Tales of StatisticiansBlaise Pascal

19 June 1623 - 19 Aug 1662Pascal is one of the most brilliant, and most tormented, figures in the history of mathematics. Forbidden by his father to study the subject, he worked out on his own, at the age of 12, the fact that the sum of the angles of a triangle is equal to two right angles (180º). From his 14th year he was included in the Mersenne circle in Paris, and his first original mathematical discovery, which laid the foundations of projective geometry, was communicated to that group when he was 16. Pascal was also an early investigator of the physical world. From observations of diminishing air pressure at different altitudes, he inferred the vacuum of outer space, a discovery which earned him the contempt (philosophy abhors a vacuum) of the more philosophical Descartes. Pascal's first formally published mathematical work was his influential Essay on the Conic Sections (1649); he was then 27. In addition to his geometric studies, he and his contemporary Fermat made foundational discoveries in number theory. His physical investigations culminated in a Treatise on the Equilibrium of Liquids (1653), which for the first time defined a complete system of hydrostatics.

Pascal was troubled by constant illness, including recurrent migraines and what proved to be cancer of the stomach. His various contacts with illness and death from 1646 on, and his own near death in a carriage accident late in 1654, together with the influence of a morbidly religious sister, turned him toward the Jansenist version of Catholicism. On this, his mental energies were increasingly expended.

The science of statistics was founded, almost as an interlude in this process, in a series of five letters between Pascal and Fermat. Close students of that correspondence give Fermat credit for the more substantial contribution, but the contribution would not have been made without the collaboration. The letters were written in the summer of 1654, only months before the traumatic carriage accident. They deal with two problems posed by the Chevalier de Méré, and already considered by Cardan. They are not the elementary combinatorics problems earlier solved by Galileo, but something more advanced: the dice problem (how many times must one throw a pair of dice before one can expect to get a double six) and the stakes problem (how shall the stakes be divided if a game of dice is abandoned while incomplete). These look not to frequencies per se, but to most likely expectations. The answer to the first involves what are called the binomial coefficients, the laws of which are embodied in diagrammatic form in the Pascal Triangle. Pascal did not invent the triangle (it was earlier known to Islamic and indeed Chinese mathematicians), but he was the first European to develop its implications systematically.

Pascal underwent his decisive religious experience on 23 November 1654, and promptly entered the Jansenist community at Port-Royal. He thenceforth devoted his talents to the Jansenists, who were then under attack for supposed heretical beliefs. He has left an enduring name in French literature as the author of the anti-Jesuit Provincial Letters of 1656-1657, written in defense of the Jansenist spokesman Arnauld.

Pascal's last mathematical gesture (1658), like his first, was geometrical: the offering of a prize for the solution of two problems connected with the curve called the cycloid. He recorded his own solutions in letters to Carcavi. His last years were given wholly to religion. During 1660-1662 he jotted down notes for a book which he intended should demonstrate the truth of Christianity. The work was never completed, and we have only the jottings, which were editorially entitled Pensées: "Thoughts." It is his other claim to immortality in French literary remembrance. Pascal's mere body was not immortal; he died in agony from the cancer which by that time had spread to the brain, on 19 August 1662.

Pascal's literarily famous "wager with God" was one of the Pensées. Like his work on statistics, it is "expectational" in character. It runs like this: However small the chance may be that God exists, the reward of Heaven, which is infinite, makes the value of belief in God also infinite. Therefore, the risk of belief will be accepted by the wise and sober gambler, in the earthly casino in which his life is set. Belief, concludes Pascal, is the rational gambler's best shot.

The wager involves multiplication by an infinity. It is not usually reckoned as an important contribution to statistics. Literarily, it does grasp something of the uncertainty which rational mankind must always feel in the presence of a random and uncaring universe. Against that cosmic uncertainty, Pascal at least comes off better than Einstein. The discovery of certainties within that large uncertainty, to which Pascal made an essential contribution, deserves to rank as one of the triumphs of the human intellect.

Bell sums up Pascal's life and distractions in this sentence:

"Pascal's difficulty was that he did not always see clearly when he was trifling, as in his wager against God, or when, as in the clearing up of the Chevalier de Méré's gambling difficulties for him, he was being profound."

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