1st Edition

Mathematics
A Minimal Introduction





ISBN 9781482216004
Published November 14, 2013 by Chapman and Hall/CRC
208 Pages 1 B/W Illustrations

USD $62.95

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Book Description

Bridging the gap between procedural mathematics that emphasizes calculations and conceptual mathematics that focuses on ideas, Mathematics: A Minimal Introduction presents an undergraduate-level introduction to pure mathematics and basic concepts of logic. The author builds logic and mathematics from scratch using essentially no background except natural language. He also carefully avoids circularities that are often encountered in related books and places special emphasis on separating the language of mathematics from metalanguage and eliminating semantics from set theory.

The first part of the text focuses on pre-mathematical logic, including syntax, semantics, and inference. The author develops these topics entirely outside the mathematical paradigm. In the second part, the discussion of mathematics starts with axiomatic set theory and ends with advanced topics, such as the geometry of cubics, real and p-adic analysis, and the quadratic reciprocity law. The final part covers mathematical logic and offers a brief introduction to model theory and incompleteness.

Taking a formalist approach to the subject, this text shows students how to reconstruct mathematics from language itself. It helps them understand the mathematical discourse needed to advance in the field.

Table of Contents

Pre-Mathematical Logic
Languages
Metalanguage
Syntax
Semantics
Tautologies
Witnesses
Theories
Proofs
Argot
Strategies
Examples

Mathematics
ZFC
Sets
Maps
Relations
Operations
Integers
Induction
Rationals
Combinatorics
Sequences
Reals
Topology
Imaginaries
Residues
p-adics
Groups
Orders
Vectors
Matrices
Determinants
Polynomials
Congruences
Lines
Conics
Cubics
Limits
Series
Trigonometry
Integrality
Reciprocity
Calculus
Metamodels
Categories
Functors
Objectives

Mathematical Logic
Models
Incompleteness

Bibliography

Index

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Author(s)

Biography

Alexandru Buium is a professor of mathematics at the University of New Mexico. He is the author of four monographs and over 70 research papers in the areas of number theory and algebraic geometry. He has held visiting positions at Columbia University, the Institute for Advanced Study, Max Planck Institute for Mathematics, University of Paris-Sud, and IHES.