some sorts of thinking are fundamental to many sorts of problems; b) there may be other methods which would work just as well but you try what you already know first; and c) there are such things as coincidences.

    I could also explain (to my satisfaction) how our thinking works and why it had to work (to a useful extent) in this universe, but that might be pushing the whole Internet concept a bit far. So I’ll just end by saying, thanks for the interesting post!

  108. Scott Says:

    Max #103: Thanks so much to you too for your detailed and interesting replies!

    f) Sorry, I simply didn’t have the right definition of “cosmological horizon.” I thought it meant the horizon beyond which a signal can never reach us, because of the dark energy (assuming of course that its density remains constant, rather than tunneling to zero). Is that called our “causal patch”?

    Of course I’ll admit claims about the light that’s going to reach us a year from now as having operational significance. :-)

    e) OK, let me be more precise: I think that whether inflation happened, and the specific form of the inflation potential, are eminently testable scientific questions (as last week’s announcement underscored). The point where I worry that maybe we’ve left science for metaphysics, is the point where we deduce from an inflation scenario that we should think of our Big Bang as having been chosen “randomly” from an infinite ensemble of Big Bangs. And no, I don’t have a good alternative way to think about the “process” (if one should call it that) that picked “our” particular Big Bang out of the inflaton. But the fact that we appear to gain zero new predictive or explanatory power by going the “random” route feels to me like a strong hint that there ought to be a better way forward.

    d) Yes, I do reject the arguments of Everett, Deutsch, Wallace, Zurek, and others that the Born rule can be “derived” from unitary QM. I would say it like this: if you’ve already decided that you want a rule for converting amplitudes into probabilities, then you can give about 20 different arguments that the Born rule is the only one that makes mathematical sense—i.e., the only one that syncs up with unitary evolution in a sane way. The arguments of Everett, Deutsch, Wallace, and Zurek are four examples; I myself gave a few others. However, nothing in unitary QM tells you that picking a basis in which decoherence happens, looking at the amplitudes of the wavefunction of the universe in that basis, and then finding a rule by which to convert the amplitudes into probabilities is something that you should do. (Tony Leggett has made the same point very eloquently.)

    Since this might seem like purist nitpicking, let me put the point more strongly: suppose we considered a scenario where decoherence didn’t happen? Suppose, for example, that we built an artificially-intelligent quantum computer, so that we could first put the QC into a superposition of thinking two different thoughts (A and B), and then apply some general 2×2 unitary that mixed the A and B branches? Since the Born rule clearly no longer suffices, what probability rule should we now use, to calculate (e.g.) the probability that the QC will switch from thinking thought A to thinking thought B when we apply the unitary? It seems to me that you have to say, either that such a QC couldn’t be conscious (e.g., that environmental decoherence is a necessary condition for consciousness), or else that its consciousness would be so different from ours that the relevant probabilities wouldn’t be defined, or else that the Born rule needs to be replaced by some other rule in this case. And whichever you choose, it seems to me you’ve conceded that the leap from unitary QM to the Born rule is not just a boring matter of manipulating some formulas. Instead, it involves tacit additional assumptions about where in the wavefunction we’re going to find these “conscious beings” whose experiences we’re trying to account for, and something about the nature of those beings.

    a) I’m completely confused about why you claim that, if there were fundamental randomness in the laws of physics (e.g., in the Copenhagen interpretation), then the MUH would be falsified. Why aren’t vectors of probabilities, which evolve via stochastic transformations, a perfectly-respectable mathematical structure—every bit as respectable as vectors of amplitudes that evolve unitarily?? It seems to me that, in order to say that the MUH is falsifiable, you need to take a weirdly limited view of what a mathematical structure can be and what it can mean for the physical world to be “isomorphic” to one.

    Now, regarding your final bit about “guilty until proven innocent”: it seems to me that both the integrated information theorist and the MUH advocate make much stronger claims to knowledge than “I don’t see what the secret sauce is.” This is particularly obvious in the case of IIT: Giulio claims to know (or be pretty sure) that a specific numerical measure Φ that he wrote down is the “consciousness-meter,” the thing that tells you whether a given agglomeration of atoms has qualia or not. And the MUH advocate claims to know (or be pretty sure) that every computably-definable mathematical structure (but not the ones that aren’t computably-definable…) gives rise to another physically-existing universe. In both cases, I’d say that admitting you don’t know what it is that makes one agglomeration of atoms conscious and not another one, or what breathes fire into one set of equations and not into a different set, is a perfectly legitimate option.

    (The case of the physicalist is different, because that one’s claim to know something about the border between life and non-life can be justified by everything we’ve learned about biochemistry over the past century.)

    Anyway, I’ve enjoyed this discussion and would be happy to continue over coffee sometime (except then no one else would get to follow it! ;-) ).

  109. Scott Says:

    Jay #104:

      If we assume supervenience, by definition differences in first-person experience are caused by physical differences.

    Right, but given that two people’s brains are different, their first-person experiences could be arbitrarily similar or arbitrarily far apart, and that’s the question at issue here.

      The possibility would always remain that there is no two persons in the universe. For the same reason we don’t care the last question, we shouldn’t care about the former (as long as predictive power etc.).

    OK, but if those are the ground rules, then why not just say “c’mon, obviously my red is the same as your red—get serious!,” without bothering to do the neuroscience experiment? Either way, then, I don’t see how doing the experiment really changes the situation.

      Nope, I specifically suggested a scenario in which you could make yourself experienced whatever you want by affecting your brain state, and so you can, at first-person, compare if first-person experience meet predictions.

    Yeah, I got that. The problem is, how would you know that the experience the neuroscientist created in your brain was actually the same as the other person’s experience? And if you’re willing to appeal to common sense—”c’mon, of course they’re the same!”—then again, why not just appeal to common sense now, and save the trouble of doing the experiment?

  110. Adam Says:

    Max:

    If I can hold Conway’s Game of Life in my mind and I am just a part of this mathematical structure called our universe, then is it fair to say that Conway’s Game of Life is contained within this universe? If so, then why do you insist that another universe physically separate from this one must exist that embodies Conway’s Game of Life?

    Cheers,
    Adam

  111. Bob P Says:

    “I thought it meant the horizon beyond which a signal can never reach us, because of the dark energy (assuming of course that its density remains constant, rather than tunneling to zero). Is that called our “causal patch”?”

    Maybe that horizon is best called the de Sitter horizon, the surface with radius sqrt(3/Lambda). The causal patch is the four-dimensional volume within that surface.

    For non-experts following along at home, horizon complementarity (HC) is the idea that it is only consistent to talk about one causal patch at a time. Talking about multiple Level I universes as if they exist at the same time is analogous to talking about the interior and exterior of a black hole as if they coexist. There’s no doubt you can jump in a black hole and experience the interior, or you can remain outside and experience the exterior. But HC argues it only makes sense to talk about one or the other existing at any moment, because otherwise the quantum no-cloning theorem would be violated (you could clone a state by throwing it into a black hole: now one copy of the state lives in the interior, and one copy lives in the Hawking radiation in the exterior. Similarly, you could clone a galaxy in the Level I multiverse by letting it drift out of our causal patch. One copy lives in our de Sitter horizon’s Hawking/Unruh radiation.)

    This complementarity between black hole interior/exterior, or between causal patches in the Level I multiverse, is a bit like the complementary between position and momentum: you can have one or the other, but not (a definite value of) both at the same time. The main assumption underlying horizon complementarity is that black hole evaporation is unitary (so that the Hawking radiation truly encodes a copy of the interior). This is unproven, but many people (including me) find the theoretical evidence compelling. I suppose a second assumption is that de Sitter horizons and black hole horizons follow the same rules, but that seems hard to avoid.

    At any rate, horizon complementarity seems like a nice way to “tame the infinities” that proliferate in multiverse discussions, because the number of allowed bits in a causal patch is finite; it is bounded by the surface area of the causal patch. (This bound is unproven, but again there is compelling theoretical evidence.)

  112. Max Tegmark Says:

    Thanks Darrell Burgan (#74) for bringing up this interesting question about the distinction between mathematics and computing! I discuss this at great length in chapter 12 of the book ( http://mathematicaluniverse.org ). There are interesting relations between *three* distinct things (see figure 12.6):

    1) Computations
    2) Mathematical structures
    3) Formal systems

    Computations can be viewed as special cases of mathematical structures, and they defined the relations of mathematical structures.

    Mathematical structures are set-theoretical models of formal systems which describe them.

    Formal systems describe computations, which can in turn probe theorems of formal systems.

    For computations, mathematical structures and formal systems,
    The Gödelesque issues relate to whether they’re halting, defined and decidable, respectively.

  113. wolfgang Says:

    @Scott

    >> it involves tacit additional assumptions about where in
    >> the wavefunction we’re going to find these “conscious
    >> beings” whose experiences we’re trying to account for

    I agree with you, but I would add that something similar is the case within relativity. Albert himself was wondering about the “now” being so special to us and it bothered him.

    Why is the year 2014 so special to us (now), compared to e.g. the year 2345 ?
    Nothing in the theory of relativity tells us anything about this specialness …

    Just as unitary quantum theory (without Copenhagen reduction) does not tell us what is so special about our branch of he universe – the one where Obama won the election and not Xorankrox 8-)

  114. Jay Says:

    Scott #109

    [Right...] So the question is, how would we falsify a theory specifying that two percepts are the same or different?

    By the same standard as anything else, e.g. by testing if we can correctly predict or cause patterns of brain activities that subjects will perceive as the same or as different, including in situations specifically selected because that’s where theory leads to its most non trivial predictions.

    [Ok...] Why bothering to do the experiment? Because the answer is not obvious at all! It could be that qualia are not commensurate, or it could be that yes they are, and I want to know the truth!

    More precisely, what I want is a theory of the mind that could among other things explain whether your red is my red or if the question makes no sense. But truth is I don’t really give a damn about what it is like to be a bat. What I want is a scientific theory of consciousness, because I want to know how this thing works.

    [Yeah...] Will we ever know for sure? No, of course not, but why this double standard?

    Whatever how good a theory is, it’s easy enough to come up with an “alternative with no teeth” (think of superdeterminism and QM). Do we know for sure the original one is the correct one? No, not on logical grounds. But that’s not the standard we use. What we ask for is the theory can do something new and impressive. Call that predictive power. Call that explanatory power. But don’t use a double standard!

  115. anon Says:

    Just a warning, I ended up in the hospital this winter after psychoanalyzing myself while reading tegmark papers. And I can say that we are our brains. It wasn’t healthy to view other people as chemicals, and automatons or just math. Scary and lonely as hell. Good discussion though, I don’t mean to derail it.

  116. Pascal Says:

    To Greg #99: When you say that observers are just ordinary quantum systems behaving according to the rules of ordinary quantum mechanics, you are in effect endorsing the many-world interpretation.

  117. Dániel Says:

    Scott #108:

    “It seems to me that, in order to say that the MUH is falsifiable, you need to take a weirdly limited view of what a mathematical structure can be and what it can mean for the physical world to be “isomorphic” to one.”

    Max, please clarify: does the MUH really rule out fundamental randomness, or can that randomness be saved by transforming it to a version of many-worlds like Scott did? I think this is an important question, because it seems to me that this kind of many-worlds model doesn’t necessarily has a much higher Kolmogorov-complexity than a similar non-stochastic (single-world) model, so we can’t refer to Kolmogorov-complexity to discredit it.

    So, Max, I am very interested in your reply. In the meantime, let me humbly suggest to Scott and others that it’s a good thing that Max’s view of a mathematical structure is “weirdly limited”. In the best case, it might turn out to be exactly the meat that you missed from the theory.

  118. Vladimir Slepnev Says:

    @Scott #108

    > And whichever you choose, it seems to me you’ve conceded that the leap from unitary QM to the Born rule is not just a boring matter of manipulating some formulas. Instead, it involves tacit additional assumptions about where in the wavefunction we’re going to find these “conscious beings” whose experiences we’re trying to account for, and something about the nature of those beings.

    We can note, without assuming the Born probabilities, that we are computers made out of classical parts, i.e. parts that happen to work reliably under the Born probabilities, rather than some other probabilities. That might be part of the explanation.

    @Scott #29

    > one now wants to know, why is the fire of existence not only breathed, but breathed so promiscuously, onto every set of equations that anyone could write down?

    I feel that might be a wrong path. A set of equations can contain creatures who feel that they exist, regardless of the set of equations itself “existing”.

  119. Fred Says:

    “Blue” isn’t an absolute – my sensation of blue is from a rich set of associations with other concepts: some of those associations are fairly general (blue is kinda greenish, blue is a cold color,..), some are more peculiar (blue reminds me of drowning in the ocean, blue reminds you of the sky in you homeland, a painter may see 12 different blues when you see only 2).
    So asking if your blue is like my blue is an exercise in graph isomorphism. One can maybe assume that if two brains are the same, …
    But this difficulty is true even for any “objective” concept we take for granted. We can often take things for granted because all our brains are roughly the same, as a result of evolution (maybe cats have objective concepts that are totally foreign to us).
    The irony is that physics, supposedly the most objective science out there, is all about mathematics – every attempt to discuss “what is the state of a particle between measurements?”, “what is a measurement?” is answered with “stick to the math! Thats all that matters!”. But even math is an endless source of confusion apparently (e.g. p!=np is a 50/50 thing… No, its 99.999/0.001!)

  120. fred Says:

    Scott #108
    “a) I’m completely confused about why you claim that, if there were fundamental randomness in the laws of physics (e.g., in the Copenhagen interpretation), then the MUH would be falsified.”

    Isn’t it because all of this is nothing but an attempt to go back to a fully deterministic view of the world?
    The space-time “block” view of reality suggested by relativity (where an observer view is just a slice) was one step in the direction of getting rid of free-will and a dynamic view of reality (past, present, future all exist at the same time).
    “Many-worlds” interpretation of QM is another step to get rid of “magical dices” and turn the space-time block into a big static space-time arborescence.
    So, you end up with a view of the universe which is basically a giant fractal snow flake (where minds are “self-similar” vortices?)

  121. Max Tegmark Says:

    Yes Scott (#108), I too have very much enjoyed this discussion, and very much look forward to continuing it over coffee! We should save time for two separate chats: one about the MUH and one about QM! Some final comments before I get back to catching up on other stuff:

    (a) Mathematical universe hypothesis: I think we’ve finally isolated the crux of the matter, and I’m optimistic that you’ll retract your “devoid of content” claim once we’ve had a chance to talk this though over coffee until we understand each other’s claims properly.

    The definition of mathematical structures that I use is rather standard (see, e.g., W. Hodges W 1997, “A Shorter Model Theory”, Cambridge Univ. Press), and yes, it’s an *extremely restrictive* definition (you’re free to call it “weirdly limited” if you prefer), and it’s precisely because it’s so restrictive that the Mathematical Universe Hypothesis has predictive power. Although restrictive, this definition (as abstract sets of elements with well-defined relations between them) includes all mathematical structures I’ve ever seen discussed in physics, from integers to complex numbers, 3+1-dimensional pseudo-Riemannian manifolds, Hilbert spaces and orbifolds). I use “isomorphism” only in the standard way: two structures are isomorphic if there’s a one-to-one correspondence between their elements respecting all relations.

