9780521631075

Practical Foundations of Mathematics

by
  • ISBN13:

    9780521631075

  • ISBN10:

    0521631076

  • Format: Hardcover
  • Copyright: 1999-05-13
  • Publisher: Cambridge University Press
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Summary

Practical Foundations collects the methods of construction of the objects of twentieth century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between 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.

Table of Contents

Introduction viii
First Order Reasoning
1(64)
Substitution
2(9)
Denotation and Description
11(9)
Functions and Relations
20(5)
Direct Reasoning
25(5)
Proof Boxes
30(5)
Formal and Idiomatic Proof
35(9)
Automated Deduction
44(8)
Classical and Intuitionistic Logic
52(8)
Exercises I
60(5)
Types and Induction
65(60)
Constructing the Number Systems
67(5)
Sets (Zermelo Type Theory)
72(9)
Sums, Products and Function-Types
81(6)
Propositions as Types
87(8)
Induction and Recursion
95(7)
Constructions with Well Founded Relations
102(4)
Lists and Structural Induction
106(6)
Higher Order Logic
112(7)
Exercises II
119(6)
Posets and Lattices
125(58)
Posets and Monotone Functions
126(5)
Meets, Joins and Lattices
131(5)
Fixed Points and Partial Functions
136(4)
Domains
140(4)
Products and Function-Spaces
144(7)
Adjunctions
151(5)
Closure Conditions and Induction
156(5)
Modalities and Galois Connections
161(8)
Constructions with Closure Conditions
169(6)
Exercises III
175(8)
Cartesian Closed Categories
183(67)
Categories
184(6)
Actions and Sketches
190(7)
Categories for Formal Languages
197(9)
Functors
206(6)
A Universal Property: Products
212(10)
Algebraic Theories
222(6)
Interpretation of the Lambda Calculus
228(7)
Natural Transformations
235(9)
Exercises IV
244(6)
Limits and Colimits
250(56)
Pullbacks and Equalisers
251(4)
Subobjects
255(6)
Partial and Conditional Programs
261(7)
Coproducts and Pushouts
268(6)
Extensive Categories
274(6)
Kernels, Quotients and Coequalisers
280(6)
Factorisation Systems
286(6)
Regular Categories
292(6)
Exercises V
298(8)
Structural Recursion
306(61)
Free Algebras for Free Theories
307(8)
Well Formed Formulae
315(7)
The General Recursion Theorem
322(7)
Tail Recursion and Loop Programs
329(10)
Unification
339(4)
Finiteness
343(9)
The Ordinals
352(8)
Exercises VI
360(7)
Adjunctions
367(59)
Examples of Universal Constructions
368(7)
Adjunctions
375(7)
General Limits and Colimits
382(9)
Finding Limits and Free Algebras
391(6)
Monads
397(7)
From Semantics to Syntax
404(8)
Gluing and Completeness
412(8)
Exercises VII
420(6)
Algebra with Dependent Types
426(43)
The Language
429(8)
The Category of Contexts
437(12)
Display Categories and Equality Types
449(7)
Interpretation
456(11)
Exercises VIII
467(2)
The Quantifiers
469(61)
The Predicate Convention
470(6)
Indexed and Fibred Categories
476(11)
Sums and Existential Quantification
487(8)
Dependent Products
495(11)
Comprehension and Powerset
506(6)
Universes
512(11)
Exercises IX
523(7)
Bibliography 530(23)
Index 553

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