# Our Mathematical Universe

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. Last Updated on This entry was posted in Book Reviews, Multiverse Mania. Bookmark the permalink. ### 125 Responses to Our Mathematical Universe 1. Conrad Albert says: if you search for the WSJ review on Google (eg ‘Our Mathematical Universe’ by Max Tegmark book review Peter Woit), you will be able to read the article for free from the Google search link – at least here in the US and hopefully anywhere else I actually read Max Tegmark arXiv paper mentioned here some years ago as I found it very interesting from a science fictional point of view and as a basis of sf world building like for example in several recent acclaimed sf novels like Anathem by N. Stephenson and the Orthogonal series by Greg Egan (Riemannian universe with metric ds^2 = dx^2 + dy^2 + dz^2 + dt^2 and lots of cool implications including speculations on what intelligent beings would look like and what the arrow of time would mean) Taking it seriously as applying to the universe we are living in is obviously a different issue, but I would not really be worried that much about this happening as the scientific method is still alive and well. 2. DLB says: He seems to confuse reality with representations of reality and forget that mathematics is a language. Though it provides rigorous objective representations and a means of standardization, mathematics remains a language and any language can be used to express lies as much as truths. Saying the reality is mathematics is like saying that language is reality. 3. Simple biologist says: Just a heads up for the ‘you can try here’ link. Worked for me and I’m an European with no subscription. 4. Max Tegmark says: Thanks Peter for taking the time to write the thought-provoking and (I thought) eloquent WSJ review, which I found significantly more nuanced than your post above! It’s not clear to me whether our views actually differ on any matters of physics as opposed to matters of speculation: what we’re guessing might be ultimately turn out to be fruitful for physicists to research. Would you agree with me that parallel universe are not a theory, merely predictions of certain theories, and that some of these theories may themselves be testable by other means? For example, would you agree that inflationary cosmology is a scientific theory that can be further tested by searching for B-modes with CMB experiments, etc., and that if we choose to take inflation seriously, then it’s not unreasonable for scientists to take seriously all of its predictions, even the untestable ones? I feel that our job as scientists is not to tell our cosmos how to be based on our emotional preconceptions (that there must be a multiverse, that a multiverse is impossible, etc.) I therefore dislike being told not to explore certain ideas. For example, I decided to meet with someone from the Templeton Foundation may years ago precisely because a famous professor told me *not* to talk with them. A key goal of the Foundational Questions Institute is to encourage people to challenge prevailing dogmas in physics, like you have yourself courageously done. I think it would be awesome if you’d join our discussions – would be be interested in attending one of our future conferences? /Max 5. Max Tegmark says: Thanks DLB for raising this interesting point about mathematics as a language, which ties in with the age-old debate about whether mathematics is invented or discovered. Personally, I think that it’s crucial not to confuse the *language* of mathematics (which we humans clearly invent) from the *structure* of mathematics, which many mathematicians feel that they discover. For example, any civilization interested in regular 3D polyhedra would discover that there are precisely 5 of them (the tetrahedron, cube, octahedron, dodecahedron and icosahedron). Whereas they’re free to invent whatever names they want for them, they’re *not* free to invent a 6th one – it simply doesn’t exist. The same applies to the mathematical structures that are popular in modern physics, from 3+1-dimensional pseudo-Riemannian manifolds to Hilbert spaces. Once we’ve classified all irreducible representations of the Poincare group, we’re not free to invent new ones! My$0.02,
Max

6. notagain001 says:

is the multiverse landscape of string theory limited to only those w/ 3 large spatial dimensions and 1 time dimension and 6–7 curled dimensions?

is there any reason the landscape could not also include any combination of large, curled and time dimensions that adds up to 11? i.e 6 large spatial dimensions, 2 curled spatial and 2 time dimensions? also why do the curled dimensions have to be spatial? why couldn’t they be curled closed time like dimensions? i.e 3 large spatial dimensions 1 large time dimension 6 closed timelike dimensions

7. Pat says:

This past summer, Tegmark was one of the featured lecturers at TASI and his entire effort would best be described as a sales pitch. He was supposed to have a few lectures on cosmology, but what he actually did was just rip some basic slides and pretty pictures from his public lecture. It’s well understood that the point of lectures at these schools is not to gain a complete understanding of the topics (especially something as fundamental, to high energy theory students at least, as cosmology), but to fill in some gaps and get somewhat comfortable with the cutting edge. Tegmark was more interested in using the grad student audience as a warm-up for his featured public lecture (there are a couple given every year at the school).

As another MIT professor (a respectable QCD guy) was one of the directors of the school, there were probably a half dozen MIT grads present. I don’t think any of them attended his public lecture because they had seen this show before. When I asked why some of them why they were not interested in going, the best response was because a running joke around Cambridge is that Tegmark is more like a “used car salesman” than a physicist when promoting some of these ridiculous ideas. After going to the public lecture, me and every last student who attended could not agree more. The local organizers (who have been running these awesome schools every summer in Boulder for the last decade, at least; note to NSF and DOE, KEEP IT FUNDED) seemed embarrassed as almost every physicist in the room cringed whenever some of the sillier aspects of Our Mathematical Universe came up. What bothered me the most was the way he would try and legitimize some of his ideas by appearing to connect them to basic principles of cosmology, while any expert (or almost any grad student) knew he was blowing smoke. For instance, he would say something like “… the universe is math, it’s all around us. We were just discussing this in Cosmology lecture today …” I really don’t mind that he makes a buck selling this drivel, I just don’t like being used as implicit endorsement of his ideas.

