Pascal was well acquainted with what could and could not be known through the mathematical method, the experimental method and reason itself. Through his philosophical investigations, he found that there were strict limits to what we as humans could know. For him, neither the scientific method nor reason more generally could teach individuals the meaning of life or the right way to live.

Pascal also wrote about how humans tried to avoid thinking about their mortality, the extent of their ignorance and their liability to error. Yet he also believed that there was nothing more important for people to consider than their true human nature. In this reasoning, without understanding who we are, it would be difficult to understand how we ought to live.

In Pascal’s view, acquiring self-knowledge was a necessary stage on the way to recognizing one’s need for living with faith and purpose in something beyond oneself.

Pascal was a mathematician of the first order. At the age of sixteen, he wrote a significant treatise on the subject of projective geometry, known as Pascal’s Theorem, which states that, if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a single line, called the Pascal line. As a young man, he built a functional calculating machine, able to perform additions and subtractions, to help his father with his tax calculations.

Pascal’s Triangle

The table of binomial coefficients known as Pascal’s Triangle

He is best known, however, for Pascal’s Triangle, a convenient tabular presentation of binomial co-efficients, where each number is the sum of the two numbers directly above it. A binomial is a simple type of algebraic expression which has just two terms operated on only by addition, subtraction, multiplication and positive whole-number exponents, such as (x + y)2. The co-efficients produced when a binomial is expanded form a symmetrical triangle (see image at right).

Pascal was far from the first to study this triangle. The Persian mathematician Al-Karaji had produced something very similar as early as the 10th Century, and the Triangle is called Yang Hui’s Triangle in China after the 13th Century Chinese mathematician, and Tartaglia’s Triangle in Italy after the eponymous 16th Century Italian. But Pascal did contribute an elegant proof by defining the numbers by recursion, and he also discovered many useful and interesting patterns among the rows, columns and diagonals of the array of numbers. For instance, looking at the diagonals alone, after the outside “skin” of 1’s, the next diagonal (1, 2, 3, 4, 5 ,…) is the natural numbers in order. The next diagonal within that (1, 3, 6, 10, 15 ,…) is the triangular numbers in order. The next (1, 4, 10, 20, 35, …) is the pyramidal triangular numbers, etc., etc. It is also possible to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series, and even to find fractal patterns within it.

Pascal also made the conceptual leap to use the Triangle to help solve problems in probability theory. In fact, it was through his collaboration and correspondence with his French contemporary Pierre de Fermat and the Dutchman Christiaan Huygens on the subject that the mathematical theory of probability was born. Before Pascal, there was no actual theory of probability — notwithstanding Gerolamo Cardano’s early exposition in the 16th Century — merely an understanding (of sorts) of how to compute “chances” in dice and card games by counting equally probable outcomes. Some apparently quite elementary problems in probability had eluded some of the best mathematicians, or given rise to incorrect solutions.

It fell to Pascal (with Fermat‘s help) to bring together the separate threads of prior knowledge (including Cardano‘s early work) and to introduce entirely new mathematical techniques for the solution of problems that had hitherto resisted solution. Two such intransigent problems which Pascal and Fermat applied themselves to were the Gambler’s Ruin (determining the chances of winning for each of two men playing a particular dice game with very specific rules) and the Problem of Points (determining how a game’s winnings should be divided between two equally skilled players if the game was ended prematurely). His work on the Problem of Points in particular, although unpublished at the time, was highly influential in the unfolding new field.

The Problem of Points - Pascal probability

Fermat and Pascal’s solution to the Problem of Points

The Problem of Points at its simplest can be illustrated by a simple game of “winner take all” involving the tossing of a coin. The first of the two players (say, Fermat and Pascal) to achieve ten points or wins is to receive a pot of 100 francs. But, if the game is interrupted at the point where Fermat, say, is winning 8 points to 7, how is the 100 franc pot to divided? Fermat claimed that, as he needed only two more points to win the game, and Pascal needed three, the game would have been over after four more tosses of the coin (because, if Pascal did not get the necessary 3 points for your victory over the four tosses, then Fermat must have gained the necessary 2 points for his victor y, and vice versa. Fermat then exhaustively listed the possible outcomes of the four tosses, and concluded that he would win in 11 out of the 16 possible outcomes, so he suggested that the 100 francs be split 11⁄16 (0.6875) to him and 5⁄16 (0.3125) to Pascal.

