The Infinite Multiverse And Monkeys Typing Pi
By Amir D. Aczel | June 6th 2014 09:49 PM | 14 comments | Print | E-mail | Track Comments

Amir D. Aczel, Ph.D., studied mathematics and physics at the University of California at Berkeley and also holds a doctorate in mathematical statistics...

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Two years ago, I expressed my doubts about the existence of a multiverse (or at least it's portrayal by some cosmologists) in a blog post in this forum. In the meantime, last March, the announcement about the discovery of gravitational waves got us perhaps closer to a multiverse--at least to one form of it, based on inflation. And then some problems with the Bicep data were discovered.
But the idea of a multiverse, once only a fantastic dream, has now still found its way into everyday conversation. Part of the reason is that cosmic inflation seems to imply the possibility of other universes, string theory's "extra dimensions" are another gateway to a multiplicity of worlds, and the very nature of quantum mechanics--at least in one of its interpretations--favors the possibility that our cosmos is not the only one around.

As physicists have come to accept the scenario of a multiverse, despite the lack of any convincing evidence for its existence both from a verifiable theoretical framework and from any kind of confirmed experimental or observational results, this concept has already been taken to its limit: infinity. But infinity has always been the scourge of physics: Any kind of "infinite" answer to a calculation has always been taken as a sure sign that a theory had something very wrong with it. Lifetimes have been spent by theoretical physicists to correct such problems, that is, to "renormalize" theories so that the nonsensical infinite answers would disappear: QED and QCD being the most well known renormalizable theories. And while many kinds of physics calculations often make use of "potential infinity" through the infinitesimals of the calculus, no actual infinity has played a major role in the field. Not until now.

The multiverse has almost invariably come to mean an actual infinite set of worlds. But what is its cardinality (or level of infinity)? The multiverse is necessary if one wants to try to explain the fine-tuning of the parameters of our universe using the anthropic principle, rather than resorting to some kind of creationism. The idea is that in a multiverse of the cardinality of the real line, there are member universes that correspond to any precise value for all the parameters (such as the fine structure constant, the strength of the dark energy, the ratio of the masses of the proton to the electron, the ratio of the strength of gravity to electromagnetism, and others) on a continuum of values. Then all you need to do is to invoke the anthropic principle and say that we live in that member of the (uncountably infinite) multiverse in which all the parameters are what they need to be to enable us to exist and observe them as what they are.

But in fact, the continuum is very problematic when used to explain the multiverse. Brian Greene, David Deutsch, Max Tegmark, and other physicists have used the metaphor of the Infinity Hotel, or "Hilbert's Hotel," to argue for other universes and (at least in the case of Greene) the infinitely many copies of you and me running around in them, sometimes doing what we do in this universe and sometimes doing things slightly or completely differently. These arguments are all mathematically dependent on the so-called "Infinite Monkey Theorem." The idea is that a monkey randomly hitting the keys of a typewriter (the example is pre-computers, but a keyboard would do just fine) would type Hamlet, and all of Shakespeare, and all of the world's published works--if only given infinite time. What happens here is that the immense force of infinity dominates and, basically, given infinite time, anything that can happen, eventually will. Thus, the thinking goes, since for God (or any "higher power" or superior being) time is eternal, there must be infinitely many worlds with any kind of possibility happening in them. The problem is that physicists take this idea and run with it without, at times, a deep understanding of mathematical infinity.

To work infinity to solve for us the very difficult "fine tuning" problem, we need a continuous-type of infinity, and here the appropriate version of the Infinite Monkey Theorem fails. Let me explain this by proposing a new theorem.

Theorem (Aczel)
With probability one, no set of monkeys, finite or infinite, forever typing numbers at random, will ever reproduce the number

Proof: If you're good with measure theory, you could prove my theorem using a more-or-less straightforward application of the first Borel-Cantelli lemma. (While a proof of the usual "Infinite Monkey Theorem" makes use of the second Borel-Cantelli lemma.) But let me give you a cleverer and more instructive proof, which I've also devised, making direct use of mathematical analysis. We have:
$\pi = 3.141592653589793238462643383279502884197169399375105820974944...$
Forget the beginning "3."--suppose that that's already written on the initial sheet the first monkey starts typing on. And assume that all monkeys use a uniform discrete distribution on the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, although this is not essential (and you can prove that if this is not true, and the distribution is different from uniform, the monkeys fail faster to get to ).