As for the ideas themselves, I think empty is the best way to describe them. While I’m not a huge proponent of the multiverse and I haven’t read the book, I just remembered thinking that he was not really saying anything when he talked about the book. Beyond issues of predictability, postulating that Math=Physics=Everything seems more like a metaphysical statement than anything that even resembles a testable theory. No to digress, but to it’s more like trying to prove that people rode around on dinosaurs by building these creation museums. In issues of faith, scientific evidence is beyond the point, so why even try? The fact that such a prominent physicist can seriously promote science fiction with the encouragement of other prominent physicists amazes me. More serious people in the community can mostly just ignore these claims, but large sums of money seem to be diverted towards such useless pursuits. I admire Tegmark’s contributions to cosmology (SDSS and WMAP efforts in particular), and I’m sure other TASI students do as well, but he will forever be the butt of our jokes when we reminisce about that summer.

8. Max Tegmark says:

Thanks Pat for sharing your views on the TASI school. I take teaching very seriously and therefore welcome criticism of my lectures. I was very happy to hear the many positive comments I got from students afterwards, so I don’t think it’s 100% accurate to suggest that you’re speaking for everyone. Just to clarify: what do you mean by “just rip some basic slides and pretty pictures from his public lecture”, when I in fact spent most of my lectures deriving equations on the blackboard, and the long and fun student discussion sessions we held involved no slides whatsoever? You’re referring to the school of 2013, right?
/Max

9. Pat says:

Max,

On second thought, I may have been too harsh in comments about your Cosmology lectures. To me they just seemed very basic, but half of the students worked on QCD, not BSM physics or cosmology. I don’t envy the task of trying to make cosmology material interesting while still having to start from scratch over only 3 or 4 lectures. And at some level what’s basic material for physics grad students will overlap with basic material in a public lecture. I thought the lectures were fun and I find The Mathematical Universe interesting and, obviously, provocative. My issue is mainly blurring the lines of science, reality and philosophy. Maybe a having such a different perspective is the point, but larger ideas are typically beyond my pay grade as a BSM phenomenology grad student.

10. Anonyrat says:

Isn’t it a matter of human psychology more than anything else that there are only 5 regular 3D polyhedra is a surprise, seems like a discovery, while 2+1=3 does not?

11. I acknowledge that Tegmark’s position is not popular and smells of being crackpot, and also that Tegmark has sharp political skills, but solely on the scientific value of the idea, it has real merit and I want to defend it.

If reality is not mathematics what is reality based on? Now we have two choices: (1) the question is not well defined (2) reality is based on “something”. For option (2) the obvious question is who created that “something”? This may lead us to the idea of “God”, but this is not necessarily true. If the idea of ontology is enlarged (like in “the mathematical universe“), software programs, even cartoon characters may be considered ontological and who is to judge that one ontology is better over another? For a software program, it is not “God” who created the program, but a software engineer, who belongs to another ontological level, and so there could be valid levels of recursions in defining reality. The recursion will either end or not (does God of the gaps ring a bell?). A natural termination criterion for this hierarchy is “reality is ultimately mathematics”.

So “the mathematical universe” idea has at least some handwaving arguments going for it, unless choice (1) is true, or maybe the ontological layers go on forever. So the question is not in the least settled. To convince the skeptics, what is needed is to derive mathematical consequences from it. For example, we know that space time is 4 dimensional, we know that nature is quantum mechanical at core, we know the gauge symmetries of the Standard Model. Can we derive this from “the mathematical universe” idea? In effect this means solving Hilbert’s sixth problem of physics axiomatization. Now this I believe it is possible in principle. But to do it, you need to turn Tegmark’s approach upside down: it is not important how reality and mathematics are similar (you are a mathematical theorem says Tegmark), but how they are different. This generates physical principles which distinguish (select) particular mathematical structures from the infinite world of mathematics. Every mathematical structure is unique, but very few are distinguished by nature. As a side note, this bypasses the Gödel’s incompleteness theorem roadblock because the final selected mathematical structures do not have to be in a closed form of a monster meta-structure. If the physical principles are correctly chosen, we should pick precisely the mathematical structures which are used by nature, nothing more nothing less. And this approach actually started to produce results. In particular I am able to derive quantum mechanics in the c*-algebra formalism! (I had submitted preliminary results for publications to serious journals).

So what about being a mathematical theorem? Am I a mathematical theorem? This is malarkey as Joe Biden would say. Mathematical structures are the ultimately building blocks of reality, but they arrange in such a way to avoid contradictions. I think we will be ultimately able to show that this leads to the concept of time, but this is a separate story. As “ugly bags of mostly water”( https://www.youtube.com/watch?v=paH97dYR6Lg) we are “entropy and information parasites” and our existence is not a mathematical necessity.

In conclusion, I think mathematical structures are the ultimate building blocks of reality, but demanding reality to be a mathematical theorem is too constraining and incorrect. But don’t take my word for it and be skeptical. In a year’s time I hope to pass peer review scrutiny in deriving quantum mechanics and win over the skeptics with valid mathematical proofs.

12. Yatima says:

In a sense. “Surprise” means there is high informational content. The default assumption of a primate brain would probably be “there is an infinite series of regular 3D polyhedra”, something which evidently can be checked by going into the street and ask people (one should do this!). This has less to do with psychology than with the fact that brains are not generators of theorems about the world of 3D geometry. Misjudgements of such an ethereal kind are luckily not deleterious to one’s capacity to procreate.

On the other hand, while 2+1=3 is indeed a theorem it is also a statement about the result of a program so simple that most people manage to implement it in their heads.

13. Laymammal says:

So let us look for a non-mathematical structure out there in reality to falsify this theory. What would that be? An impossible object. Even if it showed up, how would we be able to grasp it? Or we’d end up creating mathematics to accommodate it, which would make it a mathematical structure(like how we handle infinity). Looks like Tegmark’s theory is pretty secure.

