## 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

Denotation and Description | 11 |

Exercises I | 60 |

POSETS AND LATTICES | 125 |

CARTESIAN CLOSED CATEGORIES | 183 |

LIMITS AND COLIMITS | 250 |

Exercises V | 298 |

STRUCTURAL RECURSION | 306 |

ADJUNCTIONS | 367 |

ALGEBRA WITH DEPENDENT TYPES | 426 |

THE QUANTIFIERS | 469 |

530 | |

553 | |

### Common terms and phrases

adjunction algebraic theory argument arity axiom bijection binary calculus called cartesian closed cartesian closed category category theory closure conditions coequaliser colimits commutative composition congruence construction context coproduct Corollary defined Definition diagram displays elements elimination rule equalisers equality equation Example excluded middle Exercise factorisation fibration finitary finite forgetful functor formulae free algebra functor generalised homomorphism induction scheme infinitary interpretation intuitionistic inverse isomorphism joins kernel pair language lattice laws left adjoint Lemma logic maps mathematics mono monoid monotone function morphisms natural transformation normal form notation notion operation-symbols operations pair poset powerset predicate preorder preserves proof Proposition provable pullback quantifiers quotient recursion Remark satisfies Scott-continuous Section semantic semilattice sequence set theory Show square structure subset substitution surjective symbols syntactic syntax terminal object Theorem topology type theory unary unique universal property variables X-calculus Zermelo