# Computation, Proof, Machine

Cambridge University Press

Edition: Translation, 5/7/2015

EAN 9780521133777, ISBN10: 0521133777

Paperback, 160 pages, 21.6 x 14 x 1 cm

Language: English

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'AcadÃƒÂ©mie FranÃƒÂ§aise.

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.