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The Proof is in the Pudding - A Look at the Changing Nature of Mathematical Proof

That is all: just a computer procedure to approximate a real root. From the narrow perspective of treating mathematics as a tool to solve real life problems, this is of course sufficient. However, from the point of view of mathematics, shouldn’t a student be interested in roots of polynomials in general? Fourth degree? Odd degree? Other roots, once one is found? Rational roots? Total number of roots? Not every detail need be explained, but even the average student will have his life improved by the mere knowledge that there are such questions, often with answers, e.g., that the factor theorem and the quadratic formula predict that the above cubic. | The Proof is in the Pudding A Look at the Changing Nature of Mathematical Proof Steven G. Krantz July 25 2007 To Jerry Lyons mentor and friend. Table of Contents Preface ix 0 What is a Proof and Why 3 0.1 What is a Mathematician . 4 0.2 The Concept of Proof. 7 0.3 The Foundations of Logic . 14 0.3.1 The Law of the Excluded Middle. 16 0.3.2 Modus Ponendo Ponens and Friends . 17 0.4 What Does a Proof Consist Of . 21 0.5 The Purpose of Proof. 22 0.6 The Logical Basis for Mathematics . 27 0.7 The Experimental Nature of Mathematics. 29 0.8 The Role of Conjectures. 30 0.8.1 Applied Mathematics. 32 0.9 Mathematical Uncertainty . 36 0.10 The Publication of Mathematics . 40 0.11 Closing Thoughts . 42 1 The Ancients 45 1.1 Eudoxus and the Concept of Theorem . 46 1.2 Euclid the Geometer . 47 1.2.1 Euclid the Number Theorist . 51 1.3 Pythagoras . 53 2 The Middle Ages and Calculation 59 2.1 The Arabs and Algebra. 60 2.2 The Development of Algebra. 60 .