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On Pascal’s Wager

The Ultimate Bet

Pascal’s wager is a philosophical argument for the existence of God. Well, more accurately, it’s a philosophical argument for believing in God. The rationale is presented through analysis of a simple wager: what is the “cost” if you bet on God? Wikipedia puts it pretty clearly:

Pascal argues that a rational person should live as though God exists and seek to believe in God. If God does not actually exist, such a person will have only a finite loss (some pleasures, luxury, etc.), whereas if God does exist, he stands to receive infinite gains (as represented by eternity in Heaven) and avoid infinite losses (eternity in Hell).

Basically, you should believe in God because being wrong about it is infinitely risky:

You Believe You Don’t
He Exists $+\infty$ $-\infty$
He Doesn’t $-x$ $+y$

Here, $-x$ means you “lost” some potential, finite value to life (usually debauchery comes to mind) if you believed in God and lived life accordingly while it turned out He didn’t exist. Similarly, $+y$ would be that value gain from succesfully living in disregard of the afterlife’s expectations. We use two different variables because people behave differently under each belief system.

Pascal argues that based on this table, regardless of the probability of each row (all that matters is that both are non-zero), you should always opt to believe: the chance of infinite loss is too high-risk.

Analysis

It’s definitely an interesting argument, but it runs into problems quickly. (Obviously, Wikipedia dives deeper into all of these, but here are the cliff notes).

The “Infinite Gods” Counter

Consider, for example, a theoretical jealous god who has a different reward structure: if you believe in the previous God, this god will punish you. Under this scenario, believing as before now has infinite risk instead of infinite reward… is punishment unavoidable?

You can extrapolate this to an infinite number of theoretical gods, rewards, punishments, and wagers. Some of them will be contradictory, and since you have no way of knowing which is truly the right bet (beyond your probabilistic thought experiments), it is impossible to make a “right” choice through this logic alone. Basically, no matter your choice, you will be making a wrong one for a different “god,” and you will always run the risk of being punished.

The “Inauthentic Faith” Counter

Another flaw comes from when you consider this outcome-motivated approach to belief. Will God accept you into Heaven given that you only put your eggs into His theological basket to make the ultimate hedge? Probably not… It’s likely that you would need authentic belief in God that isn’t motivated (exclusively) by this wager to reap its rewards.

A New Fault

Most arguments against Pascal’s wager fall into these two categories. In contrast, I’d like to argue against a fundamental premise, instead:

Is Pascal’s calculation that you endure a finite loss when God doesn’t exist even logically sound?

I argue that the $-x$ above—the finite loss of potential to disregard theological consequences if God doesn’t exist—is actually infinite.

To summarize the rationale simply: if life on Earth is all that exists, not maximizing your enjoyment of it results in an infinite loss.

An individual’s life on Earth is finite, obviously. And yet, it’s all they’ll ever know and experience. In that sense, is the loss of potential not infinite? There are an infinite number of moments, choices, and memories that could’ve gone differently.

But wait, you might say, isn’t the number of new experiences limited by the person’s lifetime, and thus finite? You might think so, but isn’t there an uncountably infinite amount of time between any two moments? Sure, you might’ve started reading this sentence two “seconds” ago, but how many individual “chunks of time” passed in those seconds? Plot twist: there’s no way to measure that because time is continuous.

To reinforce this point more rigorously, and to draw a concrete analogy, we’ll need to take a quick foray into m a t h.

Aside: On Infinities

There are obviously an infinite amount of both integers and real numbers; however, Georg Cantor (no relation) showed that the set of real numbers is a “bigger infinity”: $\left\vert \mathbb{Z} \right\vert < \left\vert \mathbb{R} \right\vert$.

In fact, this relationship goes a step further: the set of real numbers between two consecutive integers is bigger than the entire set of integers. That is, there are more real numbers between 0 and 1 than there are total integers; in math terms, $$ \left\vert\mathbb{Z}\right\vert < \left\vert\{0 < x < 1: \forall x \in \mathbb{R}\}\right\vert $$

We’ve established, then, that the infinity of reals between 0 and 1 is bigger than the infinity of all integers. The final key, now, comes from the next fact: the infinity of reals between 0 and 1 is the same as the infinity of all reals. In math terms,1 $$ \left\vert\mathbb{R}\right\vert = \left\vert\{0 < x < 1: \forall x \in \mathbb{R}\}\right\vert $$

With this perspective, we can say that anything continuous that happens between two discrete moments is “equally infinite” to the entire continuous space.

Back to Reality

We live in a “continuous” universe: individual moments in time cannot be discretized2 into integers. As you read this sentence, an uncountably infinite number of moments just passed. As humans, we do discretize time into chunks like seconds or years, but there is an infinite number of “steps” within a chunk.

Thus, we can apply the same rationale as before with infinities: the “infinity” between any two moments in time (like between the beginning and ending of a person’s life) and the “infinity” of ALL time IS THE SAME.

To express the same thought in different words: the amount of time you spend on Earth is equal in size to ALL time, according to cardinality in set theory. Both are uncountably, infinitely large. It feels… strange? But it’s mathematically true.

By this logic, you suffer an infinite loss by living as though God exists when He doesn’t, since you had an infinite number of opportunities to behave differently. Whether this means hedonistically, maliciously, etc. isn’t relevant; the point is that you have just as many moments of opportunity as you do moments in eternal bliss if He does exist. Thus, the table is more accurately something like:

You Believe You Don’t
He Exists $+\infty$ $-\infty$
He Doesn’t $-\infty$ $+y$

Essentially, this nullifies the wager: though not believing is still risky (I haven’t formulated an argument for whether or not acting on your non-belief provides infinite gain), believing runs an equal risk and reward. This means taking either position is meaningless, and that, in addition with the arguments for soundness outlined above (many-gods and faith authenticity), Pascal’s wager is definitely simply not worth considering.

At the end of the day, faith should come from whether or not it’s the right thing to do, rather than an argument based on the expected reward.

Conclusion

As with all philosophical arguments, if you accept the premises, and the argument is logically sound, you must accept the conclusion. Maybe you do accept the earlier premises, in which case you probably face a conundrum. Regardless, it’s worth reiterating that Pascal didn’t intend this as a proof for God’s existence, but rather simply as a necessary pragmatic decision which is “impossible to avoid” for any living person.

While other arguments attacked soundness, this post questioned the validity of the premises: if God does not exist, is the “utility” lost during life really finite? I argue that no, it’s not, by the simple reason that there are an infinite number of things that could’ve gone differently. Thus, the reality is that while correctly believing in God leads to infinite gain (eternal life), being incorrect suffers infinite loss (an infinite number of opportunities during life).

Personally, I will continue to make the former bet, but I definitely won’t be doing it because of Pascal’s wager.


  1. It’s a little weird to use equality ($=$) here to compare infinities, but hopefully it gets the point across. More accurately, both sets have the same cardinality, so we could say $\mathbb{R} \sim \{0 < x < 1: \forall x \in \mathbb{R}\}$ instead. ↩︎

  2. Arguably, Planck time is a discretization of time, but current quantum physics only implies that quantizations smaller than Planck time have no meaning, which is different than saying they don’t exist. Furthermore, even if time is discrete, isn’t that also the case for “time” in Heaven? I guess it’s possible (if not probable) that Heaven doesn’t abide by the laws of physics, in which case it’s possible that “infinity” in Heaven is larger than “infinity” on Earth, but then I can’t directly make a mathematical analogy / argument since an “uncountably infinite set” is our “biggest” version of infinity… Anyway, you should probably stop thinking about it so much. ↩︎