Jan 15 2021

## Multiverses and the Inverse Gambler’s Fallacy

I was intrigued by an article in Scientific American by philosopher, Philip Goff, mainly because I disagree with his ultimate conclusion. He makes a very cogent logical argument, but I am having trouble with one piece of it. Here’s the quick summary:

The core enigma is the fine-tuning problem with the universe. There are a number of physical constants, such as gravity, the charge of an electron, etc., and the behavior of stuff in the universe depends on the values of all these constants. The problem is that if the values of all these constants was not pretty much exactly what they are, then complex life would not be possible in our universe. Clearly complex life is possible, because we exist, so how do we explain the fabulously improbable physical laws of the universe? To put this into perspective, Goff points out that:

The physicist Lee Smolin has calculated that the odds of life-compatible numbers coming up by chance is 1 in 10

^{229}.

The notion that this just happened by chance, and that we are incredibly lucky to exist, is not satisfying. What are some possible explanations for this highly improbable fact? One is that some powerful being (i.e. God) made the universe with these precise values so that complex life could exist. This does not solve the problem, however, it just pushes back the mystery one step – for where did God come from? I also reject this answer as an obvious “god of the gaps” argument – filling in an unknown by invoking, essentially, magic. It gets us nowhere. Another possible answer is that there is some underlying reason for the laws of physics, a metalaw, that determines that these constants must have these values. We don’t know what this could be, but at least this is something to investigate.

One solution that many scientists have found compelling is the multiverse solution – perhaps there are an infinite (or at least really big number) of universes in a grand multiverse – so many that even if the constants of each universe are determined at random, by chance alone some of them will have the values necessary for complex life. Obvious we must live in a universe with these laws, and nothing can observe universes with laws incompatible with sentience. Problem solved.

This is where Goff’s argument comes in, or rather he is reporting on the argument of several mathematicians who say that the multivere solution to the fine-tuning problem is a logical fallacy. It is, they claim, the inverse gambler’s fallacy. Regular readers here know I love logical fallacies, so this is why I am so intrigued. The gambler’s fallacy is the failure to recognize that physically independent events are statistically independent. So, if you are playing craps and hoping to roll double sixes, the odds are 1/36 that you will do so on any and every fair roll of the dice. Even if you have improbably rolled 5 double sixes in a row, the chance of doing it a 6th time is still 1/36. Past events do not affect future independent events. There are no real “streaks”, and numbers are not “due”.

The inverse gambler’s fallacy would occur if, as Goff explains, you walk into a casino and up to a craps table and the first roll you see is a double six. You might then conclude that, because the odds of this particular roll being relatively unlikely, there must be many other gambler’s rolling dice and not coming up with double sixes. While this may be true, you cannot conclude this from the one roll you witnessed. There was still a 1/36 chance of that roll being a double six, no matter how many other rolls have already occurred. In the same way, just because we observe our universe to have a highly improbable set of physical constants, we cannot conclude from this fact that there must be a multiverse of many universes. That won’t change the odds of our universe having the fine-tuning.

Goff also makes what I find to be a useful analogy, to the iconic monkey on a typewriter. If you came across one such monkey and saw that it was typing in English, would you conclude that there was something special about this monkey, or that there must be billions of other monkeys typing nonsense. But there is a flaw in this analogy, which Goff sort of addresses (and is the source of my disagreement).

I suspect that, like me, at this point most people are anticipating the major objection to this line of reasoning, and Goff does as well, addressing it head on – the selection argument. We did not just happen upon this universe at random. We evolved in this universe because it is compatible with the evolution of complex life with sufficient stability to develop sentience and ask questions about things like the inverse gambler’s fallacy.

Goff argues that the selection factor does not rescue us from the fallacy.

