INTRODUCTION: MY APPROACH TO PROBABILITY
This article is a response (of a sort) to Bob Drury’s excellent piece, “Prudence, Prejudice and Probability.” That article in turn, was a response to a comment (via email) I made on his article “Bayesian Reasoning in Religious Studies.”. Drury makes many excellent points in both articles. Rather than criticizing those of his ideas with which I disagree, I thought it easier (I’m a lazy guy) to set forth my own position. So, to some extent this piece will be a rehash of articles I’ve written before (ESSAY 2, How We Believe; How Science Works; Belief, Knowledge and Faith—Rational and Irrational; The Pearl of Great Price; Pascal’s Wager Revisited; Counterfactuals versus Faith) with a few new thoughts.
Now I will be the first to admit that my approach to probability is biased. Let me explain. As a scientist my use of probability was pragmatic: what was important were applications—thermodynamics, statistical mechanics, quantum mechanics, chemical kinetics. I was concerned with results, not theoretical justifications for the use of probability. After my retirement as an MRI research physicist 23 years ago, I did a year of graduate studies in statistics at Penn State. The statistics parts, applications, I aced; the math background—Lesbesgue integrals, set theory, etc.—not so much (maybe even the class dunce). So my bias: I’m more interested in what one can learn from probability than how its use is justified mathematically and philosophically.
Thus my approach is flexible and heuristic, which will probably disturb mathematicians, logicians and philosophers who require a sound metaphysical basis for applying probability. Nevertheless, I hope the reader can get something of value from what follows. I’ll first talk about faith, give a short lesson on probability, then discuss that most famous early application that linked probability to faith, Pascal’s Wager.
Probability also enters into discussions of the Anthropic Principle; and here I must say that there have been many erroneous applications of probability to this topic. In Part II of this post, I’ll discuss a probability interpretation (with which Bob Drury disagrees), an interpretation that follows Richard Jeffries—probability as a measure of belief, what you’re willing to bet on a belief. As a tool in discussing the Anthropic Principle, I’ll use a Bayesian probability analysis (and I’ll explain how Bayesian analysis is carried out and why it’s useful by giving some pertinent examples).
To begin then, a toast to faith as a foundation stone for belief.
FAITH, A FOUNDATION STONE FOR BELIEF
“Faith is to believe what you do not see; the reward of this faith is to see what you believe.”
“Faith is the substance of things hoped for; the evidence of things not seen.”
Let’s talk about belief first, that you think something is true. Chapters, books have been written, but my exposition will be brief. Clearly there is a difference between the statements “I believe in one God…” (the Credo) and “I believe it’s going to rain tomorrow.” An obvious difference is what one is willing to do or to pay to act on one’s belief. The Christian martyrs were willing to suffer and to die for their beliefs. You might be willing to bet five dollars that it will rain tomorrow, but not your life, no matter what the weather forecast is. Accordingly, there are degrees of belief, which in fact can be quantified using various techniques in subjective probability and decision theory (see “Probability and the Art of Judgment” and “Subjective Probability–The Real Thing” by Richard Jeffrey).
Just as there are degrees of belief, there are different paths to belief. There are varieties of rational inquiry (see here for a discussion of deductive logic, inductive, abductive and retroductive reasoning). There is science (see here again for a discussion of that). There is testimony, what we are told by others, and finally, there is Revelation, what we know as given to us by God, the Holy Spirit. And thus we come to faith—belief based on testimony and on Grace, belief inspired by the Holy Spirit.
Now, not everyone is convinced by the same testimony. And it isn’t a matter of intelligence or comprehension. I was led to faith after reading “Who Moved the Stone,” a book by Frank Morison evaluating the New Testament accounts of the Resurrection. Even though Morison had initially set out to disprove the Resurrection, he came to believe, as did I, that the accounts were credible. The point I want to make here is that belief is a matter of Grace, not intelligence or understanding. Other people, intelligent people, who have read Morison’s book did not come to believe in the Resurrection. I am grateful that I was given the Grace to believe. (A more complete account of this conversion is given in my article “Top Down to Jesus.”)
The title, “Top Down to Jesus,” suggests that one can come to faith by way of rational inquiry. This may be true in the sense that rational argument confirms faith, proof a la St. Anselm and St. Thomas Aquinas, as a crutch to faith, but it is not a complete story. The great 17th century philosopher, mathematician, and scientist, Blaise Pascal, argued that one could not prove God’s existence but one could come to faith as a prudential choice, by use of probability arguments. So before examining Pascal’s Wager, let’s discuss what probability is all about and how it’s used. I’ll do that by looking at a brief history of the development of probability and its applications.
PROBABILITY—A BRIEF HISTORY AND AN EXAMPLE.
Probability theory was born in the 17th century as a solution to a gambling question: in 1654 the Chevalier de Mere posed this question: how many throws would be required for two dice to show a total of “6” on the upper faces, and how should this influence his bet. The mathematicians Fermat (he of the “Last Theorem”) and Pascal (he of “The Wager”) responded to this question with a series of letters that laid the foundations of probability.
To see what’s involved let’s start with a simple example. If you took a single die (a cube with 1 dot, 2 dots,…6 dots on the six faces), and threw it a large number of times—600, say—how many times would 1 dot appear, 2 dots, … ? We assume that if it’s a fair die, not loaded, so then each face is equally likely to appear
How does this relate to probability? Let’s see what happened in our trial of 600 throws, with the number of times each dot appeared.