Pascal then looked for a way of generalizing the problem that would avoid the tedious listing of possibilities, and realized that he could use rows from his triangle of coefficients to generate the numbers, no matter how many tosses of the coin remained. As Fermat needed 2 more points to win the game and Pascal needed 3, he went to the fifth (2 + 3) row of the triangle, i.e. 1, 4, 6, 4, 1. The first 3 terms added together (1 + 4 + 6 = 11) represented the outcomes where Fermat would win, and the last two terms (4 + 1 = 5) the outcomes where Pascal would win, out of the total number of outcomes represented by the sum of the whole row (1 + 4 + 6 +4 +1 = 16).

Pascal and Fermat had grasped through their correspondence a very important concept that, though perhaps intuitive to us today, was all but revolutionary in 1654. This was the idea of equally probable outcomes, that the probability of something occurring could be computed by enumerating the number of equally likely ways it could occur, and dividing this by the total number of possible outcomes of the given situation. This allowed the use of fractions and ratios in the calculation of the likelihood of events, and the operation of multiplication and addition on these fractional probabilities. For example, the probability of throwing a 6 on a die twice is 1⁄6 x 1⁄6 = 1⁄36 (“and” works like multiplication); the probability of throwing either a 3 or a 6 is 1⁄6 + 1⁄6 = 1⁄3 (“or” works like addition).

Pascal’s Religion

In fact, Pascal argued that believing in the existence of God is essential to human happiness. For all of his many ideas and accomplishments, he’s probably most famous today for Pascal’s Wager, a philosophical argument that humans should bet on the existence of God. “If you win, you win everything; if you lose, you lose nothing,” he wrote. In other words, he argued, although one cannot know for certain whether or not God exists, we are better off believing in God’s existence than not.

Pascal’s Wager, Wireless Philosophy.

Pascal saw Jesus as the indispensable mediator between God and humankind. He believed that the Catholic Church was the only religion to teach the truth about human nature and therefore offered the singular route to happiness.

Pascal’s preference for Catholicism over any other religion raises a difficult question, however. For why should anyone wager on one religion rather than another? Some scholars, such as Richard Popkin, have gone so far as to call Pascal’s attempts to discredit paganism, Judaism and Islam “pedantic.”

Whatever one’s religious beliefs, Pascal teaches that all individuals have to make a choice between faith in some reality beyond themselves or a life without belief. But a life without belief is also a choice, and in Pascal’s view, a bad bet.

Human beings have to wager and to commit themselves to a worldview on which each one would be willing to bet their life. It follows that, for Pascal, human beings could not avoid hope and fear: hope that their bets will turn out well, fear that they won’t.

Indeed, people make countless daily wagers — going to the grocery store, driving a car, riding the train, among others — but don’t usually think of them as risky. According to Pascal, however, human lives as a whole can also be viewed as wagers.

Our big decisions are risks: For example, in choosing a certain course of education and career or in marrying a certain person, people are betting on a fulfilling life. In Pascal’s view, people choose how to live and what to believe without really knowing whether or not their beliefs and decisions are good ones. We simply don’t and can’t know enough to live without wagering.

The Human Condition

To properly understand Pascal’s apologetics, it’s important to recognize his motive. Pascal wasn’t interested in defending Christianity as a system of belief; his interest was evangelistic. He wanted to persuade people to believe in Jesus. When apologetics has evangelism as its primary goal, it has to take into account the condition of the people being addressed. For Pascal the human condition was the starting point and point of contact for apologetics.

In his analysis of man, Pascal focuses on two very contradictory sides of fallen human nature. Man is both noble and wretched. Noble, because he is created in God’s image; wretched, because he is fallen and alienated from God. In one of his more passionate notes, Pascal says this: What kind of freak is man! What a novelty he is, how absurd he is, how chaotic and what a mass of contradictions, and yet what a prodigy! He is judge of all things, yet a feeble worm. He is repository of truth, and yet sinks into such doubt and error. He is the glory and the scum of the universe!

Furthermore, Pascal says, we know that we are wretched. But it is this very knowledge that shows our greatness.

Pascal says it’s important to have a right understanding of ourselves. He says “it is equally dangerous for man to know God without knowing his own wretchedness, and to know his own wretchedness without knowing the Redeemer who can free him from it.” Thus, our message must be that “there is a God whom men can know, and that there is a corruption in their nature which renders them unworthy of Him. This prepares the unbeliever to hear about the Redeemer who reconciles the sinner with the Creator.

