[Topics: Logic, Mathematics, Pascal’s Wager, Philosophy of Language]
A Logical Infinite:

The Constrained Probabilistic Definitions of Chance and Infinity in Blaise Pascal’s Famous Wager



Blaise Pascal - Pascal's Wager - chance- probability - infiniteThe reasons for the lasting impact of Blaise Pascal’s writings, and specifically ‘Pascal’s Wager,’ are not difficult to discern. That piece represents at once the work of a devout Christian and a thoughtful, if self-assured, philosopher (for a work that pits a devout Christian against a thoughtful, if self-assured, philosopher, see my article on C.S. Lewis and James Rachels). In existing as such, Pascal’s Wager seems a seasoned pontification which has stood up to much historical as well as modern criticism of its mathematics and its logic, regardless of how its flaws yield a failure by scope (detailed below).

Despite being famous as an exercise in reason, Pascal’s Wager is a passage grounded on the unstable foundation of chance and built of the inherently unknowable within theology. This text’s utilization of chance is particularly fascinating due to the fact that it shares meaning between an older conception of chance as pure randomness—arising from the potentially providential turning of some wheel of fortune—and a newer conception implemented in probability theory—wherein that same purity of randomness begets a clarity of logic in cases of ever-mounting complexity.

Indeed, despite its having been written by a man supposedly holding to the tenets of fatalism under the umbrella of Jansenism, ‘chance’ is therein nearly conflated with ‘probability,’ as it would later come to be understood. In Blaise Pascal’s Wager, his use of language turns chance itself into a predictable and knowable tool in the application of logic, and in doing so presents a discourse concerning chance which remains relevant to a modern society of dubious piety to the Wager’s ultimate conclusion.

Pascal’s Wager’s Subtext:

Pascal’s use of chance in the Wager under study is as a concrete aspect of the nature of things. Shortly after establishing the nature of the decision and of the incomprehensible aspects of belief or non-belief, Pascal writes, “There is an infinite chaos which separates us. A game is being played at the extremity of this infinite distance where heads or tails will turn up.” Much as later mathematicians would establish as commonplace, Pascal is here reducing infinity in some terms to a unit of that which lies between one option and another in a binary system.

By constraining the infinite, Blaise Pascal sets himself up for the constraint of chance which shortly thereafter follows. Couched in characteristic hyperbole of surety, Pascal presents the following: “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.”

As carefully ordered as a proof (last week I tackled a different purported proof in the writings of G.E. Moore), Pascal here speaks of chance simply as a divisible set of possible occurrences, situations, or futures. By scrutinizing the quantities of heretofore unquantified abstracts such as infinity, chance, and righteousness, says Pascal, one may derive the proper logic in leading a religious life; and he says so, in addition to the affirmation of his own views in that last clause of the preceding example, via the conscious omission of any subjects on which he may be contradicted. This concealed logical flaw has been pointed out by numerous thinkers over the years.

Pythagoras - Pascal's Wager - chance - probability - infinityWhile heavily implying here, and stating elsewhere in the Pensées, that Christian doctrine is that to be followed in ascribing to Pascal’s conclusion on his Wager, within the passage Pascal advises unbelieving parties to attain belief by simply “acting as if they believed, taking the holy water, having masses said, etc. Even this will naturally make [them] believe, and deaden [their] acuteness.”

Setting aside Pascal’s assumption that one should desire to deaden one’s acuteness, in limiting the advice in this way the text seems to make itself immune to attacks which would point out the contradictory doctrines of the world’s many religions negating any attempts at approximating piety.

Rather, Pascal is vague and general on this point, dwelling in the realm of what he can rhetorically define and utilize. The chance of selecting the incorrect doctrine is irrelevant to Pascal, as chance itself is not so defined in the context of the work.

For Blaise Pascal, chance is that aforementioned divisible, quantifiable set of scenarios, to be dealt with in clear terms such as belief and non-belief, religion and atheism, never among even the convoluted factions of Christianity (let alone among all world religions). I feel, however, that the importance of this reestablishment of what chance can and can not be extends beyond the question of whether Pascal’s Wager is, as I have argued above, stronger in its manipulative rhetoric than in its pious logic.

Contemporary Chance and Infinity:

It is in this form, of the chance which contains and thereby subjugates its own randomness, that the discourse of Pascal’s Wager survives into modern society. Any new reflection on the nature of chance will be hard-pressed to be more inclusive or more logical.

Indeed, that same clarity of definition which allowed Pascal to presuppose infinities of gain and infinities of human ignorance today remains both potent and clear, though independent of those infinities. Modern discussions of chance, even in the specific context of games of chance, are concerned with probability, with quantified chance, whether infinite or finite.

Whether Pascal’s Wager, rhetorically compelling and well-composed as ever, stands up to every criticism which the thinkers of the years have levied at it (and I find it clear that it does not), the terms with which that rhetoric is crafted nevertheless vitally survive.

The ultimate irony of Pascal’s Wager seems to be that, in order to put religion into a conversation intended for true or superficial conversion of secular minds, Pascal was forced to utilize as tools secular definitions of previously theological terms such as infinity and chance, and in doing so may find a more far-reaching legacy in the fields of math and science than in popular discourse or in theology.

Detail of The School of Athens by Raphael - Pascal's Wager - chance - probability - infinity

Detail of The School of Athens by Raphael

Certainly his direct contributions to math and science are more broadly remembered and taught than much of his philosophy (with only Pascal’s Wager excluded, of course). The version of chance appearing here in the writing of Pascal thereby maintains prevalence in modern discussion of chance by way of its ability to transcend the Wager.


Pascal’s gift for logic and reason entirely belies his denouncement of pure logic and pure reason in approaching theology, and accordingly his establishment of probable chance in approaching logical issues, while certainly not unique to him alone, is a tremendous analytical legacy.

Building within the confines of the self-assured, defensive, and reflexive rhetorical structure of Pascal’s Wager, Pascal makes use of abstract nouns, such as chance and infinity, as concrete tools for his argument. This is just one instance of the countless examples of the way that the thoughts and thought processes of great geniuses provide discourses and ideas which survive far beyond the immediate context of their inceptions.

[Topics: Logic, Mathematics, Pascal’s Wager, Philosophy of Language]
A Logical Infinite:

The Constrained Probabilistic Definitions of Chance and Infinity in Blaise Pascal’s Famous Wager

was last modified: March 26th, 2020 by Daniel Podgorski
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