Max Tegmark has a new book out, entitled Our Mathematical Universe, which is getting a lot of attention. I’ve written a review of the book for the Wall Street Journal, which is now available (although now behind a paywall, if not a subscriber, you can try here). There’s also an old blog posting here about the same ideas.

Tegmark’s career is a rather unusual story, mixing reputable science with an increasingly strong taste for grandiose nonsense. In this book he indulges his inner crank, describing in detail an uttery empty vision of the “ultimate nature of reality.” What’s perhaps most remarkable about the book is the respectful reception it seems to be getting, see reviews here, here, here and here. The Financial Times review credits Tegmark as the “academic celebrity” behind the turn of physics to the multiverse:

As recently as the 1990s, most scientists regarded the idea of multiple universes as wild speculation too far out on the fringe to be worth serious discussion. Indeed, in 1998, Max Tegmark, then an up-and-coming young cosmologist at Princeton, received an email from a senior colleague warning him off multiverse research: “Your crackpot papers are not helping you,” it said.

Needless to say, Tegmark persisted in exploring the multiverse as a window on “the ultimate nature of reality”, while making sure also to work on subjects in mainstream cosmology as camouflage for his real enthusiasm. Today multiple universes are scientifically respectable, thanks to the work of Tegmark as much as anyone. Now a physics professor at Massachusetts Institute of Technology, he presents his multiverse work to the public in

Our Mathematical Universe.

The New Scientist is the comparative voice of reason, with the review there noting that “there does seem to be something a little questionable with this vast multiplication of multiverses”.

The book explains Tegmark’s categorization of multiverse scenarios in terms of “Level”, with Level I just lots of unobservable extensions of what we see, with the same physics, an uncontroversial notion. Level III is the “many-worlds” interpretation of quantum mechanics, which again sticks to our known laws of physics. Level II is where conventional notions of science get left behind, with different physics in other unobservable parts of the universe. This is what has become quite popular the past dozen years, as an excuse for the failure of string theory unification, and it’s what I rant about all too often here.

Tegmark’s innovation is to postulate a new, even more extravagant, “Level IV” multiverse. With the string landscape, you explain any observed physical law as a random solution of the equations of M-theory (whatever they might be…). Tegmark’s idea is to take the same non-explanation explanation, and apply it to explain the equations of M-theory. According to him, all mathematical structures exist, and the equations of M-theory or whatever else governs Level II are just some random mathematical structure, complicated enough to provide something for us to live in. Yes, this really is as spectacularly empty an idea as it seems. Tegmark likes to claim that it has the virtue of no free parameters.

In any multiverse-promoting book, one should look for the part where the author explains what their scenario implies about physics. At Level II, Susskind’s book The Cosmic Landscape could come up with only one bit of information in terms of predictions (the sign of the spatial curvature), and Steve Hsu soon argued that even that one bit isn’t there.

There’s only small part of Tegmark’s book that deals with the testability issue, the end of Chapter 12. His summary of Chapter 12 claims that he has shown:

The Mathematical Universe Hypothesis is in principle testable and falsifiable.

His claim about falsifiability seems to be based on last page of the chapter, about “The Mathematical Regularity Prediction” which is that:

physics research will uncover further mathematical regularities in nature.

This is a prediction not of the Level IV multiverse, but a “prediction” of the idea that our physical laws are based on mathematics. I suppose it’s conceivable that the LHC will discover that at scales above 1 TeV, the only way to understand what we find is not through laws described by mathematics, but, say, by the emotional states of the experimenters. In any case, this isn’t a prediction of Level IV.

On page 354 there is a paragraph explaining not a Level IV prediction, but the possibility of a Level IV prediction. The idea seems to be that if your Level II theory turns out to have the right properties, you might be able to claim that what you see is not just fine-tuned in the parameters of the Level II theory, but also fine-tuned in the space of all mathematical structures. I think an accurate way of characterizing this is that Tegmark is assuming something that has no reason to be true, then invoking something nonsensical (a measure on the space of all mathematical structures). He ends the argument and the paragraph though with:

In other words, while we currently lack direct observational support for the Level IV multiverse, it’s possible that we may get some in the future.

This is pretty much absurd, but in any case, note the standard linguistic trick here: what we’re missing is only “direct” observational support, implying that there’s plenty of “indirect” observational support for the Level IV multiverse.

