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Betting on God: Pascal, Probability Theory and Theology
Blaise Pascal, born in 17​th​ century France, was a mathematician and physicist who
nevertheless made surprising contributions to the field of religious philosophy. In spite of
suffering, from an early age, with an undetermined illness and dying of stomach cancer before
reaching 40, Pascal was instrumental in establishing both the mathematical theory of probability
and understanding atmospheric pressure in terms of weight (“Blaise Pascal”). Among those in
the general population, however, he is probably best remembered for his Wager. Pascal,
fascinated by mathematics, created the Wager by attempting to apply mathematical probability to
the choice of whether or not one should believe in God. His conclusion was that one should
believe because, by believing, one loses nothing if it turns out that God does not exist, but stands
to gain infinite rewards if it turns out God does exist. Choosing not to believe, in contrast,
provides no particular benefit if God does not exist, and equally provides no benefit—indeed,
makes one a great deal worse off—if it turns out one was wrong (Hájek sec. 4). Though the logic
may seem both valid and sound, Pascal's Wager has some serious issues, among them these
three: the problem of choosing between competing faiths, the question how becoming it is for a
religion to encourage converts based on greed, and the potential objection of whether or not one
can choose one's beliefs. There are also some modern responses to Pascal which attempt to
address these and other obstacles, two of which I will examine in some depth.
The most common objection—and one of the simplest, since it does not require an
extended examination of the mathematics involved in probability theory—raised against Pascal's
Wager by non-believers and philosophers is that it assumes that one's choice in religion is binary
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(Hájek sec. 5). Pascal's logic can be sound if the only choice is between believing in the
Christian God or not believing in any God, but there are myriad religions in the world, even
leaving aside the question of whether “Christianity” can truly be considered a single religion that
worships a single deity. Most religions claim to hold some ultimate and saving truth that the
others do not, meaning that an individual cannot, in most cases, choose to “wager” on more than
one set of beliefs, and certainly cannot wager on unlimited sets of belief. Given this reality, while
it may be wise to wager on some kind of religion, at best one might attempt to piece together two
or three non-contradictory creeds and attempt to believe all of them in order to maximize the
chances of whatever salvation each has on offer.
Moreover, even the conclusion that one ought to wager on a religion is suspect—in a
world with seemingly infinite numbers of gods, each with contradictory attributes, it is difficult
to make ​any assumptions about the supernatural and whether it exists at all. It is just as likely
that any god or gods that exist would prefer honest agnosticism or atheism to an attempt to
adhere to a set of religious beliefs strictly for reasons related to one's own self-interest
(Amesbury sec. 2.2.1). In the absence of any demonstrable truth, it may truly be wisest to refrain
from making any decision at all.
Currently there is only a single defense presented against “the many Gods objection.”
Authors George Schlesinger and William Lycan argue that not all deities are equally likely to
exist, and so the two scholars attempt to prescribe a method for assigning probability to various
gods. Schlesinger and Lycan compare the probability of a deity existing to the problem of
choosing between competing theories in science (Hájek sec. 5). In science, if we have multiple
theories which might explain a data-set or a collection of evidence, we always choose the
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simplest until and unless additional evidence forces us to complicate it. This approach is
preferred for two particularly important reasons: first, a simpler hypothesis is always more easily
tested and therefore more falsifiable. Second, based on simple probability, fewer assumptions
introduce fewer possibilities for error. Attempting to appropriate this reasoning without,
apparently, fully understanding it, Lycan and Schlesinger assert that the simplest deity is the
most probable (Hájek sec. 5). This assertion begins on shaky ground: one could argue that the
nonexistence of any deity is even simpler, and that their argument is therefore invalid from the
start. Ignoring this potential objection, however, they persevere, claiming the simplest concept
for a deity is one that is perfect, and that this therefore makes the Christian God both the simplest
and most probable, meaning that one ought to wager on that deity out of all those available
(Hájek sec. 5).
It hardly needs to be said that Lycan and Schlesinger's conclusion is riddled with
problems, among them a disturbing degree of ethnocentrism. Conceptions of “perfection”
necessarily vary from culture to culture, and in the argument the two authors assign primacy to
one particular traditional Western definition without even acknowledging that others might exist
in competition with it. Their definition of “simplicity” also leaves a great deal to be
desired—how, for instance, is their conception of the “perfect God” quantitatively or
qualitatively simpler than, say, the Hindu conception of reality as a single source and destination,
and individuals as mere manifestations of that underlying truth? They unfortunately make no
attempt to justify the claim further. Thus we must conclude that, while it may be true that the
existence of some deities is more or less probable than that of others, this consideration can only
do so much to narrow the field. Schlesinger and Lycan's argument fails to demonstrate Pascal's
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God as significantly more probable than any other, and even a single deity left in competition
with his, renders the Wager logically unsound.