    To see why, say, the world of Copenhagen quantum mechanics is not isomorphic to a mathematical structure, please take a few minutes and try to write down the definition of the mathematical structure! Remember that you need to write a definition that’s acceptable to a mathematician or computer, so you’re not allowed to use any “baggage” words that assume outside knowledge. For example, you asked what was wrong with talking about “vectors of probabilities, which evolve via stochastic transformations”. Baggage alert: What’s meant by “evolve”, which seems to presuppose some notion of time? Baggage alert: What’s meant by “probabilities”, which seems to presuppose some notion of observation? You’re defining a mathematical set with relations here, so your definitions must be fully self-contained.
    This is why I said that you’d falsify the MUH with an experimental demonstration of wavefunction collapse.

    (d) No-collapse quantum mechanics: Indeed we shouldn’t expect the emergence of any sort of classical world or classical consciousness from QM in the absence of decoherence. But in our actually perceived classical world and consciousness, decoherence clearly plays an important role.

  122. Scott Says:

    Dániel #117 and Max: I guess I should clarify something. I understood Max to be saying that, if an objective collapse mechanism were discovered (introducing “fundamental randomness” into nature), something that everyone agrees is an empirical question, then the MUH would be falsified.

    That’s great to know—except that, as I said, I can’t understand why such a discovery would or should falsify the MUH! For even then, we could still adopt a “many-worlds” perspective, except now with a probability vector of the universe rather than a quantum state vector. So, if this is the route Max wants to take, then my inclination would be to rename the MUH, to something like the MUAARATOPQPSH (Mathematical Universe in which All Apparent Randomness Arises by Tracing Out Part of a Quantum Pure State Hypothesis).

    On the other hand, Max also suggested in his comment that he thinks the “Copenhagen interpretation” is experimentally distinguishable from MWI—and that strikes me as not obviously true at all. It depends entirely on what one means by “Copenhagen interpretation.” If one takes it to mean, “there must be some new physical process on top of unitary evolution that causes sufficiently complicated physical entities to collapse wavefunctions,” then as I said before, of course that’s empirically distinguishable from MWI.

    The trouble is, I don’t know a single person calling him- or herself a “(neo-)Copenhagenist” who actually takes the above position. Every self-declared “Copenhagenist” I’ve met is fine with arbitrarily-complicated entities, including human brains, being placed in superposition and manipulated unitarily. They simply take a more constricted view than MWIers about what science is about: they don’t think it’s about describing the “objective state of the universe,” but about giving observers the best tools to predict the results of their observations.

    Now, you could argue that the modern “Copenhagenists” are playing a double game: that what they really want is MWI but without calling it MWI, without using the language of parallel universes. But I don’t think you can say that they’ve committed themselves to any clear empirical prediction that would separate their view from the MWIers’.

  123. Rahul Says:

    Scott:

    Post request! Any comments on last week’s announcement of B-modes in the CMB data?

  124. Scott Says:

    Rahul #123: What little I have to say, I said in this post! It’s obviously incredibly exciting if it holds up.

  125. Noah Says:

    The red vs. blue qualia thing seems totally testable to me. Design a retinal implant which switches red and blue, and implant it in some newborns. Come up with good tests which can distinguish the implant group from the control group, and then apply those tests to the stoners.

  126. Scott Says:

    Noah #125: Very cool idea! But I don’t think it works, for the following reason. Suppose you believed that half of people experienced red as Percept1 and blue as Percept2, and half the other way around, based on unknown and possibly-unknowable details of their brain organization. Then sure, you could install the retinal implant in a newborn, but you wouldn’t know whether the newborn originally belonged to the first class or the second class. So you’d know that they’d experience red things the opposite way they would have if you hadn’t installed the implant, but you still wouldn’t know whether it was Percept1 or Percept2.

    You might respond: “OK, but suppose MRI scans reveal a large brain abnormality in all of the people who get the retinal implant, but none of the people who don’t get the implant? Wouldn’t that be convincing evidence that everyone ‘naturally’ belongs to only one of these two classes, and that it’s only installing the implant that can move them to the other class?”

    Alas, I think the answer is still no. For it could be that the developing brain “expects” the sky, the sea, etc. to be associated with a retinal signal of one kind, and blood, cherries, etc. to be associated with a retinal signal of a different kind. And it could be that switching those two kinds of signals alone suffices to produce the brain abnormality—i.e., that it has nothing to do with the later conversion of the signals into either Percept1 or Percept2, a step that remains just as empirically inaccessible to you as before.

  127. John Merryman Says:

    In the skeptics corner; If the Bicep2 evidence were proof of Big Bang/Inflationary cosmology, wouldn’t it show some singular pattern? All those swirls could as well be evidence of the background radiation originating from lots of different sources, say galaxies redshifted completely off the visible spectrum? Simply finding the radiation is polarized is about as informative as finding water to be wet.
    As for Everett’s multiworlds, if we thought of time, not as a measure from one event to the next, but the process by which these events come into being and dissolve, then it isn’t so much the vector from past to future that we experience, but the process by which the future becomes past, ie. tomorrow becomes yesterday because the earth turns. Then it is not the point of the present moving, but the rate of change of what exists. Therefore the future remains probabilistic, while the past has been determined, since probability precedes actuality. So there would be no need to assume either the future must already be determined, or the past remains probabilistic, ie, multiworlds. Both being proposed solutions to how the perceived dimension of time transitions from the determined past into the probabilistic future.

  128. Peter Eckersley Says:

    A thought experiment that might be useful on this one question:

    “How would we know if we’d found something that wasn’t mathematically describable?”

    Suppose you live inside a simulation. After great effort, you and your species determine the laws of the simulation by painful application of the scientific method. All is well (and mathematically describable). Then one day the hackers running the simulation decide to mess with you, and begin injecting weird things into your universe that do not obey its usual rules. You start occasionally meeting unicorns and Cthuloid demons and talking rainbows and rifts in spacetime. When you try to study these things, their properties seem to shift inexplicably (malevolently?) under your apparatus.

    The only mathematical explanation you can find for these observations is the exhaustive bit string describing them as exceptions to the otherwise-prevailing laws of your universe (which, I would argue, is not much of a mathematical description at all in the sense that it doesn’t rely on any of the computational mechanics of mathematics).

    There may in fact be a true mathematical explanation, but it requires describing the laws and state of the parent universe that’s simulating yours. Since that is both empirically inaccessible and vastly more complicated, really the best explanation in your world is that your universe is magical. Maybe you posit the existence of mysterious deities. Those theories would be pretty accurate, non-mathematical, and as good as you can do from your place in existence.

    This suggests an alternative to the position Scott advocated in comments #95 and #97, that consciousness cannot be described at all. Instead, you could say that consciousness has two descriptions: a fuzzy, imprecise, and not necessarily mathematical one that does the best possible job of succinctly accounting for the internal experiences of the conscious being, and the much larger and more complex description of how the surrounding universe is causing those experiences, which in some cases (hallucinations, intense emotions, optical illusions, etc) are subjectively magical.

  129. fred Says:

    #126
    could we remap the visual cortex to the haptic perception cortex so that Blue feels like a kiss on the cheek and Red feels like a kick in the nuts? Then we can all agree and move on…

  130. Douglas Knight Says:

    Vladimir Slepnev:

    We can note, without assuming the Born probabilities, that we are computers made out of classical parts, i.e. parts that happen to work reliably under the Born probabilities, rather than some other probabilities. That might be part of the explanation.

    Isn’t that the very observation we are trying to explain? Isn’t the whole question of the interpretation of QM why reality juice bleeds from QM to classical theories? The fact that we have reality juice shows that the Born rule is correct, but how do we see that from just QM? You could imagine that we have very little reality juice and some other interpretation of QM has much more, but then you run into anthropic issues. And certainly we do run into anthropic issues. You can say that the Born rule is just how we care about the future, but I remember it being true in the past.

  131. Scott Says:

    fred #129: LOL! In that case, presumably we could all agree that blue was pleasant and red painful … but still, how would we know that the kind of pain I feel when my nuts are kicked isn’t analogous to the kind you feel when yours are placed in boiling water, and vice versa?

  132. Me Says:

    As far as i know, the Integrated Information theory is partly motivated by a real world phenomenon. The fact that by cutting off the links between the two brain hemispheres, it seems to be possible to split up a conscious person into two different conscious entities, is seen as evidence for the theory by Tononi. This phenomena is observed in Split Brain patients:

    http://worldsciencefestival.com/videos/two_minds_one_brain

    I don’t have an opinion about the theory but it would be fun if calculating the rate of consciousness would be linked to an interesting computer science problem of calculating Φ.

    e.g.:

    http://www.davidgamez.eu/papers/GamezAleksander11_AccuracyPerformancePhiLiveliness_PrePub.pdf

  133. fred Says:

    Peter #128
    “Suppose you live inside a simulation.”

    This line is at the crux of the matter and makes the assumption that existence and computation are intimately linked.
    But what is a computation exactly?
    We think of it as traditional electronic hardware, with bits stored as charge densities, evolving along some clock.
    At what point of the process is the “simulated” world spawning into existence and self-awareness?
    When the transistors are being updated? When the system clock ticks forward? When a higher being interprets the state of the simulation?
    The simulation could just as well be run by training an army of monkeys to update the cell in a gigantic Conway’s GoL grid by writing a one or zero in their given cell by following some local rules they’ve been thought. When is the simulated world spawning into existence in that case? Whenever a monkey writes a one or a zero? In the mind of the monkey when it figures how to apply the rule?
    What if all the possible states of the machine had been written in advance on indexed sheets of paper, and the computation just consists of computing the next index and “picking” the next sheet (a look-up).
    Does anything practical actually need to happen for the simulated world to even spawn into existence?
    This argument seems to lead to the idea that the simulated world is a mathematical structure that just exists.

  134. Scott Says:

    Vladimir Slepnev #118 and Douglas Knight #130: Here’s something that might be helpful. Without assuming the Born probabilities, I believe we can say the following:

      If, within the universal wavefunction, there are “computers made out of decohering parts,” and if those computers are programmed to behave as if the Born rule is true, then those computers will make very good predictions—assuming that we again use the Born rule to judge the success of the predictions! (In other words, to judge in what ‘fraction of universes’ the predictions succeeded, and in what fraction they failed.) Thus, assuming the Born rule gives us a nice self-consistent story about these computers. On top of that, the Born rule has immense mathematical advantages over any other rule one can write down, such as upholding no-superluminal-signalling.

    In summary, if we start from unitary evolution plus the assumption of decohering observers, then we’re “very strongly invited,” for reasons of mathematical elegance, to tack the Born rule onto our formalism. On the other hand, I’ve never seen an argument that we can prove the rationality of using the Born rule from those assumptions, that didn’t sneak in some additional assumption (or that wasn’t ultimately circular).

  135. Sid K Says:

    Scott #122:

    I feel that even with the sophisticated, neo-Copenhagen interpretation, Max’s (#121) point stands. The Copenhagenist who says that quantum mechanics is simply a set of rules for making sense of our experiences and that there is no ontology being imposed, is making an implicit claim that this is the best we can do. If indeed this is true, then it means that we can’t write down a mathematical structure and say that it is actually isomorphic to the universe. We can at best say that the mathematical structure we write down is isomorphic to our experiences of the universe; which is a much weaker claim. This will “falsify” the MUH.

    Another way put this is to say that the Copenhagenist “predicts” that we will always need to have measurement (or some human-level concept) as a primitive in quantum mechanics. The many-worlders seem to be saying that they can prove that the Born rule is what an ideal reasoner in a quantum universe should use; but that the Born rule itself isn’t a fundamental feature of the universe. Thus the wavefunction is what the universe is and therefore is isomorphic to a mathematical structure, consistent with MUH.

    (PS: I will be sad to see this discussion go offline. Update us if you reach some conclusions! Thanks.)

  136. Scott Says:

    Sid K #135: OK, good points! But

    (1) I don’t think the neo-Copenhagenists would claim to know that it’s impossible to “write down a simple mathematical structure that’s isomorphic to the universe.” They would just say that they can’t do it, and that at any rate, it isn’t the task of science: the task of science is to find theories that explain our observations.

    (2) Probably more relevant for this discussion is that, as I said in #122, I don’t see how the dispute between MWIers and sophisticated neo-Copenhagenists can ever be resolved empirically. It seems to me that the best you could ever do would be to transfer your brain onto a quantum computer, do an interference experiment on yourself, and see what it felt like! (Unfortunately, standard QM predicts that you wouldn’t remember what it felt like afterward, nor would you be able to tell anyone else.) For this reason, even assuming you’re right that Copenhagenism is incompatible with the MUH, I don’t see how that could lead to an experimental test of MUH.

  137. fred Says:

    Scott #131
    haha, you must have had a very traumatic childhood!

    But the brain doesn’t arbitrarily assign feelings to various inputs.
    It’s the characteristics of the data pattern (dynamic evolution, dimensionality, range) that creates the feel, the two aren’t dissociated.
    Experiments have been done about sensory substitution on people who’ve become blind where visual data is being applied to their tongue through a matrix of needles,
    and after a while their brain adapts to the new data and interprets the stimuli as an actual visual sensation. And the same thing has actually been achieved using sounds (scanning images and mapping them as pitch/volume).
    This plasticity of the brain seems to suggests that the same signal is being felt in a very similar way (of course emotions often enter the picture and a symbol never exists in isolation).

  138. Douglas Knight Says:

    I’m more concerned about the assumption that we should be trying to make sense of decohering observers than deriving the Born rule from that assumption. Maybe the fact that decoherence happens is enough, but I’m not sure.

    Tangentially, the point about superluminal signalling seems circular to me. We mainly care about superluminal signalling because we care about special relativity, but the topic at hand is how to get special relativity out of QM. (We might care about superluminal signalling for other reasons, such as making the Hilbert space finite dimensional, but I’m not sure how closely related those two points really are.)

  139. jonas Says:

    On the topic of red and blue, I have a question.

    Clearly even if you can’t explain to another person how you experience red, you can at least explain when you experience red, because you can point to red cars or blood or other red objects.

    Now I usually say that I rarely have headaches, but sometimes I wonder if I have headaches as often as other people only I don’t really know what other people mean by the word “headache” so I don’t realize that. Is it harder to explain what a headache is than to explain what red is?
    You could try to tell what a headache is by pointing to a particular drink and telling that headache is what you feel during getting hung over drinking it. But you could say that even with a drinking experiment with all details controlled, one person might get a headache when another one doesn’t, because their tolerance of alcohol is different. My question is, is there a difference between saying that only one of those people have a headache and saying that both of them have a headache only they experience it differently?

    Bob P #111: that’s an interesting argument about the no cloning.

