I have blogged before on infinity, which holds a certain fascination for me. For one thing, I'm working on a thesis involving self-reference and the retreat into infinite recursion. And early in my Presbyterian life, I had to defend divine sovereignty and the infinities of power, knowledge and goodness from the inroads of Arminian rationalism and free will. So with a little practice, I've gotten quite comfortable with infinity, sort of like driving 80mph in the dark in a thunderstorm--the important thing is not to think about it too long. In this post, I want to think about it long enough to show that the Multiverse doesn't save Darwin.

Georg Cantor, of course, couldn't stop thinking about it and was driven mad. But before he went into the sanatorium, he produced a most remarkable result about sizes of infinity. Some infinities are bigger than others. For example, take the number line from 1 to ∞. It's infinite of course. But now divide every number by the largest number on the line, and we have mapped the entire number line into the fractions between 0 and 1. So the rational numbers contain the entire integer number line between 0 and 1, and the rational numbers go up to infinity too. Then the rational numbers are at least ∞² bigger. (Yup, I'm being sloppy, because Cantor also showed how to map x²-->x, so instead of calling it ∞², he called it ℵ₀ cardinality where integers and rational numbers have the same size infinity.)

Now let's look at those fractions and write them as equivalent decimal fractions. Take any two adjacent fractions with long decimal extensions, and append a 1 or 2 to the end of the smaller one, or subtract it from the end of the bigger one. You have just inserted another fraction in between. But what if the number you insert has no ending or repeating? What if it is irrational? Then there can't be an "appending" and when we consider how many irrational numbers we can choose from, we can fit an infinite number of irrational numbers between any two fractions. Then the irrational numbers are at least ∞² bigger too. (This infinity really is bigger, and Cantor called it ℵ₁ cardinality).

There is then, a hierarchy of infinities, each bigger than the last. And while they are all big, some infinities can be swallowed up by other infinities but not the reverse.

Now mind you, Cantor got a lot of grief for this discussion of infinities. On the left side, those positivist philosophers who wanted no synthetic a priori, a group that tended to include materialists and atheists, couldn't stand the idea that there were metaphysical entities that couldn't be observed, couldn't be constructed, even in principle. On the right side, there were those dualist philosophers who had carved out a tidy corner for infinity as the principle right of divinity, and could not abide quantitative division of their metaphysical unities. These people included many liberal leaning theologians who liked their God neat--dry and detached. Cantor, on the other hand, thought God had given him this insight, a God who was both transcendent and personal, infinite and divisible, universal and peculiar. I think I found a kindred spirit in Cantor.

With this insight about sizes of infinity, let us now look at the Susskind, Smolin, Hawking debate about the need for infinite universes to explain us. The problem was succinctly stated by Sir Roger Penrose. In order to get the hugely ordered universe we observe today, with galaxies, stars, planets, trees, animals and Man, the Big Bang explosion had to be incredibly, stupendously, ridiculously accurate. That is, in order to avoid violating the 2nd law of thermodynamics, the universe had to begin with all this information "front-loaded" when it exploded. My favorite example due to Hawking, is that the explosive force of our universe had to match the gravity force so precisely at 1 part in 10^60, that had the universe possessed one grain of salt more or one less, we would not be here to talk about it. Yet nothing in all our physics books, nothing in all our metaphysics books, in fact, nothing in all our fairy tale books explains why there had to be this match between gravity and the Big Bang. Since there is no explanation, philosophers call it "contingent", a free choice, an arbitrary event.

Now if you are like Cantor's left wing critics, then arbitrary things must be random. It is a peculiar property of atheists that they all worship the god of Chance. It would seem possible that they might worship Lady Luck instead, but no, Xaos, Random Chance, must remain the king of the modernist pantheon. So this contingency drives them bonkers.

Sir Roger spares no feelings when he lays it out as far worse than Hawking's odds, a probability no greater than 1 part in 10^(10^123), a number so big that the zeroes can't be written out on all the sheets of paper in all the universe.  Fortunately I'm used to driving at 80mph with my eyes shut, but this is worse, this number ends with an abrupt stop sign, this number describes a specific location, this number describes me.