The interesting question is why anyone would possibly take this seriously. Tegmark first came up with this in 1997, putting on the arXiv this preprint. In this interview, Tegmark explains how three journals rejected the paper, but with John Wheeler’s intervention he managed to get it published in a fourth (Annals of Physics, just before the period it published the (in)famous Bogdanov paper). He also explains that he was careful to do this just after he got a new postdoc (at the IAS), figuring that by the time he had to apply for another job, it would not be in prominent position on his CV.

One answer to the question is Tegmark’s talent as an impresario of physics and devotion to making a splash. Before publishing his first paper, he changed his name from Shapiro to Tegmark (his mother’s name), figuring that there were too many Shapiros in physics for him to get attention with that name, whereas “Tegmark” was much more unusual. In his book he describes his method for posting preprints on the arXiv, before he has finished writing them, with the timing set to get pole position on the day’s listing. Unfortunately there’s very little in the book about his biggest success in this area, getting the Templeton Foundation to give him and Anthony Aguirre nearly $9 million for a “Foundational Questions Institute” (FQXi). Having cash to distribute on this scale has something to do with why Tegmark’s multiverse ideas have gotten so much attention, and why some physicists are respectfully reviewing the book.

A very odd aspect of this whole story is that while Tegmark’s big claim is that Math=Physics, he seems to have little actual interest in mathematics and what it really is as an intellectual subject. There are no mathematicians among those thanked in the acknowledgements, and while “mathematical structures” are invoked in the book as the basis of everything, there’s little to no discussion of the mathematical structures that modern mathematicians find interesting (although the idea of “symmetries” gets a mention). A figure on page 320 gives a graph of mathematical structures which a commenter on mathoverflow calls “truly bizarre” (see here). Perhaps the explanation of all this is somehow Freudian, since Tegmark’s father is the mathematician Harold Shapiro.

The book ends with a plea for scientists to get organized to fight things like

fringe religious groups concerned that questioning their pseudo-scientific claims would erode their power.

and his proposal is that

To teach people what a scientific concept is and how a scientific lifestyle will improve their lives, we need to go about it scientifically: we need new science-advocacy organizations that use all the same scientific marketing and fund-raising tools as the anti-scientific coalition employ. We’ll need to use many of the tools that make scientists cringe, from ads and lobbying to focus groups that identify the most effective sound bites.

There’s an obvious problem here, since Tegmark’s idea of “what a scientific concept is” appears to be rather different than the one I think most scientists have, but he’s going to be the one leading the media campaign. As for the “scientific lifestyle”, this may be unfair, but while I was reading this section of the book my twitter feed was full of pictures from an FQXi-sponsored conference discussing Boltzmann brains and the like on a private resort beach on an island off Puerto Rico. Is that the “scientific lifestyle” Tegmark is referring to? Who really is the fringe group making pseudo-scientific claims here?

Multiverse mania goes way back, with Barrow and Tipler writing *The Anthropic Cosmological Principle* nearly 30 years ago. The string theory landscape has led to an explosion of promotional multiverse books over the past decade, for instance

*Parallel Worlds*, Kaku 2004*The cosmic landscape*, Susskind, 2005*Many worlds in one*, Vilenkin, 2006*The Goldilocks enigma*, Davies, 2006*In search of the Multiverse*, Gribbin, 2009*From eternity to here*, Carroll, 2010*The grand design*, Hawking, 2010*The hidden reality*, Greene, 2011*Edge of the universe*, Halpern, 2012

Watching these come out, I’ve always wondered: where do they go from here? Tegmark is one sort of answer to that. Later this month, Columbia University Press will publish Worlds Without End: The Many Lives of the Multiverse, which at least is written by someone with the proper training for this (a theologian, Mary-Jane Rubenstein).

I’m still though left without an answer to the question of why the scientific community tolerates if not encourages all this. Why does Nature review this kind of thing favorably? Why does this book come with a blurb from Edward Witten? I’m mystified. One ray of hope is philosopher Massimo Pigliucci, whose blog entry about this is Mathematical Universe? I Ain’t Convinced.

For more from Tegmark, see this excerpt at Scientific American, an excerpt at Discover, and this video, this article and interview at Nautilus. There’s also this at Huffington Post, and a Facebook page.

