Language, Truth and Logic in Mathematics

1. Front Matter

   Pages i-x

   PDF

2. What is Elementary Logic? Independence-Friendly Logic as the True Core Area of Logic

   • Jaakko Hintikka

   Pages 1-26

3. A Revolution in Logic?

   • Jaakko Hintikka

   Pages 27-44

4. A Revolution in the Foundations of Mathematics?

   • Jaakko Hintikka

   Pages 45-61

5. Is there Completeness in Mathematics after Gödel?

   • Jaakko Hintikka

   Pages 62-83

6. Hilbert Vindicated?

   • Jaakko Hintikka

   Pages 84-105

7. Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics

   • Jaakko Hintikka

   Pages 106-129

8. Standard vs. Nonstandard Logic: Higher-Order, Modal, and First-Order Logics

   • Jaakko Hintikka

   Pages 130-143

9. The Skeleton in Frege’s Cupboard: The Standard Versus Nonstandard Distinction

   • Jaakko Hintikka

   Pages 144-173

10. An Alternative Concept of Computability

    • Jaakko Hintikka

    Pages 174-188

11. What is the Logic of Parallel Processing?

    • Jaakko Hintikka

    Pages 189-211

12. Model Minimization — An Alternative to Circumscription

    • Jaakko Hintikka

    Pages 212-224

13. New Foundations for Mathematical Theories

    • Jaakko Hintikka

    Pages 225-247

14. Back Matter

    Pages 249-249

    PDF

One can distinguish, roughly speaking, two different approaches to the philosophy of mathematics. On the one hand, some philosophers (and some mathematicians) take the nature and the results of mathematicians' activities as given, and go on to ask what philosophical morals one might perhaps find in their story. On the other hand, some philosophers, logicians and mathematicians have tried or are trying to subject the very concepts which mathematicians are using in their work to critical scrutiny. In practice this usually means scrutinizing the logical and linguistic tools mathematicians wield. Such scrutiny can scarcely help relying on philosophical ideas and principles. In other words it can scarcely help being literally a study of language, truth and logic in mathematics, albeit not necessarily in the spirit of AJ. Ayer. As its title indicates, the essays included in the present volume represent the latter approach. In most of them one of the fundamental concepts in the foundations of mathematics and logic is subjected to a scrutiny from a largely novel point of view. Typically, it turns out that the concept in question is in need of a revision or reconsideration or at least can be given a new twist. The results of such a re-examination are not primarily critical, however, but typically open up new constructive possibilities. The consequences of such deconstructions and reconstructions are often quite sweeping, and are explored in the same paper or in others.

• Area
• Gottlob Frege
• Mathematica
• Processing
• boundary element method
• computability
• evolution
• extrema
• form
• information
• logic
• mathematics
• model
• parallel processing
• predicate logic

• Boston University, USA

  Jaakko Hintikka

Jaakko Hintikka is the author or co-author of thirty volumes and of some 300 scholarly articles in mathematical and philosophical logic, epistemology, language theory, philosophy of science, history of ideas and history of philosophy, including Aristotle, Descartes, Leibniz, Kant, Peirce, The Bloomsbury Group, Husserl and Wittgenstein. He has also been active in international scholarly organizations, most recently as the First Vice-President of FISP, Vice-President of IIP and Co-Chair of the American Organizing Committee of the Twentieth World Congress of Philosophy. He has been Editor-in-Chief of the International Journal Synthese and the Managing Editor of Synthese Library since 1965.

Policies and ethics