14. srp says:

ISTM that the Level I multiverse is the most challenging because it simply extrapolates the consequences of well-known and tested theories to produce a hard-to-believe result. This extrapolation from the known to the unknown (and bizarre) is at the heart of Tegmark’s rhetorical question in the thread above about believing the untestable implications of testable (and non-falsified) theories. (I would like to know if most physicists and cosmologists accept the combinatorial arguments for the Level I multiverse; if so, why this highly “woo-woo” conclusion hasn’t been taken up by the popularizers and if not what is the flaw in the argument.)

But when people are skeptical of the foundation “theory” (e.g. M-theory) his rhetorical extrapolation point is much weaker–it’s easier to say that the bizarre consequences are just another reason to reject the foundational claim. In that case, his only real argument is a radically Platonist twist on Occam’s razor, i.e. that a theory with fewer asymmetries and free parameters is preferable even if it multiplies unobservable entities without measure. I suspect most people, however, find a theory with a small number of unobservable entities, even with many free but observable parameters, to be far superior to one with no free parameters and an unstructuredly infinite number of unobservable entities.

15. vmarko says:

Max (and others),

I think you are misinterpreting math to be a far too unique body of knowledge. Let me explain what I mean:

“I think that it’s crucial not to confuse the *language* of mathematics (which we humans clearly invent) from the *structure* of mathematics, which many mathematicians feel that they discover.”

I’ll be bold to say that those many mathematicians are deluding themselves in thinking so. Any mathematical statement consists of assumptions, conclusions and the rules of inference (the “A”, the “B” and the “=>” in a statement A=>B, respectively). The point is that *all* of them are arbitrary, most notably the rules of inference. The fact that most mathematicians are using ordinary first-order predicate logic as a “commonly accepted” set of inference rules, it is by no means unique or better than other, different logic systems. So saying that people *discover* math structures, based on axioms they *invented* and rules of inference that they also *invented*, is simply misleading. Math is completely a *language*, an *invented* thing, together with axioms and the ways we perform proofs of theorems based on those axioms. The fact that the first order predicate logic is commonly accepted as “the truth” is a consequence of the way our brains evolved, rather than that it is any “truth” per se. It’s just another (useful) language, nothing more. People who are doing research in mathematical logic know this for a few centuries already (so these are not just my wild opinions).

“For example, any civilization interested in regular 3D polyhedra would discover that there are precisely 5 of them (the tetrahedron, cube, octahedron, dodecahedron and icosahedron). Whereas they’re free to invent whatever names they want for them, they’re *not* free to invent a 6th one – it simply doesn’t exist.”

I disagree. If you build a 3D Euclidean vector space over a field of rational numbers as scalars, then you can prove (assuming standard logic) that none of those solids exist. Namely, an equilateral triangle doesn’t exist in such a space, given that its area is an irrational number. Imagine a civilization which didn’t invent the concept of real numbers — they could say that your “discovery of 5 solids” is plain false. This has actually already happened in our own history — Pythagoreans had problems with the diagonal of the square… And if you don’t like rational numbers, you can close them up in one of the p-adic fields, as opposed to reals. And if you don’t like modus ponens rule of inference, just choose some other logical system (oh yes, there indeed are logical systems where modus ponens does not hold).

Math is a language all the way, from “A” over “=>” to “B”. Plato’s world of ideas, that is being “discovered” in math, exists only for people who choose to use the same language for “A” and “=>”. There is no “existence” in math beyond the common language assumptions. Plato of course couldn’t have known this, since “other” logical systems have been shown to be possible only in 19th and 20th century. But today, nobody has an excuse for not studying and understanding mathematical logic…

And don’t even get me started on set theory (actually, theories), axiom of choice, Banach-Tarski paradox, etc. 😉

Best, 🙂
Marko

16. @Marko

I highly recommend this essay by Alain Connes on mathematics: http://www.alainconnes.org/docs/maths.pdf

To this I would only have to add that mathematics is about abstract relationships and as a nice example consider the history of imaginary numbers. People did not accept them (and the very name imaginary echos its history) for more than 200 years until someone discovered a well defined matrix representation. Those abstract relationships (and physical laws) existed well before someone discovered and named them. E=mc^2 was valid during dinosaur time too.

Mathematicians are exploring a timeless landscape of abstract relationships. Which part of this landscape is explored, the order of the exploration, and the name of various part of the landscape are historical accidents.

Set theory is not necessarily the foundation of math. Category theory is a viable option. Godel incompleteness theorem is a serious roadblock but it can be avoided.

17. vmarko says:

Florin,

Maybe you misunderstood my post. I don’t have a problem with irrational numbers. I have a problem with the uniqueness of rules we claim to be valid when we prove theorems in math. Things like rules of inference, laws of excluded middle, concepts of “true” and “false”, etc. The complete body of mathematics resides on these rules, and my point is that those rules are completely arbitrary and nonunique. There are no absolute truths in math, only the ones that pertain to the language you use.

“Mathematicians are exploring a timeless landscape of abstract relationships.”

No, they are *inventing* that landscape (typically unintentionally). If you apply a different logical system to a given set of axioms, you will reach different conclusions. One could also say that mathematicians are “exploring” some landscape using tools that actually construct that landscape. Using a different set of tools would give different landscapes.

I suggest that you take a glance at the Wikipedia article about non-classical logics, there are many of them:

http://en.wikipedia.org/wiki/Non-classical_logic

The whole math ultimately boils down to a definition of a language we accept to use. There are no mathematical “objects”, or even statements, whose existence is independent of the choice of that language, which is itself quite arbitrary.