The second objection to Pascal's Wager, which was touched on obliquely earlier, comes
primarily from other Christians. Voltaire was unimpressed by the Wager, finding the entire idea
of betting for or against God based purely on self-interest to be contrary to Christianity's stated
intention (Hájek sec. 5). William James said of the Wager: “When religious faith expresses itself
thus, in the language of the gaming-table, it is put to its last trumps,”​ and one cannot deny the
validity of his reasoning. Making an argument in favor of religion based only on practical
concerns seems to imply that the faith in question has nothing better to offer. ​Though
Schlesinger weighs in again in defense of Pascal, calling self-interest in the search for piety “a
noble greed,” one might be tempted to reply to him in the words of T.S. Eliot's Thomas Becket:
“The last temptation is the greatest treason: To do the right deed for the wrong reason” (Hájek
sec. 5; 44). While it is always possible that what begins in greed or fear may be transformed to
more noble motives in time—which is undoubtedly what Pascal intended—encouraging
nonbelievers to adopt a religion from base considerations seems more likely to corrupt the
church than to purify the false converts.
The question of false versus sincere converts moves us along to the third objection:
Pascal seems to be under the impression that one can simply choose to believe in God or not.
Philosophically, the belief that people have either direct or indirect voluntary control over at least
some of their beliefs is known as ​doxastic voluntarism. Pascal espouses the less controversial
indirect doxastic voluntarism in matters of religious belief—that is, he does not expect a
nonbeliever to simply decide to believe in God and from that point on have a firm and
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unshakable faith: “You would like to attain faith and do not know the way...Learn of those who
have been bound like you...Follow the way by which they began; by acting as if they believed,
taking the holy water, having the masses said, etc.” (Pascal 233). Pascal suggests here that one
who wants to believe for self-interested reasons ought to go through the motions of faith, and
that, eventually, that person is likely to begin believing. While this particular version of doxastic
voluntarism is certainly less controversial in the sense of whether or not it is achievable, there is
still something a bit distasteful and even dishonest about people more or less tricking themselves
into believing in a god—or anything—against their instincts and judgment. Doing so to gain a
reward or avoid punishment seems particularly shallow and manipulative.
Due to all these objections and several others that are mathematical in nature, Pascal's
Wager is generally considered to have been discredited in philosophical circles. There are,
however, more modern strategies which owe their existences to greater or lesser degrees to the
general bent of Pascal's thought. One version, put forth by Joshua Golding, sidesteps the problem
of doxastic voluntarism by discussing the value of a religious lifestyle rather than belief in God.
Golding emphasizes the importance of living in relationship with God rather than believing, and
asserts that this relationship can be built—almost regardless of one's beliefs—by living in
accordance with God's law. Unlike Pascal, Golding also does not make claims about the value of
salvation, but instead argues that there is benefit to be had merely in the living of a religious life
(Amesbury sec. 2.2.1). One of the primary mathematical objections to Pascal's Wager deals with
the fact that he sets the possible gain from choosing to believe in God as being infinite, which
leads to questions about whether the claim is logically valid—by certain constructions, all
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1
outcomes go to infinity and the math ceases to work ​ (Hájek sec. 5). Golding avoids that
particular mathematical problem by dealing only with the finite here-and-now benefits of
religious lifestyles (Amesbury sec. 2.2.1). In this way, he does manage to answer two of the
challenges which ultimately caused most scholars to reject Pascal's Wager as invalid, unsound or
both.
Unfortunately, in restricting his value to a finite number, Golding creates the flaw in his
argument: because he constructs the payoff given by a relationship with God as finite, he must
wrestle with the probability of God's existence. Pascal, by setting the value of salvation as
infinite, could argue that no matter how improbable God's existence might be, any possibility at
all makes it more rational to wager on God than on any other option (Hájek sec. 4). Golding
attempts to patch this problem by claiming that the payoff from a good relationship with God is
not simply better than the payoff from any other pursuit, but that it is qualitatively different. This
move allows him to claim that no amount of good from other pursuits could be added together to
equal the value to be found in a religious lifestyle. In claiming a qualitative difference, however,
he injects a novel mathematical problem into his claim: the probability matrix used to evaluate
these wagers is incapable of evaluating qualitative differences (Amesbury sec. 2.2.1). Golding's
math therefore does not work, and one must choose either to believe or not that a religious
lifestyle offers benefits which cannot be equaled through any other pursuit—which defeats the
purpose of attempting to give a mathematical reason for choosing religion in the first place.
