ISBN13: 9780199756568
Hardback, 226 pages
Dec 2011,
Packed with more than a hundred color illustrations and a wide variety of puzzles and brainteasers, Taking Sudoku Seriously uses this popular craze as the starting point for a fun-filled introduction to higher mathematics.
How many Sudoku solution squares are there? What shapes other than three-by-three blocks can serve as acceptable Sudoku regions? What is the fewest number of starting clues a sound Sudoku puzzle can have? Does solving Sudoku require mathematics? Jason Rosenhouse and Laura Taalman show that answering these questions opens the door to a wealth of interesting mathematics. Indeed, they show that Sudoku puzzles and their variants are a gateway into mathematical thinking generally. Among many topics, the authors look at the notion of a Latin square--an object of long-standing interest to mathematicians--of which Sudoku squares are a special case; discuss how one finds interesting Sudoku puzzles; explore the connections between Sudoku, graph theory, and polynomials; and consider Sudoku extremes, including puzzles with the maximal number of vacant regions, with the minimal number of starting clues, and numerous others. The book concludes with a gallery of novel Sudoku variations--just pure solving fun! Most of the puzzles are original to this volume, and all solutions to the puzzles appear in the back of the book or in the text itself.
A math book and a puzzle book, Taking Sudoku Seriously will change the way readers look at Sudoku and mathematics, serving both as an introduction to mathematics for puzzle fans and as an exploration of the intricacies of Sudoku for mathematics buffs.
1. Playing the Game
Mathematics as Applied Puzzle-Solving
2. Latin Squares
That Do Mathematicians Do?
3. Greco-Latin Squares
The Problem of the Thirty-Six Officers
4. Counting
It's Harder Than it Looks
5. Equivalence Classes
The Importance of Being Essentially Identical
6. Searching
The Art of Finding Needles in Haystacks
7. Graphs
Dots, Lines and Sudoku
8. Polynomials
We Finally Found a Use For Algebra
9. Extremes
Sudoku Pushed to its Limits
10. Epilogue
You Can Never Have Too Many Puzzles
Solutions to Puzzles
ISBN13: 9780199891610
Hardback, 256 pages
Mar 2012,
In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic. She argues that the fruitfulness of Frege's conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation.
The first part of the book locates the role of conceptual analysis in Frege's logicist project. Blanchette argues that despite a number of difficulties, Frege's use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege's intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals.
In the second part of the book, Blanchette explores the resulting conception of logic itself, and some of the straightforward ways in which Frege's conception differs from its now-familiar descendants. In particular, Blanchette argues that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege's position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.
Acknowledgements
Introduction
Chapter 1 - Logicism and Conceptual Analysis
1.1 Introduction to Analysis and Proof
1.2 Analysis and Proof in 1879
1.2.1 Derivation, Proof and Definition
1.2.2 Conceptual Analysis
1.3 Analysis in 1881
1.4. Analysis in 1884
1.5 Grundgesetze
1.6. The General Picture
Chapter 2 - Thoughts
2.1 Thoughts and Language
2.2 Sense and Reference
2.3 The Structure of Sense
2.4 Thoughts and Language Again
2.5 Where we are
Chapter 3 - Thoughts and Sharp Boundaries
3.1 The Issue
3.2. The Texts
3.3 Piecemeal Definition and New Objects
3.4 Ordinary Discourse
3.5 Caesar
3.6 Quantification
3.7 Conclusion
Chapter 4 - The Analysis of Arithmetic
4.1 - The Issue
4.2 Analysis as Thought-Preserving?
4.3 Reference-Preservation and Analysis
4.4 Dummett on What's Preserved
4.5 What's Preserved
4.5.i The Case of Directions
4.5.ii. Numbers
4.5.iii - Arithmetic
4.5.iv Alternative Reductions
4.6 Conclusions
Chapter 5 - Analysis and Consistency: The Case of Geometry
5.1 Introduction
5.2 Frege-Hilbert
5.3 Hilbert's Method
5.4 Frege's Objections
5.5 Consistency and Concepts
5.6 Analysis and Consistency
5.7 The 1906 Passage
5.8 Ultimate Analyses?
5.9 Concluding Remarks
Chapter 6 - Frege and Models
SC6.1 Models and Consistency
SC6.2 Models and Entailment
6.3 Implications
6.4 Summing Up
Chapter 7 - Metatheory
7.1 Frege's Metatheory
7.2 Universalism and Metatheory
7.2.i - The Issue
7.2.ii - Internal Tensions
7.3 - Soundness, Completeness, and Consistency
7.4 Categoricity
7.5 Conclusion
Chapter 8 - Conclusion
Bibliography
ISBN13: 9780199287505
Hardback, 720 pages
Dec 2011,
Since the middle of the 20th century Ludwig Wittgenstein has been an exceptionally influential and controversial figure wherever philosophy is studied. This is the most comprehensive volume ever published on Wittgenstein: thirty-five leading scholars explore the whole range of his thought, offering critical engagement and original interpretation, and tracing his philosophical development. Topics discussed include logic and mathematics, language and mind, epistemology, philosophical methodology, religion, ethics, and aesthetics. Wittgenstein's relation to other founders of analytic philosophy such as Gottlob Frege, Bertrand Russell, and G. E. Moore is explored. This Handbook is the place to look for a full understanding of Wittgenstein's special importance to modern philosophy.
