# Darwin was probably right

In Michael Ruse’s Chronicle Review piece What Darwin’s Doubters Get Wrong, Ruse reveals the surprising (to me) fact that esteemed philosopher and atheist Thomas Nagel has been flirting with Intelligent Design:

Nagel leaves the reader with the impression that Behe’s concerns are well taken. Behe, according to Nagel, argues that “widely cited examples of evolutionary adaptation, including the development of immunity to antibiotics, when properly understood, cannot be extrapolated to explain the formation of complex new biological systems.

These, he claims, would require mutations of a completely different order, mutations whose random probability, either as simultaneous multiple mutations or as sequences of separately adaptive individual mutations, is vanishingly small.“

Of course, arguing about the probability of an event after it has already happened is worthless. Maybe the probability of some event occurring is vanishingly small, but the probability of that event occurring *given* that it has already occurred is obviously one. You cannot cite the improbability of that event as evidence that it didn’t happen.

Imagine that we put 6.7 billion names in a hat and draw one of them. We are going to declare the winner “The Face of Humanity.” We all have a 1 in 6.7 billion chance of winning this contest. That is, after rounding we each have a zero percent chance of winning. But just as obviously, there is a 100% chance that *someone* will win. If Thomas Nagel were introduced to the winner, according to the above logic he would dismiss her by saying, “Sorry, the probability of your being the winner is vanishingly small, so you must not be the winner. In fact, no given person could be the winner, so there could not have been a lottery at all.”

Now replace “1 in 6.7 billion people” with “1 in 6.7 billion possible sequences of mutations,” and it should be clear why Nagel’s argument is silly.

When are arguments about the unlikeliness of an event permissible? I think mainly when you are weighing alternative explanations for a phenomenon. Imagine I am standing in your house. The question, “How likely is Carl to be standing in my house?” is now irrelevant. But you can ask if I got there by going through the door, through the window, or if a random assemblage of molecules collide and formed something that looked exactly like me. Two of these explanations can rightly be deemed unlikely relative to the other.

No matter how unlikely the emergence of the eye or human intelligence may be, *a priori* they don’t seem to me any more unlikely than the infinity of things that don’t exist: ESP? Teleportation? Immortality? Pyrokinesis? Seven arms? Wheels for feet? Skin crusted with orange rocks? Tri-sexual reproduction? Why not! These things just sound all the more outlandish *because* they don’t exist. The relevant question is: what biological mechanism seems most likely, given the entirety of biological data we see around us?

So unless Nagel (an atheist, mind you) has an explanation for the formation of complex biological systems that is more probable than natural selection, he should really stop citing probability theory as evidence for his position.

Imagine that we put 6.7 * 10 raised to the 70th power unique names in a hat. We draw a name entirely at random, and someone on this planet wins. The author of this blog post then says, “Of course, what did you expect? Even though the chances of any one person winning were vanishingly small, someone had to win. So there is nothing to explain. Besides, the lottery already happened and arguing about probability after the event is worthless.”

What is worthless, in fact, is using an analogy where the probability of someone or other winning is exactly 1.0 due to the designed match up of the count of tickets and the number of participants (more generally, if lotteries weren’t designed to produce fairly regular winners, no one would participate). Absent this contrived example, your post is quite beside the point.

But I don’t really expect you to understand this. Those who employ such arguments rarely do.

You are right: I really don’t understand your point.

“What is worthless, in fact, is using an analogy where the probability of someone or other winning is exactly 1.0 due to the designed match up of the count of tickets and the number of participants”

The Law of Total Probability states that sum of the probabilities of all possible outcomes of a trial is equal to 1. My analogy drew on the Law of Total Probability.

How is this controversial? How is this contrived? It is simply axiomatic.

If you expanded your argument, perhaps I could actually respond to it.

Does this help clarify my argument, Matteo?

Nagel is basically saying: If 10 billion years ago you asked a probability theorist, “What are the chances of the human eye evolving through a process of millions of random mutations?”, the probability theorist would respond, “Vanishingly small.”

So Nagel concludes that millions of random mutations could not have led to the human eye.

But the fallacy here is that Nagel is not conditioning his assessment on the observed fact that the human eye does exist.

Rather than asking, “10 billion years ago, would we have expected the evolution of the human eye?”, we should be asking:

“It is a given that the human eye exists. What is the most probable explanation for this observed fact?”

Because Nagel is an atheist, he is not going to cite Intelligent Design. (I understand that my argument has no power in adjudicating between Darwin and ID.) So I am criticizing Nagel for dismissing the most likely mechanism without proposing a more probable mechanism.

In short, the probability of the human eye is irrelevant here. It’s the probability of the evolution mechanisms we should be debating.

“In short, the probability of the human eye is irrelevant here. It’s the probability of the evolution mechanisms we should be debating.”

Agreed.

However, your original lottery example remains something of a red herring. The “large” number of lottery tickets has no bearing whatsoever on the logic of your example. You could have just as easily said, “Bob and Jim throw their names in a hat. Then they draw a name. That name will be either Bob or Jim. Therefore Nagel is wrong.” Or, without changing the logic at all, you could also said “Bob puts his name in a hat. He pulls it out, and lo and behold, the piece of paper has ‘Bob’ on it. Therefore Nagel is wrong.” Or how about this: “Bob’s name is Bob. Therefore Nagel is a fool.” The logic is essentially the same in every case, and equally irrelevant in every case. The use of the lottery example really adds no logical force to your post, so I question why it is there at all, except to create a certain rhetorical effect. I see such examples used quite frequently, and it does seem to tickle the ears of those of a certain ideological bent, but rationally it simply amounts to misdirection.

