Max Tegmark has written a work entitled Our Mathematical Universe, which contains many curious claims. Perhaps the most curious is this: you have infinitely many copies of yourself in the universe. These are your possible futures.1 Included in the book, as well, are descriptions of cosmological data, of some current cosmological theories, and of elementary quantum mechanics. The book is ominously without equations.
The first hundred or so pages consist of a qualitative description of the history of cosmology, as the author sees it, along with details of human interest, particularly the author’s personal experiences, his contributions to the field, and the contributions of his friends.
Tegmark then introduces readers to the theory of inflation. At the very beginning of our universe, so the theory asserts, an extremely hot plasma expanded at speeds beyond the speed of light, and rapidly cooled down. This expansion explains why the farthest regions that we can see look nearly the same. This theory also predicts that the geometry of space has no curvature; its geometry is Euclidean.2
There are many variations on this theory, and many people who do not accept it at all. Questions have been raised about whether the data from the latest space probes are consistent with it.3
The remainder of the book consists of Tegmark’s own theory. If something exists, Tegmark is persuaded, it must have had a finite, non-zero probability of existing. You too. Tegmark also believes that the total universe is infinite. He defines our local universe as the visible part of the whole universe. Whatever existed before the big bang is invisible to us. It is by no means certain that the entire universe is infinite, although measurements show that space is approximately Euclidean and potentially infinite.
Tegmark’s chief argument now follows. Our local universe arose through the process of inflation; since this inflation happened, it had a finite non-zero probability of happening. In an infinite universe, inflation must have taken place infinitely often. There must therefore be an infinite number of local universes. Tegmark then claims that, since you also have a finite non-zero probability of existing in the infinite universe, there must be infinitely many copies of you in the other local universes.
He then weakens his claim, though he does not acknowledge this, by pointing out that the physical constants we observe in our local universe, such as the ratio of proton to electron mass, might be different in other local universes. Since the laws of nature would thus be different, you could not be you. The ratio of the masses of the proton and the electron could take infinitely many values; the a priori probability that it has any particular value is zero. You would necessarily remain you. So long those other yous.
After a survey of elementary quantum mechanics, Tegmark turns to particle spin. His argument can most easily be described in terms of coin tossing. The proportion of heads in an infinite series of coin tosses approaches ½ in the limit. You are here, but you are also there, and in infinitely many other universes! Those double yous are tossing their coins as well. After you have tossed your coin, one half of the future no longer corresponds to you. Those other yous who tossed tails when you tossed heads cease to be you, but you are still guaranteed an appearance in infinitely many incarnations. If the number of copies of you performing the experiment were finite, that set would be half the size of the total. Tegmark claims that you cannot distinguish who is who. Until your coin hits the ground, both possibilities exist in an infinite number of local universes. On observing heads, your world and your future have partially collapsed, as the immediate future has vanished into the past. In a more general way, you consist of yourself and all of your possible futures, heads or tails, as the case may be. Your potential futures narrow to include whatever happens. The versions of you for whom it did not happen are no longer you.
After a short while, the yous that remain represent only a tiny proportion of the original yous; those yous that are no longer you will clutter up the universe. After a given head tossing, you become identical to the infinitely many head tosses in the universe. Head tossing is one possibility, dying is another. If you are alive, Tegmark seems to accept, then clearly you have not died. An appeal to measurement is unavailing in the case of your death, although plainly some of your near doubles could on this occasion go mad with grief.
Tegmark describes an experiment in which you are destined to kill yourself if any of ten midair coin tosses comes up tails. I do not advise attempting this. If you performed the experiment, you would survive only as someone for whom all ten tosses yielded heads. Since there would be an infinite number of yous and all youse guys would be dead otherwise, you being alive is identical with having tossed the coins and gotten all ten heads. You achieve the same result merely by being alive. It’s not clear what this experiment suggests. Tegmark does not worry that you will die in this experiment. It is enough that you who had been you will survive in an infinite number of local universes.
There are more chapters and more claims, but I think now is the time to ask how much of this makes sense. It is by no means clear.
First, Tegmark’s fundamental principle is false. It is simply not the case that if something exists, it must have had a finite, non-zero probability of existing. Second, Tegmark considers a human to be merely a set, or, perhaps, an equivalence class of sets. You may, indeed, be represented as a set of atoms, but if so, it is a set with an extremely complicated structure, one in which there are an enormous number of parameters. The probability of any one of them taking any particular value is zero.
Third, you are you in virtue of your past experiences. With no commitment to solipsism, it follows that your memories of your past experiences must provide a faithful record of your local environment. Appearing in still other universes, you would be obliged to remember just what you remember in this universe. One faithful record engenders another. Having appeared here, and having appeared there, it follows that here and there must be virtually identical. This is an unusually convenient way of bringing about comity among universes.
Recall Tegmark’s fundamental principle: if something happens, it must have had a finite probability of happening before it happened. Consider a linear function f in an interval from a to b that has the property that it is positive at a and negative at b. What is the probability that f(c) = 0 at any particular point c within that interval?
Given f(a) and f(b) for some a and b, it is easy enough to find some c such that f(c) = 0; but given only a, Pr (f(c) = 0) = 0.
