In response to: “It’s You Again” by Daniel Kleitman (Vol. 2, No. 3).


To the editors:

Max Tegmark’s Our Mathematical Universe was recently reviewed in Inference by Daniel Kleitman and I have been asked to comment on his views. Let us begin with a few words about each participant in our colloquy. Tegmark, an accomplished theoretical physicist and cosmologist at MIT, is known for developing and implementing the most effective techniques for analyzing rich data sets, such as those produced by the Sloan Digital Sky Survey (SDSS). His book gives only a sketchy account of his own substantive scientific endeavors. Instead it is a potpourri of autobiographical tidbits, scientific expositions and fanciful speculations about matters cosmological and quantum-mechanical, the most curious of which being the countless exact copies of ourselves that, he insists, exist within four “levels” of multiverse, each made up of innumerable universes both like and unlike our own.

Cosmology has evolved over the past few decades from a hand-waving qualitative discipline to what is more and more becoming an exact science. In his Chapter 4, Tegmark strives to describe, in non-technical language, both the birth of precision cosmology and his own role in this remarkable development. Among the many achievements of the SDSS was its detection of the so-called Sloan Great Wall of Galaxies, at which Tegmark claims to have been “so flabbergasted.” I wonder why, because a somewhat smaller Great Wall of Galaxies had been discovered a quarter of a century earlier by Harvard scientists Margaret Geller and John Huchra, a fact Tegmark fails to mention.

Kleitman received his doctorate in theoretical physics under the direction of Julian Schwinger at Harvard, then spent several years as a theoretical physicist while solving combinatorial problems in his spare time. Some of these he found in a collection of unsolved mathematical problems compiled by the great Hungarian mathematician Paul Erdös, who was so impressed with Kleitman’s solutions that he asked “Why are you only a physicist?” and guided him toward a position in the Applied Math Department at MIT, where Kleitman remained as a distinguished professor of discrete mathematics and chair of his department until his recent retirement. Typical of his discipline is the “traveling salesman problem,” which seeks the shortest path passing through each of N cities, a challenge becoming ever more difficult as N grows large. Nevertheless, ingeniously designed algorithms find exact solutions for N in the thousands and approximations to better that 1% for N in the millions. Puzzles of this sort are surprisingly relevant to computer science, “pure” mathematics, operations research and commerce.

As it happens, I was one of Schwinger’s doctoral students at the same time as Kleitman. Because I knew Kleitman then and know him now, he is hereafter called Danny. Long ago we collaborated on several not very important papers on particle physics. While Danny switched to math, I kept pursuing my particles. Since then we each have enjoyed a fair share of success in our chosen disciples. His review is right on! I enjoyed it thoroughly and agree with most of it, more than I can say for Tegmark’s book. I was a touch dismayed by Danny’s superfluous allegation that particle physicists and cosmologists “feel a need to step out of [their confining disciplines] to something closer to humanity.” Admittedly, the extremes of the ladder of sizes are not promising places to find socially useful applications. What we have learned about quarks, quasars, gluons, galaxies and the hot big bang is good for nothing… except to fulfill our (sacred?) obligation to understand as best we can the world we are born into.

Cosmology is the study of the origin and evolution of everything we can observe with our instruments: the visible universe, which is now known to be a 14 billion year old sphere about 46 billion light-years in radius. But there is much more out there. Most cosmologists suspect the visible universe to be only a tiny part of a universe that is likely (but by no means certainly, as Danny reminds us) infinite in extent. If so, our visible universe is just one of an infinite number of causally disjoint “local universes” wherein no one can affect or be affected by any other. They comprise Tegmark’s Level I multiverse, within which (he argues) must exist an infinite number of “copies of you [the reader!] with identical past lives and memories.” With deft mathematical metaphors, Kleitman undoes this argument: The number of local universes may be infinite, but it is countably infinite, while the number of ways the molecules of your body can be assembled is a larger infinity than that. Thus the probability that an exact copy of yourself exists anywhere within this multiverse is not one; it is zero. Tegmark would probably settle for almost identical copies, but I cannot care. In this context, a paraphrased portion of Francesco Sizzi’s famous criticism of Galileo’s discovery of Jovian moons seems pertinent: “[Other universes] cannot be detected [in any manner] and therefore can have no influence on the earth and therefore would be useless and therefore do not exist.”1