After the Level IV multiverse, it’s hard to see where Tegmark can go next. Maybe the answer is his very new Consciousness as a State of Matter, discussed here. Taking a quick look at it, the math looks quite straightforward, his claims it has something to do with consciousness much less so. Based on my time spent with “Our Mathematical Universe”, I’ll leave this to others to look into…

**Update**: Scott Aaronson has a short comment here.

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In this context it might be a good idea to read (or re-read) Peter Medawar’s review of

The Phenomenon of Man.Dear Max, it is your behavior that is having the chilling effects not only on this blog, but throughout the entire physics community.

It is quite remarkable that while you are the one with millions upon millions of dollars at your disposal, a professional publicity machine, MIT’s PR department, and legions of Ph.D.-free pop-sci-fanboys, you accuse Peter of being a “creationist” bully for merely reading your book and reflecting on its empty content as a lone individual. To pile irony upon irony, it is also remarkable that while your book does little more than promote a faith-based initiative, which is not testable science, you then have the gall to accuse scientists and objective writers of behaving like religious fanatics.

Max Tegmark’s biggest defender Orin writes, “Peter, I’m afraid I haven’t had the time to read the book yet, but I’m familiar with the material I expect to be in it.”

Max, you do realize that your career is built more on laymen who have not read your book, than on scientists who have?

Well Peter, this “Ph.D.-free pop-sci-fanboy” has given your blog a fair shake. Specimens in irony like the above only do you a disservice by lending credence to Max’s comparison. Enjoy your echo chamber.

Orin,

I hope once you do get around to reading the book under discussion, if you find a non-vacuous argument in it for the MUH or the Level IV multiverse, you return to let me know.

Max (or anyone else an expert on inflation)

Is there a lower limit on r from inflation? It seems that there are so many models of inflation (for example this ~ 400 page paper on all models of inflation http://arxiv.org/abs/1303.3787)

that given any observations (or non-observations) one can always construct any model of inflation.

Regardless of whether or not the MUH is true or not, I was wondering about your view regarding the idea of platonism/mathematical realism in general Pete. I know that a vast majority of mathematicians and a good deal of physicists would adopt a view that mathematical structures and truths are independent of human beings and that mathematical propositions are objectively true/false.

I must admit I think this view is very plausible, considering the history of mathematics and the sciences (especially fundamental physics) and the influence both disciplines have had on each other. This is not to try and bring in mysticism at all, as I am a naturalist and a physicalist, thought the second part gets increasingly harder to penetrate as you decompose “matter” into a collection of atoms that are 99.999% empty space with particles in a nucleus that decompose into further elemental particles represented as mathematical points or “vibrating strands of energy” in String Theory, whatever the hell that would even mean physically.

I mean, when modern physics points to fact that solid, physical matter is in fact vast amount of empty space linked together by interactions of profoundly small particles than seem to have an ephemeral existence all their own, is Platonism or realism about abstract structures and mathematical relations underlying the physical world really so outlandish? I agree it could probably never be tested, making it more philosophical than empirical, but do you think it a reasonable view?

By the way Frenkel’s book, Love and Math, is brilliant so far and its clear from the reading that he, along with a long list of other mathematicians, wholeheartedly embraces the Platonic view without the slightest bit of crackpot “mysticism.”

Pete,

I myself favor some form of “Platonism” or realism about mathematics (see a following posting, which has a link to something about the “Putnam-Quine indispensability thesis” and I think Quine had some of the most to the point things to say about how to think about what is “real”). The problem is the claim that this tells you anything you didn’t already know about physics, in particular Tegmark’s claim that it implies that we should think of ourselves as living in a Level IV multiverse, with no particular physical theory=mathematical structure more fundamental than any other. It is this sort of thing that I claim is empty, implying nothing about physics. It’s only use is the ideological one of promoting untestable claims about the “multiverse”.

Dear Mr. Tegmark,

Like many, I feel that dragging creationism into a scientific debate is both surprising and alarming, or to use your word “disturbing.” I don’t think there is any excuse for this, and, frankly, I think it sets a bad example for observers to see how a scientific debate should be handled. In addition, bringing in an irrelevant buzzword to the debate simply highlights a lack of a clear counterpoint to the argument.

