Sacred Probabilities — Part I

In the first of a three-part series on the theological implications of economic probability, Economist Philip Pilkington examines Pascal’s Wager.

Probabilities have not been given their theological due.

Augustin Pajou, “Pascal studying the cycloid” (The Louvre, 1785)

This is understandable, as probabilistic reasoning has long been thought to be a subfield of mathematics and, outside of Biblical numerology, mathematics seems to have little to say about theological questions. To think this, however, is misleading at every level, from root to branch.

It is a misunderstanding that has actively been peddled by the promoters of probability theory itself. Many of the promoters of probability theory are themselves mathematicians and they are often in the business of selling their skills to either researchers in other fields or even to companies and governments – modern day haruspices, reading the omens of the entrails at court. Needless to say, the marketing for such activities is much easier if the practice can remain cloaked in the language of mathematics and science.

The scientific proponents of contemporary probability theory typically think of probabilities as being objective entities.

Take the example of a coin.

Provided that the coin is balanced, there is a 50% probability of flipping a heads and a 50% chance of flipping a tails. Scientific proponents of probability theory think of this as being an objective, almost material relationship not unlike the exertion of Newtonian force on an object. Sometimes they will refer to subjective probabilities, also known as Bayesian probabilities, but this is just a method for increasing the accuracy of a given probability as more evidence becomes available. The probability is still seen as a sort of objective entity.

The alternative view was laid out in John Maynard Keynes’ now sadly neglected 1921 book A Treatise on Probability. Keynes did not treat probabilities as objective entities but rather as an assessment of a proposition rather than the as the assessment of an event. So, when we look at a coin toss, the probability does not apply to the coin toss itself, but rather our judgment of it. When we say: “The coin toss will produce heads,” it is this proposition that is being assessed in terms of probabilities. This brings probabilities out of the realm of actual objects and into the realm of our epistemological capacities. Probabilities, for Keynes, are a species of inductive argument of the sort discussed by the likes of Leibniz, Berkeley, Hume, and Kant.

It is at this point that probabilities become theologically interesting. In what follows we will examine three different cases of probabilities being of theological interest.

The first is the well-known case of Pascal’s wager in which probabilistic argument is deployed as an argument for the existence of God, although as we shall see, it goes beyond this. We will explore this in Part I.

In Part II we will move onto the second topic. This is the far less well-known of Bayes’ argument in favor of miracles. Here we will see that one of the main mathematical formulations in modern probability theory, Bayes theorem, was developed by a Presbyterian minister to argue against those, like Hume, who were sceptical of the possibility of miracles.

Finally in Part III, we will explore a novel argument that probability theory is highly relevant to the question of Extra Ecclesiam nulla salus (no salvation outside the Church) and the related matter of the specific conditions of salvation.

Put less academically, and more surprisingly, in three parts this essay argues that probability theory can shine light on that most difficult of theological questions: will my actions and beliefs damn me to an eternity in Hell?

Part One: Pascal’s Gambling Man

Pascal’s Wager is an unusual argument. It does not fit neatly into the category of a proof of the existence of God. Such proofs are metaphysical in nature. The entire argument for, say, the Ontological Proof can be laid out step-by-step and if each step is accepted then the conclusion follows. These proofs fall into the same category as mathematical proofs or Socratic syllogisms. There are other arguments for God’s existence that do not have the same logical tightness, but instead fall back on inductive evidence. Consider the argument from the fine-tuning of the universe – that is, the fact that the universe only allows for the stability that supports life because a series of physical constants possess an extremely narrow range of values. If the values that these physical constants possess were altered by even a very small amount, the universe as we know it would collapse into chaos. Adherents to the fine-tuning argument point to these constant and say they imply that an entity designed the universe for life – and that entity is God.

Pascal’s Wager relies on neither metaphysical proof nor on induction. It appears at first glance to be an offering of a wager – hence the name. The vulgarised version of Pascal’s Wager runs something like this:

• When it comes to God there are only two possibilities; either He exists, or He does not exist.

• Faced with these two possibilities, there are three options: assert belief in God (theism); deny that God exists (atheism); or say that you do not know (agnosticism).

• The vulgarised version of Pascal’s Wager states that a person has nothing to lose by believing in God because if they are correct, they gain access to Heaven and if they are wrong, they lose nothing.

