Max Tegmark is a professor of physics at MIT, a major force in the development of modern cosmology, a lively expositor, and the force behind what he calls the Mathematical Universe Hypothesis — a vision of the Universe as a purely mathematical object. Readers of The Big Questions will be aware that this is a vision I wholeheartedly embrace.
Tegmark’s new book Our Mathematical Universe is really several books intertwined, including:
- A brisk tour of the Universe as it’s understood by mainstream cosmologists, touching on many of the major insights of the past 2000 years, beginning with how Aristarchos figured out the size of the moon, and emphasizing the extraordinary pace of recent progress. In just a few years, cosmologists have gone from arguing over whether the Universe is 10 billion or 20 billion years old to arguing over whether it’s 13.7 or 13.8.
This part of the book concludes with an account of the theory of cosmic inflation — the process that triggered the Big Bang — which was spectacularly confirmed just a few weeks ago (and just a few months after this book was published) by a set of observations that Tegmark has called one of the most sensational scientific discoveries of all time. Inflation models tend to predict that our Universe is infinite (though of course we can never observe anything more than about 13.8 billion light years away). This will come up again later.
- A tour of the “landscape” — the multiverse in which, according to many versions of cosmic inflation, our infinite universe is a tiny bubble. (More precisely, though still less precisely than the explanation you’ll find in Tegmark’s book, our universe looks infinite from within but tiny to a hypothetical outside observer.) The landscape is a strange place indeed, filled with plasma and doubling in size an incredible 1038 times every second. Now and then bubbles form, and within these bubbles, the plasma cools into any of 10500 different varieties of space and time, giving rise to 10500 possible collections of laws of physics.
The theory of the landscape is respectable but controversial among physicsists. Tegmark is a true believer, largely because the landscape explains the otherwise seemingly inexplicable hospitality of our Universe, where dozens of physical constants seem to have been fine tuned to make life possible. (For example, if gravity were a little stronger, the sun would have devoured the planets long ago; if it were a little weaker, the planets would have spiraled out into space.) If there are in fact 10500 possible states of physical law, then it’s not so surprising that one of them is hospitable to life, and then not surprising at all that that’s the one we evolved in.
I have not attempted to read much of the scientific literature about the landscape (though it’s on my to-do list), and in any event I have little enough faith in my physical intuition that I’d hesitate to form my own judgment about its plausibility. But I think that the hospitality-to-life argument is quite undermined by Tegmark’s own Mathematical Universe Hypothesis (in which I do believe!). If the Universe is a mathematical object, then the great diversity of mathematical objects is enough to explain why there’s one where we can survive. We don’t need the additional diversity of the landscape.
(Of course, if the Mathematical Universe Hypothesis is true and if the landscape is ultimately a mathematical object, then the landscape certainly exists and has physical reality — as does any mathematical object sufficiently complex to contain self-aware substructures. It does not follow that we live there.)
- An account of the many-worlds interpretation of quantum mechanics, — together with a very surprising (at least to me) tie-in to the earlier material. Namely: If our Universe really is infinite, then it contains an infinite number of balls 14 billion light years in diameter. But the laws of physics allow only finitely many possible histories for such a ball, so at least some of those histories must occur an infinite number of times. So we should expect that there are, far beyond the limits of our observation, balls with histories identical to the ones we live in, as well as others that differ from ours only in minor details. (I am reminded of Father Guido Sarducci‘s bit about a far-away planet identical to earth in every way except that they hold their corn on the cob vertically instead of horizontally.) Tegmark calls this collection of balls the “Level I Multiverse”, to distinguish them from the “Level Two Multiverse” arising from cosmic expansion. So the Level I Multiverse is a tiny bubble in the Level Two Multiverse, and it contains an infinite number of these balls where history plays out more or less as it does right here.
Now, according to the many-worlds interpretation, the wave function of the Universe never collapses. It’s always seemed to me that the many-worlds interpretation should actually be called the one-world interpretation, since there’s just one wave function and hence just one history, but there’s a sense in which you can think of that history as constantly “branching”, so that the version of you who is reading this blog post and deciding whether or not to eat a cashew will soon branch into two of you, one of whom eats the cashew and one of whom doesn’t. Tegmark calls those branches the “Level III multiverse”.
The big surprise is that in some sense, there’s no reason to distinguish between the Level I and Level III multiverses. Tegmark (with a pointer to an academic paper where he works through the details) shows that when you get down to the mathematics, there’s no important difference between the various versions of yourself who have branched off the quantum wave function and the various versions of yourself who inhabit far distant versions of the Universe.
(Incidentally, Tegmark has the wonderful habit of including pointers to many of his — and others’ — academic papers, so readers can easily decide for themselves when they want to pursue topics in greater depth.)
- An account and a defense of the Mathematical Universe Hypothesis — the hypothesis that the Universe is, pure and simple, a mathematical object. I’ve reached a point where I can no longer remember why I ever thought this hypothesis even needs a defense. If the Universe isn’t made of mathematics, then what could it possibly be made of? It seems to me to be metaphysically reckless to postulate the reality of things (such as non-mathematical objects) that we have absolutely no reason to believe in.
