Intensional MathematicsEdited By
- S. Shapiro
``Platonism and intuitionism are rival philosophies of Mathematics, the former holding that the subject matter of mathematics consists of abstract objects whose existence is independent of the mathematician, the latter that the subject matter consists of mental construction... both views are implicitly opposed to materialistic accounts of mathematics which take the subject matter of mathematics to consist (in a direct way) of material objects...'' FROM THE INTRODUCTIONAmong the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.
Studies in Logic and the Foundations of Mathematics
Published: January 1985
- Chapters: 1. Introduction: Intensional Mathematics and Constructive Mathematics (S. Shapiro). 2. Epistemic and Intuitionistic Arithmetic (S. Shapiro). 3. Intensional Set Theory (J. Myhill). 4. A Genuinely Intensional Set Theory (N.D. Goodman). 5. Extending Gödel's Modal Interpretation to Type Theory and Set Theory (A. Ščedrov). 6. Church's Thesis is Consistent with Epistemic Arithmetic (R.C. Flagg). 7. Calculable Natural Numbers (V. Lifschitz). 8. Modality and Self-Reference (R.M. Smullyan). 9. Some Principles Related to Löb's Theorem (R.M. Smullyan).