  140. fred Says:

    jonas #139
    I’m experiencing ophthalmic migraines once a while, and interestingly most ppl who share this can agree on what the “aura” feels like, a sort of buzzing/butterfly effect in the visual field, and ppl have drawn it
    http://tinyurl.com/l56brou
    http://tinyurl.com/k5s4pvb
    http://tinyurl.com/l3hwclz

    so there’s a case where we can all pretty much agree that we’re experiencing the same thing, even if it’s an entirely “internal” object (but so is every concept in the end).

  141. Pascal Says:

    Scott #108: one can argue that the choice of a “preferred basis” must depend on the initial condition of the universe.
    The reason is that if you start from a random initial condition and apply a unitary transformation (the multiverse’s evolution from the big bang until now) you obtain again a random state in which no particular basis should play a special role.

  142. Urs Schreiber Says:

    A famous example that is not a “mathematical structure” in the sense of model theory are topological spaces. Hence also manifolds are not a “mathematical structure” in this sense. (Just google for these keywords..,)

  143. Jason Gross Says:

    Scott #36:

    I’m trying to make sense of what you mean by “isomorphic” in the following, and writing down my thoughts as I go.

    In first-order logic, I’d say something like

    ∃! x : Rock(x) ∧ Brown(x) ∧ StubbedMyToe(Me,x)

    Note that, in the first-order theory, I could also construct a positive integer y that encoded all the information about which predicates held and didn’t hold for the unique x above (assuming there were only finitely many predicates). And we could then say that y was “isomorphic” to x, with the isomorphism given by the encoding procedure. But that still wouldn’t cause StubbedMyToe(Me,y) to hold. I.e., I still wouldn’t have stubbed my toe against the positive integer.

    Isomorphism needs a notion of permissible objects, and permissible maps between objects; an isomorphism is a pair of composable maps which compose in both directions to the identity (in this case, you want to say that the encoding function and the decoding function are the maps). But it’s not at all clear to me what it should mean for two positive integers to be isomorphic, let alone what it should mean for a positive integer to be isomorphic to a particular rock. And, in fact, the isomorphism given by your encoding procedure is an isomorphism between rocks-in-general (or physical-objects-in-general, or whatever), and positive integers, not an isomorphism between a particular rock and a particular positive integer, whatever that would mean.

    If I’ve understood correctly, you meant to say something like “objects are isomorphic to positive integers via some encoding function, and, under this isomorphism, x is identified with y”. You point out that StubbedMyToe(Me,y) doesn’t hold—I’d strengthen it to say that it’s nonsensical. (What would it mean to have stubbed one’s toe on a number?) We can, however, construct predicates Rock’, Brown’, and StubbedMyToe’, for which it is both sensible and true to say Rock’(y) ∧ Brown’(y) ∧ StubbedMyToe’(Me,y); these alternate predicates first apply the decoding function to y, and then invoke the original predicate.

    The principle that “isomorphism is all that matters” is expressed nicely by something called the univalence axiom, which plays a key role in a fairly new branch of mathematics (homotopy type theory). The idea is that all “nice” mathematics is isomorphism-invariant, in the sense that there is a systematic transformation generalizing the priming operation above, which “transports across the isomorphism”.

    To me, MUH seems to be a generalization of the the assertion that we can import this bit of math, the idea of isomorphism-invariance, into the real world, and that whatever physical existence is, it should be isomorphism-invariant: it suggests that once we find ourselves in a perspective where any structure isomorphic to reality can (and should) be identified with it, that is, where physical existence is granted to this restricted class of structures (those isomorphic with physical reality), than it seems natural to extend this to a wider class of mathematical structures. But I haven’t read Our Mathematical Universe (yet), so perhaps someone can tell me if I’m badly misinterpreting MUH.

  144. Jason Gross Says:

    Scott #100:

    The neuroimagers could say that, but I don’t understand why it would convince anyone who wasn’t convinced already. Obviously your and your friend’s neural firing patterns will differ in various ways—so even assuming that “consciousness supervenes on the physical” (as the philosophers put it), how could you ever falsify the hypothesis that whatever physical differences there were caused your first-person color experiences to be radically different?

    There are a number of things which people like to correlate with subjective color experience, and, whatever it might “really mean” metaphysically to say “this person’s red is that person’s blue”, here is a scenario where I might use that description to convey useful information:

    Suppose we profile a large portion of the population, and discover that the time it takes to recognize a color is normally distributed. About half of the population has a peak at red, and the other half has a peak at blue. Suppose we also find that the regions of the brain which light up when recognizing color are essentially the same in everyone, and the correlation between brain region and color is also bimodal in the population, in the same way recognition speed is; that is, perhaps we’ll discover that color recognition speed is almost prefectly inferable from which region of the brain lights up with that color, independent of person. Perhaps we’ve also found that emotional associations to colors are society-independent, and we’ve run (ethically questionable) studies where we raise people in manufactured environments where there is nothing in the culture/interactions which ties emotions to colors, and we’ve found that in this situation, which emotions people tend to associate with colors is very strongly correlated with how long it takes to recognize that color, and negligibly correlated with the color itself beyond how long it takes to recognize.

    If all of these things turned out to be true, then it would probably convince me that there’s something to be said for saying “the internal experience of red and blue are swapped between these halves of the population”.

  145. Jason Gross Says:

    Scott #136:

    It seems to me that the best you could ever do would be to transfer your brain onto a quantum computer, do an interference experiment on yourself, and see what it felt like! (Unfortunately, standard QM predicts that you wouldn’t remember what it felt like afterward, nor would you be able to tell anyone else.)

    If I believed in collapse, then wouldn’t I have to predict that the interference experiment would fail? That is, I’d predict that I wouldn’t lose my memory? Or does collapse permit interference experiments to succeed on beings which cause wave-function collapse? Please correct me if I’m wrong.

  146. Jason Gross Says:

    Here is a social question that might depend on the validity of MUH: what is the moral status of duplicating, deduplicating, or shutting off computer simulations of conscious beings, and how does it compare to the moral status of instantaneously wiping all of humanity out of existence?

  147. Gasarch Says:

    it sounds like Max’s theory fails the Popper-test: there do not seem to be any experiments to falsify it. This also comes out with your `whatever happens he can claim is follows from his theory’ Does it have great explanatory power? I think this is more what you are saying it does not.

    So it seems to me that its not science.
    Glad the book is still interesting.
    (My experiment- this comment has no unusual capitol letters, lets see if it still gets spam filtered.)

  148. Dani Phye Says:

    I prefer the opposite: The only universes that exist are the ones that don’t contain intelligent life.

    Also I feel like the best way to explain existence is to pretend we’ll live forever, then try to figure out how to make that meaningful.

  149. Dani Phye Says:

    (Also best characterized by this question: If you lived forever, what would you do?)

  150. Noon Says:

    @Max #121:

    If I’m only required to write down sets and relationships between them, can I not write down strings, comment on which sets they fall into (maybe it’s only one set) and then trivially “describe” the entire universe in this way? Is the MUH not devoid of content (i.e. it says nothing more than “this is the list of things that happen.”) if this is how I choose to apply it? (Notably, I’m not predicting things.)

  151. Philip Thrift Says:

    “Copenhagen-style randomness” may be excluded from “mathematical structures” but I don’t think it is excluded from the semantics of quantum programming languages.

  152. Scott Says:

    Jason #144: See my comment #126 for my argument that such observations still wouldn’t get directly at the qualia, but “only” at the relationships between different measurable color-correlates and each other.

  153. James Cross Says:

    Scott 78

    If we take King Lear and translate it into French, German, and Spanish, the bits representing the play would be different in each language. To a person fluent in those languages, the play would be the same.

    There could be an infinite number of translations of Lear but any two or more would be recognizable as the same to anyone fluent in the languages.

    Doesn’t that mean there is some “essence” of Lear not describable finitely?

  154. Scott Says:

    James #153: Actually, I don’t think King Lear is the same when translated into other languages! As Robert Frost said, “poetry is that which is lost in translation.” Having said that, yes, there will certainly many different mathematical objects that are acceptable encodings of the text of King Lear (some will use bit-strings and others integers, some ASCII and others UTF-16, some will have a few typos, etc.)—nothing I said contradicts that. And yes, there might be aspects of the subjective, first-person experience of reading or watching King Lear that are not conveyed by just giving someone the text. On the other hand, presumably another person can generate the appropriate qualia in their own mind, after you do give them the text and they read it! And even if you wanted to describe all the measurable aspects of that person’s reaction to King Lear, presumably you could do so, in principle, by giving a complete description of the quantum state of the person’s brain after they’d read it (to some suitable precision).

  155. Urs Schreiber Says:

    @Noon #150,

    indeed, the word “mathematical structure” in model theory, as referred to in #121,

    (see http://ncatlab.org/nlab/show/structure+in+model+theory)

    is, as discussed at the above link, essentially a term for “model in sets of a first-order formal theory”, where “theory” refers to theories as they are formalized in mathematics, given by a formal language in some ambient logic together with a list of formal axioms.

    Hence saying that the world is described by (or just “is”, if one insists) a “mathematical structure”, just means: it is described by some formal language with some axioms. (In fact the “mathematical structures” in the sense of model theory referred to in #121 are only very restrictive such, which for instance do not even include basic concepts such as topological spaces and manifolds. It will be hard to encode something close to quantum field theory in these model theoretic terms.)

    So the claim is that the world is describeable by some mathematical theory. (And, if you rellay insist, conversely that every theory describes some world. ) This kind of statement is commonly attributed to Galileo.

  156. Scott Says:

    Jason Gross #143:

      To me, MUH seems to be a generalization of the the assertion that we can import this bit of math, the idea of isomorphism-invariance, into the real world…

    Yes, I think that’s basically right! But a crucial question, which I feel Max doesn’t sufficiently grapple with, is what we mean by isomorphism. After all, under powerful enough isomorphisms, everything is isomorphic to everything else! Now, Max would reply “but I give a specific definition of isomorphism! I mean there’s a one-to-one correspondence between the elements and relations, etc.” But then the problem is that the definition only makes sense in a formal context where the objects he’s talking about are defined! In the case of the physical world, what should we take to be the basic “elements,” and what should we take to be relations between them? Even supposing we had a final theory of physics, there might be many different ways of carving things up…

  157. Urs Schreiber Says:

    @Scott #156,

    so that’s the whole point that there is a formal definition of “mathematical structure” (in model theory) with a formal definition of isomorphism, and indeed — this is not a new claim or even an insight — every type of mathematical structure in this sense is (the class of models over) a theory, in the sense of formal logic.

    Any formalized theory of physics is (or will be) a theory in this sense. That’s just the basics of mathematical logic.

    Whether every theory in the sense of mathematical logic should be called a theory of physics, as is the claim made here, is a different question.

    More interesting would seem to be characterization of those theories in formal logic which might qualify as theories of physics. One person who has worked on this kind of question throughout his life is William Lawvere, see here.

    Lawvere talks for instance about Toposes of laws of motion for certain infinitary theories that admit a formulation of equations of motion of the kind encountered in continuum mechanics. This can be refined a bit to also capture local quantum field theory (“Higher toposes of laws of motion“).

    In any case, these are types of formal theories, hence of “mathematical structures”, in which a large chunk of modern physics may be formalized. (They are not however just first-order theories as in classical model theory as in #121 above.) The concept of isomorphism here is clear and uncontroversial. The question is which piece of physics is being formalized.

  158. fred Says:

    Scott #156
    If that mathematical structure of reality can be “written” wouldn’t it have to be able to include itself in the description? (similar to a quine program)

  159. E. S. Says:

    I predict that in simulated multiverses, Berkley PhD. students dating Australians appear infinitely often. Wouldn’t a partial confirmation of such a prediction be impressive?

  160. Scott Says:

    E. S. #159: LOL!

    fred #158: Yes, but I don’t see any particular problem or paradox there. For example, the set of all binary strings ({0,1}*) certainly contains many coded descriptions within it of the set of all binary strings.

  161. Adam Says:

    Scott, #160

    Re: quine… what of my related question of self-referential problems for the MUH?

    Why do we need to posit a multiverse?

    More specifically, why do we need to posit that there is another universe synonymous with Conway’s Game of Life if mathematician’s in this universe hold in their minds a model of Conway’s Game of Life?

    In other words, if this universe contains physical manifestations of mathematical structures in the form of the neurons firing in mathematician’s brains, then why do we need to posit a separate universe for these mathematical structures?

    I don’t see the need for the multiverse. Why not just hypothesize that *this* universe contains a physical representation of every known mathematical structure?

    This is what I mean with the MUH having a self-referential problem and so far no one has commented. Does anyone see what I’m getting at?

  162. Scott Says:

    Adam #161: Well, for one thing, “the set of all mathematical structures that human minds have or will come up with” is an arbitrary and historically-contingent set. And there are countless structures that humans could come up with but won’t, merely for lack of time or interest.

    For example, consider the positive integer 12387095382957180389452: I’ll wager that this integer has never been thought about by humans before right now—but not because there was any obstruction to thinking about it, just because there wasn’t a reason to. So, did it only become part of the “mathematical multiverse” when my fingers happened to produce it? Was “that which breathes fire into the math” anxiously watching my keyboard, withholding its universe-creating fire until it saw which integer I typed?

    And anyway, why did it have to be a human? If I programmed my computer to spit out a pseudorandom list of integers, wouldn’t those integers also have the fire of existence breathed into them by your lights, because they were physically stored in the computer’s memory?

    But wait: why would I even need to run the program? Why wouldn’t writing the program be enough—since the code already suffices as a mathematical description of the integers?

    But wait: why would I even need to write the program? Why not just look at my brain, and consider the set of all possible programs that my brain had some nonzero quantum-mechanical probability of generating? (And keep in mind that, in QM, I have some nonzero probability of continuing to exist for as long as you wish, and generating any possible finite string of 1′s and 0′s…)

    Anyway, as I said in my post, this is the slippery slope that Max simply rides all the way to the bottom.

  163. Adam Says:

    Scott #162,

    I would answer that you should ride that slippery slope all the way to the bottom. So where is the bottom? I think it is that this Tegmark multiverse must be populated with other universes composed of mathematical structures which we can’t even fathom or can not be produced in a finite time by some brain/computer/program/writing machine in this universe. And if that is so, then this multiverse is absolutely empty of any content that matters in any way to us or anything in this universe.

    Tegmark might as well hypothesize a multiverse populated by “everything not in this universe.” That is a much easier formulation and I would say equally empty of content that any reasonable mind in this universe should care about.

  164. fred Says:

    Scott #161
    It’s the exact same argument I’ve been wondering about minds or simulated worlds (#133).
    If a mind is nothing but a computation, what is it about the activity of the neurons in my brain that makes it “real”.. then the slippery slope can be ridden all the way down, going from a traditional computer to merely writing symbols, or just picking symbols. And at that point then physical matter itself is no longer necessary (all hardware is itself software in nature).
    If not, then the mind has to be tied to physical matter in a way that’s totally different from hardware running software.