In probability theory, we prove that a result is true "with probability one" (or "almost surely," which is analogous to "almost everywhere" in measure theory) by showing that it fails only on a set of arbitrarily small probability (measure). Arbitrary means: give me any small number you like, , and I will show you that I can make the error probability smaller than that number.

Suppose now that the monkeys perform their tasks successively. The first of the infinite (if it's finite, of course, they fail much faster) set of monkeys takes to the keyboard and strikes away, producing a random sequence of digits. I will now take the arbitrarily small you've given me and halve it. Now, the probability that Monkey 1 gets the first decimal of  (which happens to be "1") is 1/10. The probability that the monkey gets the first two digits of pi correctly (the 1 and the 4) is (1/10)(1/10). And so on. Now, given the:

I can find a number (representing the decimal digits of  counted from the beginning), N1, such that:

At that point, Monkey 1 has failed with probability one. Monkey 2 now steps to the keyboard, and continues at the point N1, where Monkey 1 has failed (I give my monkeys a break--not making them start from the beginning). I now halve my epsilon again, and through it I find N2, defined such that:

Continuing in this way, and assuming an infinite set of monkeys, who can type "forever" (meaning as long as necessary), the probability that they succeed is:

The right-hand equality follows from the sum of the infinite series (1/2)i
This sum is equal to 1. This proves the theorem.

What does it all mean? It means that if you create universes that are countably infinite then, yes, you could say that things will happen (maybe something like you and me will materialize in other universes--maybe), similarly to how a monkey might reproduce Hamlet after a really, really long time. But you can't really say anything about parameters and fine tuning. If you think that you can somehow "create" finely-tuned parameters for your universe, ones that live on the continuum of numbers (such as pi!), then you can forget about it: With probability one (that is, except for on a set of measure zero), this will never happen! Put another way, there is a zero probability that you could ever recreate finely-tuned parameters that would replicate those of our universe. What does this imply about our own universe?

Front page image credit: istock.com

"the very nature of quantum mechanics--at least in one of its interpretations--favors the possibility that our cosmos is not the only one around." I think MWI is a useful fiction for, say, quantum computation. I didn't think anyone believed they actually exist.
I wholly agree with you, Steve!! Many worlds is preposterous, in my understanding! But Brian Greene seems to like it (see how he tried to ridicule me for doubting it):https://www.youtube.com/watch?v=fJqpNudIss4

And I know David Deutsch believes in it, and--I think--also Max Tegmark in his new book.
Amir D. Aczel
Greene has a book out (The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos, 2011) so, he seems to be invested in defending MWI or the multiverse. Isn't postulating the multiverse like postulating the aether? But there is a difference. There at least was some method of demonstrating or disproving the existence of the aether--the Michelson-Morley experiment. I've built a tabletop version in this article: DIY Laser Interferometer. It was built using a child's toy but one can substitute a cat toy or a green laser pointer (even better). With the mutiverse, however, there is no method of demonstrating or disproving its existence. That is, demonstrating the existence of just one other universe (besides the one we observe) cannot be done "by any craft we here possess" (any LOTR fans?) because the distance between the observable universe and this other universe is so vast that no signal could ever reach us.
This is really cool, but is it enough? I agree that multiverse theories predict in general a countable number of universes, and that countable random choices of countable options can't reproduce a continuum. In particular, all of the decimals of pi. But there is no need that the fine tuning gives all the decimals of the constants. It is enough to reproduce them up to the point when the universe is as we see it now, including human beings like us. The fine tuning may be within a tolerance, which can even grow exponentially in time, but the deviation is not yet visible. Maybe it will become important after many years, possibly billions. In this case we can always imagine a large enough but finite number of monkeys which are able to randomly type the needed number of decimals of pi. Of course, if we would ask the anthropic principle and multiverses to ensure the possibility of intelligent living beings like us forever, then the monkeys will need to type forever. But even so, this is possible. Say even that for each extra second the number of needed monkeys have to be n times larger than it was up to that point (or the number of calculations grow exponentially with each extra decimal of pi). The number will still be countable. In the multiverse, you don't need to actualize all the decimals of pi simultaneously, just to keep adding one at a time. The monkeys only need to buy another second for intelligent life. With the expense of another branches which are sterile or in which humanity is extinct because the fine tuning was not fine enough to add that second. And if the multiverse explanation is true, perhaps already billions of billions of alternative universes failed to save life up to the present day, for every one in which it survived until now, like ours. So I think that your nice theorem is not enough. However, say we would predict that to add another day, the chances are infinitesimal. Could we check this? Unfortunately no, because in most universes we will not survive to prove the universe is no longer good for life. Only those in the lucky branches will survive, and they will witness that the monkeys succeeded typing another decimal of pi. Now, I don't support any of the multiverse explanations for the fine tuning, however, we can't say whether God picked the fine tuned constant, or the multiverse picked it. But since two explanations can't be distinguished, maybe they are equivalent. Maybe God is the Multiverse. At any rate, multiverse theories seem to me very similar to religion, by the ability to explain anything but predicting nothing. Maybe each fine tuned constant has something interesting to say, that we don't hear, because we consider that that constant has to be what it is to ensure life (we replace causal explanations with teleological, both in theological, and in multiverse theories).

