## 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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adjoint algebra algebraic theory applied argument binary called cartesian Chapter 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 holds homomorphism induction interpretation isomorphism joins language lattice laws Lemma limits logic maps mathematics means meets 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 variables write