My post on infinite Earths inspired some discussion with folks who have thought about it, and I thought I’d consolidate my ideas.
The argument, as Derek Fox points out to me, boils down to two assumptions: infinity and stochasticity (or randomness). If the Universe is random and if everything in it is basically governed by the same natural laws, subject to true randomness, then it must be true that there is some (very large) distance at which there is an arbitrarily close copy of you. Either assumption may be wrong, but I think if they are both correct then the argument holds.
Julia Kregenow reminds me that we really don’t know that the Universe is infinite, and in fact we may never know. This may seem obvious, but it’s actually a recent development. The Universe has a global curvature that we can measure; if it has a “closed” geometry then it must be finite, and if it’s open then it must be infinite.
As an analogy, imagine that you are a sailor on the equator of Kevin Costner’s Waterworld (i.e. almost no land on the whole planet), and you want to know if the ocean goes on forever. You drop a buoy anchored to the sea floor and then set your compass point for true north and head out some distance until you’re really far North, near the North Pole. You then turn 90 degrees left and head out exactly the same distance (so you’re back at the equator), then turn 90 degrees left again and that same distance later you find the buoy! Two 90 degree left turns, and you are back where you started (try it on a globe). If you turn left 90 degrees you could do the trip again, forming an equilateral triangle with three right angles. You can conclude that the ocean is not flat; in fact you must have gone a quarter of the way around the planet each time. It must be that you could fill the ocean with a finite number of buoys.
But now imagine that the ocean is flat and infinite: no matter how far you sailed, you could never do an experiment like I described. Any time you tried to make a closed path, the angles you turn would add up to 180 degrees (not 270, as in the example above). Either the ocean has an edge, or the ocean is infinite; it can’t be finite, unless it is so huge that you just can’t get far enough to detect the curvature.
The Universe is, as best as we can tell, “flat” (in a 4-D spacetime sense). Either it is infinite, or it is finite but so huge we can’t tell the difference. So we might get out of the infinite Earths argument by assuming the Universe is finite, but there’s no way we can ever tell if this is correct (even if we get so good at measuring curvature that we eventually find some, it could just be a local wrinkle, a swell in the cosmic ocean fooling us into thinking it’s curved).
The next component is that it is random. If I write out the number pi in binary, I get an apparently random collection of 1’s and 0’s that go on forever and never repeat. I can also look at the binary representation of the Complete Works of William Shakespeare as a big string of 1’s and 0’s. As I look at more and more bits of pi, then the probability that I will find exactly this string grows with every set of numbers I look at, if the numbers are truly random (which, I will note, they are not). There must be some number of bits so large that the probability is so close to 1 that I will be satisfied that the Romeo and Juliet is, indeed, encoded in pi. Of course, also encoded in pi is a version of Romeo and Juliet where all of the Capulets are named “Beavis”, all of the Montagues are named “Butthead”, and all of the swords are whoopie cushions.This metaphor maps (almost) perfectly onto the Universe; in fact, it’s already been done. In The Library of Babel Jorge Luis Borges1 imagines an infinite library filled with identical rooms filled with identical books filled with identical numbers of pages with identical numbers of lines each with an identical number of characters from a set of 25. The twist is that each book is filled with jibberish, and each book is different. The implication is that the library in infinite (or effectively so) and that every possible book is somewhere in the library. Like the pi example, there must be Shakespeare in there somewhere.
But this is only true if the books are random and the library is infinite, or if the books are ordered (in a complex way, perhaps) and every single possible book is present (in which case the library could be finite). Likewise, if the Universe is random and infinite then eventually you have to find Shakespeare (the actual guy, writing the actual plays) over and over, but if there is some underlying order then this needn’t be true.
Or, as Derek pointed out to me, it is fallacious to say “I have an infinite number of teapots, one of them must be orange” (or, more precisely here, “one of them is orange so there must be an infinite number of orange ones”). This would only be true if the colors of the teapots are random and include orange as a possibility.
To get back to my pi example, the bits of pi are definitely NOT random, they just seem that way. I don’t think anyone has proven that Shakespeare is or isn’t in those bits, and it may be unprovable for all I know. Similarly, we don’t know that the Universe is truly stochastic in the way necessary for the infinite Earths argument to work. (We know that it is fundamentally random at the quantum level, but we don’t know how the laws of physics might be different very far away from us — for example if c increases along the z axis, e along the y axis, and h along the x axis, then the Universe would be ordered in such a way that there is certainly no repeat of us anywhere).
I actually find the finite Universe to be a more satisfying way out. If the Universe is finite in the spatial dimension, then infinity is a purely human concept; there is nothing infinite about the Universe (or infinitessimal; the Planck length means you cannot even infinitely subdivide things). Yes, the Universe might go on “forever” in time in principle, but it did have a definite beginning, so at any point in the future the age of the Universe is finite, and the future is still unwritten (the Universe is not deterministic).
Science teaches us a lot; but the fact that the Universe is very flat means the answer to the infinite Earths question one isn’t one of the things it can teach us (but it’s profound to me that it took an actual physical experiment to figure that out!)
[Image from this link to the Library of Babel].
1Borges was not unaware of this parallel. The first words of The Library of Babel are “The universe (which others call the Library)…”.