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Computation, Proof, Machine: Mathematics Enters a New Age
Computation is revolutionizing our world, even the inner world of the “pure” mathematician. Mathematical methods – especially the notion of proof – that have their roots in classical antiquity have seen a radical transformation since the 1970s, as successive advances have challenged the priority of reason over computation. Like many revolutions, this one comes from within. Computation, calculation, algorithms – all have played an important role in mathematical progress from the beginning – but behind the scenes, their contribution was obscured in the enduring mathematical literature. To understand the future of mathematics, this fascinating book returns to its past, tracing the hidden history that follows the thread of computation. Along the way it invites us to reconsider the dialog between mathematics and the natural sciences, as well as the relationship between mathematics and computer science. It also sheds new light on philosophical concepts, such as the notions of analytic and synthetic judgment. Finally, it brings us to the brink of the new age, in which machine intelligence offers new ways of solving mathematical problems previously inaccessible. This book is the 2007 Winner of the Grand Prix de Philosophie de l'Académie Française.
- GenresPhilosophyMathematics
158 pages, Hardcover
First published May 5, 2015
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Displaying 1 - 2 of 2 reviews
November 8, 2015
Interesting text that could be used in a special topics class in both math and computer science
The idea of resolving mathematical uncertainties via computation has been in mathematics for centuries. The most famous original expression of that idea was by Gottfried Wilhelm Leibniz in the seventeenth century when he was describing disputes among persons and he said, “... we can simply say: Let us calculate, without further ado, to see who is right.”
Adding machines and computers were used from the moment of their creation to perform numeric computations and resolve some outstanding issues, but the machine did nothing but perform glorified arithmetic. Since the operations were arithmetic, verification was tedious, but still possible.
That changed in 1976 when Kenneth Appel and Wolfgang Haken announced their proof of the four color theorem. The proof was revolutionary in that the rigor of the proof was provided by a computer program that evaluated nearly 2,000 different map structures to reach the conclusion. This was the first major mathematical result done by computers where there was no human verification. The announcement literally changed the definition of the phrase “rigorous proof.”
Dowek reviews the history of computing within the context of mathematics, how the art of computation is changing mathematics and how more and more of mathematical progress is defined by the improvement in and development of new computational algorithms. The author begins with mathematical prehistory and ends with Turing machines, Church’s thesis and the growing length and complexity of proofs.
This is a book that could be used in a special topics class in both mathematics and computer science. It is a look at the past as well as some very logical speculations as to how both fields will simultaneously advance in the future. There are also some major philosophical points. For example, in chapter fourteen, “The End of Axioms?” Dowek discusses the idea whether computation rules will replace traditional axioms.
This book was made available for free for review purposes and this review also appears on Amazon
The idea of resolving mathematical uncertainties via computation has been in mathematics for centuries. The most famous original expression of that idea was by Gottfried Wilhelm Leibniz in the seventeenth century when he was describing disputes among persons and he said, “... we can simply say: Let us calculate, without further ado, to see who is right.”
Adding machines and computers were used from the moment of their creation to perform numeric computations and resolve some outstanding issues, but the machine did nothing but perform glorified arithmetic. Since the operations were arithmetic, verification was tedious, but still possible.
That changed in 1976 when Kenneth Appel and Wolfgang Haken announced their proof of the four color theorem. The proof was revolutionary in that the rigor of the proof was provided by a computer program that evaluated nearly 2,000 different map structures to reach the conclusion. This was the first major mathematical result done by computers where there was no human verification. The announcement literally changed the definition of the phrase “rigorous proof.”
Dowek reviews the history of computing within the context of mathematics, how the art of computation is changing mathematics and how more and more of mathematical progress is defined by the improvement in and development of new computational algorithms. The author begins with mathematical prehistory and ends with Turing machines, Church’s thesis and the growing length and complexity of proofs.
This is a book that could be used in a special topics class in both mathematics and computer science. It is a look at the past as well as some very logical speculations as to how both fields will simultaneously advance in the future. There are also some major philosophical points. For example, in chapter fourteen, “The End of Axioms?” Dowek discusses the idea whether computation rules will replace traditional axioms.
This book was made available for free for review purposes and this review also appears on Amazon
May 14, 2015
A wonderful exploration of what mathematical proof means and is, and how the future of proof in mathematics may begin to more closely what one wold think of as 'computation' rather than axiomatic reasoning.
Displaying 1 - 2 of 2 reviews