But this is not really Pascal’s argument. Pascal’s actual argument is addressed to a deeper layer of rationality. Implicitly, Pascal is arguing not against atheism, but against agnosticism. Agnostics will typically say that their position is the rational one. Both atheists and theists, they say, make positive statements about something – God’s existence – which is unknowable. The agnostic claims that in recognising that God’s existence is unknowable and not deciding either way, a person is behaving rationally.

Pascal’s Infinite Casino

Pascal denies this. He starts by distinguishing between the nature of God and man’s own existence. Man is obviously a finite creature. He is of a finite nature in every way: he possesses a finite amount of intelligence, he is finite in size, he eventually dies. God, on the other hand, is infinite. This is by definition: the monotheistic God is defined as an infinite entity that is unlike any finite entity – indeed, the infinite entity is the creator of all finite entities. The Christian religion states that if a man believes in the Christian God and obeys His laws then he will eventually be joined with that God, first in the afterlife and then in a new body after the Resurrection. Judaism and Islam offer somewhat similar ‘terms’, although Judaism’s position on the Resurrection is unclear and Islam denies it. The Abrahamic religions therefore introduce the need on the part of mankind to make a choice – something that was not required when mankind was pagan.

Pascal’s Wager in its true form is about determining which choice is rational. He accepts the terms of the agnostic’s argument: that is, there is no true or false choice, but rather a rational and an irrational choice. Accepting these terms he then interrogates which choice is rational. Here is the key passage in the third section of the Pensées:

For it is no use to say it is uncertain if we will gain, and it is certain that we risk, and that the infinite distance between the certainty of what is staked and the uncertainty of what will be gained, equals the finite good which is certainly staked against the uncertain infinite. It is not so, as every player stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty to gain a finite uncertainty, without transgressing against reason. There is not an infinite distance between the certainty staked and the uncertainty of the gain; that is untrue. In truth, there is an infinity between the certainty of gain and the certainty of loss. But the uncertainty of the gain is proportioned to the certainty of the stake according to the proportion of the chances of gain and loss. Hence it comes that, if there are as many risks on one side as on the other, the course is to play even; and then the certainty of the stake is equal to the uncertainty of the gain, so far is it from fact that there is an infinite distance between them. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain. This is demonstrable; and if men are capable of any truths, this is one.

We might rephrase this argument in clearer terms as such:

• Either God exists or He does not.

• In the Abrahamic religions, it is stated that believing in God and following His laws will result in eternal life.

• Eternal or infinite life is to be contrasted with our own finite life.

• Likewise, eternal life is characterised by infinite joy and happiness which must be contrasted with our finite joy and happiness.

• Pascal accepts that we cannot know whether God exists or not.

• Therefore, we assign a 50% probability to His existence and His non-existence – in probability theory this is known as the “principle of indifference”.

• Knowing the probabilities, we can now examine the stakes.

• We have our finite existence regardless of everything else, but if we choose to not believe in God we can act as we please – some might argue we will be able to engage in more pleasurable finite activities.

• If we choose to believe in God, however, we might be able to access infinite life and infinite joy and happiness.

• With a 50% probability assigned to each, we must make our bet in line with what is at stake.

• We have finite pleasures on one side and infinite life and joy on the other; it is obvious to the rational person that infinite life and joy are not just better than finite pleasures, but they are infinitely better.

• Therefore, the rational person will wager that God exists while the irrational person will be content to hold onto their finite pleasures.

Pascal’s argument is purely rationalistic insofar as it deals (a) with probabilities and (b) with quantities, in terms of the stakes. The stakes can be measured quantitatively relative to one another. Since an infinity is infinitely larger than a finity, and since the probabilities for gaining either are equal, it can be stated with absolute certainty which choice is rational. Pascal’s Wager should not, therefore, be characterised as a simple argument that appeals to a person’s self-interest. Rather it should be seen as akin to a utility problem in economics in that it constitutes a decision that a person is required to make that can either be made rationally or irrationally. In fact, we might say that Pascal’s Wager is the ultimate utility problem.

Stay tuned next week for Part Two on why Thomas Bayes “miraculous calculator” stands against David Hume’s skeptical view of miracles, and what that means for us.