Yes, of course you are right, it's really an academic point, not an actual one. What got me to write this was a comment in David Deutsch's book "The Beginning of Infinity," where he says that we need a continuous distribution. That's what got me to formulate and prove the (abstract, in this sense) theorem. Of course Deutsch (smart as he is) is wrong on another count: You don't strictly need a continuum because in every neighborhood of an irrational number there are infinitely many rationals, which can thus approximate it to any pre-specified level of accuracy. But if you think long and hard about this problem (as I have) keeping this theorem in mind, you may reach the conclusion that even a countably infinite multiverse is very hard to accept. Food for thought...
Amir D. Aczel
One thing I have thought about (and this would be in layman's as I am not nearly as brilliant as you) is that the concept of infinite attempts is foiled by infinite possibilities. I think this is similar to the fine-tuning that you are talking about, how on an infinite spectrum it would be impossible to hit an exact number, where all numbers no only include 3, 79, 1 billion and Pi, but also includes an infinite set of possible numbers, an infinite number of numbers like Pi. Referring to the infinite monkey theorem, we are already starting with a false premise, that there are a limited number of possible keys and not an infinite number. The theory is based within the limitations of our universe, and wouldn't apply to a universe able to obtain true infinity. Of course, then the infinite monkey theorem wouldn't work.

I have sort of thought about infinite possibilities within a single instance. If infinite universes exist, then on an infinite spectrum there would be an infinite number of universes where everyone does 'exactly' what they do on this universe, but also an infinite number of universes where everyone does 'exactly' what they do in this universe except one person moves their toe an extra billionth of a millimeter at a single moment in time, and everything else about the universes are perfectly similar.

Of course, how many different possibilities are there for that person moving their toe? Theoretically there is an infinite number of possibilities in a single action (like how there is an infinite number of 'numbers' between 1 and 2; 1.1, 1.2, 1.21, 1.2121212, etc.), and so even given an infinite number of attempts you will never achieve the same exact result. If you had a number with an infinite decimal, and you try to duplicate this number via randomness, you could very well get a number that is practically the same, but no matter how far you got there would be another number that you would have to randomly duplicate, meaning that given an infinite number of attempts you would eventually fail every single time, as each attempt is a 9/10 chance of failure and you would never achieve a perfect duplicate since the number you are trying to duplicate is infinite. You might have an infinite number of attempts, but concerning fine tuning (getting the millionth billionth decimal exactly the same, and the one after that, and after that, and after that, etc.) probability necessitates that you eventually fail.

The problem is that people think of the infinite monkey theorem with limits, not infinite possibilities (as there is nothing but preconceptions that mandate any arbitrary number of keys), as if there was an infinite number of letters on the keyboard somehow.

With an infinite number of possibilities, even the first attempt would fail with probability one. If the possible numbers were not just 0-9, but rather 0-infinity (including decimals), then typing the number 1 would be impossible because of an infinite number of alternative scenarios. Every attempt of these infinite numbers of attempts would almost immediately fail.

There can be no other universe exactly like this one because, like you said, achieving perfect fine tuning requires hitting a target infinitely small. It is impossible, as one person's toe could move 1 extra millimeter, or 1.1 extra millimeters, or 1.21212121 extra millimeters. Every single check as to whether or not a universe would be exactly like this one would immediately fail because of the infinite number of possibilities, such as this one. It is a false premise to assume that possibilities are finite, and not infinite.

There will always be another number that you must type, so there will always be another 9/10 possibility of failure (even when typing with limited numbers) no matter how far you get, implying an assured eventual failure. Not only this, but assuming a limited number of numbers/letters to type with is based on our own systems, rather than whatever system the multiverse would work with, and if it is able to accomplish a truly infinite number of universes it is logical to believe that it is able to have an infinite number of unique building blocks to construct those universes out of. No matter which type of multiverse theory you try to use, it fails because of one or both of these problems.