Perhaps the most successful offspring of Pascal's Wager is William James' much more
modest claim: that, in the absence of sufficient evidence, it is permissible to believe based on
1
For background information on probability theory and the equation used by Pascal, see Hájek sections 1-4.
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faith. The connection between the Wager and James' thesis is not immediately obvious, but
James, in spite of his recognition of the limitations of and problems with Pascal's Wager,
nevertheless sees value in arguing matters of faith from the standpoint of probability and
mathematics. The argument he makes draws on the philosophical traditions surrounding the
Wager even if he does not attempt to construct a wager himself. Moreover, the question he is
driving at is the same: what does reason tell us about what we should or shouldn't believe? Pascal
was convinced that it told us we ought to believe, if for no other reasons than prudential ones.
James makes the argument that it says we ​may believe, at least in the absence of evidential
reasons to avoid doing so. The answers to the question may be different and may be arrived at in
different ways, but both men are still addressing the same question.
James laid out his claim in his essay, “The Will to Believe,” a response to an argument
made by 19​th​ century philosopher and agnostic W.K. Clifford, who asserted that there can be no
justification for ever believing anything without sufficient evidence (Amesbury sec. 2.2.3).
Clifford saw all of civilization as being founded upon skepticism, and believed that no progress
could be made without it. His conclusion was that believing anything in error was, in essence, a
moral transgression that regressed society, and so every attempt should be made to avoid making
such errors. James, on the other hand, saw situations in which missing out on potential truths
would be a greater evil than occasionally being wrong. James' example of how this might
function practically was social cooperation—one must extend faith that others are acting with
good intentions before relationships can be built (Jordan sec. 3). James acknowledged that
avoiding error might be of primary importance in science, but argued that in other circumstances
it is necessary to put the fear of error aside lest one miss out on important truths (Amesbury sec.
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2.2.3). Here again we see a connection to Pascal, for whom missing out on a positive
good—salvation—was infinitely more important than simply avoiding an inaccuracy.
In terms of religion, James saw faith as more akin to social cooperation than science. He
outlined two claims made by religion: first, that things are better the “more eternal” they are,
and, second, that believing confers immediate benefits (25-26). James contends that choosing to
believe is a perfectly rational choice for three reasons: in the first place, the claims made by
religion can, to some degree, only be evaluated experientially. In the second, there is no
consensus on whether there is enough evidence to validate or invalidate any major religion. In
the third, refraining from making a choice amounts to choosing not to believe. He argues, in fact,
that choosing to refrain from believing is driven just as much by emotion as the choice to
believe—the former is a choice made in fear of being wrong, and the latter a choice made in the
hope of being right (Amesbury sec. 2.2.3).
Of course, since James considers it permissible to believe as one pleases only in the
absence of evidence, the claim that religion is one such case is open to criticism. There are many
non-believers who would argue that there ​is sufficient evidence for reaching the conclusion that
there are no gods, and that believers, in defiance of James' principle, willfully ignore the
evidence. James does not, indeed, attempt to define what “sufficient evidence” means, which is
problematic. If one were arguing against Christianity, for example, one might cite the actual age
of the Earth against the genealogical calculations to be gleaned from the Bible as a factual error
that makes suspect the reliability of all other Biblical accounts. Physical facts ought to be easier
to get right than metaphysical facts, after all. This one error would be enough evidence for some
to say that Christianity, at least, can be dropped from the roster of potentially true religions. This
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example demonstrates the difficulty in determining just what “sufficient evidence” means, since
it can vary wildly from person to person—though this is a criticism which can certainly be
leveled against Clifford as well as James.
Other than that single critique, James' more modest answer to the question presented by
Pascal's Wager seems largely sound. Reason may not demand that one believe in God—or in any
other deity—but James makes a good case for reason failing to demand that one reject religion.
Could Pascal have foreseen this humbling of his original claim, he might have been
disappointed. Even so, it seems unfair to expect a purely natural phenomenon like reason to be
able to pass definitive judgment on issues which lie outside the realm of nature. If reason can be
used to defend supernatural beliefs as permissible, that is likely the best outcome that a believer
can legitimately hope for.