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An explanation of the value of Hilbert's program of axiomatic completeness

  • Jul 17, 2010
There are two broad categories of major players in any field of intellectual human endeavor; the people that pose the problems and the ones that (re)solve them. Unfortunately, history often allocates greater praise to the solvers rather than the equally essential proposers. In the 1920's, the great German mathematician David Hilbert proposed an approach that would place mathematics on a sound axiomatic foundation. The goal was to prove the consistency of mathematics, in other words that it is not possible to ever properly deduce a contradiction. Like so many such programs, it was one whose time had come as there were several earlier discoveries and issues that pointed the way. In this book, Franks explains the program, some of the work done on it and the primary consequences of what Hilbert initiated.
The program went to the very essence of what mathematics and proof are, in some ways it is one of the most complex areas of philosophy. Very few question the existence of physical matter and the laws that govern its' behavior, but in mathematics the objects can only be approximated, they are the product of thought. So too are the reasoning techniques used to manipulate them, the primary reason that so many people find mathematics difficult is that you are manipulating abstract ideas represented by unusual symbols. Hilbert's goal was to formalize these ideas as much as possible so that whenever an object was declared and processed, each step in the route was formally understood. Furthermore, the rules regarding what you could do were rigidly traced back to a solid foundation of understanding.
Franks does an excellent job in describing this process, giving Hilbert much deserved credit for putting forward the program. While Kurt Gödel's Incompleteness Theorems meant that there were some limits beyond which even mathematics could not cross, Hilbert helped initiate a mindset that drove the mathematical community towards those barriers.
Most mathematicians simply do their mathematics without regard to the deep metamathematics underlying their actions. In most cases, this really does not matter, yet it is something that all practitioners should consider from time to time. Franks does a sound job in revisiting what Hilbert started and with few exceptions it is a book that all mathematicians can understand and appreciate.

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About the reviewer
Charles Ashbacher ()
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Charlie Ashbacher is a compulsive reader and writer about many subjects. His prime areas of expertise are in mathematics and computers where he has taught every course in the mathematics and computer … more
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'Franks' book is remarkable for the clarity of the prose and the originality of the arguments. It is a beautiful introduction to Hilbert's thought on the nature of mathematics and a natural extension of his work.' Jeffrey Barrett, University of California, Irvine
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ISBN-10: 0521514371
ISBN-13: 978-0521514378
Author: Curtis Franks
Genre: Mathematics
Publisher: Cambridge University Press
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