Thirty-five brand-new essays written by an excellent line-up of international contributors
Breaks new ground in the interpretation of Wittgenstein's philosophy
The most extensive and authoritative resource on the subject
Essential reading for anyone interested in Wittgenstein
ISBN13: 9780199640089
Paperback, 256 pages
May 2012,
A quantity can be made smaller and smaller without it ever vanishing. This fact has profound consequences for science, technology, and even the way we think about numbers. In this book, we will explore this idea by moving at an easy pace through an account of elementary real analysis and, in particular, will focus on numbers, sequences, and series.
Almost all textbooks on introductory analysis assume some background in calculus. This book doesn't and, instead, the emphasis is on the application of analysis to number theory. The book is split into two parts. Part 1 follows a standard university course on analysis and each chapter closes with a set of exercises. Here, numbers, inequalities, convergence of sequences, and infinite series are all covered. Part 2 contains a selection of more unusual topics that aren't usually found in books of this type. It includes proofs of the irrationality of e and ?, continued fractions, an introduction to the Riemann zeta function, Cantor's theory of the infinite, and Dedekind cuts. There is also a survey of what analysis can do for the calculus and a brief history of the subject.
A lot of material found in a standard university course on "real analysis" is covered and most of the mathematics is written in standard theorem-proof style. However, more details are given than is usually the case to help readers who find this style daunting. Both set theory and proof by induction are avoided in the interests of making the book accessible to a wider readership, but both of these topics are the subjects of appendices for those who are interested in them. And unlike most university texts at this level, topics that have featured in popular science books, such as the Riemann hypothesis, are introduced here. As a result, this book occupies a unique position between a popular mathematics book and a first year college or university text, and offers a relaxed introduction to a fascinating and important branch of mathematics.
Introduction
I Approaching Limits
1. A Whole Lot of Numbers
2. Let's Get Real
3. The Joy of Inequality
4. Where Do You Go To, My Lovely
5. Bounds for Glory
6. You Cannot be Series
II Exploring Limits
7. Wonderful Numbers
8. Infinite Products
9. Continued Fractions
10. How Infinite Can You Get?
11. Constructing the Real Numbers
12. Where to Next in Analysis? The Calculus
13. Some Brief Remarks About the History of Analysis
Further Reading
Apendices
1. The Binomial Theorem
2. The Language of Set Theory
3. Proof by Mathematical Induction
4. The Algebra of Numbers
Hints and Selected Solutions
ISBN13: 9780199651443
Hardback, 320 pages
May 2012,
David Bostock presents a critical appraisal of Bertrand Russell's philosophy from 1900 to 1924--a period that is considered to be the most important in his career. Russell developed his theory of logic from 1900 to 1910, and over those years wrote the famous work Principia Mathematica with A. N. Whitehead. Bostock explores Russell's development of 'logical atomism', which applies this logic to problems in the theory of knowledge and in metaphysics, and was central to his philosophical work from 1910 to 1924. This book is the first to focus on this important period of Russell's development, examining the three key areas of logic and mathematics, knowledge, and metaphysics, and demonstrating the enduring value of his work in these areas.
The first book on the most important period in Russell's career
A clear and insightful study of key aspects of his thought
Illuminates the development of analytic philosophy
Shows the continuing philosophical interest of logical atomism
Preface
Abbreviations
Part I: Logic and Mathematics
1. Elementary Logic
2. The Contradiction (i): The Problem
3. On Denoting
4. The Contradiction (ii): A Simple Solution
5. The Contradiction (iii): A Ramified Solution
6. Problems
Part II: Knowledge
7. Acquaintance
8. Knowledge and its Foundations
9. Logical constructions (i): matter
10. Logical constructions (ii): minds
11. The Demise of Neutral Monism
Part III: Metaphysics
12. Propositions
13. Universals
14. Facts
Epilogue
Concordance to Reprints of Russell's Journal Articles
References
Index
ISBN13: 9780199588480
Paperback, 144 pages
Jun 2012,
Making good decisions under conditions of uncertainty requires a sound appreciation of the way random chance works. It requires, in short, an understanding of probability. In this Very Short Introduction, John Haigh introduces the ideas of probability--and the different philosophical approaches to probability--and gives a brief account of the history of development of probability theory, from Galileo and Pascal to Bayes, Laplace, Poisson, and Markov. He describes the basic probability distributions and discusses a wide range of applications in science, economics, and a variety of other contexts such as games and betting. He concludes with an intriguing discussion of coincidences and some curious paradoxes.
Explores ideas of probability and the different philosophical approaches to it
Provides a brief account of the history of development of probability theory
Considers the work of some of the big players; from Galileo and Pascal to Bayes, Laplace, Poisson, and Markov
Discusses a wide range of applications of probability theory in science, economics, and a variety of other contexts
Part of the bestselling Very Short Introductions series - over five million copies sold worldwide
1. Fundamentals
2. The workings of probability
3. Historical sketch
4. Chance experiments
5. Making sense of probabilities
6. Games people play
7. Applications in science and operations research
8. Other applications
9. Curiosities and dilemmas
Appendix - Answers to questions posed