Pascal says that people know deep down that there is a problem, but we resist slowing down long enough to think about it. He says:

Rick Wade examines the contemporary relevance of the apologetics of Blaise Pascal, a 17th century mathematician, scientist, inventor, and Christian apologist. Man finds nothing so intolerable as to be in a state of complete rest, without passions, without occupation, without diversion, without effort. Then he faces his nullity, loneliness, inadequacy, dependence, helplessness, emptiness. And at once there wells up from the depths of his soul boredom, gloom, depression, chagrin, resentment, despair.

Pascal says there are two ways people avoid thinking about such matters: diversion and indifference. Regarding diversion, he says we fill up our time with relatively useless activities simply to avoid facing the truth of our wretchedness. “The natural misfortune of our mortality and weakness is so miserable,” he says, “that nothing can console us when we really think about it. . . . The only good thing for man, therefore, is to be diverted so that he will stop thinking about his circumstances.” Business, gambling, and entertainment are examples of things which keep us busy in this way.

The other response to our condition is indifference. The most important question we can ask is What happens after death? Life is but a few short years, and death is forever. Our state after death should be of paramount importance, shouldn’t it? But the attitude people take is this:

Just as I do Rick Wade examines the contemporary relevance of the apologetics of Blaise Pascal, a 17th century mathematician, scientist, inventor, and Christian apologist. not know where I came from, so I do not know where I am going. All I know is that when I leave this world I shall fall forever into oblivion, or into the hands of an angry God, without knowing which of the two will be my lot for eternity. Such is my state of mind, full of weakness and uncertainty. The only conclusion I can draw from all this is that I must pass my days without a thought of trying to find out what is going to happen to me.

Pascal is appalled that people think this way, and he wants to shake people out of their stupor and make them think about eternity. Thus, the condition of man is his starting point for moving people toward a genuine knowledge of God.

Knowledge of the Heart

Pascal lived in the age of the rise of rationalism. Revelation had fallen on hard times; man’s reason was now the final source for truth. In the realm of religious belief many people exalted reason and adopted a deistic view of God. Some, however, became skeptics. They doubted the competence of both revelation and reason.

Although Pascal couldn’t side with the skeptics, neither would he go the way of the rationalists. Instead of arguing that revelation was a better source of truth than reason, he focused on the limitations of reason itself. (I should stop here to note that by reason Pascal meant the reasoning process. He did not deny the true powers of reason; he was, after all, a scientist and mathematician.) Although the advances in science increased man’s knowledge, it also made people aware of how little they knew. Thus, through our reason we realize that reason itself has limits. “Reason’s last step,” Pascal said, “is the recognition that there are an infinite number of things which are beyond it. Our knowledge is somewhere between certainty and complete ignorance, Pascal believed. The bottom line is that we need to know when to affirm something as true, when to doubt, and when to submit to authority.

Besides the problem of our limited knowledge, Pascal also noted how our reason is easily distracted by our senses and hindered by our passions. “The two so-called principles of truth*reason and the senses*are not only not genuine but are engaged in mutual deception. Through false appearances the senses deceive reason. And just as they trick the soul, they are in turn tricked by it. It takes its revenge. The senses are influenced by the passions which produce false impressions. Things sometimes appear to our senses other than they really are, such as the way a stick appears bent when put in water. Our emotions or passions also influence how we think about things. And our imagination, which Pascal says is our dominant faculty, often has precedence over our reason. A bridge suspended high over a ravine might be wide enough and sturdy enough, but our imagination sees us surely falling off.

So, our finiteness, our senses, our passions, and our imagination can adversely influence our powers of reason. But Pascal believed that people really do know some things to be true even if they cannot account for it rationally. Such knowledge comes through another channel, namely, the heart.

This brings us to what is perhaps the best known quotation of Pascal: “The heart has its reasons which reason does not know. In other words, there are times that we know something is true but we did not come to that knowledge through logical reasoning, neither can we give a logical argument to support that belief.

For Pascal, the heart is “the `intuitive’ mind” rather than “the `geometrical’ (calculating, reasoning) mind. For example, we know when we aren’t dreaming. But we can’t prove it rationally. However, this only proves that our reason has weaknesses; it does not prove that our knowledge is completely uncertain. Furthermore, our knowledge of such first principles as space, time, motion, and number is certain even though known by the heart and not arrived at by reason. In fact, reason bases its arguments on such knowledge. Knowledge of the heart and knowledge of reason might be arrived at in different ways, but they are both valid. And neither can demand that knowledge coming through the other should submit to its own dictates.