Dear Max,

Many thanks for replying and for your kind words. I was unaware of your version of the argument you make, but I did comment on Garriga and Velinken’s version in a footnote in Time Reborn. To quote again from there, p 284,

“23. Jaume Garigga and Alex Vilenkin have pointed out, in “Anthropic Prediction for Lambda and the Q Catastrophe,” arXiv:hep-th/0508005v1 (2005), that a particular combination of the two constants does better when applied to Weinberg’s argument: It happens to be the cosmological constant divided by the fluctuation size cubed. But this leaves two issues: First, what sets the size of the fluctuations? Second, we already knew that the argument did all right when only the cosmological constant was con- sidered. There are many combinations of the two constants that could be tried; the fact that one combination does better than the others is not surprising and, even if there is an argument for it, this does not constitute evidence for the hypothesis that our universe is one world of a vast multiverse.”

I would think that this would also apply to your W which depends also on the dark matter density per photon. Should we be surprised that there is a combination of powers of three constants that is extremized in nature? Given that there are many hypotheses and scenarios and many combinations of constants that could be tried, how strong of a case does this make for the hypothesis that our universe is not unique?

Also, Weinberg’s paper did predate the measurement, but yours and Garriga and Vilenkin’s did not. Even if we accept that your argument is a better version of Weinberg’s, it was made after the observation and, in any case, you agree that Weinberg’s logic was wrong.

Raphael Sorkin also published a paper before the observation of dark energy, predicting correctly the magnitude that was observed. He did this on the basis of causal set theory, an approach to quantum gravity. No one disagrees with his logic or prediction. It seems that the case here is stronger than for Weinberg as the argument did not have to be improved after the observation. So if you are being logical, and responding to the evidence, shouldn’t you be writing a book promoting causal set theory rather than the multiverse? Instead, Sorkin’s correct prediction is almost never mentioned, except by specialists in quantum gravity.

Thanks,

Lee

Thanks for the response Pete. I’m pretty much in total agreement with you as far as that’s concerned. And by the way I’m hoping that Frenkel/Holt debate you mentioned in that recent post continues as well. Always good getting some philosophy of mathematics out there in the open.

Does Tegmark talk about issues like inverted spectra and Mary in her black and white room? I know these are usually presented as problems for physicalism, but it seems to me they also present a problem for the idea that “everything is maths”.

Lee or anyone else,

could you point me to Sorkin’s paper where he predicted the value of cosmological constant using causal set theory.

Many thanks

Shantanu,

R.D. Sorkin, “Spacetime and Causal Sets”, in J.C. D’Olivo, E. Nahmad-Achar, M. Rosenbaum, M.P. Ryan, L.F. Urrutia and F. Zertuche (eds.), Relativity and Gravitation: Classical and Quantum (Proceedings of the SILARG VII Conference, held Cocoyoc, Mexico, December, 1990), pages 150-173 (World Scientific, Singapore, 1991);

R.D. Sorkin, “Forks in the Road, on the Way to Quantum Gravity

”, talk given at the conference entitled “Directions in General Relativity”, held at College Park, Maryland, May, 1993, Int. J. Th. Phys. 36: 2759–2781 (1997), eprint: gr-qc/9706002

Maqbool Ahmed, Scott Dodelson, Patrick B. Greene, Rafael Sorkin, {\it Ever present Lambda}, astro-ph/0209274.

Sorkin was mentioning this in talks over many years.

Lee

Tegmark should address Peter’s most important point, which is that he doesn’t really have any inner understanding of mathematics as a discipline in itself.

-drl

Hi Peter,

I read the book (I thought it was delightful), and I have to say I’m left even more perplexed than before at your characterization of the MUH as “empty”. Max says (Chapter 12):

“If the theory that the Level IV multiverse is correct, then since it has no free parameters whatsoever, all properties of all parallel universes (including the subjective perceptions of self-aware substructures in them) could in principle be derived by an infinitely intelligent mathematician.”

Of course this is fleshed out in the book, but I think it is fairly self-evident. Examples of the kind of strategy for falsifiability begin as early as Chapter 6, where he writes:

“If we’re living in a random habitable universe, the numbers should still look random, but with a probability distribution that favors habitability. By combining predictions about how the numbers vary across the multiverse with the relevant physics of galaxy formation and so on, we can make statistical predictions for what we should actually observe.”