Yes, everything you say is indeed descriptive of the infinite multiverse and its problems: all these number examples. In "Many Worlds" you do have everything 'the same' except that one quantum event went one way instead of the way it went in 'our' world. This is the nub of the problem: you don't need to have uncountable infinity to see the very difficult problems you get into when you allow infinity into a physical system. For, what is truly infinite in the physical universe? The stars and galaxies we see are very large in number but finite. And even if inflation tells us that our universe is a tiny fraction of a larger collection of stars and galaxies we cannot see--why would it be 'infinite'? When infinity enters you get all kinds of paradoxes, such as a set being able to be mapped 1-1 with a proper subset of itself (some take this as the definition of infinite) and a lot more. I chose pi because it is transcendental, and so naturally belongs to the uncountable set R. Even the irrational number the square root of two is a member of a countable set because it is algebraic. You can see more on the distinction between such numbers here:http://blogs.discovermagazine.com/crux/2014/06/06/century-old-math-mystery-solved/

Amir D. Aczel
Can you really say that our current universe is not infinite with an infinite (uncountable) number of galaxies? We don't see the entire universe even with the biggest and brightest telescopes, just those in our light horizon. Beyond our visible horizon there can be ever more galaxies that we will never see because their light will never reach us. Even when the darkest, seemingly emptiest, segment of sky is viewed long enough, what appears is not an empty patch but yet another picture of hundreds or thousands more galaxies.

Our light horizon isn't the universe's horizon, it is merely our horizon.

Thank God you didn't invoke the Fermi Paradox but you did overlook the classic answer to "what is the shortest way to denote Pi?"  If written in Base Pi then Pi would be 10.  Hence, any monkey has a pretty reasonable chance of typing out all of Pi.
The clear problem is that sequence of Pi is infinite. It looks like you are essentially trying to divide infinity by infinity - so sure enough you're coming with an answer of "impossible"... However, physical parameters/constants necessary for somewhat pseudo-stable-ish universe like ours are quite finite. Change some of those parameters/constants - and a universe won't survive past inflation. As those parameters deviate more from ours' (lets say, striving to infinity, like Pi), the lifetime of those newly-created universes with those parameters strives to 0, getting infinitely short. Moreover, the shorter their lifetime is, the smaller their maximum size would be. Ultimate majority of those universes with different parameters/constants perhaps would never emerge past the Planc scale. Which would make possible values of those constants essentially constrained to being somewhat similar to what we observe in our pseudo-stable universe. Unless, of course, each of those values has an emergence chance of 1 in Infinity. I'm not a mathematician and I'm not sure whether 1/Infinity is even a mathematically possible number...
Another possibility is that speed of light, rate of expansion and density of quantum fluctuations infinitely uniform through space-time. That would make multiverse inevitalble...

Well, infinity is a very big concept. Conservation laws in physics are all based on finiteness. Just imagining the stars and galaxies going on and on 'forever' doesn't an infinity make. Either the universe is finite or infinite: we don't know. But 'uncountably infinite' means (infinitely) more than infinite.
Amir D. Aczel
This post is a bit oversimplified for a number of reasons. The first is that no measuring device could ever measure a pure number like pi to arbitrary precision, it invariably hits up against various uncertainty principles and renormalization group flows and things of that nature.

So the point is the monkeys don't have to be perfect when they are finetuning their fundamental constants, they just have to be more perfect than the accuracy that any conceivable operator/natural process would need.

The bigger problem with infinity involves how to *choose* a physical measure over the set of inflating regions (which is known as the measure problem in cosmology)

We should bear in mind though that multiverse theories do not necessarily preclude an infinite 'verse.

Furthermore, the proof invoked assumes a continuum of values available. That's how we typically treat our Math, but we're not absolutely certain at the moment that this is a trivial assumption. It may well be that the parameters of our universe can only vary by incredibly minuscule interval values, and any further "accuracy" beyond this is meaningless or doesn't exist. Consider the notion of plank lengths as a modest example.

I do agree that if such an infinite multiverse, or heck a multiverse period, turned out to be true that such is quaint and hard to believe. I agree as well that I would be evoking the "extraordinary claims require extraordinary evidence" notion. Still, the more we discover about our universe, the more common intuition is being bedazzled, and we may need to rethink some of the most "trivial" assumptions to get at the truth.

Interesting, I have created a close comparison to the random monkeys typing, but for a closed universe. This universe is solving a sudoku puzzle by purely random means (also called the brute force method) and I found some interesting results about operating systems as well as pseudo-number generators.