Look. To exist is to have properties. If statements about a thing are objectively either true or false, then that thing exists. Statements about (say) the natural numbers are objectively either true or false; for example, it is objectively true that every natural number is a sum of four squares. Therefore the natural numbers (and similarly other mathematical objects) exist. Mainstream physics accounts for all physical phenomena as mathematical phenomena (for example, an electron is a disturbance in a field, and a field is a mathematical function). These physical theories are surely only approximations to the truth, but as the approximations get better, they get no less mathematical. Therefore mathematical objects not only exist; they are the only things that we have any particular reason to believe in. To suggest that there’s anything else is to raise the apparently unanswerable question of why. In that sense, physical substance, apart from mathematical substance, is no different from God — a desperate pre-scientific attempt to postulate our way out of anything we don’t immediately understand.
That, in essence, is my story, as told in The Big Questions. Tegmark’s got a good story too. He starts by presuming that there is such a thing as truth, independent of the way humans happen to think — and proceeds to demonstrate (at least to my satisfaction!) that any such truth must be purely mathematical. The basic point is that most of what we think of as reality — rocks, for example — appear to have individual identities only because of the way our brains work — we think of a certain collection of elementary particles as a “rock”, even though they fundamentally have nothing more in common with each other than with other elementary particles that we don’t consider part of the rock. Get rid of those artificial human distinctions, and all you’re left with are the particles. But what’s a particle? It’s a disturbance in a field, which our human brains find it convenient to conceptualize as a “particle”. Take away the human brain, and all you’re left with is a disturbance in a field. And what’s a field? It’s a bit of mathematics.
- A more detailed description of Tegmark’s version of the Mathematical Universe Hypothesis. Here he sometimes loses me. Obviously, he’s very smart and he’s thought very hard about this stuff, so perhaps the fault is mine. But Tegmark says a number of things that seem to me to make no sense. For example, he says that “to define the mathematical structure of [the ring of integers], we need to specify just the shortest computer program that can read in arbitrarily large numbers and add and multiply them.” But this is circular. How can you determine the length of a computer program without counting its lines? And how can you count its lines if you don’t already know a whole lot about the integers?
Indeed, we know that the Peano axioms for arithmetic admit non-standard models — that is, structures that obey all the axioms of arithmetic, but are in fact very different, to the extent that their addition and multiplication rules cannot be implemented by any computer program of finite length. Suppose you managed to mistake one of these non-standard models for the standard integers. Then you could easily fool yourself into believing you’d written a finite computer program to implement your model’s arithmetic — but only because you’d be confused about what “finite” means. So we cannot rely on the existence of such computer programs to assure us that we’re talking about the standard natural numbers. For that and related reasons, the integers arguably form an extraordinarily complex structure.
Tegmark prefers to believe that our Universe is not just any mathematical structure, but a computable mathematical structure. He goes on at some length about this. But I cannot figure out exactly what “computable” is supposed to mean in this context. I know what it means for a function to be computable, or for a set of axioms to be recursive, but what does it mean for a structure to be computable? My first guess would be that a structure is computable if the set of true statements about that structure (in some formal language) all follow from recursive set of axioms. But Tegmark seems to want to count the integers as computable, which rules out that interpretation. What other interpretation is there?
It wouldn’t entirely surprise me to learn that Tegmark’s got good answers to these questions, but based on what’s in this book (and the bits of his papers that I’ve worked through) I don’t understand what those answers could be. In the scheme of things, though, who cares? The Tegmark vision of a mathematical universe is, I think, a final solution to the single greatest philosophical conundrum of all time, namely “Why is there something instead of nothing?”. If he’s got a few details wrong, that accomplishment is not significantly diminished.
- A scientific autobiography, vividly portraying the triumphs and frustrations of research, and the excitement of being on the scene just at the birth of “The Age of Precision Cosmology”. We get a lot of good anecdotes about Tegmark, the people he’s worked with, and his relations with the rest of the scientific community, including a remarkable email from a senior scientist warning him that if he keeps thinking so far outside the box he risks being labeled a crackpot. Tegmark’s enthusiasm, for science, for life in general, and above all for ignoring such warnings, is palpable and infectious.
- A bunch of self-indulgent asides of the sort to which every author of an excellent book (and this book is surely excellent) is well entitled, though we need not take them too seriously, except of course when we agree with them. So I’m happy to laugh off some of his ideas about what the most important issues should be in the next election, and happier still to see that he shares my equally inexpert (and hence equally self-indulgent) sense of the unlikelihood of intelligent life (see also here) elsewhere in the 14-billion light-year ball that constitutes our observable Universe.
Bottom line: A terrific book, really. The idea he really wants to sell is the mathematical universe idea, an idea that puts to rest (at least for me) the greatest philosophical problem of all time, namely “Why is there something instead of nothing?”. That is an astonishing accomplishment and one that, all by itself, makes Tegmark a towering figure in intellectual history. But even if you don’t buy that, or don’t care about it, I am not aware of any more lively or readable account of the history and current state of cosmology, including widely accepted ideas like cosmic expansion and more controversial ideas like the landscape. All lightened up with charming personal anecdotes and deepened with appropriate links to the academic literature.
I’m glad he wrote this book and glad I read it. You should read it too.