Mathematics: A Minimal Introduction, 1st Edition (Paperback) book cover

Mathematics

A Minimal Introduction, 1st Edition

By Alexandru Buium

Chapman and Hall/CRC

208 pages | 1 B/W Illus.

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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

About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT028000
MATHEMATICS / Set Theory
MAT037000
MATHEMATICS / Functional Analysis