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Advanced essays on the philosophical underpinnings of mathematics

  • May 30, 2010
This is not a book about mathematics; the content is about the philosophy of mathematics, which is as old as abstract mathematics. When humans were limited to using mathematics to measure land and count sheep, there was no need to ask questions about where the knowledge of mathematics resided, in fact the measurers and counters would have no doubt considered the questions silly. That changed after the amazing flowering of abstract mathematics that took place in ancient Greece. Once that occurred, then the questions about the residence of the abstractions of circles, lines and planes were logical consequences of the mathematical work being done.
These ponderings have at least occasionally occupied the minds of the greatest mathematicians of all time and this book contains a series of short essays about the deep underpinnings of mathematics written by some of the greatest mathematical minds. The essays and authors are:

*) "Mathematics and thinking mathematically," by Mary Cartwright
*) "Mathematical invention", by Henri Poincare
*) "Thoughts on the heuristic method," by Jacques Hadamard
*) "Mathematical proof," by G. H. Hardy
*) "The unity of knowledge," by Hermann Weyl
*) "Mathematics and the arts," by Marston Morse
*) "Intuition, reason and faith in science," by George David Birkhoff
*) "Logic and the understanding of nature," by David Hilbert
*) "The cultural basis of mathematics," by Raymond Wilder
*) "Presidential address to the British Association," by J. J. Sylvester
*) "The mathematician," by John von Neumann
*) "The community of scholars," by Andre Lichnerowicz
*) "History of mathematics: Why and how," by Andre Weil
*) "Does God exist?" by Paul Levy
*) "Goethe and mathematics," by Wilhelm Maak
*) "Leonardo and mathematics," by Francesco Severi
*) "The highest good," by Norbert Wiener

Despite the lack of equations, theorems and proofs, this is a difficult book to read, for the content goes right to the heart of what mathematics truly is. Very little formal mathematics is really needed to understand it, however you have to be a philosopher at heart, willing to read carefully and think about how some of the most complex ideas were created and applied. This would be an excellent source of material for an upper level undergraduate or graduate course in the philosophical underpinnings of mathematics.

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Charles Ashbacher ()
Ranked #76
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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About this book


' ... a collection of articles written by renowned mathematicians of the 20th century. An important criterion ... is that the articles should be accessible to the literate reader who may or may not have a technical knowledge of mathematics.' L'enseignement mathematique
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ISBN-10: 0883855496
ISBN-13: 978-0883855492
Author: Raymond Ayoub
Publisher: Mathematical Assn of Amer

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