  165. J Says:

    what do you mean by “how to say whether you’re more likely to be living in the first 1010^120 digits of π, or the first 1010^120 digits of e. ”

    Pi and e are fixed numbers. What is probablistic about findiong patterns in fixed numbers? Are you implying the pattern occurring more than once in the 1010^120 digits of pi versus e?

  166. Scott Says:

    J #165: In order to do such a calculation, as a first step you would need to know what kinds of patterns in the digits of π or e count as “instantiating you.” I.e., which encoding schemes are we allowed to use in converting a sequence of digits to your experienced life history? And what if we can decode your history from the digits, but only by applying a rather convoluted transformation (e.g., take all the digits at prime-numbered positions, concatenate them into a single positive integer, then square it, then re-express the result in base 7, then…)? Does that count? Or does it count, but with “less weight,” the weight by which it counts falling off rapidly with the complexity of the decoding procedure?

  167. Jay Says:

    > keep in mind that, in QM, I have some nonzero probability of continuing to exist for as long as you wish

    You believe in quantum suicide? I thought you wrote against it.

    Would you then count finding yourself >>2^7 as a confirmation of MWI?

  168. fred Says:

    Scott #167
    “as a first step you would need to know what kinds of patterns in the digits of π or e count as “instantiating you.””

    Is that asking whether unique mathematical structures can all have an essence that is independent of any symbolic scheme used to write them?

    Like this
    z-> z^2 + c
    and this
    http://tinyurl.com/co6a9r
    represent the same “thing”.

    I still wonder whether graph isomorphism isn’t a key tool.

  169. fred Says:

    One of my professors used to say “everything is a graph”…

  170. Scott Says:

    fred #168: Well, it’s obvious that the same mathematical structure can often be represented in many different ways, and that there might be no unique canonical choice. (We even have choices about how we’re going to encode positive integers as strings of bits…)

  171. Scott Says:

    Jay #167: No, I wasn’t talking about quantum suicide. I said that for every finite T, I have some nonzero probability of living for T years—not that I have probability 1 of living forever!!! The former is just standard QM, while the latter requires not only MWI, but a bizarre way of calculating probabilities over worlds (where you get to postselect on remaining alive, no matter what you do).

  172. fred Says:

    Scott #171,
    Right, but the question is whether there’s in principle a common way to re-encode every possible representation of many different structures (e.g. as graphs?) such that there is also a common tool (e.g. graph isomorphism?) to then identify the ones that represent the same thing.
    Even the concept of “same mathematical structure” seems questionable, but without it I don’t think there can’t be any MUH.

  173. fred Says:

    We know of at least one mathematical object that is definitely real in both MUH and non MUH theories: the human mind (your own consciousness).
    And there is one mathematical representation that is flexible and simple enough to both realize the mind and encode all mathematical objects known to men: the brain. It’s both a static encoding structure and a dynamic one (isomorphism search engine). And by necessity and evolution it’s isomorphic to our environment (the known universe) since it’s primary goal is to simulate it and make predictions to increase chances of survival.
    So the brain is probably a good starting candidate for a universal math representation.
    One would think that any sort of graph/subgraph isomorphism algo would be really handy for applied cognitive science (when trying to compare brain activity).

  174. Jay Says:

    Scott #171 : That makes sense if we interpret probabilities as a measure of knowledge, but I’m not sure to understand your point under MWI interpretation.

    Would you say there are two kinds of MWI? The reasonable one would assume that, yes, a lot of world must exists, but none in which a Gladstone Gander can live forever by pure luck. The desirable one would assume that every measurement basis is valid (Good news! Everyone willing to terminate some branchs can have the universe he desserves!).

    In other words, what do you think bizarre in post-selecting the measurement basis, assuming MWI?

  175. Scott Says:

    Jay #174: Here’s a more concrete way to phrase your question. If you believe MWI, then should you jump off a bridge, expecting to miraculously survive since you’re actually immortal? Empirically, I observe that basically no one who says they believe MWI actually lives their life that way. For that reason, I assume that the view called “MWI” does not entail belief in the “quantum suicide experiment.” And rationally, I don’t see why it should: if you like living, then why shouldn’t you be trying to maximize the total probability mass of the universes to your causal future in which you remain alive?

    I’m tempted to say that any MWI proponent who disagrees with me about this is welcome to try the experiment, but actually, I don’t want that on my conscience…

    In any case, all of this is completely unrelated to the point I was making before: that regardless of which interpretation of QM you believe or don’t believe, quantum tunneling gives you some nonzero (but ridiculously tiny) probability for just about anything you can imagine, including a blue whale spontaneously materializing out of the air above your head, and also including you staying alive for 10,000 years.

  176. Mike Says:

    “. . . regardless of which interpretation of QM you believe or don’t believe, quantum tunneling gives you some nonzero (but ridiculously tiny) probability for just about anything you can imagine, including a blue whale spontaneously materializing out of the air above your head, and also including you staying alive for 10,000 years.”

    More than one person has been called crazy because of this. ;)

  177. fred Says:

    Scott #176
    I’ve been wondering – if we were to discover that not only intelligent life but that life itself only exists on earth, wouldn’t that make MWI more likely, since then the chance of our own existence would be ridiculously low in a non-MWI world? (but our existence even if very improbable would always be realized in a branch of MW)

  178. Darrell Burgan Says:

    Fred #133 – “We think of it as traditional electronic hardware, with bits stored as charge densities, evolving along some clock.”

    Is this really the accepted definition of computation? Because from my perspective computation has nothing at all to do with the underlying implementation, which I consider to be a fairly irrelevant detail. Computation, at its heart, is nothing more than rigorous logic.

    Further, it assumes all computation is imperative, with a predefined thread of execution. This flies very much in the face of both declarative programming in its many forms, as well as distributed computing, which is increasingly looking more and more like high density matrix mathematics.

    Maybe I’m biased because I’m a software head, but it seems to me that conflating computing with the computer it runs on is like conflating math with the paper it is written on.

  179. Jay Says:

    Scott #175: Ok to give this question a break, but no that’s not the same question. The question was, if you assume MWI, what is the logic* against quantum suicide? My own provisional answer was there may well be no consistent measurement basis in which quantum suicide can works, e.g. it’s not obvious any conceivable event has non zero probability.

    So, if we stick to quantum tunneling, don’t you think measurement basis can restrict the possibilities, e.g. if we could look carefully enough, we could exclude blue whale spontaneously materializing because that would implicate violating no cloning for reasons non trivial to compute but simple to hand wave as: decoherence prevents high level magic.

    *as you mentionned, there are obvious practical reasons for not trying even if you strongly believe you’d survive, including you don’t want your decisions to hurt most versions of your beloved relatives.

  180. Sid K Says:

    Warning! Long, rambling, non-rigorous comment by non-expert.

    Scott #29:

    You ask: “For one thing, this “solution” seems merely to push the riddle somewhere else: one now wants to know, why is the fire of existence not only breathed, but breathed so promiscuously, onto every set of equations that anyone could write down?”

    You have pushed the argument one level deeper and asked why all these mathematical structures exist. Thus we get into the rabbit-hole of what we mean by “existence.” Let us consider the various brands of existence:

    (1) Do we exist? I think we all agree that Yes, we do exist; otherwise, who is doing the agreeing?

    (2) Does the Solar System exist? We agree, Yes. It seems clear from the motion of the planets and from space probes.

    (3) Does the Andromeda Galaxy exist? Also, I think we agree, Yes. We can only be sure that it existed 2 million years ago; though we know of no likely process that might’ve caused it to cease existing. Also, we can in principle, travel there and check.

    (3) Did the Universe exist 10 billion years ago? Also, I think we agree Yes, even though, there is no in-principle experiment we can do to go back in time and see if it did exist. We can only look at the present and posit the existence of the far past to parsimoniously explain what we see.

    (4) Does Sand exist? We agree, Yes; even though sand is simply made of fundamental particles and you may say that only fundamental particles exist. But it would be insanity to specify the wavefunction of the quantum fields every time you want to say: “I can’t get this sand out of my ears!”

    (5) Does Love exist? I think we all agree, Yes; it is a most singular experience. This is true even though, at best, we can point to patterns of human behavior and say that it is caused by Love. Or we could, if you like, point to a set of electrical signals in the brain and show it is correlated with certain behaviors associated to Love.

    (6) Do Unicorns exist? I think we all agree, No. Postulating Unicorns adds no explanatory power for our experiences. It may help in explaining why there are people who believe unicorns; though, we all agree that there are better explanations for belief in unicorns than unicorns.

    ———————————————————————————————-

    From these examples we can discern the sense in which the word existence is used: basically we use it to mean that our experiences become much more easier to understand, predict and think about if we deem that certain aspects of our experience ‘exist’ independent of our experiences of them.

    It seems then that asking whether something exists or not isn’t the right way to frame the question. The right way to frame it is to ask whether admitting a certain hypothesis would improve the predictive power, simplicity compared to previous theories and explanatory power. (We seem to have approximately user-independent notions of simplicity and predictive power; I don’t know of any good user-independent notion of explanatory power.)

    Armed with this intuition let us tackle the more trickier questions.

    (5) Do branches of a wavefunction containing conscious entities exist? Some say Yes, some say No. Here the existence is also in principle directly uncheckable (or at least it seems so). How is this different from the question of the existence of the far past? It’s not that there isn’t any evidence: MWI seems to be completely consistent with quantum mechanics. Also, people like David Deutsch claim that quantum computers are the evidence left by the many-branches. The problem here is that there are competing hypotheses that aren’t that much more complicated (though it seems that MWI is the simplest). In the case of the existence of the far past, there is no other hypothesis that comes close in simplicity to explaining all that we see around us. But with quantum mechanics, one can hold other reasonable positions, such as the Copenhagen position. Thus, it seems that for people like David Deutsch it adds explanatory power, whereas for people like Chris Fuchs, it doesn’t. Thus, can we conclude that the branches exist for David Deutsch, but don’t for Chris Fuchs? Maybe yes. Here you see a user-dependent notion of explanatory power kicking in.

    (6) Do the inflationary bubbles exist? If inflation is indeed the best theory we have and indeed this predicts these bubbles, then we should increase our prior that assuming the existence of these bubbles will add predictive & explanatory power and also be simpler. If indeed these bubbles don’t add any predictive & explanatory power and can be replaced by something simpler, then their existence again becomes user-dependent.

    (7) Do the mathematical universes of Tegmark exist? Has it increased our predictive power? I think that is a clear no. Maybe work on the measure problem holds promise. Is there some notion by which it is simpler? I think yes, because it removes the requirement for some “fire” to be “breathed.” One might say that the ontology becomes much larger because we permit all these other universes making the theory much less simple. But notice that Tegmark demands only consistency and finiteness for existence [actually, he may demand other things; I really should read the book]. Thus all the universes are (hopefully) derivable from these conditions. Does it improve explanatory power? This is largely user-dependent as of yet. Maybe attempts to formalize can yield fruit. So as yet, the status of the existence of Tegmark universes is dubious. But there is hope (and I’m one of the hopefuls).

  181. Scott Says:

    Sid K #180: That’s an extremely nice summary, thanks!! (Except your (5), (6), (7) should be (7), (8), (9))

  182. Scott Says:

    Fred #177:

      I’ve been wondering – if we were to discover that not only intelligent life but that life itself only exists on earth, wouldn’t that make MWI more likely, since then the chance of our own existence would be ridiculously low in a non-MWI world? (but our existence even if very improbable would always be realized in a branch of MW)

    Even under these rules (where our uniqueness is a puzzle that needs to be explained), it seems to me that you’d have many alternatives available. For example:

    (1) The Level I or Level II multiverses. I.e., you could simply believe that reality in “our” branch extends much further than you can see.

    (2) Life is a rare event, but not ridiculously rare: in fact, you expect it to occur about once per causal patch, on average.

    (3) The universe is a computer simulation, and the simulators tuned the parameters so that life would arise on exactly one planet. (More traditionally-minded people might have a different name for this hypothesis… :-) )

    It also seems to me like your logic could be pushed even further. If you find that you’re the only copy of you in the visible universe, then why shouldn’t you interpret that as evidence for a multiverse—since the probability of you, specifically existing would be ridiculously tiny otherwise?

  183. fred Says:

    Scott #183
    ” If you find that you’re the only copy of you in the visible universe, then why shouldn’t you interpret that as evidence for a multiverse—since the probability of you, specifically existing would be ridiculously tiny otherwise?”

    Only if you happen to be a blue whale spontaneously materializing out of the air!

  184. wolfgang Says:

    >> Do Unicorns exist? I think we all agree No

    There is a nice puzzle/paradox philosophers discuss(ed) about the fact that you cannot really make such a statement.

    In short: If unicorns do not exist then by definition there can be nothing this sentence refers to. Or in other words, if unicorns do not exist then there can be no (direct) evidence your statement refers to.

    Your statement is really about a certain hypothesis which exists (in your head) about “non-existent unicorns” or about a search for unicorns which came up empty,but not about unicorns.

    I am never sure if this is a really deep puzzle or just trivial. It was of course part of the long philosophical back and forth if God exists or not and it means that atheists literally do not know what they are talking about 8-)

  185. wolfgang Says:

    >> there are competing hypotheses that aren’t that much more complicated (though it seems that MWI is the simplest)

    But when we talk about the Born probabilities then Copenhagen is the simplest imho.
    It is at least simpler than Bohm-deBroglie and simpler than mwi (by far).

  186. Scott Says:

    wolfgang #184: I vote for “trivial.” I don’t feel like I have difficulty parsing the statement “there does not exist an x such that x is a naturally-occurring flying, winged horse with a horn on its head”—and I don’t feel like the statement becomes meaningless by virtue of being true. I also feel like, while I don’t have a proof of the statement, I have excellent circumstantial evidence from physics, biology, searches that came up empty, and general knowledge about human myths.

  187. fred Says:

    I read the Fabric of Reality a long time ago, but I recall that the multiple world alternatives weren’t always evolving independently of each other but actually “influencing” each other to explain quantum interference (double slit experiments), that part was a bit murky.
    I’ve read recently about some experiment showing entanglement reaching back in time
    http://arstechnica.com/science/2012/04/decision-to-entangle-effects-results-of-measurements-taken-beforehand/
    (although I could not figure what’s stopping Victor to base his decision to entangle the photons on the actual measurements of Alice and Bob, it seems he could then bias the statistics arbitrarily)
    I wonder if there could be a way to combine the two ideas to find some evidence for or against MWI – e.g. flipping a quantum coin a million times in a row, then only entangle some particles if the quantum coin has landed on 1 a million times, and not do the entanglement in all other cases (a sort of milder version of quantum suicide where the existence of that one branch where the entanglement is done would “bleed” into all the other alternatives).

  188. Adam Says:

    Scott 175:

    If you believe MWI, then should you jump off a bridge, expecting to miraculously survive since you’re actually immortal? Empirically, I observe that basically no one who says they believe MWI actually lives their life that way.

