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The problem, first posed by an Italian monk in the late 1400s, had remained unsolved for nearly two hundred years. The issue in question was to decide how the stakes of a game of chance should be divided if that game were not completed for some reason. The example used in the original publication referred to a game of balla where six goals were required to win the game.
If the game ended normally, the winner would take all. But what if the game stopped when one player was in the lead by five goals to three? In seeking a solution to the problem, Pascal entered into correspondence with the lawyer and mathematician Pierre de Fermat. Between the two of them they laid the foundations of modern probability theory. What Fermat and Pascal realized was that the solution came from listing all of the possibilities and then counting the proportion of the time that each player would win.
From this approach, Pascal derived more general results and developed rules of probability. While Pascal's contribution to probability theory was undoubtedly substantial, it was just the beginning. Importantly, his analysis did not stretch to more realistic situations where a finite number of equally likely possible outcomes could not be listed (Stigler 25). In the early 18th century, Jacob Bernoulli, who stressed the role of statistical sampling in dealing with uncertainty, addressed problems with a potentially infinite number of outcomes.
Through his law of large numbers, Bernoulli sought to provide a formal proof of the idea that uncertainty decreased as the number of observations increased. Another key development of that century was the discovery by Abraham De Moivre of the normal curve the fact that random drawings would distribute themselves in a bell shape around their average value. Although theories relating to risk and uncertainty have continued to develop, the contributions of Pascal, Bernoulli and De Moivre remain pivotal to our understanding of risk.
The work done by Fermat and Pascal into the calculus of probabilities laid important groundwork for Leibniz's formulation of the infinitesimal calculus (http://www.math.rutgers.edu/courses/436/Honors02/leibniz.html).After a religious experience in 1654, Pascal mostly gave up work in mathematics. However, after a sleepless night in 1658, he anonymously offered a prize for the quadrature of a cycloid. Solutions were offered by Wallis, Huygens, Wren, and others? Pascal, under the pseudonym Amos Dettonville, published his own solution. Controversy and heated argument followed after Pascal announced himself the winner.
Philosophy of Mathematics
Pascal's major contribution to the philosophy of mathematics came with his De l'Esprit geometrique ("On the Geometrical Spirit"), originally written as a preface to a geometry textbook for one of the famous "Les PetitesEcoles de PortRoyal" ("Little Schools of PortRoyal").
The work was unpublished until over a century after his death. Here, Pascal looked into the issue of discovering truths, arguing that the ideal of such a 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, therefore, 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. Pascal also used De l'Esprit geometrique 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.
In De l'Art de persuader ("On the Art of Persuasion"), 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.
Connor, James a. Pascal's Wager. New York: Harper Collins, 2006.
Hald, Anders. A History of Probability and Statistics and Its Applications before 1750. Indiana:
Wiley Publications, 1990.
Kisacanin, Branislav. Mathematical Problems and Proofs. New York: Kluwer Academic
Pattanayak, Ari. The…[continue]
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