So I continue to not understand why you insist on using the word “empty.” I think Max makes a very persuasive case in the book for the potential of the MUH to be predictive, and even falsifiable (he discusses this extensively in Chapter 11, giving examples of falsification using naive measures, ultimately concluding that the major hurdle to be overcome is a solution to the measure problem).

Orin,

Sure, all properties of all universes could be “derived”, just look at “all properties of all mathematical structures”. Fine, but this predicts nothing at all about any particular property of our particular universe.

The part you quote from the book is referring to the sort of “Level II” multiverse of the anthropic string theory landscape: there’s some specific fundamental physical law that determines the probability distribution he invokes. There are plenty of problems with this, but it’s not what I’m referring to as empty, the Level IV business of the title of his book.

I’ve gone over here carefully what is in his book that claims to be a prediction of Level IV, and I’ve explicitly challenged him here and at the Scientific American site where he wrote a response to this, asking him for a falsifiable prediction of “Level IV”. The best he could come up with is “if we find a physical phenomenon not describable mathematically” that would do it. See

http://www.math.columbia.edu/~woit/wordpress/?p=6669

and the Scientific American site

http://blogs.scientificamerican.com/guest-blog/2014/02/04/are-parallel-universes-unscientific-nonsense-insider-tips-for-criticizing-the-multiverse/

for my comments about why this is empty. If there are all sorts of great examples of how to falsify Level IV in the book that I missed, how come Tegmark is not invoking them when asked about this issue?

You could argue that all you have to do is “put a measure” on the space of mathematical structures, but this is again an empty statement until you give some indication of what such a measure should look like. The only thing he discusses in the book is some sort of counting measure, but obviously you get more and more examples of mathematical structures as you increase complexity, so this sort of thing, besides the “measure problem” due to infinity, has the obvious problem that it predicts your mathematical structure will be as complicated as possible, whereas we know that fundamental physical laws are based on remarkably simple mathematical structures.

On a related note, Tegmark likes to claim that he just has a “measure problem”, not knowing how to relatively count things, when he has something much worse: he doesn’t know how to characterize the space he is trying to put a measure on (the people trying to use string theory at Level II have a much simpler version of this problem: not only do they not know how to compute relative weights of string vacua, they don’t know what the set of string vacua is).

Peter, you seem to be setting the mark for what is not empty just high enough to fit your definition. I think Max has outlined the beginnings of a theory that is clearly not empty in principle. You point out that it is currently empty in practice. That is fine (if you make it clear what you mean), but obviously theories have periods of gestation before predictions are made, and I don’t think it is fair to be quite so dismissive of the fact that they are not yet mature. There is a Chicken/Egg problem here; you won’t be satisfied until the theory is predictive in practice, yet the theory won’t have an honest shot of being predictive in practice until more people work on it.

On your point about the measure problem (second-to-last paragraph) and the apparent simplicity of physical laws, I think your argument is not as strong as you think it is. For one, your measure must be “anthropic aware”, a feature Max harps on quite a bit (although I don’t think he uses that wording), and it is not at all clear to me that the “pruning” afforded by such a measure would not alone be a sufficient counter-argument. But additionally (and this is a point Max does not make in the book, but nonetheless his ideas should be discussed independently of this particular book written for lay-audiences), a strong case can be made that as you go up the latter of increasing complexity, there are correspondingly increasing numbers of approximate equivalences between structures of greater and lesser complexity. There are also fairly strong counter-arguments in algorithmic complexity theory (I linked to one example earlier in this thread) that clearly indicate to me that your intuition is off. So no, I don’t think the measure situation is nearly as dire as you think it is.

Orin,

I’m not talking about “in practice”, but “in principle”. All the arguments I’ve given about the emptiness of “Level IV” are arguments of principle. To show it is non-empty you need to produce a non-empty prediction, even if it is only “in principle”. Tegmark hasn’t been able to do this (and you don’t seem to either). And you can’t hide behind every crackpot’s favorite excuse “OK, I haven’t been able to get anything out of my wonderful theory of everything, but if only lots of people would work on it, maybe they would find something”.

