(paper)
ISBN-10: 0-19-921562-6
ISBN-13: 978-0-19-921562-1
Estimated publication date: May 2007
296 pages, 240x168 mm
Series: Oxford Texts in Logic number 3
Description
Based on the authors' extensive teaching on the subject
Practical examples are given for each idea as it is introduced
Methods and concepts are introduced intuitively in terms of
actual mathematical practice, but then developed rigorously
Extensive exercises are presented along with selected solutions
Assuming no previous study in logic, this informal yet rigorous
text covers the material of a standard undergraduate first course
in mathematical logic, using natural deduction and leading up to
the completeness theorem for first-order logic. At each stage of
the text, the reader is given an intuition based on standard
mathematical practice, which is subsequently developed with clean
formal mathematics. Alongside the practical examples, readers
learn what can and can't be calculated; for example the
correctness of a derivation proving a given sequent can be tested
mechanically, but there is no general mechanical test for the
existence of a derivation proving the given sequent. The
undecidability results are proved rigorously in an optional final
chapter, assuming Matiyasevich's theorem characterising the
computably enumerable relations. Rigorous proofs of the adequacy
and completeness proofs of the relevant logics are provided, with
careful attention to the languages involved. Optional sections
discuss the classification of mathematical structures by first-order
theories; the required theory of cardinality is developed from
scratch. Throughout the book there are notes on historical
aspects of the material, and connections with linguistics and
computer science, and the discussion of syntax and semantics is
influenced by modern linguistic approaches. Two basic themes in
recent cognitive science studies of actual human reasoning are
also introduced. Including extensive exercises and selected
solutions, this text is ideal for students in Logic, Mathematics,
Philosophy, and Computer Science.
Readership: Undergraduate students in Logic, Mathematics,
Philosophy, and Computer Science
(Hardback)
ISBN-10: 0-19-928594-2
ISBN-13: 978-0-19-928594-5
Estimated publication date: May 2007
240 pages, 20 diagrams, 234x156 mm
Description
Original work on a fascinating topic
Multi-disciplinary approach; draws on the latest research in
cognitive science, psychology, mathematics education, and
philosophy
Wide range of mathematical examples from geometry, algebra,
arithmetic, and real analysis
Offers solutions to long-standing philosophical problems
Visual thinking - visual imagination or perception of diagrams
and symbol arrays, and mental operations on them - is omnipresent
in mathematics. Is this visual thinking merely a psychological
aid, facilitating grasp of what is gathered by other means? Or
does it also have epistemological functions, as a means of
discovery, understanding, and even proof? By examining the many
kinds of visual representation in mathematics and the diverse
ways in which they are used, Marcus Giaquinto argues that visual
thinking in mathematics is rarely just a superfluous aid; it
usually has epistemological value, often as a means of discovery.
Drawing from philosophical work on the nature of concepts and
from empirical studies of visual perception, mental imagery, and
numerical cognition, Giaquinto explores a major source of our
grasp of mathematics, using examples from basic geometry,
arithmetic, algebra, and real analysis. He shows how we can
discern abstract general truths by means of specific images, how
synthetic a priori knowledge is possible, and how visual means
can help us grasp abstract structures.
Visual Thinking in Mathematics reopens the investigation of
earlier thinkers from Plato to Kant into the nature and
epistemology of an individual's basic mathematical beliefs and
abilities, in the new light shed by the maturing cognitive
sciences. Clear and concise throughout, it will appeal to
scholars and students of philosophy, mathematics, and psychology,
as well as anyone with an interest in mathematical thinking.
Readership: Scholars and advanced students of philosophy,
mathematics, and psychology. Anyone with an interest in the role
of visual thinking in mathematical thought.
Contents
1. Introduction
2. Simple Shapes: Vision and Concepts
3. Basic Geometrical Knowledge
4. Geometrical Discovery by Visualizing
5. Diagrams in Geometric Proofs
6. Mental Number Lines
7. Visual Aspects of Calculation
8. General Theorems from Specific Images
9. Visual Thinking in Basic Analysis
10. Symbol Manipulation
11. Cognition of Structure
12. Mathematical Thinking: Algebraic v. Geometric?
(hardback)
ISBN-10: 0-19-857061-9
ISBN-13: 978-0-19-857061-5
Estimated publication date: June 2007
184 pages, 234x156 mm
Series: Oxford Lecture Series in Mathematics and Its Applications
number 35
Description
Highly significant mathematical work
Authored by leading researchers in the field
Provides a complete account of the construction of 3-fold and 4-fold
klt flips
Presents the work of Shokurov on 3-fold flips in a more
accessible manner
10-page glossary provided
This edited collection of chapters, authored by leading experts,
provides a complete and essentially self-contained construction
of 3-fold and 4-fold klt flips. A large part of the text is a
digest of Shokurov's work in the field and a concise, complete
and pedagogical proof of the existence of 3-fold flips is
presented. The text includes a ten page glossary and is
accessible to students and researchers in algebraic geometry.
Readership: Graduates and researchers in algebraic geometry