The Probability of Peirce

It’s times like last Friday morning that fuel my determination to stick with such a monumental reading plan. The rewards of moments like that one repay the inevitable disappointments and tedium with abundant interest.

It began with Basil of Caesarea. But I didn’t know it at the time, so I’ll jump ahead in my story and come back to the Cappadocian Father later. My sense of the onrushing confluence of wonderfulness began when Charles Peirce, exploring the meaning of possibility, told me that odds mean nothing in a single case. As a game player, I’ve been fascinated with the mathematics of probability since I was a tot. And I’ve puzzled over this same conundrum many times. As much as I know how to calculate odds, what do they actually mean in a single instance? Here’s an example. I once went to the pound to find a dog for my kids and saw a beautiful cocker spaniel. I went to claim the dog, but the fellow at the front desk told me it wasn’t clear for adoption yet and that I would have to come back that Saturday. “So I can have it then?” I asked. “If you’re the only one who shows up. Otherwise, you’ll participate in a drawing.” So I showed up Saturday (without telling my children where I was going – thank goodness!) to find three other people hoping for the pretty puppy. One-in-four, I cleverly told myself (displaying my dazzling grasp of the math). And yet, I thought, what difference does that make? The agent is going to dip his hand into the jar and touch one slip of paper. Just one. That paper will either have my name on it or not. It won’t contain one-fourth of my name. I won’t return home with one fourth of a cocker spaniel. My kids won’t be one-fourth happy. In a minute, there will be a Fact, and that Fact will be either black or white.

Peirce illustrates his idea for a while with dice. But his explanation becomes clearest when he proposes a scenario involving cards. Suppose, he says, you were shown a deck of twenty-five red cards plus one black card and a different deck with twenty-five black cards plus one red card. Then suppose that you had to draw the top card from one deck and that a red card would immediately take you to a life of bliss. (I would add that drawing a black card dooms you to execution.) Of course you choose the deck with twenty-five red cards. But still that top card is what it is, even though you can’t see it. Touching it and drawing it won’t change its color. So what do odds mean in this case? Peirce’s form of the question is, Is it logical to choose the deck with twenty-five red cards? If you think only about yourself and you happen to draw the one black card, thinking that you acted logically provides absolutely no comfort as you are led to the scaffold. Your end is what it is. However, if you think of a large number of people given the same torturous option, you can begin to say that the thinking is logical.

Thus, he concludes, "logic is rooted in the social principle." This is the logic, he says, that convinces a man to storm an enemy fort with the rest of his unit. “I will probably die,” he reasons, “but enough will make it through to achieve the goal I desire.” So we have to be ready to think of others ahead of self if we are to call our thinking logical. He even goes so far as to say that logical thinking requires three virtues that look suspiciously like Charity, Faith, and Hope. When this (in my experience) unique approach ends up suggesting that logical thinking equals Christian thinking, it garners my fullest attention.

So probable logic makes no sense in the individual case, only in the long run, and probable logic is where we do the vast majority of our thinking. (Of how many outcomes can we be 100% sure?) So to think logically, we must be in the habit of thinking about the long run. But now go back to the die. The odds of rolling a 3 or a 6 are 1 in 3. If a particular game comes down to a final throw by your opponent, and he must roll a 3 or 6 in order to win, probability doesn’t mean anything when he succeeds. Your saying that he shouldn’t have won makes no difference in the world. He has won, and that’s that. Probabilities mean nothing in a single case; the 1-in-3 odds only make a difference in the long run. But the run can’t be too long, either. If your concept of odds rests on imagining the outcome of an infinite number of throws, then all odds (other than 0) become the unhelpful infinity-in-infinity.

The final fascinating facet: A chance of failure will, given enough opportunities, come about. Every gambler, he says, will come to ruin if he continues; with even the best strategy, his losses are bound to exceed his resources at some point. Similarly, no insurance company can last forever; there is a chance that widespread disaster resulting in claims exceeding the company’s assets will occur, so given enough time, that chance will come about. Peirce then applies the lesson to even grander scenarios. Every civilization is bound to fall. A human with the power to cheat death will one day eventually find that every earthly thing he has ever trusted in will have failed him. He’s on the verge of stating the Second Law of Thermodynamics, and I’m amazed (perhaps as a result of my ignorance of physics) that he doesn’t see it. Since every atom has a chance of coming apart, there will eventually come a time when they will all have fallen apart. But he doesn’t see his principle through to this bitter end. In fact, he says “there can be no reasons for thinking that the human race, or any intellectual race, will exist forever. On the other hand, there can be no reason against it.” Now maybe he specifies “intellectual race” in order to allow for the eternity of the soul and God’s restoration of the material world. But it sure seems like he misses the golden opportunity to discover the (barring divine intervention) inevitable decline of the universe into utter entropy.

The idea of the dissolution of the elements probably stood out so prominently in my mind Friday morning because of two of the coincidences that constantly attend my reading. I had just read, a few days earlier, one of several passages by C. S. Lewis in which he critiques some of his materialist contemporaries’ belief in a gloriously perpetual evolution of mankind. But even more astonishingly, just thirty minutes earlier I had read Basil the Great’s fourth-century anticipation of the Second Law. It is foolish to believe in the eternity of the material universe, says Basil. The philosophers who teach it contradict themselves when they also teach that all material things are subject to corruption and death. Don’t they then see, he asks, that the time will surely come when corruption and death will have happened to everything at once?

Three references in one week to the inexorable fate of the Long Run! I mean, what are the odds?