Rebuttal of Pascal's wager

Teaching Pascal's wager

It is legitimate to put Pascal's wager on the school curriculum. But it happens that some teachers, with little respect for secularism, develop this theme beyond what is required by the culture to make it a missionary tool, the aim being to prepare the pupils to welcome the faith ¹. When ideology prevails over the critical sense, the pupil must perceive it clearly. Reason then requires a counterweight to be opposed to it.

Pascal's wager

The reasoning behind Pascal's wager is circular

Let us temporarily assume the value of one chance in two for the probability that God exists. If this is the case, one gains eternal life in Paradise, and the gain is infinite. If not, one loses nothing. The choice seems easy to make.

However, one must be wary of hidden assumptions. First of all, in the object of the wager, there is not only the existence of God, but also that the Catholic religion would be true and that religious practice would lead to Paradise. Secondly, it is prudent to examine what is covered by the term "infinite".

In mathematics, infinity appears as the limit of sequences. Consider for example the following suggested sequence:

• in a game with a zero bet, every time you try, you win a thousand euros randomly every other time;
• in a game with a zero bet, every time you try, you win a million euros randomly every other time;
• in a game with a zero bet, every time you try, you win a billion euros randomly every other time,
• and "so on".

However, the earth's resources are limited. To pronounce the "so on", one must admit that the supernatural exists. In other words, Pascal implicitly assumes the existence of God, which constitutes a vicious circle, a circular reasoning.

Generalised formulation of Pascal's wager

Initially, Pascal's wager was supposed to support the Catholic faith. But its central element - the possibility of a gigantic gain - is not specifically Christian and can be adapted to any doctrine that promises much. Its versatility even allows its principle to be exploited far beyond the religious realm. Its general formulation is: "The more wonderful the promise, the more justified it is to bet on it".

Variations on Pascal's wager

Beyond charlatanism

An unverified hypothesis remains a hypothesis whose confirmation or rebuttal is postponed to the future. On the other hand, an "unverifiable hypothesis" loses its status as a hypothesis to become a fable or an ideology.

The principle of Pascal's wager puts the gullible to sleep by the immediate comfort provided by the hope of a miraculous payoff. The huckster is indifferent to true and false, for he is concerned only with pleasing, to his greatest advantage. While the promises of charlatans can be invalidated by the absence of expected results, those of religious propagandists, being absolutely unverifiable, go further than charlatanism.

For lovers of mathematical expectation

In the context of Pascal's wager, the bet, which is the Christian commitment, is fixed, or at least capped. In what follows, we assume it to be constant. Two variables remain: the winning and the probability of winning. In all games of chance, the more you aim for a high winning, the lower the probability of winning. For example, if you bet 1 euro, it is a fair game to be able to win 1000 euros with a probability of 1/1000; in another game, if you bet 1 euro, it is a fair game to be able to win 1,000,000 euros with a probability of 1/1,000,000. In this context, we can affirm that, when the winning tends towards infinity, the probability of winning tends towards 0.

What happens if the mathematical expectation of the net winning E of the game is non-zero? The formula to be considered is as follows:

\[ p = \frac{E+bet}{winning} \]

While the players to whom the wager is addressed expect a net winning expectation close to zero, i.e. a game that is not too biased, believers imagine an immense net winning expectation. But this doesn't change anything: even if E is worth a billion, when the winning tends towards infinity, the probability of winning tends towards 0.

If the probability of winning is positive, to make the winning tend towards infinity is tantamount to admitting the supernatural. But this cannot be hypothesised, since that is precisely what we want to prove. In the context of games of chance, the two assertions "the winning is infinite" and "the probability of winning is a real positive" are incompatible.

The above principle can now be corrected : «The more wonderful the promise, the less likely it is. And, in the end, it is implausible.»

To reinforce by another argument that " the probability of obtaining an infinite winning is null", we can refer to the document On the likelihood that a given religion is true, which brings us to the following situation:

\[ E = -bet + \underbrace{\overbrace{winning}^{\to \infty} \cdot \overbrace{p}^{\to 0}}_{\text{indefinite}} \]

We are faced with an indetermination of the type "infinite times zero". Thus the mathematical reasoning comes to an impasse, and the conclusions drawn by Pascal are unfounded.