Let’s Make a Deal

I’m a fan of Michael Shermer, a prominent atheist and skeptic who writes a column on skepticism for Scientific American. But in an article entitled “How Randomness Rules Our World and Why We Cannot See It” he has fallen for a mistake common among atheists today. The mistake involves misunderstanding the relationship between probabilities and the physical world—specifically assuming that the physical world is probabilistic in nature. This is wrong. In fact there is no relationship between probabilities and reality at all. Probabilities relate solely to our knowledge or lack of knowledge of something, and as such can tell us nothing at all about the nature of reality.

On its surface, this may not appear to have much to do with atheology. But if you stick with me you will see that the surface is misleading. Underneath the surface this is all about the nature of reality and therefore it is about what sort of natural worldview, if any, fits with the facts of our existence.

Most atheists today, I would guess, assume a version of naturalism based on scientific realism.  The underlying assumption of scientific realism is that correct scientific knowledge is possible and when obtained that knowledge uncovers the fundamental nature of the physical world. I for one think that scientific realism is off base. I don’t think it’s compatible with an evolutionary explanation of the origin of the mind. Since as I have argued elsewhere, naturalism is the proposition that mind was not present from the beginning but came into existence later, it follows that naturalism requires an evolutionary explanation for the mind’s advent. One purpose of this blog is to try to make the case that scientific realism should be rejected by atheists and advocates of naturalism.

My version of naturalism is based on neurological constructivism, the view that knowledge is a model of the world constructed by the brain simply because it’s useful for survival. As such, knowledge is about usefulness, not truth. Our minds evolved to develop knowledge models of the world based on the application of pragmatic empiricism. If I were to give a one sentence explanation of pragmatic empiricism, I would say that it is the idea that there is no way to verify our knowledge of the world against the world itself, other than to observe its usefulness.

I will write more about neurological constructivism and pragmatic empiricism in the future. I mention them here only to give the reader a bit of context for what follows. If we adhere to scientific realism, we assume that probabilities are inherent in things, and we may even conclude that randomness is inherent to reality.

Imagine that you are a contestant on the classic television game show Let’s Make a Deal. Behind one of three doors is a brand-new automobile. Behind the other two are goats. You choose door number one. Host Monty Hall, who knows what is behind all three doors, shows you that a goat is behind number two, then inquires: Would you like to keep the door you chose or switch? Our folk numeracy—our natural tendency to think anecdotally and to focus on small-number runs—tells us that it is 50–50, so it doesn’t matter, right?

Wrong. You had a one in three chance to start, but now that Monty has shown you one of the losing doors, you have a two-thirds chance of winning by switching. Here is why. There are three possible three-doors configurations: (1) good, bad, bad; (2) bad, good, bad; (3) bad, bad, good. In (1) you lose by switching, but in (2) and (3) you can win by switching. If your folk numeracy is still overriding your rational brain, let’s say that there are 10 doors: you choose door number one, and Monty shows you door numbers two through nine, all goats. Now do you switch? Of course, because your chances of winning increase from one in 10 to nine in 10. This type of counterintuitive problem drives people to innumeracy, including mathematicians and statisticians, who famously upbraided Marilyn vos Savant when she first presented this puzzle in her Parade magazine column in 1990. —Michael Shermer, “How Randomness Rules Our World and Why We Cannot See It”

Shermer’s explanation of why the contestant should switch is incomplete and inadequate. Surprisingly, Marilyn vos Savant and before her Martin Gardiner, who presented an earlier version involving cards in his long-running column on mathematical games in Scientific American, have misled Shermer. Their conclusion about the probabilities that apply in this situation only holds if two unstated (and potentially false) assumptions are in fact true. To arrive at Shermer/vos Savant/Gardiner’s calculation of probabilities, we must (1) assume that Monte knows the winning door and (2) assume that Monte intended to reveal a losing door no matter which door the contestant initially choose.  In other words, so long as we can safely assume that Monty will reveal a losing door but not the winning door before offering the chance to switch, the contestant should switch. The problem is, our experts here seem to be unaware of these underlying assumptions, or have failed to consider the possibility that they are false.