The Knowledge of God

If reason is limited in its understanding of the natural order, knowledge of God can be especially troublesome. “If natural things are beyond [reason],” Pascal said, “what are we to say about supernatural things?”

There are several factors which hinder our knowledge of God. As noted before, we are limited by our finitude. How can the finite understand the infinite? Another problem is that we cannot see clearly because we are in the darkness of sin. Our will is turned away from God, and our reasoning abilities are also adversely affected.

There is another significant limitation on our knowledge of God. Referring to Isaiah 8:17 and 45:15, Pascal says that as a result of our sin God deliberately hides Himself (“hides” in the sense that He doesn’t speak}. One reason He does this is to test our will. Pascal says, “God wishes to move the will rather than the mind. Perfect clarity would help the mind and harm the will.” God wants to “humble [our] pride.

But God doesn’t remain completely hidden; He is both hidden and revealed. “If there were no obscurity,” Pascal says, “man would not feel his corruption: if there were no light man could not hope for a cure.

God not only hides Himself to test our will; He also does it so that we can only come to Him through Christ, not by working through some logical proofs. “God is a hidden God,” says Pascal, ” and . . . since nature was corrupted [God] has left men to their blindness, from which they can escape only through Jesus Christ, without whom all communication with God is broken off. Neither knoweth any man the Father save the Son, and he to whosoever the Son will reveal him. Pascal’s apologetic is decidedly Christocentric. True knowledge of God isn’t mere intellectual assent to the reality of a divine being. It must include a knowledge of Christ through whom God revealed Himself. He says:

All who have claimed to know God and to prove his existence without Jesus Christ have done so ineffectively. . . . Apart from him, and without Scripture, without original sin, without the necessary Mediator who was promised and who came, it is impossible to prove absolutely that God exists, or to teach sound doctrine and sound morality. But through and in Jesus Christ we can prove God’s existence, and teach both doctrine and morality.

If we do not know Christ, we cannot understand God as the judge and the redeemer of sinners. It is a limited knowledge that doesn’t do any good. As Pascal says, “That is why I am not trying to prove naturally the existence of God, or indeed the Trinity, or the immortality of the soul or anything of that kind. This is not just because I do not feel competent to find natural arguments that will convince obdurate atheists, but because such knowledge, without Christ, is useless and empty.” A person with this knowledge has not “made much progress toward his salvation. What Pascal wants to avoid is proclaiming a deistic God who stands remote and expects from us only that we live good, moral lives. Deism needs no redeemer.

But even in Christ, God has not revealed Himself so overwhelmingly that people cannot refuse to believe. In the last days God will be revealed in a way that everyone will have to acknowledge Him. In Christ, however, God was still hidden enough that people who didn’t want what was good would not have it forced upon them. Thus, “there is enough light for those who desire only to see, and enough darkness for those of a contrary disposition.

There is still one more issue which is central to Pascal’s thinking about the knowledge of God. He says that no one can come to know God apart from faith. This is a theme of central importance for Pascal; it clearly sets him apart from other apologists of his day. Faith is the knowledge of the heart that only God gives. “It is the heart which perceives God and not the reason,” says Pascal. “That is what faith is: God perceived by the heart, not by the reason. “By faith we know he exists,” he says. “Faith is different from proof. One is human and the other a gift of God. . . . This is the faith that God himself puts into our hearts. . . . Pascal continues, “We shall never believe with an effective belief and faith unless God inclines our hearts. Then we shall believe as soon as he inclines them.

To emphasize the centrality of heart knowledge in Pascal’s thinking, I deliberately left off the end of one of the sentences above. Describing the faith God gives, Pascal said, “This is the faith that God himself puts into our hearts, often using proof as the instrument.

This is rather confusing. Pascal says non-believers are in darkness, so proofs will only find obscurity. He notes that “no writer within the canon [of Scripture] has ever used nature to prove the existence of God. They all try to help people believe in him. He also expresses astonishment at Christians who begin their defense by making a case for the existence of God.