    I don’t think that is a fair assessment. Just because I might believe I am ultimately immortal doesn’t preclude me from believing that I am also subject to sickness, disease, physical impairment and suffering. Maybe I don’t jump off a bridge becaus e while I am not afraid of dying, I would rather not live out my days in a wheelchair.

    That is the big problem I have with quantum suicide arguments. My life and my self condition can not be represented as a binary state. Like just because I believe in reincarnation doesn’t make me want to jump off a bridge to get to my next life.

  189. Adam Says:

    Scott 170:

    fred #168: Well, it’s obvious that the same mathematical structure can often be represented in many different ways, and that there might be no unique canonical choice. (We even have choices about how we’re going to encode positive integers as strings of bits…)

    I wonder if Tegmark’s MUH can be interpreted as the prediction that every mathematical structure does have a unique preferred canonical representation and it is only in this form that fire is breathed into the equations.

    That would provide an answer to the slippery slope we’ve Ben talking about I think. But it would present all kinds of new problems of course.

  190. Scott Says:

    Adam #188: If you were actually going to live forever, then you’d have infinite time to search for a cure for whatever had put you into the wheelchair—or even to be “miraculously” cured because of some quantum tunneling event. So this particular malady would only affect you for finitely many of your infinitely many years.

  191. Max Tegmark Says:

    Hi Noon (#150), Urs (#155), & Scott,

    Noon (#150):
    You ask: “If I’m only required to write down sets and relationships between them, can I not write down strings, comment on which sets they fall into (maybe it’s only one set) and then trivially “describe” the entire universe in this way?”
    Excellent question. Answer: no, since you’re not allowed to do the “comment” part! This is also what’s flawed about the decimals-of-pi critique from Scott’s post – you’ll never get a 3D space etc out of the digits of pi without adding more structure. The mathematical structure is defined only by the abstract sets and relations without any human comments, and they have no intrinsic properties whatsoever (I hope you’ll find that the many examples in chapter 10 clarify this point). What’s so unique about certain mathematical structures from physics (say Minkowski space with quantum field theory) is that a natural interpretation *emerges* from them rather than having to be put in by hand: bound states that it’s convenient to coin names for such as protons, atoms, and stars.

    Urs (#155): you say that “Hence saying that the world is described by (or just “is”, if one insists) a “mathematical structure”, just means: it is described by some formal language with some axiom”.
    As I mention in #112 and discuss in detail in chapter 12, it’s important not to conflate the formal system (language) with the structure, and it’s well-known that some formal systems can describe multiple non-equivalent mathematical structures. For example, the Löwenheim-Skolem theorem theorem (with which I’m sure you’re familiar) implies that the axioms for real numbers also describe a mathematical structure that’s countably rather than uncountably infinite.

    Scott: how about coffee next week? I’m not sure what your plan is for this thread now – have you switched to answering all posts except mine in anticipation of our meeting? Your new claim that I’m “riding a slippery slope all the way to the bottom” sounds like a conflation of mathematical structures with computation (#112), and of my claims with those of Konrad Zuse and Jürgen Schmidthuber.

  192. Adam Says:

    Scott #188: That assumes quite a bit:

    * That I will be happy to live in pain while I am looking for a cure
    * That I will even have the conditions to be able to look for a cure
    * That the future will present itself with conditions for a cure to be developed
    * That I’ll have the necessary means to actualize the cure even if it is developed and found

    In short, the quantum suicide thought experiment assumes an optimistic future which there is no reason to suppose. Perhaps I will live forever, but my personal state will devolve into ever greater levels of misery and pain. Perhaps a few million years from now I’ll be in a semi-vegetative state with no arms or legs and living on a planet in the Andromeda galaxy with no friends or family within a few light years and no means to reach them. Perhaps because of the MWI I’ll still keep on miraculously living, but not in a very happy way.

  193. Scott Says:

    Adam #192: No, I don’t have to assume most of those things, because eternity is a long time and the universe is quantum! As I tried to explain, in an infinite time, tunneling gives a nonzero probability for essentially anything to happen to you, including your being miraculously cured, then injured again, then swallowed by a whale, etc. So if you expect to live forever, then you should really only care about the equilibrium probabilities of being in one state rather than another—what happens to you in the next decades or century is completely irrelevant. That might sound silly, but it’s certainly no sillier than the starting assumption.

    But maybe I should ask: do you actually expect that you’ll live forever, on the basis of MWI? If you don’t, then this whole discussion is sort of surreal: who am I arguing against?

  194. Scott Says:

    Max #191: Sure, let’s get coffee next week! I’ll email you.

  195. Adam Says:

    Scott #193:

    No, I don’t personally believe I’ll live forever on the basis of MWI. I do believe in reincarnation though, fwiw. Regardless, I don’t understand how appealing to the length of eternity resolves the fact that I still don’t want to experience unhappiness for even a finite amount of time and therefore would not consciously choose to subject myself to the pain of jumping off a bridge. Perhaps because I’ll live forever and there exists a non-zero probability of it that I’ll be forced off a bridge or that the atoms in my brain might spontaneously form to cause me to choose to jump off… but I still maintain that even if I were to believe in MWI immortality I am not compelled to go jumping off bridges or firing guns into my mouth.

    Regarding, only caring about equilibrium probabilities… this whole question about conscious decisions presupposes that I might have a chance at changing such probabilities with my choices whereas your argument seems to be that in an eternity my choices won’t really change the outcome of what I will experience. Is that correct?

  196. Sandro Says:

    wolfgang #185:

    But when we talk about the Born probabilities then Copenhagen is the simplest imho.

    I’m not sure what gives you this impression. Copenhagen merely assumes the Born rule and the measurement postulates. At least de Broglie-Bohm (dBB) can derive the Born rule by assuming a simpler axiom, and no measurement postulates are needed. Therefore at least dBB is more parsimonious than Copenhagen.

  197. Urs Schreiber Says:

    @Max, #191,

    one doesn’t need to appeal to Löwenheim-Skolem to see that in general a theory has more than one model. Imagine the theory of groups had just one model…

    (In the 90s there was a wide-spread subconcious belief that string theory has only one, or just a handful, of models… the shattering of that belief is — by a curious course of history — the reason why some fundamental physicists these days turned to philosophy.)

    Anyway, what saying that the world is modeled by a “mathematical structure” in the sense of model theory means is that it is a model of a first-order theory — that’s the source of the word “model theory”, after all.

    Now what is interesting is to characterize those theories whose models (whose “mathematical structures”, if one insists) exhibit physical properties, hence theories inside which one may find something like quantum field theory (or maybe a refinement thereof), so that its models (the “mathematical structures” over it) are systems of quantum fields.

    There is a remarkable simple characterization of quantum gauge field theoretic structures in a version of homotopy type theory that, following Lawvere, may be called “cohesive”. Using this one can pinpoint those “mathematical structures” (though in a more general sense than that of just classical model theory) which exhibit properties of local Lagrangian quantum gauge field theory, which exhibit quantum anomaly cancellation as seen in nature, etc.

    There is a lot of interesting things to be said in founding modern physics in formal logical theory. (But classical model theory is unsuited, as far as I can see.)

    From discussions like this here it almost feels as if there is suddenly a wide public interest in foundations of physics in formal logic (and maybe classical or modern model theory). That used to be a topic that besides William Lawvere few people were working on. On the other hand then it seems most of the interest here is in witty chat, not so much in mathematical or physical theory.

    I am wondering if there is a way that the occasion that model theory has been sneaked into the public debate through the backdoor could be used to increase scientific interest in it that goes beyond lay chat.

    Our host, Scott, for instance might be interested in questions such as: what might characterize those formal theories such that their models (the “mathematical structures” over them) exhibit a form of quantum computation? (Would he not be distracted by debating suicide of MWI proponents — maybe there is a psychologically interesting Freudian aspect here, but none of mathematical or physical interest). For instance I claim that within linear homotopy type theory one may say something interesting about this, which looks relevant to the question of whether the world has an origin in formal mathematical theory.

  198. Magic Numbers Says:

    Ok, so Sid has insisted on dragging unicorns into this. As if that weren’t bad enough, it prompted Scott to attribute them to having wings. Please bear in mind, that I am merely an enthusiastic amateur in regards to Mythology.

    We all know by virtue of platonism that indeed Unicorns do exist, and in fact they possess an attribute orthoganal to winged equines (aka Pegasi) that attribute is their velocity.

    Remember: Unicorns are the fastest land equines. On land, they run so fast, their velocity is not even a number anymore. It makes C look like nothing. If you never get done counting how far a unicorn moves on land in unit time, when you don’t get there (but you’re tired of counting) you realize that sucker is going Aleph-null units of length per unit time. (REALLY FAST!)

    Ok, so unicorns are so fast on land. What’s the big deal?

    Well, when the afforementioned winged equines (Pegasi) happen to be in motion, they are the fastest flying equines. They travel so fast, that in the time that a unicorn moves one unit of length, they travel Aleph-null units of length (WOW! How fast is that, you ask?) Those bad mamajammies are whizzing around, traveling at 2^(Aleph-null) units of length per unit time.

    A common question that frequently comes up in conversation is: “Hey. I wonder if there exists an equine who’s velocity is strictly lower-bounded by the unicorn’s speed and upper-bounded by the pegasusses’s speed. (aka the ‘Corn Hypothesis (CH))

    Whilst visiting Leibniz’s “Labyrinth” Theme-park, I was considering this conjecture while poking my head into a hall-of-mirrors called “Many Worlds”. I took out a picture of my favorite Mythology Professor “Kurt” who embarassed himself before all the other mythologists when (on a little run of pain-pills) he concocted a “This is not a Proof” of CH.

  199. wolfgang Says:

    @Sandro

    deBroglie-Bohm is perhaps conceptually simpler than Copenhagen for non-relativistic QM but as far as I know deBroglie-Bohm does not even exist yet for QFT (or Lorentz invariant qt in general).

  200. Douglas Knight Says:

    Does QM really say that there is a nonzero chance that you’ll live for every duration? Maybe QFT does (though I’m skeptical), but if you believe that the universe is a finite quantum computer, doesn’t that bound your life?

  201. fred Says:

    Douglas #201
    Whenever the state of the “universe computer” is being updated we all die a little bit, but we’re being fully reborn once the new state has stabilized. And this repeats every tick.
    (see “discrete Zeno paradox”)

  202. Max Tegmark Says:

    Urs (#197): I fully agree with you that more rigorous work on these model-theoretic topics is interesting and worthwhile. Please let me know if travel takes you from Utrecht to the Boston area, since I’d love to meet up and dig into these topics at a more technical level! Like you, I’ve found that many physicists simply aren’t that interested in the details. Yet, the devil is of course in the details.
    Indeed, Even Scott, one of the most rigorous thinkers I know, sometimes makes sloppy statements, like above (#156) where he writes “under powerful enough isomorphisms, everything is isomorphic to everything else” – which is of course false. The empty set is not isomorphic to the integers, period. In the same spirit, the Mathematical Universe Hypothesis can’t be seriously discussed and evaluated if one disregards its detailed definition.

  203. Scott Says:

    Douglas #200:

      Does QM really say that there is a nonzero chance that you’ll live for every duration?

    Well, roughly speaking, the probability for T particles to do something really crazy goes like 1/exp(T). So, if there are maybe ~1028 atoms in your body, then each second, there’s maybe a ~10-10^28 chance that you’ll be miraculously healed of everything that ails you.

    (Note: The above is OK as long as you treat our causal patch as an open quantum system, whose quantum state can “collapse” by leaking information beyond its de Sitter horizon. If, instead, you were to regard our causal patch as a closed system evolving unitarily, then the question becomes much murkier. For then, there will just be some wavefunction of the universe rotating eternally through Hilbert space. And that wavefunction will almost always have some nonzero component in which you remain alive. But do we get to interpret that as saying that you “live forever”?)

      … if you believe that the universe is a finite quantum computer, doesn’t that bound your life?

    If the Hilbert space accessible to you has only a finite number of qubits, then that does bound how large you can ever become, but it doesn’t bound how long you can live for.

  204. J Says:

    Scott Comment #166

    I was thinking this way. Say every possible string hypothetically occurs in e or pi. Then say there is a universal string S which is the program. There is a string H_i which is the input to S and from SH_i we can recover history for user i. Then SH_i either occurs in first 10^10^120 bits of e or pi or does not occur. So this is deterministic correct?

  205. Me Says:

    MUH?
    Meh!

  206. Scott Says:

    J #203: Well, which are the strings Hi from which you can recover the history for user i? My point was that they can be completely different depending on which “interpreter program” S you choose.

  207. Scott Says:

    Urs #197: If you frame the choice as “witty chat” versus “rigorous axiomatic formalization,” then you’ve pretty much guaranteed that I’m going to choose the former… :-)

    Seriously, I’m happy to prove rigorous theorems (I do it for a living), but only once I’m convinced that there’s a sufficiently meaty subject-matter to prove theorems about. And in the case of MUH, it’s not clear to me that there is. The problem is not just the philosophical one of whether you want to regard every mathematical structure as a “physically-existing universe,” and it’s not just that we don’t have a final theory of physics. Rather, it’s that even for the physical theories we already know—like general relativity and the Standard Model—I’m skeptical that there’s any one mathematical structure (in the formal sense) that everyone could agree that they correspond to. Rather, it seems like the basic ideas of those theories can be represented formally in many different ways, and those ways will give rise to different structures.

  208. Douglas Knight Says:

    Sure, the Hilbert state might always maintain positive amplitude of my current state, but to live for N years, shouldn’t we mean a state that is, in some sense, an N year evolution of my current state? I don’t think getting stuck in an infinite loop should count as living forever. Maybe that’s another debate, but I think it is separate from the relevance of QM.

  209. Sid K Says:

    Scott #181:

    Thanks! I realized that the numbering was off the one second after I hit the Submit button.

  210. fred Says:

    Scott #207
    Wouldn’t you say that general relativity, the standard model, or whatever field is your expertise are represented,.. they’re all represented in your brain in a manner that captures all the subtleties and richness of the field? The representation might not be “formal” in a traditional sense, but it’s complete enough for your brain to then come up with arbitrary formal representations.

  211. Adam Says:

    Max #202:

    In the same spirit, the Mathematical Universe Hypothesis can’t be seriously discussed and evaluated if one disregards its detailed definition.

    Well, please know that Scott’s review did convince at least one more person to buy your book even while being highly skeptical of the idea. I don’t agree with Peter Woit that even if the MUH is ultimately empty of content that the book shouldn’t be read. I know that Scott takes exception to many of the ideas in Penrose’s various books, but would still highly recommend them and I would enthusiastically agree. Even failed ideas can cause the conditions to lead to true insights by helping to map out the problem space!

