Computation, Proof, Machine
Mathematics Enters a New Age

To understand the future of mathematics, this fascinating book returns to its past, tracing the hidden history that follows the thread of computation.

Gilles Dowek (Author), Pierre Guillot (Translated by), Marion Roman (Translated by)

9780521133777, Cambridge University Press

Paperback, published 5 May 2015

160 pages, 6 b/w illus.
21.7 x 13.9 x 0.8 cm, 0.2 kg

'Dowek's book is a superb overview of the transformation of mathematics toward becoming a computational science. It is historically rich, philosophically inquisitive and mathematically rigorous.' Andrew Arana, Metascience

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.

Part I. Ancient Origins: 1. From the prehistory to the Greeks
2. Two thousand years of computation
Part II. The Age of Reason: 3. Predicate logic
4. The decision problem
5. Church's thesis
6. Lambda-calculus
7. Constructivity
8. Constructive proofs and algorithms
Part III. Crisis of the Axiomatic Method: 9. Intuitionistic type theory
10. Automated proof
11. Automated proof checking
12. News from the field
13. Instruments
14. The end of axioms?
15. Conclusion: as we near the end of this mathematical voyage.

Subject Areas: Computer science [UY], Computing & information technology [U], Philosophy of science [PDA], Philosophy of mathematics [PBB], Mathematics [PB]