Suppose you draw a triangle in the plane. This triangle has three angles, two of which, in flat space, determine the third. That particular triangle had a zero probability of being drawn before you drew it. But you did, and it came into being. Suppose that randomly you draw another triangle. The probability that it will be similar to the first, with the same set of angles, is zero. Suppose an infinite, but countable, number of yous each randomly drew an infinite number of triangles. The probability that any two are similar to one another is still zero.
Tegmark’s inference from the assumption that you have a non-zero probability of existing to the conclusion that you are infinitely everywhere is plainly incorrect. The same claim when made on behalf of local universes is incorrect.
For all anyone knows, you may be endlessly repeated throughout the universe, and the universe endlessly repeated throughout the multiverse. But this is not a conclusion that Tegmark’s argument justifies.
Nor does it do much good to argue by analogy. If an infinite number of identical triangles exist because one triangle does, this still says nothing about you. However much you may be a blockhead, you are certainly not a triangle.
Tegmark’s argument is weaker than its weakest version. A human being has certain memories—obviously so. To be me, you would need my experiences, and vice versa, of course. If you were me, the enveloping sense of identity between us ramifies until it includes everyone that you have encountered over the years. My Aunt Bee and your Aunt Bee turn out to be one and the same Aunt Bee. What a surprise!
For this to be so, any two local Bee-able universes must be identical. If identical, then by Leibniz’s law, identical at creation, and thus following identical paths until now.4 These are not universes with a vibrant channel of communication, or any channel at all. What happens in one must be completely determined by what happens in the other, and vice versa. There could be no chance at all in either—no quantum mechanical uncertainty either.
And would you look at this now. Identical for billions of years, two universes suddenly become distinct after you toss a coin. What your other you is doing is anyone’s guess. I find this hard to believe. What I have called tossing a coin is described by Tegmark as a quantum phenomenon. Having been quiet at every previous moment, it seems distinctly odd that quantum uncertainty should pop up now just as you are tossing a coin.
Tegmark believes that his interpretation of quantum mechanics was inspired by Hugh Everett. In quantum mechanics, measurements are inevitably probabilistic. Such is the fabled collapse of the wave function. Tegmark seems to believe that, on Everett’s views, there is no collapse, the moving stream of the universe bifurcating into two equally real possibilities whenever a measurement is made. It is easy to see why Tegmark and Everett should be the best of friends. You are destined to appear in Everett’s infinitely bifurcating universes.5
The number of local universes containing you is countable. It follows that there is a minimum distance between you and you, and this means that each of you can be described by a distinct point. The living, breathing, bestseller-writing you may be found in a spatial grid happily isolated from all other yous. The number of yous loitering around is countable. Nothing like this could possibly be true of quantum mechanics.
Tegmark generously admits that there are flaws in his theory. He wonders how our local universe could have lasted billions of years when the different local universes outnumber the identical ones. This seems to involve a contradiction. Tegmark is also perturbed by the fact that the list of outcomes of a countable set of coin tosses does not have a specific average value, since reordering the sequence can make that average anything. For example, if you place all the heads in front, you will never reach any tails at all, as the number of heads is infinite; placing all the tails in front of all the heads will similarly cause you to see only tails, all with the same initial set of results.
So what are we to make of all this?
There is a real world out there, and there is the world as I see it in my mind. The real world has limitations. And vice versa, of course. We cannot make an infinite number of choices in the real world, no matter the axiom of choice. You and I are both free to imagine an infinite number of us, as Tegmark does, even if that has zero probability of being correct. Logical inconsistency is no bar to enjoyment.
One of the problems of cosmology, as well as particle physics in its present state, is distance from every other human interest: life and love, and biology or chemistry or many-body physics, or politics or economics or engineering of any practical kind. I can imagine that those who devote themselves to such fields feel a need to step out of the confined set of facts and problems that their choice of field has restricted them to, into something closer to humanity. Delving into alternate worlds is a natural fulfillment of that need.
Tegmark’s own scientific work involves developing and applying ingenious data processing techniques so as to be able to draw conclusions from massive amounts of raw astronomical data in which there are only very weak signals. There is not a word in this book about these methods or how they represent methodological advancements. He notes that in his professional work he similarly has made no mention of his ideas. He describes himself as an intellectual Jekyll and Hyde, and this book is the work of his Hyde personality.
I do not disagree.
- This is not exactly what he claims. His possible futures actually correspond to possible outcomes of quantum mechanical measurements, but the distinction only muddies his arguments. ↩
- Tegmark defines inflation in terms of negative energy, which is curious. When two objects approach each other and are affected by gravitation, their gravitational potential energy becomes kinetic energy. Tegmark describes this as the potential energy of the system becoming more negative. ↩
- The Wikipedia article on cosmological inflation is a concise description of the subject and its many variations. See Wikipedia, “Inflation (Cosmology).” ↩
- Except Tegmark contradicts himself, at one point asserting that local universes start out differently. That they should later become identical seems impossible to me. Differences anywhere affect everything around them and make them different as well. ↩
- When Tegmark first talked about these matters at a meeting in Maryland in 1997, he took a poll of the audience and found that 13 accepted the Copenhagen picture of quantum mechanics, and eight the Everett view. When he spoke at Harvard in 2010 about the subject, he found that 16 held the Everett view and none the Copenhagen one. I sense that he tells us this to convince us that the Everett view is the now dominant one, or at least one of growing popularity. I interpreted the result to mean that serious physicists aware of the first lecture did not attend the second, while science fiction buffs were then more anxious to hear from him. ↩