    Urs #197:

    I’m one of the lay chatters, but I find this whole discussion and many of the highly off-topic tangents fascinating (*ahem* MWI/suicide arguments *ahem*). I wouldn’t have found out about your work and Lawvere’s work on grounding modern physics in a formal system without it! I think this is very relevant for Max’s MUH idea and if there is something there it might likely come from such work. And now I’m going to read Max’s book because of it. Can you recommend any other easier works on formal type theory? Anything that might help a programmer by trade understand what is going on?

  212. Scott Says:

    Max #202: “Isomorphic,” as I understand the word, only makes sense once we’ve specified a class of isomorphisms that interests us. E.g., a coffee cup is “isomorphic” to a donut topologically, but not geometrically. I indeed don’t know of any notion of “isomorphism” in common use in mathematics, under which the empty set is isomorphic to the set of positive integers. But it’s easy to make up a contrived notion. For example, if I can permute the 1′s and 0′s in the code arbitrarily, then I can surely convert a program that outputs the set of positive integers to a program that outputs the empty set, and vice versa. And who are you to tell me that something like that isn’t the notion of “isomorphism” relevant for the mathematical multiverse? Where does your knowledge of this multiverse’s laws come from?

    I know you say you have a specific notion of “mathematical structure” in mind, which comes with a well-defined notion of “ismorphism.” My question is, why that particular notion? For one thing, different logical formalisms will give rise to different notions of “mathematical structure” (e.g., some will have k-ary relations for arbitrary k, others will only have binary relations, with higher-arity relations built up out of those). So, given that there are different notions of mathematical structure, all rich enough to encompass (in some sense) all of mathematics, how do you know the “true” version relied on by the multiverse?

    (Also, to me strings of 1′s and 0′s seem even more basic than “structures”—and since the former can encode the latter, presumably there’s no loss of generality in taking bit-strings as the basic building blocks of the Mathematical Muliverse. By what argument can you show me that I have to look instead at the “structures” encoded in some particular way by those bit-strings?)

    Anyway, even after you’ve settled on a notion of “structure,” as I said in comment #207, a further difficulty is how to map the physical universe onto a structure of that kind. Even classically, what should the elements be: the spacetime points? (But then what about diffeomorphisms?) The values of the fields at spacetime points? The connection? And should there be a relation between every two points? Or only every two “neighboring” points? And in passing to quantum, should we use the Schrödinger, Heisenberg, or Feynman pictures? etc.

    Let me stress that, in my view, the fact that we don’t have a final theory of physics is the least of the problems! Even if you take a theory that’s already known, like general relativity—or even Conway’s Game of Life, for that matter!—and even if you fix a notion of mathematical structure, there will still generally be many different ways to encode the theory as a mathematical structure.

    In the end, then, I completely reject your claim (expressed in comment #191) that there can exist an “interpretation-free” way of moving between formal mathematical structures and the informal meanings of those structures. At best, I’d say the mapping is many-to-many: the same informal concept can generally be formalized in many different ways, and the same formal structure can have many different informal interpretations.

  213. Scott Says:

    fred #210:

      Wouldn’t you say that general relativity, the standard model … they’re all represented in your brain in a manner that captures all the subtleties and richness of the field?

    Represented in someone else’s brain, not in mine—but basically yes. ;-) Again, though, that doesn’t give you unique formal representations of these fields. Different experts would probably formalize their knowledge of the fields in different ways (some using Lagrangians, others using Hamiltonians, etc.), and even the same expert would probably formalize differently depending on what the purpose was.

  214. Ian Says:

    +1 for Scott #212!

  215. fred Says:

    Scott #214
    Yes, different experts have different representations, but I’m looking one level above that, saying that the brain is modelling all those representations using the same trick, i.e. neural pathways. And that is the universal structure to consider and try to formalize (very graph-like obviously). And then try to formalize isomorphisms that apply between brain structures.
    By definition, there can’t be any other structure better suited to encompass efficiently all human knowledge (which is all we can possibly ever talk about), not just statically but dynamically as well – that structure has been perfect to get us from this
    http://tinyurl.com/q27pxq6
    to that
    http://tinyurl.com/ony4eo2

  216. Greg Kuperberg Says:

    Max: The empty set is not isomorphic to the integers, period.

    Actually, they are both measure zero subsets of the real numbers. So, they are isomorphic in the category of measurable subsets of ℝ.

  217. Scott Says:

    Yeah, what Greg said!

  218. Jason Gross Says:

    Scott #152:
    I’m coming at qualia from the perspective of “I don’t know what people mean by ‘qualia’, so I’m going to try to make the best approximation I can”. The most generous interpretation I’ve managed so far is the one that comes from dropping the “qualia have no physical effects on reality” requirement that seems to sometimes come with definitions of “qualia”. My point was that if you’re looking for a definition of qualia, and you discover the experimental results that I’ve mentioned, then it seems appropriate to say “I’ve found something that looks a lot like qualia, and, look, there’s a red-blue swapping of it between these people”. (Personally, I’ve not yet really made sense of the concept of ‘qualia’.)

  219. Mike Says:

    “Again, though, that doesn’t give you unique formal representations of these fields. Different experts would probably formalize their knowledge of the fields in different ways (some using Lagrangians, others using Hamiltonians, etc.), and even the same expert would probably formalize differently depending on what the purpose was.”

    But isn’t the point that the human brain is capable in principle of modeling an accurate representation of these fields?

  220. Jason Gross Says:

    Greg #216, Scott #217:
    What are the maps in the category of measurable subsets of ℝ? They’re certainly not functions, if you end up with functions to the empty set.

    Of course, you can define the trivial mapping to the terminal category where all objects are isomorphic. It’s not clear to me why you should consider the objects that you started with to be the same as the ones that you ended with. When you say “the empty set”, I hear “the initial object in the category of sets”. It’s not clear to me that you get to call it the same empty set when you switch to another category. (Of course, you can say that they’re equal as sets, if your category theory is backed by set-theory. Then you can also ask questions like “is 2 an element of π?” and the answer is “it depends on what encoding you used of numbers” when it should really be “mu” or “type error”. So I find myself working internally to homotopy type theory instead.)

  221. Jason Gross Says:

    Scott #212:

    And who are you to tell me that something like that isn’t the notion of “isomorphism” relevant for the mathematical multiverse? Where does your knowledge of this multiverse’s laws come from?

    This might be an empirical question, should we find a natural way to solve the mathematical measurement problem; if there are sufficiently few free parameters (and one of them is which notion of isomorphism to use), then perhaps we ask which notion of isomorphism gives rise to the correct anthropic probabilities.

    Alternatively, it might be the case that we will find that our hand is forced; if we have a way of interpreting various structures, this interpretation should be isomorphism invariant (in the sense that for any structure which seems to be interpretable into our physical reality, any isomorphic structure should also be interpretable; some structures might not have enough information to obtain an interpretation).

    However, I’m straying into the land of things I don’t understand very well; I should go read some books on model theory or take a class on it.

  222. Jason Gross Says:

    Max #191:

    As I mention in #112 and discuss in detail in chapter 12, it’s important not to conflate the formal system (language) with the structure, and it’s well-known that some formal systems can describe multiple non-equivalent mathematical structures.

    I’ve been wondering for a while about whether or not there is a way to construct a theory with a unique model, if your ambient logic has multiple models. From reading Andrej Bauer’s The elements of an inductive type, I came away with the impression that non-standard models are unavoidable in general. (My hunch is that this has something to do with whether you interpret ∀ and ∃ (or, in type theory land, ∏ and ∑) into your model or into your metatheory.)

    I have the following questions:

    (1) Is categoricity possible for models of type theories?

    (2) If categoricity isn’t possible for type theories, how should I go about understanding what is a mathematical structure when I’ve lost the perspective of living inside a model, and only have the perspective of living inside a type theory? (It might be that the answer to this question is “learn to think in NuPRL rather than Coq” or “go read chapter 12″, and, if so, that’s fine with me.)

    (3) Benedikt Ahrens has told me that the strength of the internal logic of a topos depends on the ambient logical strength in ill-understood ways. Is there a similar dependence in cohesive ∞-topoi (or whatever structures might let us do synthetic physics), and, if so, is there a way to probe the “logical strength of the internal logic of physical reality” and thereby determine what “ambient logic” we expect to be “living in”? (Could this an example of a result that should strongly bias us towards something like MUH? Or is there some way to remove any such dependence in general?)

  223. fred Says:

    I think there is a fundamental characteristic for a candidate universal structure – it has to be discrete in nature. Any continuous structure has to be excluded. Continuity would lead to zeno paradoxes, i.e. there could be no evolution, no dynamic possible. This I think also explains the struggle I had with the connection between consciousness and computation. The dynamic of the system has to enter the picture and that’s why a computation is just not a lookup table. Our mind is not just a series of snapshots of states but a connected structure that evolves both in time and space. This reduces to the key conservation and symmetry concepts (like energy) which are nothing more than limits on how fast and how wide things can grow. You simply can’t have that limitation with continuous structures.
    The simplest illustration of this is Conway’s game of life (or any automata along x dimensions) – for complex structures to appear you need “room” on every dimensions for them to “grow”. Connected structures don’t arbitrary appear but they “bloom out” outwards and they need empty cells to do so, and can only grow at a certain rate along all dimensions. That simple requirement captures any fundamental limitations on dynamics and complexity that we observe in our universe, and is a universal pattern – expansion of the universe since the big bang, formation of galaxies and solar systems, evolution of life, growth of individual organisms (from a egg to a full adult), neurons forming connections in a learning brain, etc. The brain itself can be modeled with such a structure, and the brain can represent all human knowledge ( it is isomorphic to our universe by definition).

  224. Greg Kuperberg Says:

    Jason – The morphisms are measure-preserving maps, which are certain equivalence classes of functions.

  225. Max Tegmark Says:

    Apologies for repeating myself, but the Mathematical Universe Hypothesis can’t be seriously discussed and evaluated if one disregards its detailed definition. I’m using the mathematical structure definition given in the book and in Appendix A in http://arxiv.org/pdf/0704.0646v2.pdf which, as you’ll see follows standard definitions in the literature but is slightly more general, to allow multiple sets and k-ary relations for all k (this should please you Scott as a programmer!), and I explicitly specify the class of isomorphisms used to define the equivalence for structures. And no, the empty set is not isomorphic to any mathematical structure with non-zero cardinality with this definition. I look forward to our coffee, Scott!

  226. fred Says:

    Now whether we’ll ever discover that structure, I doubt it – it doesn’t seem possible to describe a system fully from the inside if the system is discrete and finite, the description would have to be fully isomorphic to the whole system but also to itself and possibly recursively so. Probably why we’ll never fully understand and explain consciousness either, the tools lie beyond our realm.

  227. wolfgang Says:

    @Jason #218

    >> Personally, I’ve not yet really made sense of the concept of ‘qualia’.

    I have now witnessed many debates, with one side usually arguing that qualia are indeed a “hard problem” – let’s call them type Q, and this includes myself btw.

    The other side insists that they cannot make sense of the concept of qualia and that there is no “hard problem” that would need an explanation – let’s call them type N.

    I take this as strong evidence that Q and N people indeed experience the same sensory input quite differently.
    I think this actually settles the question if some people experience the “redness of red” different than other people: I think that Q people experience some “redness of red”, but the N people do not (if we can trust the verbal information we hear from both – as we should according to Daniel Dennett).

    But this also settles the question if qualia are a “hard problem” – obviously there must be something Q people experience which N people do not; But this is some difference we will not see under a microscope, thus I conclude that qualia are indeed a “hard problem”.

  228. Noon Says:

    Max #191,
    > You ask: “If I’m only required to write down sets and
    > relationships between them, can I not write down
    > strings, comment on which sets they fall into (maybe
    > it’s only one set) and then trivially “describe” the
    > entire universe in this way?”
    >
    > Excellent question.
    >
    > Answer: no, since you’re not allowed to do the
    > “comment” part!

    Okay. Let me be state my intentions – I want to understand what are the allowable “descriptions” in the MUH. I also want to know what it means, in this theory, to “describe something mathematically”. I’d further like to know why this is saying something *interesting*; i.e. I’d like to know specific examples of things I can *not* describe “mathematically”.

    In order to understand this, an idea of the most trivial “mathematical description” seems like a good starting point.

    My question was trying to guess what this most trivial description would be – namely the listing of outcomes into one big set of “Things That Happen”. This, to me, fits into the pattern of a “abstract sets and relations”, because I have only one set, and no relations.

    You suggest:
    > The mathematical structure is defined only by the
    > abstract sets and relations without any human
    > comments, and they have no intrinsic properties
    > whatsoever

    This is something I actually just don’t understand. How does a thing have no intrinsic properties and is yet somehow defined and categorised (placed into an abstract set) without humans? Who decides what set the thing goes into? Who decides the relationships? (Was the word “intrinsic” here a mistake? In the next paragraph, you say that a “natural interpretation” emerges; unless I’m misunderstanding I don’t see how this can happen *without* the structures containing some intrinstic property – namely “mathematical describability” in this sense.)

    Regarding properties emerging – you claim that there is a natural way to interpret the mathematical structures that are apparent in physics. How exactly does this relate to the MUH? Does the MUH claim that there is always a natural mathematical structure for anything we observe? What does “natural” mean? How does it specify would should be considered “in” a particular mathematical structure and not in another? (Does it talk about where classical physics ends and quantum physics begins, or does it say we don’t yet have a natural description? (I think you said earlier that it basically says we don’t yet have a natural description – so how do we know when we do? what makes a natural description unique and non-trivial?)

  229. fred Says:

    Max #226
    Thanks for the article link!
    Btw, Scott’s review prompted me to get your book! (kindle edition)

  230. Overwhelmed Says:

    “Okay. Let me be state my intentions – I want to understand what are the allowable “descriptions” in the MUH. I also want to know what it means, in this theory, to “describe something mathematically”. I’d further like to know why this is saying something *interesting*; i.e. I’d like to know specific examples of things I can *not* describe “mathematically”.”

    Is that all? Somebody should write a book on this. ;)

  231. Jason Gross Says:

    Greg #224 – As I understand it, there are no functions from a non-empty set to the empty set, measure-preserving or otherwise. This would imply that there are no morphisms from the integers to the empty set, because all equivalence classes of such functions are empty. Have I misunderstood something?

  232. Jason Gross Says:

    Wolfgang #227 – I am not arguing that there is no hard problem of qualia. I am saying that no one has yet given me a means of pinpointing the ‘qualia of color’ (or even ‘qualia in general’) nor the problem. As an example of what I’m looking for, Yudkowski’s Where Physics Meets Experience and the subsequent Where Experience Confuses Physicists crystalized a quantum measurement problem for me, and gives me something to point to as something like “the hard problem of the qualia of experience”—it gives me something concrete that I believe (that probability of outcome is proportional to the norm squared of the wave-function), and describes a way in which my best understanding of the world fails to account for that fact (why should “probability of which branch of the wave-function I find myself in” be part of a fundamental theory of physics, which is supposed to be independent of me?!). So I accept, tentatively, that “qualia of wave-function-branch” is a hard problem. I’ve yet to be convinced that there is any more to qualia than “which branch of the wave-function am I in”.

