How To Really Understand Pascal’s Wager

Pascal’s wager is a philosophical and mathematical argument in favour of faith in God. There is nothing special about such arguements in religious or philosophical contexts; everyone has an opinion about such things, after all. But Pascal’s wager is truly unique and special for two specific reasons:

1. The argument involves a “wager” or a “bet” as its name implies.

2. Blaise Pascal, who happens to be one of the most iconic science figures of all time, proposed it.

In order to start appreciating the thought that went into constructing this argument, we have to start with who Blaise Pascal was. That is precisely where we will start in this essay. Following this, we will cover the core argument of Pascal’s wager.

Then, we will go on to analyse the argument and cover certain opposing points of view. Finally, I will present my own opinion/stance on the topic, should you be interested. Let us begin.

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Who was Blaise Pascal?

The Prodigy

Blaise Pascal is one of the all-time French science heavyweights. Born to a tax-collector father with a keen interest in science and math in 1623, Pascal lost his mother when he was just three years old. His father, Étienne Pascal, chose to raise Pascal and his two sisters on his own.

As a result, Pascal had heavy exposure to the scientific method very early in his childhood. To his credit, he turned out to be a child prodigy. As a 16-year-old, he wrote a mathematical essay on conics that was so advanced that René Descartes (another French science-heavyweight) was convinced that it had to be the father and not the son who wrote it.

The Inventor and the Philosopher

Just a few years later, Pascal went on to invent one of the very first mechanical calculators (just to relieve his father of his tax calculation burdens) — the Pascaline. He made crucial contributions to the field of physics as well. He studied the principles of hydraulic fluids and was involved in proving the concept of “vacuum” (back then, people believed in “Ether”).

As he matured as an adult, he became a philosopher (a dark one at that), and later, a theologian. Throughout his life, he suffered from frail health and died at a very young age of 39. It sounds like he did an awful lot in just 39 years, right? Hold on, “an awful lot” would be a terrible understatement!

The Man Whose Legacy Lives On

The S.I. unit of pressure (Pascal) is named after him; the programming language Pascal is named after him; Nvidia’s Pascal GPU-series that started in 2014 is named after him; the minor planet 4500 Pascal is named after him. And the list goes on. It would be counter-productive to cover it all in this essay.

In short, Blaise Pascal is no ordinary science figure; humanity’s gratitude for his work centuries later is proof of his genius. In short, the man could think!

The Precursor to Pascal’s Wager

In 1654, Pascal was trying to solve a mathematical gambling problem to help out his friend Antoine Gombaud, Chevalier de Méré. On de Méré’s suggestion, Pascal started corresponding with Pierre de Fermat (a French math heavyweight).

What these two Gentlemen were trying to solve is what we know today as the problem of expected value (if you are interested in this topic, check out my essay on how to really understand expected value). So, they were essentially laying the foundations of probability theory.

However, progress on this topic came to an abrupt halt when Pascal had a divine experience one night (the same year). Immediately after, Pascal decided to give up on his scientific work and took-up theological work in service of his faith. When Fermat tried to woo him back in 1660, Pascal famously refused by saying:

“I can find little difference between a man who is nothing else but a geometrician and a clever craftsman…my studies have taken me so far from this way of thinking, that I can scarcely remember that there is such a thing as geometry.”

— Blaise Pascal.

Just two years after this, Pascal passed away. However, a collection of his notes and short essays which were apparently meant to be a book appeared 8 years after his death. This collection was called “Pensées” (Thoughts) and revealed Pascal’s deep thoughts surrounding theology and philosophy. The entire book was made up of (almost aphoristic) numbered points.

While the book has become a classic in literature, it was point number 233 that became the most famous. This point made a mathematical argument in favour of faith in God. Point number 233 from the Pensées came to be known as Pascal’s wager.

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What is Pascal’s Wager?

Pascal’s Argument

On the surface, using mathematics to construct a theological argument sounds absurd. But remember, Pascal is a man who could think!

He starts elegantly by mentioning the (unknowably) vast difference between unity and infinity. He mentions that we, as finite human beings can wrap our heads around unity, but have no way of truly understanding what infinity is.