1 dot: 85; 2 dots: 104; 3 dots: 115; 4 dots: 97; 5 dots: 98; 6 dots: 101;
If we apply an empirical measure of probability, we would say the probability of event X, say, X = 3 dots appearing, would be the number of events X/ total number of trials, or in our example 115/600 = 1.15/6 . If we did a very large number of trials, 600,000 say, we would find the ratio, number with X=3 / 600,000, to be very close to 1/6.
What does this signify? Consider the following conventions and rules. If an event or combination of events is certain to happen, its probability is 1 (exactly). If an event is certain NOT to happen, then its probability is zero. If you have a series of possible events, as in our example: X=1 (1 dot appearing on the face), X=2, X=3, X=4, X=5, X=6, then on a single event, one or the other of the faces must show. This condition implies that
P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1 (P(X), where X goes from 1 to 6, is the probability that the number of dots X will be on the uppermost face.)
This says that the sum of the probabilities for the individual faces, the total probability is 1, certainty; in other words, one or another of the faces will turn up. Since it is equally likely that any of the six faces turns up, we can say P(1)= P(2)= P(3)= P(4)= P(5)= P(6), so each of these probabilities is 1/6. This is what’s called an “a´ priori” probability, a probability set beforehand, not based on trials, but purely on deductive reasoning.
Let’s go now to the case of two dice. To make the explanation simpler, let’s distinguish between the two dice: one die is red, the other green. The number that turns up on the red die won’t depend on which number turns up on the green die. Here’s how would we get a total of six on a throw of two dice:
Green(1) and Red (5); Green(2) and Red(4); Green(3) and Red(3); Green(4) and Red(2); Green(5) and Red(1),
or altogether five different ways to get a total of 6. How many different ways can the faces turn up? Well, each die can show six faces, so there are then 6×6 = 36 possible combinations of faces that can turn up on a throw of the two dice. Accordingly, the probability of a total of six turning up is 5/36 = (the number of ways a total of six can turn up)/ (the total number of ways the faces can turn up)=0.14 (rounding off).
THE EXPECTATION VALUE: WINNING AND LOSING IN THE LONG RUN
What does this tell the Chevalier de Mere about betting on a total of 6 for the next throw of the two dice? He wants to win over many trials, so he wants the “Expectation Value,”( his likely gain or loss over many trials) to be greater than zero. What is this “Expectation Value?” His probability of losing is 1- 5/36=31/36 (probability of winning + probability of losing has to add up to 1—either winning or losing has to occur). So if he bets one franc, then over a large number of trials, (call it N), he’s likely to lose (31/36)N francs. His probability of winning is 5/36; so he wants the amount if he wins the bet, W, times the probability of winning, (5/36), times the number of trials, N, to be greater than (31/36) N, or (5/36)NW to be greater than (31/36)N or W to be greater than 31/5.
The Expectation Value, the net gain or loss over many trials—call it E(X)—is given by
E(X)= – (amount bet) [ 1- P(X)] + (W) P(X)
probable loss probable winnings
If W= 31/5 francs (the amount won if a total of 6 turns up) the Marquis will break even in the long run [E(X) = 0]. If the stakes are greater than 31/5, he’ll wind up ahead in the long run; if they’re less, he’ll wind up a loser.
This expectation value is what enters into Pascal’s Wager, which I’ll discuss next.
First, some background (see the article linked above for a detailed account). We should keep in mind that although Pascal was a mathematician and physicist of the first order, he did not believe it was possible to show from reason alone that God exists (so much for Anselm and Aquinas!) :
If there is a God, He is infinitely incomprehensible, since having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is.
—Blaise Pascal, “Pensees,” #233.
On the other hand we can know God by faith:
But by faith we know His existence; in glory we shall know His nature.
The last part of this quote shows the route Pascal wants us to follow: there is an afterlife, and its benefits are infinite. This being so, the odds for following God are infinite; whatever one might lose in believing, even if there were no God, is finite, whereas that which one can gain from belief, if there is a God, is infinite:
But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite.
Pascal is saying that even though the probability that God exists might be very small, the expectation value, E(Belief), is finite since the win if God exists is infinite:
E(Belief)= – (Loss) [ 1-P(God exists)] + (Infinity) P(God exists) = Infinity
since infinity times a finite number, however small, is still infinity.
There are problems with this approach as a number of philosophers have shown (see references cited in the linked article). The problems arise becaue of the use of infinity as a quantity rather than a limiting situation, which use leads to paradoxes, for example the St. Petersburg Paradox.
In the linked article I give a Decision Analysis approach, a “mini-max regret” analysis to justify Pascal’s Wager. The goal in such a decision process is to choose the option that gives the least regret. Clearly, if an afterlife—heaven and hell—exist, the choice with least regret would be heaven.
In the linked article I also discuss how, according to Pascal, such a prudential approach might lead to true faith, even though belief in God is not present initially. (There is a Twelve Step saying about how to achieve sobriety even if you don’t think the Twelve Steps will work: “fake it until you make it.”)
MORE TO COME
In Part II of this article I’ll discuss Bayesian probability analysis, probability as a measure of belief, and how this might be applied to consideration of the Anthropic Principle.