The lottery analogy directs our attention to the separability of the data-generating mechanism (lottery:natural selection) and an outcome (winner: evolved trait):

Nagel is caught up in judging the probability of the outcome. Because the probability of any one outcome (viewed from time t = 0) rounds to 0, Nagel (at t = 1) looks at this unlikely outcome and decides that the lottery itself doesn’t exist.

This is a fallacy (and needing that probability to round to 0 is why the large number of participants in the lottery is relevant).

Nagel should be asking: Is there a mechanism that better explains our observed outcome? If he can propose a mechanism that is more likely than the lottery to have produced our observation, then he has a valid probabilistic argument.

But his argument that “the probability of the eye evolving is vanishingly small” is the red herring that distracts us from discussing the likelihood of the lottery itself.

—

Familiarity with the statistical theory of Maximum Likelihood Estimation makes my argument much clearer. Where theta is a parameter (equivalent to a mechanism for our purposes): P(theta | data) is proportional to P(data | theta).

P(data | theta) will most likely be vanishingly small. In fact, the more data you collect, the smaller P(data | theta) becomes. BUT, the more data you collect, the stronger is your confidence in your inference of theta.

In other words, the only relevant inference criteria is the ordering of P(data | theta) for different values of theta. The absolute value of any given P(data | theta) is entirely irrelevant, even when P (data | theta) approaches 0.

—-

Anyway, the logic underlying maximum likelihood estimation is very real, and I did not just include it for rhetorical effect. I still think the lottery analogy works well, and that it is not at all reducible to “Bob’s name is Bob. Therefore Nagel is a fool.”

“If he can propose a mechanism that is more likely than the lottery to have produced our observation, then he has a valid probabilistic argument.”

Unfortunately, you’ve given an analogy where there *is* no mechanism more likely than the lottery, because you have set up a rhetorical example wherein the lottery probability is precisely 1. With n tickets, and n players, it is 100% guaranteed that your lottery will have a winner contained in the set of n. Therefore, your example is a tautology with no bearing on the argument.

You say “His argument that “the probability of the eye evolving is vanishingly small” is the red herring that distracts us from discussing the likelihood of the lottery itself.”, yet you keep defending a lottery example with a fixed probability of one. And then you cast blame on Nagel for not coming up with something that beats 100% odds.”

Look, I don’t necessarily disagree with your main point, but I do disagree that your lottery example has any bearing on it. Obviously the probability of the eye being here is 100%, given that it is here. But what does this have to do with the question of whether it got here via an undesigned process?

“Obviously the probability of the eye being here is 100%, given that it is here. But what does this have to do with the question of whether it got here via an undesigned process?”

Exactly. P(Eye) = 1. Not that interesting. But Nagel has instead stated that P(Eye | Evolution) approaches 0, therefore P(Evolution) = 0.

Granted that P(Eye | Evolution) is very small, the eye has nonetheless been observed. So the relevant test statistic is the ratio L(Evolution | Eye) – L(Other Mechanism | Eye) which is proportional to P(Eye | Evolution) – P(Eye | Other Mechanism).

Until Nagel frames his argument in these terms, he should not be making a probabilistic argument about evolution.

But you mainly seem hung up on the lottery analogy. Yes, in the analogy I assumed that the lottery actually took place. But that’s what analogies are for: you create a model, you let the model run, and then you see what outcome the model produces. WE know the true data-generating process; the characters in the analogy do not. So for them, P(Lottery) does not equal 1. The question is: do their inferential processes lead them to a reasonable conclusion?

Running simulations is a very standard way to gauge the bias, precision, and consistency of statistical estimators.

All I have shown is that Nagel’s logic leads him to the wrong conclusion in the situation where the lottery is the true data-generating mechanism. By the same token, Nagel’s logic will lead him to the wrong conclusion if evolution is the true data-generating mechanism. Thus Nagel’s logic is ill-suited for the question at hand.

Because (in the real world) Nagel is not comparing the likelihood of two different evolutionary models and (in analogy world) Nagel is not comparing the likelihood of two different contest winner models, Nagel cannot make any valid inferences as to the likelihood of any single model.

This is an axiom of maximum likelihood estimation. The true probability of a data-generating mechanism is – and is necessarily – unknown. The only inference we can make is to which of a set of models is the most likely, given the data.

—

“With n tickets, and n players, it is 100% guaranteed that your lottery will have a winner contained in the set of n. Therefore, your example is a tautology with no bearing on the argument.”

Look, if you think it matters, pretend I wrote that there are 100 billion tickets but 994 billion go unclaimed. Then Thomas Nagel is introduced to someone he is told is the winner, and he concludes that this person can’t be the winner, because the probability that this person would be the winner was 0%. So he concludes there was no lottery.

The 994 billion unclaimed tickets are irrelevant. Nagel’s probabilistic reasoning is still flawed. The analogy still soldiers on.