He then likens the concept of infinity to God. He extends it further by arguing that because of our limited nature, we cannot know whether God exists or not. He then concedes that we can never use reason to answer the question of whether God exists or not.

But then, he runs a thought experiment. We know that the notion of infinity exists even though we cannot narrow it down to a number or a range. Similarly, what if we may be able to know that God exists without knowing what the God is?

What if we had to use reason to answer the question of whether God exists or not, and there was no other option? It is under these circumstances that Pascal makes his wager. He treats the situation as if it were a probabilistic problem.

He models two possibilities for the truth: Either God exists or God does not exist. He then models two possibilities for an arbitrary person’s belief: either the person believes in God or the person does not believe in God. Given these variables, he (sort of) models a decision matrix that is the basis of what is known today as utility theory:

He then makes a few assumptions. If a person believes in God, he or she would lead a pious life that gives up worldly pleasures. On the other hand, a person who does not believe in God would lead a life where there is no constraint on enjoying worldly pleasures.

Pascal’s Wager

At this point Pascal goes into the core of the argument. He says that worldly pleasures, no matter how enjoyable, are a finite phenomenon. On the other hand, he goes to reasonable lengths to describe how when one gives up worldly pleasures with belief in God and if God exists, one gains an infinite upside by having nothing to lose anymore.

“When I had nothing to lose, I had everything.”

— Paulo Coelho

As a result, every other combination of a person’s state of belief and absolute truth results in either a finite gain or a finite loss. According to Pascal, in the face of infinity, the finite might as well amount to nothing.

So, purely based on the expected value of the utility offered, Pascal argues that it is rationally prudent to believe in God regardless of the absolute truth. Back then, intellectuals were into either proving or disproving God’s existence. Voltaire (who opposed Pascal’s views), for instance, suggested that the beauty of our world would not be possible without a God.

The true beauty of Pascal’s wager is that he does not try to prove God’s existence. Instead, he models a decision based on the expected value of a bet. He tries to optimize for infinite upside and finite downside.

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Arguments Against Pascal’s Wager

At this point, I am pretty sure that you have your own views on this discussion. Hold on to those thoughts just for a little bit longer. Historically, there have been a lot of pushbacks against Pascal’s wager. Similarly, there have been numerous misunderstandings as well.

The first thing to note is that Pascal knew that “reason” could not answer the question. Yet, he just wanted to run a thought experiment to see what comes out.

His argument is easily thwarted by tweaking the assumptions he makes. For instance, what if God is against belief/faith and punishes anyone who is pious. Then, the decision chart would look like this:

In this model, with an expected utility value of infinite loss, it then makes no rational sense to believe in God.

Pascal’s wager also takes a beating when we try and model sub-states. For instance, what if an arbitrary person does not believe in God, but acts as if he/she believes in God? In such a model, almost every single human action has a finite downside and an infinite upside.

As you can see, it is fairly easy to come up with arguments against Pascal’s wager. So, it makes sense to ease up a little, and respect the way Pascal approached the problem knowing his limitations.

Author’s Opinion

I have nothing but respect for Pascal for his deep thoughts and meticulous approach to hard problems. Pascal’s wager is an argument that definitely has its place in philosophy and theology. But in my opinion, the mathematical side of Pascal’s wager was ahead of its time.

At the same time, Pascal’s wager is also a good example of the serious limitations of expected utility theory. The issue I see is with building a rational model of the typical human being. The reality is that no such thing as a ‘rational human being’ exists. The relatively recent work done by scientists such as Daniel Kahneman and Amos Tversky has thwarted this notion.

Furthermore, in the field of quantitative finance, it was often the case that when I spotted some market player running an expected utility model, I ran arbitrage against that model. It turned out to be very effective. In the game of bets, you do not get sure bets very often. If you do, they do not last very long. So, you better make them count.

That concludes my opinion. Remember when I told you to hold on to your thoughts on this topic a little earlier? Well, if you’d like to share those thoughts, do so in the comments section now. Discussions on such hard topics often turn out to be interesting and enlightening!

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Further reading that might interest you: The Thrilling Story Of Calculus and Logarithms: The Long Forgotten Story Of Scientific Progress.

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