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Knoxville - David M. Bressoud Macalester College, St. Paul,...

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Unformatted text preview: David M. Bressoud Macalester College, St. Paul, MN MathFest, Knoxville, August 12, 2006 The task of the educator is to make the childs spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide. Henri Poincar The task of the educator is to make the childs spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide. Henri Poincar 1. Cauchy and uniform convergence 2. The Fundamental Theorem of Calculus 3. The HeineBorel Theorem 1. Cauchy and uniform convergence 2. The Fundamental Theorem of Calculus 3. The HeineBorel Theorem A Radical Approach to Real Analysis , 2nd edition due January, 2007 A Radical Approach to Lebesgues Theory of Integration, due December, 2007 What Weierstrass Cantor did was very good. That's the way it had to be done. But whether this corresponds to what is in the depths of our consciousness is a very different question What Weierstrass Cantor did was very good. That's the way it had to be done. But whether this corresponds to what is in the depths of our consciousness is a very different question Nikolai Luzin I cannot but see a stark contradiction between the intuitively clear fundamental formulas of the integral calculus and the incomparably artificial and complex work of the justification and their proofs. I cannot but see a stark contradiction between the intuitively clear fundamental formulas of the integral calculus and the incomparably artificial and complex work of the justification and their proofs. Nikolai Luzin Cauchy, Cours danalyse , 1821 explanations drawn from algebraic technique cannot be considered, in my opinion, except as heuristics that will sometimes suggest the truth, but which accord little with the accuracy that is so praised in the mathematical sciences. Niels Abel (1826): Cauchy is crazy, and there is no way of getting along with him, even though right now he is the only one who knows how mathematics should be done. What he is doing is excellent, but very confusing. Cauchy, Cours danalyse, 1821, p. 120 Theorem 1. When the terms of a series are functions of a single variable x and are continuous with respect to this variable in the neighborhood of a particular value where the series converges, the sum S ( x ) of the series is also, in the neighborhood of this particular value, a continuous function of x . S x ( 29 = ( 29 =1 , S n x ( 29 = ( 29 =1 , ( 29 = ( 29 - ( 29 ( 29 . < . S...
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Knoxville - David M. Bressoud Macalester College, St. Paul,...

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