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Jan 26 2011 Lecture Note

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PHIL 1050
Mark Mc Cullagh

Pg 8189thJan26 LectureThe Binomial Theorem One important contribution to deductive argument was Blaise Pascals Treatise on the Arithmetic Triangle It indirectly furthered the understanding of logicdeductive reasoningst It contained 1 of the 1 important calculations in the theory of probabilityit helped provide the foundations of modern decision theory Pascals work on the arithmetic triangle provideda systematic connection between the theory of combinations ordinary algebra Descartes has succeeded in connecting geometry w algebra by showing that geometrical figures such as the linecircle could be viewed as the collection of points that satisfy different algebraic relations Therefore any 3 s call them A BC determine a straight line if AB are not both 0 The line is the set of all points x y such that AxByC0 NOW Pascal showed the number of ways of choosing r things from a collection of n things concerned numbers that also have purely algebraic significance Pascals result is called the Binomial TheoremFor a positive integral n the bionomical coefficient ofis exactly the number of ways of choosing r things from n thingsthe binomial coefficient is nrnr Remember that 01 Pascals Treatise helped to make it evident that the analysis of combinations arising from the AristotelianPlatonic traditions was an aspect of algebraic relations among numbers Leibnizthe Mathematics of Reasonst Pascals Treatise was published in 1665 the following year Leibniz 19 years old published his 1 work a Latin treatise on logiccombinatorics De Arte Combinatoria Leibniz over his lifetime wrote a long series of unpublisheduncompleted papers on logic These show the formation of some of the key modern ideas about deductive inferenceproofShow how difficult the issues were for 1 of the greatest philosophersmathematicians of the century Leibniz viewpoint can be thought of as what you get if yo
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