Best, 🙂
Marko

18. S says:

Leaving aside the (very important and forceful) questions about testability, etc., it’s always seemed to me that Tegmark’s “everything in math” stuff would run into some very strange consequences for our own universe, right now.

If “all mathematical structures” exist, then I guess that means (maybe?) that *every* set is somehow instantiated in reality, *every* topological space is a “spacetime” somewhere (whatever that means), and so on. But the thing is, most mathematical objects aren’t very nice. There are far more discontinuous functions than continuous ones (in a reasonable topology); there are far more non-Hausdorff topologies than Hausdorff ones; and so on.

If every mathematical object exists, then why shouldn’t our universe be one of the ones that stops being nice tomorrow? It’s basically just David Hume’s induction-skeptical argument, of course, but it seems to me that it gets very awkward for physics once you actually assert that, yes, you believe in all these universes, and that for every spacetime that acts in a nice regular way like ours up to today, and then keeps doing so tomorrow, there’s an uncountable infinity of them that acts nice and regular up to today, and then lapses into awful, unmeasurable incoherence tomorrow.

What would be the rationale for believing that *our* spacetime is one of the structures that’s nice globally?

Again, these are questions that *every* physicist and every person has to address (at least in theory), since Hume; but Tegmark, it seems to me, has given a negative answer, without drawing attention to it: we can’t. We’re probably in a universe where the laws of physics stop working tomorrow. End of story.

That seems like a big problem with the theory. But possibly I’m missing something.

19. S says:

“everything in math” should read “everything is math.” *sigh* Sorry.

20. Marko,

You state: “There are no absolute truths in math, only the ones that pertain to the language you use.” I completely agree.

“If you apply a different logical system to a given set of axioms, you will reach different conclusions.” agree again.

“One could also say that mathematicians are “exploring” some landscape using tools that actually construct that landscape. Using a different set of tools would give different landscapes.” this becomes repetitive. I agree 100%

“There are no mathematical “objects”, or even statements, whose existence is independent of the choice of that language, which is itself quite arbitrary.” ditto

So how can I agree on all this end yet hang on the “math landscape”? I think we differ on the semantics of “existence” (As Bill Clinton put it, “It depends upon what the meaning of the word ‘is’ is.”). Let me repeat my prior observation: math is about abstract relationships with the emphasis on abstract. The same abstract mathematical structure can have very different and distinct concrete realization. In math the notion of truth is not definable inside an axiomatic system (Tarski theorem) and there is no “context independent” notion of truth. In nature any two observers agree on an event and the notion of truth is context independent. That is why physics is an experimental science and we settle disputes between competing theories (e.g. classical vs. quantum mechanics) by experimental evidence. In physics true means agreement with nature.

When I say that nature is ultimately made out of mathematical relationships I mean there is nothing unexplained which cannot be described as a mathematical relationship (here I agree with Tegmark). This is a restatement of Wigner’s position: “unreasonable effectiveness of mathematics”. Where I don’t agree is that reality is one giant mathematical theorem/structure. This is because nature and reality is not abstract and in nature truth is context independent. One funny way to state this is: sticks and stones may break my bones, but when was the last time you read in the news that someone was hurt by Pythagoras theorem?

Let me present an analogy (to be taken with a grain of salt like any analogy). Suppose that you want to understand ice and water, you don’t know any chemistry, but a friend of yours who is a chemist tells you that ice and water are made out of the same building blocks, H2O. You are very familiar with ice but never seen liquid water before (ok this is far fetched, but please play along). What would you do? Trusting your friend would you look at the similarities, or at the differences to understand liquid water? If when you have a hammer, everything looks like a nail, you will pick the similarities and pretty soon you will run into great difficulties. If you look at the differences, you can learn something new. In this analogy the water molecules are like the abstract mathematical structures. The ice corresponds to the concrete realizations of the abstract mathematical structures into what you object that are historical accidents: set theory, groups, a specific type of logic, etc. Just like ice, those mathematical structures are immutable or “frozen”. Liquid water corresponds to nature and another way to have a realization of the abstract mathematical structures: arranged in such a way to give rise to a context independent notion of truth. You don’t need to agree with me on this, but from looking at the differences I can derive hamiltonian mechanics (both classical and quantum mechanics) and I am in the process of publishing this. What differences gives you turn out to be very useful physical principles. Here is a key new physical principle which I am using: the laws of nature are invariant to how we partition in our mind a physical system into subsystems. This is trivial but unbelievably powerful because the laws of nature must be invariant to tensor composition and there are only 3 possible ways this can be achieved (one of which is quantum mechanics).

Back to the math landscape. The fact that in flat 2D space the sum of the angles in a triangle is 180 is a “timeless” statement true in its context. It is the very existence of the abstract math landscape which makes proofs possible.

When we talk math which describes nature we are talking about an infinitesimal small amount of mathematical structures which are selected by nature. For example, why nature prefers SO(3,1) for example? Why not SO(p,q) with p and q arbitrarily large? Why is SO(3,1) distinguished from the infinite number of possibilities? The standard way of physics is to look at the experimental evidence and state: with so and so precision, in its domain of validity, this or that theory/mathematical structure describes nature. “Why” is not a scientific question, but a nebulous philosophical blabbering for crackpots or Nobel prize winners gone soft. What I do want to show is that this question “why” is actually very scientific and falsifiable in Popperian sense, and moreover the answer can be given in the most rigorous mathematical way. On key ingredient however is that reality is ultimately made out of mathematical relationships. To the approach of Tegmark, I have to say: show me the money. What can you derive from this hypothesis?