It is of course true that this selection effect exists, but it makes no difference to whether or not the fallacy is committed. We can see this by just adding an artificial selection effect to the monkey and typewriter analogy of the last paragraph. Consider the following story:

He gives a Joker (from Batman) analogy – Imagine you wake up in a room tied to a chair in front of a monkey on a typewriter and the Joker. The Joker tells you that if the monkey typed English words then he would set you free, otherwise he would have killed you already. So clearly you are only awake and being set free because the monkey typed in English, otherwise you would have died with no knowledge of the whole setup. In this case would you consider yourself lucky, think there must be a separate explanation for the monkey’s prose, or would you conclude that the Joker must have done this to many people who died? Goff argues that the latter makes no logical sense, and that in any case you are still lucky or there is some other explanation. He writes:

Given how unlikely it is that an ordinary monkey would come up with “I love how yellow bananas are” just by randomly bashing away, you might suspect some kind of trick. What you would not conclude, however, is that there must be many other monkeys typing rubbish. Again, what you need explaining is why

Joeyis typing English, and the postulation of other monkeys doesn’t explain this. (“Joey” in this story is the monkey.)

Here is where I disagree with Goff and by extension the mathematicians. In trying to avoid the inverse gambler’s fallacy they are, I think, committing the lottery fallacy. Let’s say you buy a single lottery ticket for a game that has a 1 in 100 million chance of winning, and you win. Would you conclude that there needs to be some special explanation for why you won? No such explanation is needed – that is the lottery fallacy, thinking that because you are the beneficiary of an unlikely event there must be a special explanation, failing to consider all the losers out there. Put another way, it’s thinking of the odds of you (or any specific individual) winning rather than anyone winning. It’s partly a post-hoc fallacy – you are asking the question after you know who won.

Therefore, in considering the odds of there being any winner to the lottery, knowing that millions of people buy lottery tickets is certainly relevant. Likewise, if there are millions of universes, that one of them won the cosmic lottery and has an assortment of physical constants compatible with life seems less unlikely than if there were only one universe. What scientists need to figure out is why there are any universes (or at least one) compatible with life, not necessary why our particular universe is compatible with life, otherwise they are committing the lottery fallacy. This is also where the selection argument is relevant – we only observe the winning universe.

To extend this to Goff’s Joker fallacy – imagine you wake up in a room with four other people. All of you experienced waking up with the monkey and a typewriter, and all of you “won” your life in that the monkey typed enough English words for the Joker to let you live. The Joker then put you back under and transported you to this room with the other winners. After exchanging stories and realizing that you all won the monkey lottery, what would you conclude?

I think in this case there are only two viable hypotheses. The first is that the Joker is lying, and the monkey test was rigged (the laws of the universe are not random). The second is that the Joker (and his minions – forget the logistics for now, and consider only the statistics) did the monkey test to very many people, and you are the five who survived. You can rule out on statistics alone that the Joker did the test fairly five times and all five of you won. You might consider as an outside hypothesis that the Joker unknowingly sourced his monkeys from a research lab that was genetically engineering them for super intelligence (there are some unknown metalaws of physics out there).

The point is, you five were selected because you won. The five of you were not chosen at random from tested individuals, in which case most would be dead. It is not viable to conclude that a fantastically unlikely event occurred.

So the real relevance to the fine-tuning dilemma is this – what can we conclude from the fact that the particular arrangement of the laws of physics, if they are randomly determined, are fantastically unlikely to be compatible with life? I think we can rule out by statistics alone that there is only one universe, and that the physical constants of that universe are determined entirely at random. Also, any “god of the gaps” argument is not helpful.

So I think we are left with two possibilities. One is that the laws are not determined at random, and there is some reason in the deeper laws of physics for why they are what they are. The second is the multiverse hypothesis – that we may have won the cosmic lottery, but there are so many universes it is highly probable at least one universe would win. We don’t need to explain why our universe won, only that some universe would win (otherwise that’s the lottery fallacy).

I acknowledge that I could be wrong in all this, and that serious thinkers clearly disagree. But sometimes statistics and logic can be extremely counterintuitive and even experts make mistakes (including me). So if you think I made an error somewhere, please point it out. Either way this should be a good learning experience.