Blaise Pascal

Blaise Pascal (June 19, 1623–August 19,1662) was a French mathematician, physicist, and religious philosopher. Pascal was a child prodigy, who was educated by his father. Pascal's earliest work was in the natural and applied sciences, where he made important contributions to the construction of mechanical calculators and the study of fluids, and clarified the concepts of pressure and vacuum by expanding the work of Evangelista Torricelli. Pascal also wrote powerfully in defence of the scientific method.

Contributions to mathematics

In addition to the childhood marvels recorded above, Pascal continued to influence mathematics throughout his life. In 1653 Pascal wrote his Traité du triangle arithmétique in which he described a convenient tabular presentation for binomial coefficients, the "arithmetical triangle", now called Pascal's triangle. (It should be noted, however, that Yang Hui, a Chinese mathematician of the Qin dynasty, had independently worked out a concept similar to Pascal's triangle four centuries earlier.)

Related Topics:
1653 - Binomial coefficient - Pascal's triangle - Yang Hui - Chinese

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In 1654, prompted by a friend interested in gambling problems, he corresponded with Fermat on the subject, and from that collaboration was born the mathematical theory of probabilities. The friend was the Chevalier de Méré, and the specific problem was that of two players who want to finish a game early and, given the current circumstances of the game, want to divide the stakes fairly, based on the chance each has of winning the game from that point. Pascal later (in the Pensées) used a probabilistic argument, Pascal's Wager, to justify belief in God and a virtuous life. The work done by Fermat and Pascal into the calculus of probabilities laid important groundwork for Leibniz's formulation of the infinitesimal calculus.

Related Topics:
1654 - Fermat - Probabilities - Pascal's Wager - God - Leibniz - Infinitesimal calculus

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After a religious experience in 1654, Pascal mostly gave up work in mathematics. However, after a sleepless night in 1658 he offered, anonymously, a prize for the quadrature of a cycloid. Solutions were offered by Wallis, Huygens, Wren, and others; then Pascal, under a pseudonym published his own solution. A controversy followed in which the competitors, including Pascal, behaved less than philosophically.

Related Topics:
1658 - Cycloid - Wallis - Huygens - Wren

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Philosophy of mathematics

Pascal's major contribution to the philosophy of mathematics came with his De l'Esprit géométrique ("On the Geometrical Spirit"), originally written as a preface to a geometry textbook for one of the famous "Little Schools of Port-Royal" (Les Petites-Ecoles de Port-Royal). The work was unpublished until over a century after his death. Here Pascal looked into the issue of discovering truths, arguing that the ideal such method would be to found all propositions on already established truths. At the same time, however, he claimed this was impossible because such established truths would require other truths to back them up—first principles cannot be reached. Based on this, Pascal argued that the procedure used in geometry was as perfect as possible, with certain principles assumed and other propositions developed from them. Nevertheless, there was no way to know the assumed principles to be true.

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In De l'Art de persuader, Pascal looked deeper into geometry's axiomatic method, specifically the question of how people come to be convinced of the axioms upon which later conclusions are based. Pascal agreed with Montaigne that achieving certainty in these axioms and conclusions through human methods is impossible. He asserted that these principles can only be grasped through intuition, and that this fact underscored the necessity for submission to God in searching out truths.

Related Topics:
Axiomatic method - Montaigne

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Pascal also used De l'Esprit géométrique to develop a theory of definition. He distinguished between definitions which are conventional labels defined by the writer and definitions which are within the language and understood by everyone because they naturally designate their referent. The second type would be characteristic of the philosophy of essentialism. Pascal claimed that only definitions of the first type were important to science and mathematics, arguing that those fields should adopt the philosophy of formalism as formulated by Descartes.

Related Topics:
Definition - Essentialism - Formalism - Descartes

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