21. Peter Woit says:

Hi Max,
Thanks for writing in here, and graciously dealing with a rather hostile crowd.

I think you’re right that we likely agree on what’s physics and what’s speculation. But we also I think disagree on how to handle different sorts of speculation. In my book I used the Bob Dylan line “But to live outside the law you must be honest” to try and make the point that if you’re going to make yourself immune from being kept honest by experimental data, you need to take great care to not fool yourself.

Sure, inflationary cosmology is a theory (in lots of versions…) that you can extract predictions from. Like any theory, if you figure out how to test distinctive parts of it and it passes those tests, you can have some confidence that other, untestable parts of it correspond to reality. I really don’t have a simplistic view of testability: for instance, I’m happy to agree that a theory can be so beautiful and mathematically compelling that I’d have confidence in it given only minimal experimental evidence. What I have no confidence in is the combination of a very complicated, uncompelling mathematical structure with no experimental evidence (I’m thinking M-theory…).

I also don’t have a simplistic attitude about Templeton, and honestly would have loved to hear more in your book about the story of them and how FQXi happened. Templeton has an agenda of a peculiar kind that people should be aware of, but all money comes with some sort of agenda. I’ve attended one Templeton-sponsored workshop, and that was interesting. As for FQXi events, often the topics seem to be things I’m not particularly interested in (for instance your latest on information), but for some topics I’d be happy to join your discussions. As a general rule, I do think you and many other physicists would probably benefit from more interaction with mathematicians who work on core research mathematics. Math is a subject that takes on different aspects when you start to see what the structures are that the best research mathematicians are now struggling with.

Thanks again for writing in!

22. Peter Woit says:

Florin and Marko,
Enough, I don’t see this going anywhere or having much at all to do with Tegmark.

23. Orin says:

S,

Yes you are missing something. If the MUH suggests that we are probably in a universe where the laws of physics stop working tomorrow (and it’s not at all clear to me that it does), then it also suggests that there is a universe where the laws of physics start working tomorrow. From an anthropic standpoint no conscious observer would ever take notice of their “swapping” between such universes. This kind of logic can be extended to arbitrarily small time slices, of universes jumping in and out of existence, and I think ultimately leads to the conclusion that conscious observers should exist, blissfully unaware of their nature, in even the most chaotic and unseemly of mathematical soups. This is one form of the Bolzmann brain paradox, which I do think needs to be better addressed in “multiverse-aware” physics, although it is a really difficult problem of defining a suitable probability measure. And that’s the rub of it. It may be that (for example due to some morphisms between/onto some mathematical structures being far more common than others in the set of all mathematical structures) that simpler , “nicer,” more contiguous universes are far more probable than those ugly ones that give you pause.

Orin

24. S says:

Thanks, Orin, but I don’t think I find that convincing. I see no reason why the concious observer would happen to jump into a spacetime where the laws were about to start working tomorrow (or jump into another spacetime at all, for that matter). It’s one thing if we’re talking about the past, at which point that might be an explanation for why we haven’t noticed such a thing (though none is needed); but, given an a-priori belief in a level-IV multiverse, it completely fails to give any reason to believe that, in the FUTURE (i.e., even tomorrow), we will still find ourselves in an ordered universe.

In fact, I don’t even know what “tomorrow” means in another spacetime. We’re only in a single spacetime, and “tomorrow” is a point only in ours — if there’s another universe, then none of the points of time in it can rightly be called “tomorrow” by us, here, now. My point is just that there’s good reason to believe that, of all the mathematical structures whose “initial part” (i.e., that part which we’ve already experienced or can observe) is consistent with our universe so far, very, very few of them stay good for us for another day.

Of course, these questions would be easier to discuss if the theory in question were better defined, and Peter’s point is perhaps therefore the most relevant; but I do think that it’s already well enough defined to raise this fairly serious concern; unless, of course, I am missing something (else).

25. S says:

Orin,

As I think more, I suppose you are saying simply that somebody else will begin who happens to remember what the past was of the person in the doomed universe that went crazy. That would make sense as far as it went, except that I think far too much is assumed in applying the anthropic principle in such a way.

If every mathematical structure exists as a spacetime, then there will also be those where YOU exist, but everything ELSE goes crazy; and I would argue again that, of those, we should expect that in almost all of them, everything else will stop working tomorrow. (This argument of course cannot be made rigorous beyond the level of analogy to simpler mathematical contexts, but I find the analogies quite powerful).

As you say, this is getting close to a Boltzmann brain paradox, except that the concern instead is with the persistence of physics, beyond the observer, into the future. And yes, finding suitable probability measure is a problem, but the best analogies we have suggest that the answer won’t look very good. I think I fail to see the motivation for adopting a theory that asserts the existence of such problems (At the very best, on a level-IV multiverse, we would have to say we have no reason in the world to believe that the laws of physics WILL persist outside us for another day), when there is rather little argument for the theory in the first place. Science, after all, is based (isn’t it?) on the search for regularities, for laws. What is the motive or justification for adopting a theory that implies their probable nonexistence?

26. Jesper says:

First: Peter, thank you so much for your persistency. I truly enjoy following what might appear – but I think, is not – a work of Sisyphus.

Second: I have always wondered about what to me appears as a complete shift in the attitude of theoretical physics. Before string theory I think that the history of science and physics has a clear arrow towards simplicity. Maxwells equations, Einsteins Relativity, Quantum Mechanics … – even the Standard Model. To me, it all seems to point, perhaps not in a completely straight line, towards something very simple.

And then comes string theory and claims that its all an “accident”, and the multiverse stuff, which seems to be infinitely worse. I never understood this.