Their enterprise would cause me no surprise if they were addressing the arguments to the faithful, for those with living faith in their hearts can certainly see at once that everything which exists is entirely the work of the God they worship. But for those in whom this light has gone out and in who we are trying to rekindle it, people deprived of faith and grace, . . . to tell them, I say, that they have only to look at the least thing around them and they will see in it God plainly revealed; to give them no other proof of this great and weighty matter than the course of the moon and the planets; to claim to have completed the proof with such an argument; this is giving them cause to think that the proofs of our religion are indeed feeble. . . . This is not how Scripture speaks, with its better knowledge of the things of God.

But now Pascal says that God often uses proofs as the instrument of faith. He also says in one place, “The way of God, who disposes all things with gentleness, is to instil [sic] religion into our minds with reasoned arguments and into our hearts with grace. . . .

The explanation for this tension can perhaps be seen in the types of proofs Pascal uses. Pascal won’t argue from nature. Rather he’ll point to evidences such as the marks of divinity within man, and those which affirm Christ’s claims, such as prophecies and miracles, the most important being prophecies. He also speaks of Christian doctrine “which gives a reason for everything,” the establishment of Christianity despite its being so contrary to nature, and the testimony of the apostles who could have been neither deceivers nor deceived. So Pascal does believe there are positive evidences for belief. Although he does not intend to give reasons for everything, neither does he expect people to agree without having a reason.

Nonetheless, even evidences such as these do not produce saving faith. He says, “The prophecies of Scripture, even the miracles and proofs of our faith, are not the kind of evidence that are absolutely convincing. . . . There is . . . enough evidence to condemn and yet not enough to convince. . . .” People who believe do so by grace; those who reject the faith do so because of their lusts. Reason isn’t the key.

Pascal says that, while our faith has the strongest of evidences in favor of it, “it is not for these reasons that people adhere to it. . . . What makes them believe,” he says, ” is the cross.” At which point he quotes 1 Corinthians 1:17: “Lest the cross of Christ be emptied of its power.

The Wager

The question that demands to be answered, of course, is this: If our reason is inadequate to find God, even through valid evidences, how does one find God? Says Pascal:

Let us then examine the point and say: “Either God exists, or he does not.” But which of the alternatives shall we choose? Reason cannot decide anything. Infinite chaos separates us. At the far end of this infinite distance a coin is being spun which will come down heads or tails. How will you bet? Reason cannot determine how you will choose, nor can reason defend your position of choice.

At this point Pascal challenges us to accept his wager. Simply put, the wager says we should bet on Christianity because the rewards are infinite if it’s true, while the losses will be insignificant if it’s false. If it’s true and you have rejected it, you’ve lost everything. However, if it’s false but you have believed it, at least you’ve led a good life and you haven’t lost anything. Of course, the best outcome is if one believes Christianity to be true and it turns out that it is!

But the unbeliever might say it’s better not to choose at all. Not so, says Pascal. You’re going to live one way or the other, believing in God or not believing in God; you can’t remain in suspended animation. You must choose.

In response the unbeliever might say that everything in him works against belief. “I am being forced to gamble and I am not free,” he says, “for they will not let me go. I have been made in such a way that I cannot help disbelieving. So what do you expect me to do? After all, Pascal has said that faith comes from God, not from us.

Pascal says our inability to believe is a problem of the emotions or passions. Don’t try to convince yourself by examining more proofs and evidences, he says, “but by controlling your emotions.” You want to believe but don’t know how. So follow the examples of those who “were once in bondage but who now are prepared to risk their whole life. . . . Follow the way by which they began. They simply behaved as though they believed” by participating in various Christian rituals. And what can be the harm? “You will be faithful, honest, humble, grateful, full of good works, a true and genuine friend. . . . I assure you that you will gain in this life, and that with every step you take along this way, you will realize you have bet on something sure and infinite which has cost you nothing.

Remember that Pascal sees faith as a gift from God, and he believes that God will show Himself to whomever sincerely seeks Him. By taking him up on the wager and putting yourself in a place where you are open to God, God will give you faith. He will give you sufficient light to know what is really true.

Scholars have argued over the validity of Pascal’s wager for centuries. In this writer’s opinion, it has significant weaknesses. What about all the other religions, one of which could (in the opinion of the unbeliever) be true?

However, the idea is an intriguing one. Pascal’s assertion that one must choose seems reasonable. Even if such a wager cannot have the kind of mathematical force Pascal seemed to think, it could work to startle the unbeliever into thinking more seriously about the issue. The important thing here is to challenge people to choose, and to choose the right course.

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