Practical Foundations of Mathematics, Volume 59Practical Foundations of Mathematics explains the basis of mathematical reasoning both in pure mathematics itself (algebra and topology in particular) and in computer science. In addition to the formal logic, this volume examines the relationship between computer languages and "plain English" mathematical proofs. The book introduces the reader to discrete mathematics, reasoning, and categorical logic. It offers a new approach to term algebras, induction and recursion and proves in detail the equivalence of types and categories. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries across universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work. |
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Contents
TYPES AND INDUCTION | 65 |
POSETS AND LATTICES | 125 |
CARTESIAN CLOSED CATEGORIES | 183 |
LIMITS AND COLIMITS | 250 |
STRUCTURAL RECURSION | 306 |
ADJUNCTIONS | 367 |
ALGEBRA WITH DEPENDENT TYPES | 426 |
THE QUANTIFIERS | 469 |
| 530 | |
| 553 | |
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Common terms and phrases
adjoint algebra algebraic theory applied argument axiom binary calculus called Chapter choice closed closure colimits commutative complete composition condition connectives consider consists construction context coproduct corresponding defined Definition dependent diagram directed displays elements equality equation equivalence Example Exercise exists expressions extend fact finite formulae founded function functor given gives homomorphism induction interpretation introduced isomorphism joins language lattice laws Lemma limits logic maps mathematics means mono monotone morphisms names natural normal notation notion objects operation-symbols operations pair partial particular poset predicate presented preserves proof Proposition prove pullback quantifiers recursion relation Remark respectively result rules satisfies Section semantic sense Show space square structure subset substitution surjective symbols Theorem theory topology unique universal variables write
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Bibliographic information
| Title | Practical Foundations of Mathematics, Volume 59 Volume 59 of Cambridge Studies in Advanced Mathematics Practical foundations of mathematics Issue 71 of Symposia of the Zoological Society of London |
| Authors | Paul Taylor, Taylor Paul |
| Editors | Nigel Dunston, Martyn L. Gorman, B. Bollobas |
| Contributors | Nigel, Zoological Society of London, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon |
| Edition | illustrated, reprint |
| Publisher | Cambridge University Press, 1999 |
| ISBN | 0521631076, 9780521631075 |
| Length | 572 pages |
| Subjects | › Mathematics / General Mathematics / History & Philosophy Mathematics / Logic |
| Export Citation | BiBTeX EndNote RefMan |