I can’t help mentioning Alain Connes here. His approach to the Standard Model is, in my reading, a natural continuation of the quest for simplicity. In his formulation he sees the Standard Model as unique. As a “has to be” theory. This kind of thinking appeals to me infinitely more than all this “lets just give up on science before we give up on our own ideas”.

Also, about this Tegmark book: I haven’t read it, but my immediate thought was – as you also points out – that he appears to need some kind of measure over the space of mathematical structures. Such a measure must, I guess, also be some kind of mathematical structure (which, of course, is completely absurd) and thus the idea appears to be self-contradictory – but perhaps I just haven’t got the right attitude? Vibes …

27. Jesper says:

One more comment:

Max Tegmark writes: “…I therefore dislike being told not to explore certain ideas. For example, I decided to meet with someone from the Templeton Foundation may years ago precisely because a famous professor told me *not* to talk with them. ”

I think its great to be a rebel and in a way I think its a problem for theoretical physics that we don’t have enough of them – especially among young people (perhaps the rebels just don’t survive, in our system today). But I think the real challenge is to be a scientific rebel. To come up with sci-fi ideas, such as “the universe is an atom in some huge being” is great while smoking pot, its easy – but to come up with truly new bold ideas is much harder – and that, I believe, does take a certain measure of ‘rebelness’.

28. Seth Lloyd has a post on edge.org [ http://www.edge.org/response-detail/25449 ] that relates to this “mathematical universe” idea that makes sense to me:

“Suppose that everything that could exist, does exist. The multiverse is not a bug, but a feature. We have to be careful: the set of everything that could exist belongs to the realm of metaphysics rather than of physics. Tegmark and I have shown that with a minor restriction, however, we can pull back from the metaphysical edge. Suppose that the physical universe contains all things that are locally finite, in the sense that any finite piece of the thing can be described by a finite amount of information. The set of locally finite things is mathematically well-defined: it consists of things whose behavior can be simulated on a computer (more specifically, on a quantum computer). Because they are locally finite, the universe that we observe and the various multiverses are all contained within this computational universe.”

29. Georg Mayer says:

I don’t think that is far off to be astonished about how well math fits reality – and especially how some areas of math evolved without any real-world triggers into something, that later on did perfectly fit to empirical phenomena. What is fascinating to me is that math can be thought and developed independently from the natural world and later on the two still fit together.

Therefore I think it is highly simplified to say that we use math only as a language to talk about the natural world. If so, it is at least a very (very very) well designed language, as it works surprisingly fine, even if we only use the language on its own. And I do not regard math as a language – using that term makes the subject too blurry.

From that perspective, the relation between math and reality are interesting. And they are not esoteric, as some of the most important natural sciences use math as a given and could not work without it. It seems to me therefore not far off or funny, if this relationship is examined a bit more – and that people come up with theories.

I also find it strange, that Max Tegmarks background (financing, name, family) is used (besides other, more reasonable arguments) to criticize his theory. It’s to me not even wrong to talk about money and family when discussing scientific theories.

Seems like some people think of Max Tegmarks ideas as metaphysical and therefore throw the old stones, which are used since 2500 years in this fight. Natural science needs people who strive in the area which is currently thought of as metaphysics, in order to move scientific knowledge deeper into this realm, to find certainty where so far we only had mystery.

And no, I am not subscribing to multiverses or mathematical structures being the real reality – I agree that this sounds too easy (although it does not sound too empty to me). But I subscribe to respect and the diversity of ideas – out of that we get discussions and real scientific progress.

30. Peter Woit says:

Georg Mayer,
Tegmark’s father, FQXi, his name change, etc. are all topics he discusses himself in the book. A major topic of the book (which I think is interesting and relevant) is the issue of how he has gone about working on and getting attention for things like the “Level IV multiverse”.

31. Joe Riley says:

Since Dr. Tegmark is reading this blog (and his willingness to reply is most appreciated) perhaps he will clarify his comments that lead up to the statement (location 5796 in the Kindle version) that “The way I see it, inflation has logically self-destructed.”

That would seem to have implications….

32. Sandro says:

I’m still though left without an answer to the question of why the scientific community tolerates if not encourages all this.

Because some people are Platonists, though you likely are not. If you’re already convinced of mathematical realism, then Tegmark argues for some rather natural implications of this view; for instance, it explains why even the strong anthropic principle is true, which is pretty compelling.

If you’re already unconvinced of mathematical realism, then you’re not likely to find anything in Tegmark’ work to change your mind, but arguing the philosophy of mathematics wasn’t his goal.

33. Peter Woit says:

Sandro,
Actually I am a Platonist, have no problem at all with the idea that physical reality is in some sense “mathematical”. What I don’t believe is that this statement by itself gives you any new useful insight beyond the already obvious fact that mathematics is our best and very powerful way of describing fundamental physics. In particular I don’t believe it means Tegmark’s “all mathematical structures exist” is anything other than a completely empty statement.

34. Apostolos Syropoulos (@asyropoulos) says:

The idea of a mathematical universe is similar to the idea that the universe is a enormous computer that computes its next state. In fact, Fotini Markopoulou has published a paper where she “proves” that the universe is a quantum computer! More generally, Seth Lloyd has already “showed” in Programming the Universe that the universe is a computer. And of course this goes back to Konrad Zuse who, to the best of my knowledge, was the first who put forth this “idea”. Thus, in a sense, Max Tegmark is not saying something new but he repeats, maybe in a different way, an old idea. Personally, I think this idea is pure mysticism and has nothing to do with computing and mathematics.

35. Sandro says:

What I don’t believe is that this statement by itself gives you any new useful insight beyond the already obvious fact that mathematics is our best and very powerful way of describing fundamental physics.

