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:

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. 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.