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Fine tuning, likelihood, and "specialness"
« on: March 26, 2016, 07:07:34 am »
Imagine you buy a lottery ticket. What's the probability that you will win?

To answer, there are several ways that you might approach this question. The most obvious, perhaps, is simply to look and see what the space of possible winning numbers is. A pretty standard lottery set-up looks like this:

Six balls are drawn in order without replacement from an effectivelyy randomized hopper containing 50 uniquely marked balls (marked, of course, 01 through 50). This means there are 50*49*48*47*46*44, or 50!/(50-6)! if that makes the math clearer, possible winning numbers.

The probability of you winning, then, is



Well, sort of. This is the probability of you winning *given* a certain set of assumptions: namely, that the mechanism actually does work as described, that the hopper is properly randomized, and that no-one is cheating. In the "fair" lottery, this is your chance of winning.

Incredibly low.

So, let's say you win.

Now. The probability of you winning, given that the game was fair, is incredibly low: 1/11,441,304,000. Does it mean, however, that we should conclude that the game was *not* fair? Should we conclude, instead, that the game was rigged?

No. Of course not. That's essentially the objection that many opponents of the Fine Tuning Argument try to get at, and it seems intuitively true: just because something is extremely unlikely to have happened as a result of "random chance" doesn't mean that we should conclude that some agent was looking out for you.

The interesting question, though, is why? This might seem intuitively obvious, but our intuitions are often wrong when it comes to matters of logic and probability. Is this conclusion actually right? Would it be more rational of us to conclude, when we win the lottery, that the lottery was rigged in our favor?

And, if not, why not?

The answer lies in considering the space of alternative hypotheses. After all, you *did* win the lottery. It had to happen...somehow. So what is the alternative to a fair lottery? Well, basically an unfair lottery.

What is the probability of winning the lottery if the lottery was *unfair?* Here, obviously, we run into a problem. Unfair in what way? Is it unfair because the hopper is more likely to drop ball 50 than it is to drop ball 35? Is it unfair because the company faked the results? Because a clever engineer programmed the hopper to produce a specific result?

Let's consider the last one for a second. Assume that there exists a clever engineer capable of programming the hopper to produce a specific result. What is the probability of you winning the lottery given this hypothesis?

Well, again, of course, the probability is 1/11,441,304,000. Why? Because this hypothesis doesn't give us any reason to think that the engineer would prefer *your* number to any other. Your number isn't special in any particular way. Even though our hypothesis specifies the engineer's *ability* to rig the game, it tells us nothing about what result our engineer would rig the game *for.*

And, of course, this probability of winning given some hypothesis--this term is called the "likelihood" or, equivalently, P(E|H) where H is the hypothesis and E is winning (or, generically, the evidence in question) is not the only term in consideration. We also have to consider the prior probability of our hypothesis, P(H).

Prior to knowing that you won, which would you have said is more likely--that the lottery is fair, or that some engineer is secretly picking the results?

Obviously, the fair game. With a higher prior and the same likelihood, then, the "fair" hypothesis actually comes out ahead of the "cheating engineer" hypothesis.

What does this say about the Fine Tuning Argument, then?

In the fine tuning argument, we have two hypotheses that are compared against each other:

On the one hand, we have the hypothesis that some unspecified designer selected the physics of our universe from the vast space of possible physics.

On the other hand, we have the hypothesis that our physics were selected at random from that vast space of possible physics.

(This is not at all the complete hypothesis space, and that is the focal point of my own favored objection, but it's not the issue, here.)

The probability of getting life supporting physics on this second "random chance" hypothesis is astronomically low, but--just like in the case with the lottery--the probability of getting life supporting physics on the "designer" hypothesis is equally low.

This is where the "specialness" objection raises its head. In order to actually produce a "design" hypothesis that is any better than a "random chance" hypothesis, one must complicate his "design" hypothesis with details which specify not just why "life" is special to *us* but why life is special to the hypothesized designer.

The generic, unspecified "there exists some designer of the universe" hypothesis simply doesn't produce a higher likelihood of an LPU than the "random chance." hypothesis. Some arbitrary assertion of "specialness" is required. 
P((A => B), A) = P(A => B) + P(A) - 1