Tegmark’s Level II multiverse was twice conceived: first from inflationary cosmology, then via string theory. The two versions may or may not coincide. Back in 1980, Alan Guth showed how the seemingly inexplicable flatness and isotropy of our universe could be understood if the big bang had been followed by a short period of exponentially rapid growth now called cosmic inflation. Although its underlying mechanism is not yet understood, the notion is consistent with all observations and (despite Danny’s doubts) is accepted by the vast majority of physicists, astronomers and cosmologists. Its deniers are probably fewer than those of anthropically caused climate change. Paul Steinhardt, in 1983, showed that inflation could be eternal, a feature soon found to characterize most inflationary schemes. Eternal inflation leads to the creation of a vast multitude of bubble universes, each displaying different physical properties but none in causal contact with any other. They comprise the first version of the Level II multiverse.2 The second version evolved from particle physics.

We particle physicists are justly proud of what we modestly call our “standard model”—a quantum field theory providing, in principle, a complete, correct and consistent description of virtually all observed particle phenomena, including the prediction of the recently discovered Higgs Boson. But our theory leaves many questions unanswered. Here are two of our Big Problems: Gravity, although too weak a force to affect particle phenomena, must be reconciled with quantum mechanics, a task that seems impossible in the context of quantum field theory. Furthermore, the standard model involves dozens of parameters (like quark masses) whose values must be set by experiment, just like the many knobs on early television sets which had to be carefully adjusted to produce decent pictures. Modern TV sets have done away virtually all of these knobs, but particle theorists have been struggling for decades to cut down the number of independent parameters in our theory… with no success whatsoever. As we shall see, the Level II multiverse provides an unsatisfying explanation for our failure.

String (or, superstring) theory began in the 1970s as an attempt to understand the force holding neutrons and protons together. Its advocates found, to their great surprise, that their theories imply and include an apparently consistent quantum theory of gravity, thereby revealing a possible solution to one of our Big Problems. Enthused, they came to believe that quantum strings might provide the one and true unified theory describing all of nature’s particles and interactions. It would answer all the questions vexing particle physicists. Alas, that was not to be. Several quite different variations of string theory turned up and, in 1995, Edward Witten showed that these theories and many more correspond to different possible vacuum states of an overarching structure he called M-theory. Soon afterward, Lenny Susskind proposed a “landscape” consisting of very many different and causally distinct universes, one for each version of string theory. This was the second conception of the level II multiverse. Neither version can answer any of our many questions about the structure and parameters of the Standard Model, or determine the value of the cosmological constant which governs the expansion rate of the universe (which Einstein regarded as his biggest blunder).

In each universe of the Susskind landscape the laws of physics may be the same, but what physicists regard as fundamental constants of nature are different. Many of these universes are short-lived, empty or contain unrecognizable forms of matter. In only a tiny fraction do long-lived stars with habitable planets evolve. In how many of these does life arise? ...or intelligence? …or people like us? …or Danny’s Aunt Bea?

Tegmark would claim copies of Aunt Bea inhabit an infinite number of them. Tegmark would be wrong. The parameters defining our sciences (such as the ratio of proton and electron masses) can take any real positive values, the number of which, Danny reminded us, is a greater infinity than the number of universes. So much for copies of you, me and Aunt Bea in Level II universes.

String theory can say nothing about the constants of nature because they are mere accidental properties of our particular universe. Our unanswered questions are moot. All we can know is that the fundamental constants of nature must be such as to have allowed us to evolve and to ask such questions. This is the so-called Anthropic Principle: “Tout est pour le mieux dans le meilleur des mondes possibles,” as Dr. Pangloss assured us. For phenomenological particle physicists like me, the multiverse yields a philosophy of despair.

Quantum mechanics leads Tegmark to his Level III multiverse. He much favors Everett’s “many-world” interpretation to the Copenhagen “collapse of the wave-function” view, no matter that these two (among other) interpretations of quantum mechanics provide identical descriptions of all observable phenomena. According to Tegmark via Everett, with each flip of a coin our universe splits into two coequal universes: one where the result is heads, another where it is tails. He insists both universes to equally real, even though neither can affect the other. Consider what several of my distinguished scientific colleagues say about Everett’s many worlds:

Arthur Fine: There is, I think, no sense at all to be made of the splitting of worlds.3

John Bell: The many worlds interpretation seems to me an extravagant, and above all an extravagantly vague hypothesis.4

Murray Gell-Mann: Everett’s ideology that there are many worlds that are equally real is operationally meaningless.5