  233. Jason Gross Says:

    Max, regarding http://arxiv.org/pdf/0704.0646v2.pdf:

    In all cases, there are many equivalent ways of describing the same structure, and a particular methematical structure can be defined as an equivalence class of descriptions. Thus although any one description involves some degree of arbitrariness (in notation, etc.), there is nothing arbitrary about the mathematical structure itself.

    How do you pick which foundations of math or ambient logic to use? For example, what if two structures are equivalent only if you assume the axiom of choice or the continuum hypothesis, or if some structure is only definable if you assume the law of excluded middle, or if the finiteness of some structure is independent of the axioms of Peano arithmetic (e.g., something based on the hydra game of Goodstein’s theorem).

  234. Sandro Says:

    @wolfgang #199:

    1. You’re changing the goalposts. The original claim was about the parsimony of QM, not QFT.

    2. Even so, there are many relativistic extensions of dBB (and a few field theory formulations). Most of them include an unobservable preferred foliation, although no particular theory has gathered enough support to be considered canonical.

    3. There are dBB extensions that do not require preferred reference frames:
    http://arxiv.org/abs/1002.3226

    And formulations in which the unobservable reference frame is derived from the wave function, thus not requiring any addition axioms:
    http://arxiv.org/abs/1307.1714

    4. What this shows is that John Bell was correct: the real problem of QM that needs serious investigation is non-locality. Waving it away by denying realism is just a cheat.

  235. Philip Thrift Says:

    From a programmer’s perspective, it seems that in considering what constructively could be in a MUH one could find some connection to infinitary programming languages, e.g. λ_ZFC: faculty.cs.byu.edu/~jay/static/toronto-2012flops.pdf, which “contains infinite sets as values, in which to express exact mathematics and gradually change infinite calculations to computable ones.”

  236. Mike Says:

    “What this shows is that John Bell was correct: the real problem of QM that needs serious investigation is non-locality. Waving it away by denying realism is just a cheat.”

    Agreed, but perhaps not such a problem. :)

    http://arxiv.org/abs/1109.6223

    http://arxiv.org/abs/0909.2673

    http://arxiv.org/abs/quant-ph/9906007

  237. Greg Kuperberg Says:

    Jason – As I understand it, there are no functions from a non-empty set to the empty set, measure-preserving or otherwise.

    You’re right, so the definition is a little different from what I said. A morphism between two measurable sets is a partial function, defined everywhere except on a set of measure 0 (and which is also a measurable function). Such a partial function is equivalent to what you get if you erase its values on a measure 0 subset.

  238. T H Ray Says:

    David Brown # 76 says

    “’… anything that can be described at all can be described mathematically.’ This is an interesting conjecture. What is the mathematical description of Thomas Hardy’s novel ‘Jude the Obscure’ or Shakespeare’s ‘King Lear’?”

    It’s an excellent question, and I answered it in a technical end note to an essay that responds directly to Scott Aaronson’s question of computing “using the resources of the universe” under the title “Well Ordered, Totally Ordered, Partially Ordered or Random?”:

    “Mathematicians have many ways to speak of order. We have tried to make clear in these pages that we have taken our terms from mathematical set theory and applied them to the theory of computation.

    “When we speak of computing ‘using the resources of the universe,’ we have assumd a parameter outside of classical computability using the rules of arithmetic — numbers aren’t physical, though counting is. I.e., physical quanta are countable by definition.

    “‘Countable’ doesn’t necessarily imply well ordered, though — cardinality of sets {a,b,c …} might be represented by labels that are well ordered (1,2,3 …) independent of the orderedness or randomness of the set. For example, when we speak of the set of all books, the information contained in Tolstoy’s *War and Peace* differs from Vonnegut’s *Slaughterhouse Five*, yet both have the cardinality of the continuum (uncountably infinite) — even though the set N of all books is countably finite. As strange as it may seem to speak of an finite set of infinite things, the case is true: a comprehensible explanation may be found in Hermann Weyl’s 1918 classic, *The Continuum: a critical examination of the foundation of analysis.*

    “In our essay, we have proposed that global information is totally ordered and finite, while our ways of computing are partially ordered and infinite. Gregory Chaitin’s application of his Algorithmic Information Theory would seem to bear this out in an experimental way; Chaitin’s Omega number is produced from an algorithm that outputs different and random results depending on which computer language is ‘speaking.’ Metaphorically, one program would output *War and Peace*; the other, *Slaughterhouse Five*, from the same algorithm.

    “So it goes.”

    If Max Tegmark is correct, it’s more than a metaphor; it’s a mathematical identity. I’ve always been a big fan of Tegmark’s premise, and I am reading the book now.

  239. T H Ray Says:

    Addendum: I should have made clear the context:

    “Scott Aaronson asks a seminally important question – “Can NP-complete problems be solved in polynomial time using the resources of the physical universe?”6 – arguing that NP-complete problems themselves possibly constrain physical theories. We find, however, that infinite self similarity in any finite time interval promises self organized, and therefore self limiting, maps of short (local) intervals to long (global) intervals. (Cf. Perelman’s proof strategy for Thurston’s geometrization conjecture.7)”

    (The references are to Aaronson’s paper, “NP Complete Problems and Physical Reality,” and to Anderson, M.T. “Geometrization of 3-Maniolds via the Ricci Flow.” Notices of the AMS (vol. 51. No. 2, February 2004)

  240. Shmi Nux Says:

    Scott,

    Re #108d:

    > I do reject the arguments of Everett, Deutsch, Wallace, Zurek, and others that the Born rule can be “derived” from unitary QM.

    What about your friend Sean Carroll? (Admittedly, he does not claim deriving the Born rule.) In his recent blog post The Many Worlds of Quantum Mechanics he unapologetically supports MWI. Here is slide 34 from his presentation. Note the line it’s tested every time we observe interference, so, presumably, deriving the Born rule is not even necessary.

  241. Mark Schnitzius Says:

    I can’t wait to read this book — MUH has been a pet “gedanken-hobby” of mine ever since I independently (no, really!) came up with it a decade or so back. I was gobsmacked when I first came across Tegmark’s formulation of it, and while I even consider it a likely explanation of the cosmos, I largely agree with Scott’s objections here in regards to evidence. So anyway, I have been pondering the implications of MUH for a long while, and attempting to poke holes in it. Here are two holes that I can’t seem to get past — perhaps they are addressed in the book, or have obvious flaws, but I will mention them here anyway.

    1. A single universe in a Level 4 Multiverse consists not just of a set of laws, but also an initial state. In fact, in MUH, every possible combination of initial state state and set of laws would define a different universe. So there would be universes with our own set of physical laws, but with all manner of initial states, of ALL sizes/complexities (including ones which look very much like “last Thursday”, say with a minor tweak). The odds of us finding ourselves in universe with a highly complex initial state would have to be considerably higher than if our beginning were simple. So why is our initial state so simple (or why does it seem to be)? (This is exactly analogous to the low entropy problem in standard cosmology, but just because MUH has the same flaw doesn’t mean it’s dismissable.)

    2. Where are the really oddball laws in our universe? A personal preference for simple and beautiful laws is fine, but consider ugly, discontinuous functions (such as F= [ma if m=41.3] where 41.3 is just some number I picked out of the air). These functions are just as valid in the mathematical definition of a universe, and would seem to be at least as numerous (again with the measure problem) as the simpler functions. It wouldn’t be hard to devise some that wouldn’t threaten the existence of life, I would say, so we can’t fall back on the anthropic principle. So why haven’t we found any such functions? Why are all our laws so seemingly simple?

  242. Scott Says:

    Shmi Nux #240: I’m not exactly the world’s biggest MWI opponent, nor is Sean its biggest cheerleader (“I’m happy to bring up the outstanding issues with the approach, but I do want people to know it should be taken seriously”). But yes, he and I do have somewhat different perspectives about MWI. As for Sean’s derivation of the Born rule, I heard him give a talk about it at MIT and found it quite nice. I’d place it in the same class as Deutsch’s, Zurek’s, etc. arguments: it’s another way to see why the Born rule is “the only rule that’s mathematically sane.” No, it doesn’t get you all the way from unitary QM to the experiences of observers without an extra “anthropic”/metaphysical assumption to help it along, but as long as you’re clear about what your extra assumption is (and Sean is always clear), more such arguments can only lead to more insight.

  243. Shmi Nux Says:

    Scott #242: Thanks, I must have misunderstood Sean’s position.

  244. Greg Kuperberg Says:

    Scott – I do reject the arguments of Everett, Deutsch, Wallace, Zurek, and others that the Born rule can be “derived” from unitary QM.

    As I see it, the full Born rule can be derived from “unitary” QM plus a far weaker version of the Born rule.

    First of all, even in “unitary” QM, you do have a specific version of decoherence from locality. If Alice and Bob have a joint state, then the state of Alice alone is for all quantum purposes described by a density matrix. In conjunction with other reasonable physics, you do expect to see decoherence, thermalization, approximate Maxwell-Gibbs distributions, and ultimately physical objects which have classical behavior expressed in density matrices.

    The only catch is that if you are a strict unitarian, you would say: “Sure, this is called a density matrix, but if that is meant as a reference to a probability distribution, that’s misleading. We only know it as a partial-trace entity; the density aspect is purely formal.” Still, let’s take the interpretation that if almost all of a wave function or density matrix is concentrated at a certain condition, then we can, qualitatively, call that condition approximately true. After all, if almost all of the wave function of an electron is near an atomic nucleus, then even without any quantitative Born rule, you would informally say that the electron is near the nucleus. You would have to be a truly reactionary unitarian to say, “We can’t say anything about the location of the electron if even some infinitesimal bit of its wave function is far away from the nucleus.” Actually, little bits of wave functions are assigned to all sorts of wild possibilities for any physical object.

    So, if Alice is a quantum object who is a human being or an intelligent computer who has witnessed many quantum experiments, then her own density matrix is highly concentrated at a perception that the Born rule is true and that wave functions “collapse”. In this sense, the Born rule is almost a corollary of “unitary” QM and really plays the same role as Bayesian “collapse” in classical probability.

    Not to mention that much of the calculus of density matrices is mathematically identical to classical probability, even if those density matrices happen to come from partial trace and physical locality rather than from the Born rule. It is dubious to reject that coincidence as philosophically meaningless. (Again, even if an external philosopher rejects the point, an “Alice object” still perceives this mathematical agreement to make complete sense.)

  245. wolfgang Says:

    @Greg #244

    The universal wavefunction of mwi follows objectively a deterministic evolution. Therefore probabilities have to be subjective – and this is indeed what you use in your derivation: Alice does not know the state of the environment.

    But once you go the subjective route, why not use Copenhagen, which basically uses the fact that you know the state of your mind (by definition) – therefore you reduce the wavefunction every time it affects your state of mind (i.e. during a measurement) ?

    The latter seems much simpler to me.

  246. Greg Kuperberg Says:

    wolfgang – Alice may not know the environment, but she does know something about it: She knows that 99.9999% of the wave function of the entire universe is concentrated in an endorsement of subjective Copenhagen. The point is that she may have tested subjective Copenhagen with millions of repeated trials; it is then endorsed by the law of large numbers.

    Thus a form of the Copenhagen interpretation follows as a corollary from these considerations:

    (1) Physical locality and mathematical validity of density matrices.
    (2) De facto decoherence via massive entanglement between an object and its environment.
    (3) Concentration: If a wave function is very highly concentrated at a conclusion, the conclusion is approximately correct.
    (4) Subjective experience of an intelligent physical object.

    So, if you are anti-Copenhagen, which of these considerations should be deprecated? (1) and (2) follow from the other laws of physics. (3) is what allows us to discuss the locality of particles, atoms, molecules, etc. (4) is the basis of scientific experiment. I don’t see any basis to deprecate any of them.

    When you say “But once you go the subjective route, why not use Copenhagen”, I sort-of agree and sort-of don’t understand. On the one hand, what does it mean to “go the subjective route”? Does it mean anything other than to “conduct scientific experiments”? On the other hand, once you do “go the subject route”, whatever that means, it’s not just “why not use Copenhagen”, it’s that you have no choice but to use exactly Copenhagen.

  247. fred Says:

    #246
    “you know the state of your mind”?
    Jeez, that’s really sweeping under the rug tons of hard questions about the notions of identity, time, self-references – how does one probe/measure his own state? Is the probing is part of the state?
    It’s a recursive process going on at several levels simultaneously – can each level be considered a seperate mind/observer?
    Is there a computation analogy? Turing machine capturing its current own state as data?

  248. wolfgang Says:

    @Greg #246

    I don’t think we disagree (but perhaps my English is not good enough, I am not a native speaker).

    >> what does it mean to “go the subjective route”
    I mean the assumption that probabilities are subjective – as opposed to interpretations where they are objective (e.g. deBroglie-Bohm or ‘objective collapse’ a la GRW).

    @fred #247

    >> how does one probe/measure his own state?

    This is not necessary. I do not have do measure what I experience; I am always aware of what I experience 8-)

  249. MH Says:

    @Scott S #86

    ‘… were all developed decades or even centuries before anyone thought of any applications to physics, but then turned out to be exactly what physicists needed.’

    The fact that existing mathematical theories are usefull for physics does not mean that they are ‘exactly’ what physicists need. They may contain a lot of fat and there may exists simpler theories that perform exactly the same service. The fact that these theories are seldom stripped of their fat (related to a specific application) has to do with the fact that you don’t get the Nobel or Fields prize for simplifying things, only for finding new things.

  250. Greg Kuperberg Says:

    wolfgang – “I mean the assumption that probabilities are subjective”

    But my argument doesn’t assume that probabilities are perceived. It only makes the much weaker assumption that high concentration of quantum state is perceived. It follows as a corollary that the exact Born rule is perceived as well.

  251. Greg Kuperberg Says:

    wolfgang – Sorry, I think I misunderstood your latest remark. Bohmian “mechanics” is a separate story with its own problems. I am critiquing “unitary only QM”, which takes the position that the Born rule can somehow be ignored or rejected as not even subjective, or otherwise somehow peripheral or optional. This is a very confusing position because, as I want to argue, you can practically derive subjective Copenhagen from unitary QM.

  252. Greg Kuperberg Says:

    Max #225:

    I’m using the mathematical structure definition given in the book and in Appendix A in http://arxiv.org/pdf/0704.0646v2.pdf

    Okay, I looked at the definition in Appendix A. I’m not sure that this issue of what counts as an isomorphic is so easily settled.

    If you want to be rigorous about all of this, then I think you appeal to a model of isomorphism in a subcategory of the category of sets. I.e., in the category of sets, an isomorphism is a bijection; if you decorate the sets, then some of the bijections may survive as isomorphisms and others won’t.