It’s not possible that “my intuition is off” about “all mathematical structures”, because I have no intuition at all about what that even means. As far as I can tell it’s a concept every bit as empty as saying “all sets” or some such. There’s no there there. The counting measure I was quoting was Tegmark’s intuition, not mine. Again, I don’t think his problem is a measure problem, his problem is that he doesn’t know anything non-trivial about the space he wants to put a measure on.

From your comments, you seem to have your own Tegmarkian theory, since you’re making claims not in his book. If you’ve written it down anywhere, and you have non-trivial implications of such a theory that he doesn’t have, let us know what they are. So far, neither you nor he are able to point to anything in his book that gives, in principle, an implication of this MUH that is not on its face empty.

Peter, the quote I provided from Max’s book is exactly such an example, even including the words “in principle.” Apparently you need a specific “in principle” prediction (this seems like an oxymoron; I’m not positive what you mean) rather than a general one. Fine, but the fact that the theory can in principle make specific predictions is the reason I am pestering you about your use of the word “empty.” The theory is not empty. It clearly can in principle make specific predictions, and the practical reasons why it currently cannot are made as plain as day in Max’s book.

Orin,

This has now become a waste of time, you’re just ignoring whatever I write here. I’ll just cut and paste the relevant part.

“Sure, all properties of all universes could be “derived”, just look at “all properties of all mathematical structures”. Fine, but this predicts nothing at all about any particular property of our particular universe.

Peter you are playing dumb. Obviously that is not the only implication of what Max wrote when he discusses the prediction of “the subjective perceptions of self-aware substructures” in a multiverse in which he has already established that in principle there are “probability distributions that favor habitability,” with which one can use to falsify the theory. He is not just saying one can in principle derive the properties of all mathematical structures. He is clearly saying that if a reasonable measure is found then one can in principle derive the probability for us to live in a universe with an effective Standard Model lagrangian and General Relativity with a finely tuned cosmological constant, etc, and that if this probability is found to be small then the theory is falsified.

Orin,

Now we’re back to “all I have to do is find the right measure on the space of all mathematical structures”, and the problem that “all mathematical structures” is an empty concept. From nothing you get nothing. Tegmark gets nowhere with this in his book because it’s inherently empty and can’t go anywhere. If you point to anything other than absurd wishful thinking about explaining everything from nothing, please do so, but so far you’re just wasting your and my time.

Suppose some relatively slightly more advanced alien civilization living in Omega Centauri has already solved, for instance, their metabolic syndrome and counteraffects of ageing, integrated successfully with their quantum computing power, essentially have achieved a form of immortality. There’s probably a Hollywood movie playing this out, or at least there should be.

What gives when an alien from this civilization visits you, Peter (after all, they originate in our own galaxy) and tells you Everettian postulates about decoherence, many worlds, and ultimately MUH are all true? As a matter of fact, the alien even takes the time to demonstrate to you on his quantum hand-held device how he has uploaded his “essence” innumerable times into his planet’s hosted quantum supercomputer, and in fact is himself living out “many lives” … any and all of which can and do seem no more or less “real” to him.

Theoretical emptiness, perhaps.

Impossible or improbable?

billandturk,

If a space alien or a guy with gold tablets appears magically in front of me, and explains to me about how string theory really is the TOE, the multiverse works, the space of all mathematical structures carries a natural measure that explains everything, etc. I am happily going to agree that yes, these ideas are testable science and have been tested.

However, if this possibility is the argument from proponents about why their ideas really are non-empty and scientific, I think they have a problem….

Thank you Peter taking the time replying.

I am young, dumb, and novice with this subject matter and even more “rookie” with the math underpinning it all; however, your response to my post actually helps me better contextualize and appreciate your points.

I like the saying: There are known unknowns – things we know we know; There are known unknowns – things we know we don’t know; and There are unknown unknowns – things we don’t know we don’t know.

After having just finished Tegmark’s book, I was inclined to believe his ideas fell into that middle category. I see where you’re coming from, and it also makes sense to me now, that perhaps (not trying to suggest my words are coming out of your mouth and hopefully not to be taken offensively), these ideas are more the latter, “unknown unknowns”, and the fundamental flaw with this category is presuming you can articulate “science” attributes upon it?

Oh well, genuinely thanks again!