History and Philosophy of Constructive Type Theory
Free download. Book file PDF easily for everyone and every device. You can download and read online History and Philosophy of Constructive Type Theory file PDF Book only if you are registered here. And also you can download or read online all Book PDF file that related with History and Philosophy of Constructive Type Theory book. Happy reading History and Philosophy of Constructive Type Theory Bookeveryone. Download file Free Book PDF History and Philosophy of Constructive Type Theory at Complete PDF Library. This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats. Here is The CompletePDF Book Library. It's free to register here to get Book file PDF History and Philosophy of Constructive Type Theory Pocket Guide.
Contents:
Edit this record. Mark as duplicate. Find it on Scholar. Request removal from index. Revision history. From the Publisher via CrossRef no proxy doi. Configure custom resolver. Intuitionistic Type Theory.
Howard - - Journal of Symbolic Logic 51 4 Conditional Theories. Rambaud - - Studia Logica 45 3 - Syntactic Calculus with Dependent Types.
A comprehensive survey of Martin-Löf's constructive type theory, considerable parts of which have only been presented by Martin-Löf in lecture form or as part of. History of Constructive Type Theory (). 4. Philosophical and Technical Prehistory of Constructive Type Theory (). Conclusion.
Type Theory and Homotopy. Steve Awodey - unknown. Their project suggests that mathematical finance, like mathematical economics, may be a rich source of elegant, practical constructive theorems. The traditional route taken by mathematicians wanting to analyse the constructive content of mathematics is the one that follows classical logic; in order to avoid decisions, such as whether or not a real number equals 0, that cannot be made by a real computer, the mathematician then has to keep within strict algorithmic boundaries such as those formed by recursive function theory.
In contrast, the route taken by the constructive mathematician follows intuitionistic logic, which automatically takes care of computationally inadmissible decisions. This logic together with an appropriate set- or type-theoretic framework suffices to keep the mathematics within constructive boundaries.
Constructive Mathematics
Thus the mathematician is free to work in the natural style of an analyst, algebraist e. As Bishop and others have shown, the traditional belief promulgated by Hilbert and still widely held today, that intuitionistic logic imposes such restrictions as to render the development of serious mathematics impossible, is patently false: large parts of deep modern mathematics can be, and have been, produced by purely constructive methods.
Moreover, the link between constructive mathematics and programming holds great promise for the future implementation and development of abstract mathematics on the computer. Brouwer, Luitzen Egbertus Jan logic, history of: intuitionistic logic logic: intuitionistic mathematics, philosophy of mathematics, philosophy of: intuitionism set theory: constructive and intuitionistic ZF type theory: intuitionistic. Introduction 2. The Constructive Interpretation of Logic 3. Varieties of Constructive Mathematics 3.
The Axiom of Choice 5. Constructive Reverse Mathematics 5. Constructive Topology 7. Constructive Mathematical Economics and Finance 8. Introduction Before mathematicians assert something other than an axiom they are supposed to have proved it true. The Constructive Interpretation of Logic It should, by now, be clear that a full-blooded computational development of mathematics disallows the idealistic interpretations of disjunction and existence upon which most classical mathematics depends.
Varieties of Constructive Mathematics The desire to retain the possibility of a computational interpretation is one motivation for using the constructive reinterpretations of the logical connectives and quantifiers that we gave above; but it is not exactly the motivation of the pioneers of constructivism in mathematics. Brouwer was not the clearest expositor of his ideas, as is shown by the following quotation: Mathematics arises when the subject of two-ness, which results from the passage of time, is abstracted from all special occurrences.
We feel about number the way Kant felt about space.
The positive integers and their arithmetic are presupposed by the very nature of our intelligence and, we are tempted to believe, by the very nature of intelligence in general. The development of the positive integers from the primitive concept of the unit, the concept of adjoining a unit, and the process of mathematical induction carries complete conviction.
In the words of Kronecker, the positive integers were created by God. This is the Fall and original sin of set theory, for which it is justly punished by the antinomies. It is not that such contradictions showed up that is surprising, but that they showed up at such a late stage of the game. Bibliography References Aberth, O.
- Similar books and articles.
- Brain Edema XIII.
- History and Philosophy of Constructive Type Theory - Giovanni Sommaruga - Google книги;
- Per Martin-Löf.
Aczel, P. Bauer, A.
Intuitionistic Type Theory (Stanford Encyclopedia of Philosophy)
Beeson, M. Berger, J. Beckmann, U. Berger, B. Tucker eds. Bishop, E. Bourbaki, N. Meldrum trans. Bridges, D. Cederquist, J.
Homotopy type theory
Constable, R. Coquand, T. Zalta ed. Curi, G. Dent, J.
Barendregt, H. Landry, ed. Keywords argumentation conditional logic connexive logic contradiction cut elimination decidability information logic mereology modal logic modal logics natural deduction paraconsistency paraconsistent logic propositional logic self-reference sequent calculus temporal logic three-valued logic truth vagueness. On the concept of number. Normative ethics. By Nathan Sobo at Thu, login or register to post comments.
Diaconescu, R. Diener, H.
Artemov and A. Nerode eds. Crossley ed. Boffa, D. McAloon eds. Friedman, H. Goodman, N. Hayashi, S. Hendtlass, M.
2. The Constructive Interpretation of Logic
Heyting, A. Berlin , 42— Hilbert, D. Putnam and G. Benacerraf and H. Putnam eds. Hurewicz, W. Ishihara, H. Crosilla and P.
1. Overview
Schuster eds. Johnstone, P. Joyal, A. Julian, W.