The Search for Certainty : A Philosophical Account of Foundations of Mathematics: A Philosophical Account of Foundations of MathematicsClarendon Press, Jun 6, 2002 - 298 pages The nineteenth century saw a movement to make higher mathematics rigorous. This seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics. The Search for Certainty examines this foundational endeavour from the discovery of the paradoxes to the present. Focusing on Russell's logicist programme and Hilbert's finitist programme, Giaquinto investigates how successful they were and how successful they could be. These questions are set in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all G--ouml--;del's underivability theorems. More than six decades after those discoveries, Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response. |
Contents
| 3 | |
Numbers and Classes | 15 |
The Class Paradoxes | 37 |
Freges Logicism and his Response to Russells Paradox | 49 |
Type Theory as a Response to the Class Paradoxes | 58 |
The Definability Paradoxes and the Vicious Circle | 69 |
Principia Mathematica | 85 |
Paradoxes of Truth | 98 |
Hilberts Finitism | 142 |
Hilberts Programme | 158 |
Incompleteness and Undefinability of Truth | 167 |
Underivability of Consistency | 182 |
Paradise Restored? | 201 |
Solving the Class Paradoxes | 214 |
Outlook | 230 |
| 267 | |
Other editions - View all
The Search for Certainty: A Philosophical Account of Foundations of Mathematics Marcus Giaquinto No preview available - 2002 |
The Search for Certainty: A Philosophical Account of Foundations of Mathematics Marcus Giaquinto No preview available - 2002 |
The Search for Certainty: A Philosophical Account of Foundations of Mathematics Marcus Giaquinto No preview available - 2004 |
Common terms and phrases
A₁ absolutely infinite analysis and set argument attribute Axiom of Choice Axiom of Infinity Axiom of Reducibility Axiom of Replacement Axiom of Separation axiomatic bound variable Burali-Forti paradox Cantor's paradox cardinal number class paradoxes concept condition consistency proof Dedekind defining predicate definition derivable doctrine of types domain entity enumerable expression extension false Feferman finitary finitary arithmetic finitary proof finitary reasoning finite finitist first-order follows formal system formula free variable Frege given Gödel Hence Hilbert Hilbert's Programme impredicative induction intuitionistic language least upper bound Liar paradox logic mathematics natural numbers negation non-empty non-finitary order type ordinal positive integer Principia Mathematica propositional function quantifier Ramsey real numbers reliability Russell's paradox schema Second Underivability Theorem second-order semantic sentence sequence set theory stage statement subclass subset symbols T-derivation tion transfinite true truth type theory universe of sets Vicious Circle Principle Whitehead and Russell Zermelo's
Popular passages
Page xi - I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new kind of mathematics, with more solid foundations than those that had hitherto been thought secure.
References to this book
Representation and Productive Ambiguity in Mathematics and the Sciences Emily R. Grosholz No preview available - 2007 |