The Proof is in the Pudding: The Changing Nature of Mathematical Proof
Steven G Krantz
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This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. Most of the proofs are discussed in detail with figures and equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.
August 1875, seems to indicate that Sylvester had expressed at least a willingness to share in forming the tone of the young university; the authorities seem to have felt that a Professor of Mathematics and a Professor of Classics could inaugurate the work of a University without expensive buildings or elaborate apparatus. It was finally agreed that Sylvester should go, securing besides his travelling expenses, an annual stipend of 5000 dollars “paid in gold.” And so, at the age of sixty-one,
constant. In fact today it is twenty. But in 1960 it was nineteen. And all the nineteen positions were filled. And nobody was about to step down or abandon his professorship so that H¨ormander might have a job. Recall that Isaac Newton’s teacher Isaac Barrow did indeed give up his Lucasian Chair Professorship so that the brilliant young Newton could assume it.4 But nothing like this was going to happen in socialistic Sweden in 1962. So H¨ormander, never a wilting flower, quit Sweden and moved to
and many drafts, for a new Bourbaki book to be created. The first Bourbaki book, on set theory, was published in 1939; Bourbaki books, and new editions thereof, have appeared as recently as 2005. ´ ements de So far there are thirteen volumes in the monumental series l’El´ Math´ematique. These compose a substantial library of modern mathematics at the level of a first or second year graduate student. Topics covered range from abstract algebra to point-set topology to Lie groups to real analysis.
which is 192 pages, has this avowed intent: The purpose of these notes is to provide the details that are missing in  and  [these are [PER1] and [PER2] in the present book], which contain Perelman’s arguments for the Geometrization Conjecture. It is not clear as of this writing that the world has accepted this contribution as a bona fide proof of the Geometrization Conjecture. The Morgan/Tian book, comprising 473 pages, has been completed and submitted to the Clay Mathematics Institute
was needed for a mathematical “fact”. Sometimes one argued by analogy. Or by invoking the gods. The notion that mathematical statements could be proved was not yet an idea that had been developed. There was no standard for the concept of proof. The logical structure, the “rules of the game”, had not yet been created. If one ancient Egyptian were to say to another, “I don’t understand why this mathematical statement is true. Please prove it.”, his request would have fallen on deaf ears. The