    The problem is that it is not at all clear that a subcategory of Set is a very good model for our universe. For instance, quantum probability is closely related to measure theory, and measure theory is the basis for the “counterexample” that I gave. Again, in the category of measurable subsets of \R^n, the integers and the empty set are in fact isomorphic. In general, a lot of categories are most naturally constructed as subquotient categories of Set rather than as subcategories, or even by more general constructions than subquotients. For instance, the homotopy category of topological spaces is naturally a quotient of Top, which is a subcategory of Set.

    Now, yes, there is an elementary result in category theory that every category embeds as a category of Set. But not very naturally, and not uniquely either. If you emphasize mathematical isomorphism, and if you also emphasize (in your Appendix A) morphisms in the category Set, then presumably the specific implementation of the witnessed universe in Set should matter!

    For instance, if you do implement some other category as a subcategory of Set, a typical construction would be to turn “the integers” into some totally different object, maybe the empty set, maybe a one-element set, maybe an uncountable set. It may not count for very much that the strict empty set is not isomorphic to the strict integers, if in our universe the perceived “empty set” and the perceived “integers” are implemented by some totally different sets after all.

    But also, if you want to predict that other universes exist, then this could be even more speculative than you may have expected, if our own universe’s implementation as an object in Set (or a subcategory of Set) is so utterly negotiable.

  253. fred Says:

    Wolfgang #249
    “This is not necessary. I do not have do measure what I experience; I am always aware of what I experience ”

    Is an atom aware of what it experiences? How about two atoms? Three thousand? All the atoms in your brain?
    “state of a mind”, “awareness of experience”… What are the definitions of all those vague concepts?
    It all starts with the patterns of the data streams running along the nerves connecting the brain to the sensory organs.
    But going from there to drawing conclusions about consciousness and QM is quite a jump.
    For example, what if awareness was the wave-function of the brain? Someone then is gonna say something about the brain being highly decoherent… Sure, so is a stone. But a stone isn’t an evolving structure highly isomorphic/correlated to the past, present, and future of itself and its environment.

  254. wolfgang Says:

    @fred #253

    Perhaps you want to describe a stone using a superposition of quantum states.
    Perhaps you want to describe me as being in such a superposition.
    However, I certainly cannot describe myself as being in a superposition in any reasonable way.
    So this is where the Copenhagen reduction necessarily happens.

  255. fred Says:

    wolfgang #255

    Well, I’m certainly feeling like my mind is nothing but an endless series of superpositions!
    By evolution the main task of the brain is to simulate its environment and itself, so at a very basic level the mind is a series of feedback and feed-foward loops, mixing past states and predicted states.
    More primitive lower level systems also constantly feed signals to higher levels (sub-conscious).
    It’s also very probable that at every given level multiple “consciousness” work/compete at the same time.
    Whether any of those superpositions “are” or “feel” quantum or classical, I don’t know, and I doubt anybody does.

  256. fred Says:

    wolfgang #255

    The following suggestion is a stretch, but when considering extreme superpositions like “me” being alive and dead, who’s to say that the brain isn’t perceiving this to some degree?
    You can certainly find plenty of people claiming they’ve experience odd sense of dread/imminent disaster or connections to distant event (like death of a relative).
    Maybe, with consciousness, nature has found a practical way to leverage various alternative the MW tree to increase chances of survival… isn’t that what we’re hoping to achieve with QC anyway?

  257. Greg Kuperberg Says:

    wolfgang – However, I certainly cannot describe myself as being in a superposition in any reasonable way.

    But that’s because your sense of higher perception is in the classical limit. As far as we know (and actually Max has written a paper about exactly this topic), virtually all higher aspects of multicellular biology are non-quantum.

    Certainly we all accept that many lower-level aspects of ourselves are in a superposition. We are made of atoms and molecules, and the electronic orbits are in superposition. If they weren’t, there would be no Fermi pressure to make approximately incompressible solids and liquids.

    If biology had evolved quantum computation and not just classical computation, then we would have a much better sense of existing in quantum superposition. Actually, our sense of perception does have one near miss. The photosensitivity of a retina cell is about 10 photons. If it were easy to see one photon, then we would have a clear perception that photons exist in quantum superposition, even if we wouldn’t sense ourselves in quantum superposition. Quantum mechanics would probably have been discovered much sooner.

  258. Sandro Says:

    The photosensitivity of a retina cell is about 10 photons. If it were easy to see one photon, then we would have a clear perception that photons exist in quantum superposition, even if we wouldn’t sense ourselves in quantum superposition.

    There are proposed experiments designed to entangle humans in precisely this way.

  259. fred Says:

    Maybe we should add a “noise” knob to our brains.
    http://spectrum.ieee.org/tech-talk/computing/hardware/quantum-computing-adds-control-knob-for-dwave-machine?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+IeeeSpectrum+%28IEEE+Spectrum%29

  260. quax Says:

    Greg Kuperberg#49, I am a bit late to catch up on this thread but must admit this blog never ceases to amaze. Find myself for the first time in full agreement with you on an issue.

    To me it seems obvious that quantum probability as a generalization of the classic case is precisely what’s needed to undo the Gordian Knot of the QM measurement problem (despite parsimony apparently aligned against it).

    And kudos for comment #216 – what an awesome comeback. Made me laugh out loud, and that really doesn’t happen all that often.

  261. Greg Kuperberg Says:

    quax – Well, thanks for the compliment!

  262. fred Says:

    The big boys investing in neocortex simulation companies:
    http://blogs.wsj.com/digits/2014/03/21/zuckerberg-musk-invest-in-artificial-intelligence-company-vicarious/

  263. David Wheeler Says:

    I’m just a guy, no special qualifications, a “true” layman, if you will. Nevertheless, this is a speculative issue I find fascinating. I think almost everyone agrees: the universe “has structure”, and where we disagree is whether or not the universe “is structure”. To be fair, these two statements are using “structure” in two distinct ways, the latter referring to a logical (=mathematical) structure, which is a much more narrow use of the word.

    I think secretly (perhaps unknowingly?) many physicists “hope” something like the MUH is true, that the rules of the universe DO exist, and that they CAN BE discovered, and we will (at the very least) understand all PHYSICAL phenomenon. I think a limited form of the MUH is eminently scientific, and could have one possible consequence of, at some future date, causing most fields of knowledge to be subsumed under its study (provided, of course, our species survives that long, which is perhaps doubtful).

    That said, although I find personally the MUH provocative, I do not feel it is “the truth”. As my friend Steve Kangas used to say (quoting Korzybski), “the map is not the territory”. I think if the universe DOES possess a structure, it is rather like a Riemann surface, and what we model it as is like a projection of that surface. I feel we will always be in the situation of the blind men and the elephant: competing maximal theories of everything will continue to persist. I firmly believe “accounting for everything” cannot be done CONSISTENTLY.

    This is not because I believe the universe (or multiverse, or whatever) is itself inconsistent, but rather reflects our limitations as consistent constructs within it. Put another way: I don’t think we have “quantum brains”.

    Here is why I think the MUH is a valid scientific hypothesis: if it is true, we ought to be able to develop “enough mathematics” to model enough of what is going on to create a virtual universe good enough to fool anyone experiencing it into believing it is real (we only need to maintain faithfulness down to the Planck scale, and to the observable limits of the universe). It seems likely we might actually be able to test this within a few centuries or less. And if we CAN’T, it serves as a kind of falsification for the MUH, that there is something missing, some magic the “algebra doesn’t capture”.

  264. The Mathematical Universe Hypothesis | Logic Matters Says:

    […] “Our external physical reality is a mathematical structure.” That’s how Wikipedia sums up the cosmologist Max Tegmark’s mathematical universe hypothesis. Looks as if some conceptual untangling is needed. Scott Aaronson makes a great start in a wonderful blogpost here. […]

  265. asffsa Says:

    That the Universe is completely described by mathematics is indeed an old idea, however Pythagoras and Galileo did not provide enough arguments why it is so! It is more like a postulate in their philosophy. It is easy to say that the Universe is mathematical but we need epistemological and ontological basis for such claims. I saw in your website the link to ontic structural realism (OSR) so I guess you are familiar with it. OSR indeed provides good arguments about the underlying invariant structure of all our theories.

    I think Immanuel Kant is the first philosopher to provide the strong arguments why the Universe is described by mathematics. If you are familiar with history of philosophy, Kant reacted to the famous debate between Rationalists (Leibniz, Descartes, Spinoza) and Empiricists (Locke and especially David Hume). Rationalists claim that the source of knowledge is reason and innate ideas, while empiricist claim that the source of knowledge is experience through the senses. Both are right from their perspective. Kant said that to speak about innate ideas in our mind which ground mathematics, metaphysics (a priori knowledge) as rationalists did is lazy business. David Hume has shown that everything comes from experience but he had problems with establishing mathematics on firm ground because maths speak of experience a priori. He could not explain how mathematics is possible! Kant tried to defend this a priori knowledge (mathematics, theoretical physics) and so-called synthetic a priori judgments. That’s why I have used Kant to model our cognitive framework (and the Universe as it appears to us) as a quantum computer defined on a grid of cells. I claim that this grid is invariant structure within which all our thoughts, knowledge and theories originate. The structure OSR seeks.

    Kant had influenced such mathematicians as Henri Poincaré and David Hilbert. In philosophy of mathematics Kant belongs to intuitionist school. It is also interesting to study the logicist school, that is Frege, Russell. I know that you are involved with FQXi. I claim that we will not understand ultimate reality unless we view everything as a system of mathematics, theoretical physics, philosophy of science and cognitive science. Cognitive science is of absolute importance in understanding ultimate reality because all our thoughts about the world originate in our brain. I know that you come from strictly scientific background but philosophy of science, philosophy of mind cannot be left out if you want to understand the ultimate reality.
    It seems that you have buried the philosophy of corporeal nature. This is the true purpose of proper metaphysics of corporeal nature – to assist mathematics and physics. They should go together. It does not matter that people did not know about the Higgs boson or the mathematical description of general relativity 200 years ago. What Kant and Hegel knew is fundamentals – how our knowledge about the world in general is possible. If you know the roots of your knowledge, the epistemological basis of mathematics and physics, everything else is just details. To understand ultimate reality we must understand how we understand things in the first place! That is, we must have the picture of our cognitive faculties in general. This yields the big picture of the Universe how it appears to us.
    That’s why I took Kant who asked and provided answers in his work to the questions: ”how is mathematics possible?”. ”How is physics possible?”. ”How is metaphysics as science possible?”.

    https://www.academia.edu/7347240/Our_Cognitive_Framework_as_Quantum_Computer_Leibnizs_Theory_of_Monads_under_Kants_Epistemology_and_Hegelian_Dialectic

  266. 31428571J Says:

    I think the author here is slightly frustrated that MWI (and Max Tegmark’s MUH) answers almost ‘all the questions going’ but that it is also entirely theoretical.

    So what?

    Theoreticians (by definition) build upon experiment and observation and push us to the next logical stage.

    Since the macroscopic is fundamentally microscopic (and many of our physical quantities are dimensionless, including Pi:-), Platonic ‘reality’ is here to stay I think.

    (I doubt if you think much of Julian Barbour’s ‘Platonia’ either I suppose then:-)

  267. Raoul Ohio Says:

    I assume that by now you have all heard that the BICEP2 results turned out to be a mistake. Forgot to account for dust in the galaxy. Bummer; I hate it when that happens. Anyway, don’t forget to readjust your confidence in IC to the former level.

    Because it was about 3 months from the press conference until the refereed version appeared (basically claiming nothing), I propose a new unit. The BICEP2 (= 91 days) is used for measuring the lifetime of supposed proofs of far fetched theories, such as MUH, IC, MWI, String Theory, etc.

  268. Scott Says:

    Raoul #267: My understanding is simply that the situation is back up in the air (in the galactic dust?) right now. The dust won’t settle (man, I crack myself up) until Planck releases its results, probably in the fall. Or was there some new announcement within the last week or so that clarified things further?

    I suppose I can feel vindicated in my reticence about “celebrating” this result. But really, I’d just like to know what the truth is.

  269. Raoul Ohio Says:

    Here is a (6 week old) update from “Nature”:

    http://www.nature.com/news/gravitational-wave-discovery-faces-scrutiny-1.15248

    Here is my grip about the entire IC situation:

    1. It is not obvious if any test can confirm IC as opposed to theory X.

    2. There is no reason to think QM, GR, or any other theory holds “at or near” a singularity. They might, and they might not. The only reasonable answer for anything at a singularity is “don’t know”. If you favor IC, you need to acknowledge that it is your favorite guess, not “what happened”.

    3. The BICEP2 group gets worldwide attention from a press conference announcing proof of IC, and it turns out that a key piece of data is from a scanned photograph of a slide presented at a conference.

    Is this a joke? It is like a CS paper announcing the discovery that in a ten year computational experiment, we have shown that numbers bigger than 2147483647 never turn up.

    4. Four decades ago, physicists were having a lot of fun analyzing tachyons (remember them?). But no one thought they actually were likely to exist. Everyone was in on the joke. What has happened to common sense and standards in Physics?

  270. Nick Greaves Says:

    My interest is in the mechanism behind memory, since the fact that there is no viable explanation for this crucial quality is a woeful lapse in our understanding. Without it we cannot even start to have any notion of mathematics, since in my memory is about 80% of the mind’s operation. To my surprise I found that my hypothesis, which relies on a self ordering principle resulting from a resonance through time effect, has implications for cosmology, in that it appears to make possible a redefinition of inertia and gravitation which qualifies Mach’s Principle.

    In order to qualify and describe this link I have to had to study as much of cosmology as is possible to a non mathematician, and this is not an easy task. I know two physicists, both eminent enough but retired, and to whom I have sent a copy of my paper on Gravitation, Mach’s Principle and matter distribution. One of them recommended last week that I read Max Tegmark’s book, which I have now done and have amended a few minor points in my paper as a result.

    Max’s book to someone like me is a god send of clarity, especially the first six chapters: wonderful stuff not that I believe for one moment in the multiverse, but his book gives a fascinating account of how such clever people’s minds’ work and can justify something that can never be tested: absolutely held me in thrall. It will take a great deal to persuade me away from my faith in the axiom of William of Occam (Merton College Oxford, c 1320) and that when there is enough known about surrounding circumstances of any phenomena, it will appear simple enough to be self evident, but meantime the explosions of highly complex intellectual activity incurred on the way to that end are terribly absorbing and of course in a very good cause.

    I have to say I came across this website by chance and am amazed by the prolixity and rapidity of the comment: very good.

  271. Simon Says:

    > For one thing, this “solution” seems merely to push the riddle somewhere else: one now wants to know, why is the fire of existence not only breathed, but breathed so promiscuously, onto every set of equations that anyone could write down?

    I think about it in a sort of informational complexity way. If the choice is simply for every set to exist or not, that’s a single bit of information. In order to describe some subset that exists, you need a “set of equations” (or some mathematical object) to describe which ones – which introduces another level. The only way to reach a grounding is if the final level is existence for all objects describable at that level (or the trivial case of nothing existing at all, which we assume with some good reason not to be the case!).

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