Shermer, Gardiner, and vos Savant have erred in this fashion because they mistakenly believe that probabilities are inherent to physical situations. They fail to realize that probabilities are not “discovered” in the physical world around us, but are the result of judgements we make about what we do or do not know—and even, we shall see, about what we believe others do or do not know.

Consider this virtually identical situation, in which there are still exactly 3 possible door configurations: (1) good, bad, bad; (2) bad, good, bad; (3) bad, bad, good.  Monte lets you pick a door, and you pick door #1, exactly as in the original. But then Monte brings another contestant up on stage and lets her pick a door—she picks door #3. Monte now reveals that door #2 has a goat behind it, just as in the first case. Now what should you do if Monte offers you and the other contestant the opportunity to switch doors?

Well, according to these experts, your “folk numeracy” misleads you into thinking it doesn’t make a difference—that your chance is the same with door 1 or door 3. But that would be wrong, according to “expert numeracy”.  After all, to quote Shermer again,

Our folk numeracy—our natural tendency to think anecdotally and to focus on small-number runs—tells us that it is 50–50, so it doesn’t matter, right?

Despite your intuition that two doors remain and one has as good a chance as the other, according to Shermer et. al. you should in fact jump at the opportunity to switch your choice from door 1 to door 3 because, since door 2 has been revealed as a loser, door 3 has twice the probability of being the winner than does your original choice.

But wait—this analysis holds not just for you but for the other contestant as well. According to “expert numeracy” both of you should jump at the chance to switch. The door on the other side really is greener—each of you has a two-thirds chance of winning if you swap choices.

Obviously, there is something wrong with “expert numeracy”. Nothing materially has changed about what’s behind the doors in this second scenario. In both cases, Monte knows which door is the winning door (or more pertinently, as contestants, you assume he does). In both cases, Monte reveals the middle door to harbor a goat. In both cases, there are “three possible three-doors configurations: (1) good, bad, bad; (2) bad, good, bad; (3) bad, bad, good”, but the conclusion that there is an advantage in switching is now false.

In short, the experts have missed something here. They have failed to realize that probabilities differ based on what each individual doing a probability calculation knows. What has changed between the two scenarios is Monte’s motivation in revealing door 2, and the option he had (or might not have had) in following that motivation.

In the first scenario, Monte is presumed by the experts to deliberately want to reveal one of the “bad” doors. If he only has one contestant, there will always be a “bad” door to reveal. But if there are two contestants, then 1/3 of the time there will NOT be a “bad” door to reveal, so Monte can only make the switch offer 2/3 of the time. That materially changes the odds.

But even this is an insufficient analysis. Imagine that a Professor somewhere has carefully studied Let’s Make a Deal, and in that study has observed that 75% of the time when Monte makes a switch offer, it is to a contestant who has chosen the winning door. So let’s go back to our first scenario with the single contestant—but with one difference: our contestant happens to have read about the Professor’s observation. Does that change the odds for that contestant? You bet it does! (But only if the contestant considers the Professor’s study reliable.)

I hope my point is clear: probabilities only pertain to our knowledge (or lack of knowledge) of the physical world—probabilities do not pertain to the physical world itself, never have and never can.  If one wants the simplest possible proof of this, it is found in the fact that probabilities can differ for each observer. Consider Let’s Make a Deal again. Up on that stage, Monte knows which door is the winning door—so let’s ask a question: what is the probability that the door Monte knows to be the winning door is in fact the winning door? For the contestant faced with picking a door and knowing nothing beyond the fact that there are 3 doors, each door must be assigned a 1/3 chance. But for Monte, two doors have virtually no chance and one door is a virtual lock. (It’s not 100%, however, because Monte may have remembered incorrectly, or been misinformed by the show’s producer, or a rare snafu may have resulted in the prize being put behind the wrong door). Monte and the contestant have different sets of knowledge, and so the probabilities differ depending on whose perspective we choose.

Different observers have different knowledge and therefore properly assign different probabilities. And this means simply that probabilities are not inherent in things. “Randomness” does not rule our world, any more than “certainty” rules it.

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