Steven Weinberg: I find the many worlds interpretation repellent.6

In fact, quantum mechanics is an effective and easy-to-use vehicle offering a consistent and unambiguous description of the physical world. Its interpretation is another matter and has been argued about for almost a century, but today, I believe, should be left to historians of science, philosophers or retired physicists.7 Evidently, Tegmark disagrees. Of an imagined copy of yourself far far away, he writes: “The life of this person has been identical to yours in every respect — until now, that is, when your decision to read on signals that your two lives are diverging.” Danny (suspending his own disbelief in such copies) finds it “distinctly odd that quantum uncertainty should suddenly pop up.” Of course, it doesn’t! Quantum mechanical uncertainty is all-pervasive, with or without observers. It pertains to how particles scatter, whether molecules react, which mesons are produced by a cosmic ray, when a radioactive nucleus decays, et cetera ad infinitum. Upon each such unwatched “measurement” the wave function splits, thereby spinning off hosts of brand-new Level III universes in each lower level universe… or not, if we choose the perfectly equivalent Copenhagen interpretation of quantum mechanics.

Chapter 9 is devoted to what can only be called mathematical epistemology. Tegmark alleges the plausible hypothesis that there exists an external physical reality completely independent of us humans to imply a rather more startling Mathematical Universe Hypothesis: that our external physical reality is a mathematical structure.8 The Level IV multiverse, it turns out, is nothing other than the set of all mathematical structures, each of them constituting a universe unto itself.

According to Tegmark, our universe is (rather than merely “is described by”) the long sought

Theory of Everything, or ToE, from which all else can be derived… [S]uch a complete description must be devoid of any human baggage. This means that it must contain no concepts at all! In other words, it must be a purely mathematical theory… [An] infinitely intelligent mathematician should be able to derive the entire theory tree [including all of science, engineering, sociology, psychology etc.] from these equations alone, by deriving the properties of the physical reality that they describe, the properties of its inhabitants, their perceptions of the world, and even the words they invent. This purely mathematical theory of everything could potentially turn out to be simple enough to describe with equations that fit on a T-shirt.9

And our ToE is just one among an infinity of mathematical structures, each of them its own universe. If Tegmark is correct, there must exist a slightly different mathematical structure, whose equations are emblazoned on another T-shirt, wherein I am Tegmark’s psychiatrist rather that a physicist. I do not believe a word of it. Paraphrasing Danny, I may be a blockhead but I am certainly not a mathematical structure akin to a triangle.

Sheldon Glashow


Sheldon Glashow is professor of Mathematics and Physics at Boston University and professor emeritus of Physics at Harvard University. He received the Nobel Prize in Physics in 1979.

  1. Francesco Sizzi, Dianoia astronomica, optica, physica, qua Syderei Nuncij rumor de quatuor planetis à Galilaeo Galilaeo mathematico celeberrimo recens perspicillì cuiusdam ope conspectis, vanus redditur. Auctore Francisco Sitio Florentino (Venice: apud Petrum Mariam Bertanum, 1611). 
  2. Steinhardt, whose eternal inflation led to the multiverse, has renounced his creation:

    To me, the accidental universe is scientifically meaningless because it explains nothing and predicts nothing ... Scientific ideas should be simple, explanatory, predictive. The inflationary multiverse as currently understood appears to have none of these properties.
    John Horgan, “Physicist Slams Cosmic Theory He Helped Conceive,” Scientific American, December 1, 2014. 
  3. Maximilian Schlosshauer, ed., Elegance and Enigma: The Quantum Interviews (Berlin: Springer, 2011), 69. 
  4. John Bell, “Six possible worlds of quantum mechanics,” Proceedings of the Nobel Symposium 65: Possible Worlds in Arts and Sciences (1986). 
  5. Murray Gell-Mann – Quantum Mechanics Interpretations – Feynman Sum over Histories – EPR Bertlemann’s,” YouTube video, May 31, 2014. 
  6. Questions and answers with Steven Weinberg,” Physics Today, July 8, 2013. 
  7. Quantum theory’s “spooky action at a distance,” which so befuddled Einstein, was explained in the 1960s by John Bell. Now referred to as quantum entanglement, it is neither problem nor paradox but a phenomenon being studied with promising applications to quantum computation and communication. 
  8. Tegmark provides a short list of “mathematical structures,” including the empty set, the octahedral group and Boolean algebra. The particular mathematical structure that we inhabit is likely to be much further down the list and a trifle more elaborate. 
  9. Max Tegmark, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality (New York: Knopf, 2014), 174.