If you believe that mathematical objects exist, then that implies that a mathematical object that completely describes our universe also exists. Why then would you insist that *our* physics/reality is not exactly this mathematical structure?

Furthermore, this implies that mathematics isn’t our “best” tool, it’s the *only* tool in the end. It also explains why the anthropic principles scientists often appeal to in order to explain existence are *necessarily* valid, as I previously said.

Tegmark’s position addresses a number of outstanding metaphysical and epistemic questions in philosophy and science. While this perhaps doesn’t provide much mathematical insight, at least, not the kind of mathematics that interests you, that’s not the only kind of insight of interest.

In particular I don’t believe it means Tegmark’s “all mathematical structures exist” is anything other than a completely empty statement.

Tegmark is saying that all *finite* mathematical structures exist. That’s not an empty statement, and is a restriction on unbounded Platonism which is meaningful given what we now know from computer science.

36. Orin says:

S,

When you say “but the best analogies we have suggest that the answer won’t look very good.” What analogies are those? For analogies my mind first wanders to Feynman’s path integral formalism of quantum mechanics. One could look at these many crazy paths that far outnumber the classical ones, and ask how is it possible that all of these contribute, and yet we find ourselves in a classical-seeming trajectory? So I think there is precedent. And I think that it is logical that we try to extend good/successful ideas to their absolute limits.

I do not see how you jump to the conclusion that the MUH predicts lawlessness. That is far from clear. It may be that once a good measure is found that simpler laws are far more common. In a similar vein to the MUH there are various arguments in algorithmic information theory (I don’t know, see here for example) that indicate that lawful algorithms are far more common than unlawful ones in the set of all algorithms. This may be at first counter-intuitive but that doesn’t mean it is wrong!

Speaking to your point about motivation: for me the motivation is ultimately philosophical. Putting aside the argument about intrusion of philosophy into physics, philosophy for its own sake is not such a tragic endeavor. I personally will never be satisfied with a TOE unless it answers every last “why” question. Ideas like the MUH seem to come closest to actually being able to do that in a satisfying way (and as a class of ideas seem to be pleasantly unique in this regard). But of course to each his or her own.

37. Peter Woit says:

Sandro,
I still just don’t see Tegmark’s “MUH” as telling me something non-empty about physics (or math), and adding “finite” as a modifier doesn’t change this.
He may have something non-empty to say about philosophy, but, honestly, instead of paying attention to his philosophical views I think my time and other people’s would be better spent paying attention to good philosophers who have thought deeply about the subject and are working within a long tradition of others doing this. In other words, I just don’t have time for amateur philosophers, life is too short.

All,
Please, don’t use this as an excuse to post comments on what you think about general issues of math, physics and philosophy. If it’s not about Tegmark’s specific views or book, it doesn’t belong here. I can’t moderate a general philosophical discussion on these topics, and an unmoderated one rapidly becomes unreadable.

38. Sandro says:

This will be my last post on the philosophical aspect of this, as per Peter’s request above.

I still just don’t see Tegmark’s “MUH” as telling me something non-empty about physics (or math), and adding “finite” as a modifier doesn’t change this.

Does de Broglie-Bohm and Many Worlds tell you something non-empty about quantum mechanics? Tegmark’s work is in the same vein. Perhaps you don’t find it compelling, but that puts you in the Copenhagen camp that dismissed the utility of such alternate interpretations for years, ie. “shut up and calculate!”. Alternate interpretations don’t say anything interesting if you’re only interested in empirical data or calculated predictions, since all interpretations are formally equivalent. They do have significant explanatory power though, a power that Copenhagen completely lacks.

I also think you are too dismissive of scientists doing philosophy. Philosophers aren’t as well versed in science and mathematics as you seem to imply, so their views aren’t necessarily so well-informed as you think. Furthermore, “long tradition” doesn’t count for a damn in selecting a scientific theory, or relativity and QM would never have been adopted, and it’s not a meaningful metric in philosophy either. What matters is its axiomatic parsinomy and the set of resolved and unresolved questions it answers.

If you’re not interested in what Tegmark’s work addresses in the philosophical domain, that’s fair enough, but that shouldn’t be taken as a value judgment of the work across all domains. Physicists often unknowingly do metaphysics, like the aforementioned work on interpretations of QM, and while this sometimes seems to have questionable value at the time, plenty of abstract reasoning has led to breakthroughs before.

Sandro, […] what exactly do you mean by “exist”? Until such a statement is substantiated into something that can in principle be measured, I think it is pretty close to being empty.

This seems like a category error. Measurements are defined within universes, not across them. The MUH as a whole will forever be unobservable. You might as well ask whether the people in Sim City could ascertain that you exist.

At best, the MUH might eventually predict some interesting properties about our theories, on an epistemic or ontological level. Its real utility at the moment is in its explanatory power via its unification of computation, mathematics and physics.

39. lun says:

Max Tegmark, the statement “there are 5 and only polyhedra” is true given the assumptions of 3D flast space geometry (which isn’t even physically correct), and a given a formal definition of polyhedron.
Given different assumptions, the statement will not be true. Further more, as Godel taught us, our ability to follow from assumptions to conclusions is limited in principle, and (perhaps even more importantly), as Tarski taught us, even our ability to define things formally is limited.
Of course there are sensible and nonsensical mathematical structures, but, as your collegue Chomsky taught us, there are also levels of grammatical consistency, with a continuum of grammatically consistent sentences that make no sense.
So, to be perfectly honest, your distinction between language and structure might not be so clear. And if Mathematics is more of a language, saying “the universe is mathematical” is a vacuus statement, in principle and not just in practice.