We can't write it off as metaphysical nonsense, as Cantor's left wing critics tried. We can't dismiss it as metaphysical contradiction as his right wing critics tried. It just sits there in the mirror and mocks us every time we look. What is a respectable agnostic going to do? (And we all know what disrespectable atheists like Dawkins do, but that is a different blog.)

In a previous blog we discussed the Penrose version of the Hoyle solution that brought back infinite time and infinite space, much like the Greek solutions of Plato and Democritus. Even these solutions are too ordered, too lawlike for the worshippers of Xaos, and so they proposed a new kind of infinity: an infinity of attempts, an infinity of dice rolls, an infinity of Random Chance. In the Smolin, Susskind, Hawking solution, new universes get born every second, even every attosecond. And not just copies of our universe, with its gorgeous galaxies and precise explosion, but random universes, where all the constants of physics, all the contingencies of nature are randomized. (Xaos, we worship you, we adore you, we give our theories to you. Please accept the humble sacrifice of our brains.)

So you see, they gleefully cry, even [1 / 10^(10^123)]  x  ∞ = 1! Even the most improbable events can be certain if you have an infinite number of tries.

Ahh, but does it? I mean, zero divided by zero is not one, nor is 1/∞ x ∞ = 1. Why? Well for starters, it assumes that the two infinities have the same cardinality. And Sir Roger didn't say that the information in the Big Bang was exactly 10^(10^123), he said it had to be at least that much. Why? Because that's the number you get if you take all the subatomic particles and all the photons in the universe and permute them, rearranging them into all possible orderings. But this does not take into account the non-local interactions, the long-range QM entanglements between them. Nor does this take into account the possible discretization of space-time itself, and its own entanglements, the Christoffel connections, the holographic projections. And given the ever-enlarging size of the "front-loaded" information, it might be the better part of valor to extrapolate over the past 100 years to get an idea of how much bigger it might appear to future physicists. In fact, it might be infinite.

Let's come at this from another angle. In one of the multiple interpretations of quantum mechanics, the many world interpretation due to Hugh Everett III, we ask what is the meaning of the QM wave that can turn into a particle when we look for it. Everett argued that the wave was real, so that all its many potentialities were real as well. In Schroedinger's famous example, we have a box with a cat and a cyanide capsule which may or may not have been broken, depending on the wave function of a radioactive atom. In Bohr's Copenhagen interpretation, the cat is located in Limbo, half-dead and half-alive. Now that the Pope has banished Limbo, Everett's interpretation is gaining acceptance that there are two worlds, one with the cat alive and one with the cat dead. Every time a QM wave has a choice, says Everett, the universe splits into to real actualities. Since a wavefunction of an electron is morphing every femtosecond or so, and the universe has around 10^80 wavefunctions, we have this exponentially growing number of parallel universes, all based on exactly the same initial conditions.

So when SSH create another universe by tweaking the initial conditions, we can bin it by taking all 2 dozen or so physical constants and adding them together to make a Cantor mapping to the number line. That is, we can bin these variable "initial conditions" of all those baby universes into an ℵ₀ infinity. But to make trees, and people, and my irascible QM teaching assistant, there has to be an exponentially growing number of tries from exactly the same starting condition, there has to be another infinity tucked between every baby universe, there has to be ℵ₁ infinity. So multiplying ℵ₀ x 1/ ℵ₁ will never equal one, and in fact, must be obviously equal to zero. Darwin's incredibly improbable origin-of-life, Darwin's incredibly improbable progress of evolution, Darwin's incredibly improbable consciousness, cannot be the product of ℵ₀ attempts to win the ℵ₁ lottery.

Can SSH rejigger their model to include an ℵ₁ infinity? Only by making their physical constants into irrational numbers, or perhaps, making their baby universes into irrational sets. But such rejiggering has its own consequences, not least that they are products of their irrational physical constants, which by my metaphysics, makes SSH irrational themselves.