Blaise Pascal: Mathematician, Physicist, and Philosopher

Blaise Pascal was a French mathematician, physicist, and religious philosopher. As a person, Pascal integrated different qualities in a nearly inconsistent manner. He held a position of basic skepticism directed against that of Descartes, who employed philosophical doubt only to obtain a secure basis for his philosophy. At the same time, Pascal contributed path-breaking ideas to applied mathematics. He laid the foundation of probability theory and invented the first mechanical calculator (Rooney 43). He was a child prodigy educated by his father, a public servant. The first commercial attempt at a calculating machine was made by Pascal to help his father, an administrator in Rouen, France, who had to deal with complicated tax figures (Rooney 41–42). It caused a sensation all over Europe, as a mechanical device that seemed capable of performing tasks of human intelligence was remarkable for the time.

Pascal's earliest work was in the natural and applied sciences, where he made significant contributions to the construction of mechanical calculators, the study of fluids, and the understanding of pressure and vacuum by extending the work of Evangelista Torricelli. Pascal also wrote in defence of the scientific method. He was a mathematician of the first order who helped form two important new areas of research: he wrote an extensive treatise on projective geometry at the age of sixteen, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science. Following Galileo and Torricelli, in 1646 he refuted Aristotle's followers who denied the existence of a vacuum, and his results provoked considerable controversy before being accepted.

In 1646, his family converted to Jansenism, and his father died in 1651. Following a spiritual experience in late 1654, he underwent a second conversion, set aside his scientific work, and dedicated himself to philosophy and theology. His two most well-known works date from this period: the Lettres provinciales and the Pensées, the former situated within the dispute between the Jansenists and the Jesuits. He also wrote an important treatise on the arithmetic of triangles. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids. Pascal was in poor health throughout his life, and his death came just two months after his 39th birthday (Hald 44).

Among the contemporaries of Descartes, none demonstrated greater natural genius than Pascal, but his mathematical reputation rests as much on what he might have done as on what he actually accomplished — for during a considerable part of his life he felt it his duty to devote his whole time to religious exercises. Born in Clermont, France, Blaise Pascal lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal (1588–1651), was a local judge and member of the noblesse de robe who also had a keen interest in science and mathematics. Pascal had two sisters, the younger Jacqueline and the elder Gilberte. In 1631, following the death of his wife, Étienne Pascal traveled with his children to Paris.

The newly arrived family soon hired Louise Delfault, a maid who in due course became an influential member of the household. Étienne, who never remarried, decided that he alone would educate his children, for they all showed extraordinary intellectual ability — above all his son Blaise. The young Pascal showed a remarkable aptitude for mathematics and science. At the age of eleven, he composed a short treatise on the sounds of vibrating bodies, and Étienne responded by forbidding his son to pursue mathematics further until the age of fifteen, so as not to neglect his study of Latin and Greek. One day, however, Étienne found Blaise writing an independent proof that the sum of the angles of a triangle equals two right angles, drawn with a piece of coal on a wall.

From then on, the boy was allowed to study Euclid. Perhaps more importantly, he was permitted to sit as a silent observer at gatherings of some of the greatest mathematicians and scientists in Europe — including Roberval, Desargues, Mydorge, Gassendi, and Descartes — in the monastic cell of Père Mersenne. Of particular interest to Pascal was a work by Desargues on conic sections. Following Desargues's thinking, the sixteen-year-old Pascal produced a short treatise on what was called the "Mystic Hexagram," titled Essai pour les coniques ("Essay on Conics"), and sent it — his first serious work of mathematics — to Père Mersenne in Paris. It is still known today as Pascal's theorem.

Pascal's work was so impressive that Descartes, when shown the manuscript, refused to believe it was the work of a teenager and assumed it had been written by the elder Pascal. When Mersenne confirmed it was indeed the son's work, Descartes simply dismissed it.

In France at that time, offices and positions could be bought and sold. In 1631, Étienne sold his position as second president of the Cour des Aides for 65,665 livres (Connor 42). The money was invested in a government bond that provided, if not a lavish, then certainly a comfortable income, allowing the Pascal family to enjoy life in Paris. But in 1638, Richelieu, desperate for funds to continue the Thirty Years' War, defaulted on the government's bonds. Suddenly Étienne Pascal's assets had dropped from nearly 66,000 livres to less than 7,300.

Like many others, Étienne was eventually forced to flee Paris because of his opposition to the fiscal policies of Cardinal Richelieu, leaving his three children in the care of his neighbor Madame Sainctot, a woman of renowned past who kept one of the most scintillating and scholarly salons in all of France. It was only when Jacqueline performed well in a children's play attended by Richelieu that Étienne was pardoned. In time he returned to the cardinal's good graces, and in 1639 was appointed the king's commissioner of taxes in the city of Rouen — a city whose tax records, thanks to civil uprisings, were in utter chaos.

In 1642, in an effort to ease his father's endless and exhausting tax calculations, Pascal — not yet nineteen — constructed a mechanical calculator capable of addition and subtraction, called Pascal's calculator or the Pascaline. The Musée des Arts et Métiers in Paris and the Zwinger museum in Dresden, Germany, each exhibit one of his original mechanical calculators. Though these machines are early forerunners of computer engineering, the calculator was not a great commercial success. Because it was extraordinarily expensive, the Pascaline became little more than a curiosity and status symbol for the wealthy in France and across Europe. Nevertheless, Pascal continued to make improvements to his design over the following decade and built fifty machines in total.

In addition to the childhood achievements previously mentioned, Pascal continued to influence mathematics throughout his life. In 1653, Pascal wrote his Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle"), in which he described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle. He defines the numbers in the triangle by recursion: calling the number in the (m+1)st row and (n+1)st column t_mn, then t_mn = t_{m−1, n} + t_{m, n−1}, for m = 0, 1, 2… and n = 0, 1, 2… The boundary conditions are t_{m, 1} = 0 and t_{1, n} = 0 for m = 1, 2, 3… and n = 1, 2, 3… with the generator t₀₀ = 1. In 1654, prompted by a friend interested in gambling problems, Pascal corresponded with Fermat on the subject, and from that collaboration the mathematical theory of probabilities was born.

Italian writers of the fifteenth and sixteenth centuries — notably Pacioli (1494), Tartaglia (1556), and Cardano (1545) — had discussed the problem of dividing a stake between two players whose game was interrupted before its conclusion. This problem was proposed to Pascal and Fermat, probably in 1654, by the Chevalier de Méré, a gambler said to have possessed unusual mathematical ability. The correspondence that ensued between Fermat and Pascal was fundamental to the development of modern concepts of probability.

The specific problem involved two players who wished to end a game early and, given the current state of play, wanted to divide the stakes fairly based on each player's chance of winning from that point forward. From this discussion, the notion of expected value was introduced. Pascal later used a probabilistic argument in the Pensées — known as Pascal's Wager — to justify belief in God and a virtuous life.