40. Abraham says:

Prof. Tegmark’s book is clearly not a physics ,or a mathematics or a philosophy ,or a theology textbook.
Still,the book provides good and highly needed scientific entertainment which is so rare today

41. Tom says:

I am reminded of Stephen Wolfram’s book “A New Kind of Science”, wherein he also posits the “universe=math”, or more specifically, the universe is just a bunch of ‘cellular automata’ working to simple “rules”. Long time since I attempted to read book, so I could be mis-remembering.

Anyway, Wolfram’s book was impenetrable and ultimately seemed little more than navel-gazing. I’m not aware of a single illuminating explanation, prediction, or retrodiction that ever arose from thinking of the “universe = cellular automata” in Wolframs book

I am very skeptical — no, disbelieving — of any attempt to claim the material, physical universe literally *is* “math structures” , unless one is twisting the conventional meaning of words the way Humpty Dumpty did.

How about this: with all respect (ahem) and as a “child” of the 70s and a still practicing physicist. my opinion is that what Tegmark “preaches” is nonsense. Such nonsense used to be confined to the “crackpot sessions” of the APS meetings.

43. S says:

(I think this comment is about the MUH in a sufficiently direct way not to run afoul of your above request, Peter — I apologize if not).

Orin,

An example that would leap quickly to mind to suggest that, in an MUH, most spacetimes might not be very nice, would be the space of all continuous functions on R^k with its canonical topology: the space of functions that are ANYWHERE differentiable is meager in this space, and prior differentiability (for example, differentiability for t < t_0) doesn't buy you any more than your hypothesis.

Another example would actually be algorithmic information theory, which you bring up: overwhelmingly most computable bit strings are not computable by short programs, and as the length n goes to infinity, the ratio goes sharply to zero if "short" means anything close to strong.

You raise the example of Feynman's sum over paths; but that sum has a measure with a great deal of content, the action. Where would one find such a measure in the space of all mathematical structures, since it is itself just another mathematical structure and another one could be supplied? You say you want every "why" question answered. Well, if the MUH does work only because one chooses a particular measure on it, doesn't that just mean that it has completely failed to answer the biggest question of all? How that is any better than simply looking for the laws that describe our own universe, I can't imagine.

I did skim with interest the paper that you linked, but it had a lot of very strong (I thought) hypotheses, none of which would apply to a space as big as the space of all mathematical structures. (Indeed, I am reminded of the NFL theorems, which again are relevant by close analogy if not directly).

At the end of the day, this perhaps comes down to taste — what is a convincing analogy, what is a strong assumption, what is an exceedingly non-parsimonious explanation relative to the phenoma it explains, etc. Perhaps we can both agree in the hope that such a structure as the Level-IV multiverse/MUH, should it ever gain currency, will be supported somehow by actual experimental evidence or devastating argument that simply defeats skepticism, and not by mere taste?

44. Maurice Carid says:

A question to those who have read Max’ book: does it contain any major new insight by him that is not already contained in a Scientific American article he published more than a decade ago (http://arxiv.org/abs/astro-ph/0302131)?
Or is the book just an inflated write-up of that article?

45. Peter Woit says:

Maurice Carid,
In terms of the multiverse levels, and Math=Physics claims, the book is basically an extended version for a popular audience of the same material you can find in Tegmark’s articles and other material (some linked here, more on the arXiv and available from his website). The book does cover other topics, for instance some of it is a memoir of his career, and at the end he writes about a variety of topics with no connection to math/physics issues.

All,
Please, no more comments that are not directly and specifically about Tegmark’s work, and no more attempts to use this to promote your own ideas that have nothing much at all to do with Tegmark.

46. Orin says:

S,

I wrote you what I thought was a thoughtful reply, but apparently Peter removes only the one-side of a discussion that does not support his own opinion, even when the two sides are engaged in discussing the same thing (namely Max Tegmark’s work).

47. Peter Woit says:

Orin,
I’m cutting off your discussion with S just as I’ve cut off a bunch of others. This has nothing to do with agreeing with S vs. you. My writing about Tegmark’s empty claims does not mean I’m willing to host empty general discussions about math and physics inspired by him here.

This topic has become a crank magnet (I’ve deleted 2-3 times the number of comments you see here), making it even more clear to me the danger Tegmark’s efforts pose (that of turning a once great subject into crank city). As I point out in the posting, I think the only interesting question his work raises is why it is getting attention from usually serious quarters of the community, and what can be done about the problems this raises.

48. Neil says:

Personally, I think it is useful to have brilliant thinkers like Tegmark proposing highly speculative ideas. The problem is when highly speculative ideas absorb a large fraction of researchers in the discipline, particularly younger researchers.

49. Peter Woit says:

Neil,
“Highly speculative” is one thing, “meaningless” is another…

50. Max Tegmark says:

> Sure, inflationary cosmology is a theory (in lots of versions…) that you can extract
> predictions from. Like any theory, if you > figure out how to test distinctive parts
> of it and it passes those tests, you can have some confidence that other, untestable
> parts of it correspond to reality. I really don’t have a simplistic view of testability:
> for instance, I’m happy to agree that a theory can be so beautiful and
> mathematically compelling that I’d have confidence in it given only minimal
> experimental evidence.
So is it fair to say that you agree with me that what I call the Level I multiverse (the existence of at least some spatial regions the size of our observable universe that unobservable because light from them hasn’t yet had time to reach us) is a prediction from some inflation models that you in turn consider to be within the purview of science (testable scienfic theories/models)? Is it fair to say that you therefore view the Level I multiverse as a topic appropriate for scientific (as opposed to merely philosophical) discussion?