Series: Cambridge Introductions to Philosophy
Hardback (ISBN-13: 9780521854337)
Paperback (ISBN-13: 9780521670265)
This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area.
* Second edition includes sections on tense logic and logics of constrictible
negation * Provides new material on relevant logics and fuzzy logics *
Much expanded and now in a larger format
Preface to the first edition; Preface to the second edition; Mathematical prolegomenon; Part I. Propositional Logic: 1. Classical logic and the material conditional; 2. Basic modal logic; 3. Normal modal logics; 4. Non-normal modal logics; strict conditionals; 5. Conditional logics; 6. Intuitionist logic; 7. Many-valued logics; 8. First degree entailment; 9. Logics with gaps, gluts, and worlds; 10. Relevant logics; 11. Fuzzy logics; 11a. Appendix: many valued modal logics; Postscript: an historical perspective on conditionals; Part II. Qualification and Identity: 12. Classical logic; 13. Free logic; 14. Constant domain modal logics; 15. Variable domain modal logics; 16. Necessary identity in modal logic; 17. Contingent identity in modal logic; 18. Non-normal modal logics; 19. Conditional logics; 20. Intuitionist logic; 21. Many-valued logics; 22. First degree entailment; 23. Logics with gaps, gluts, and worlds; 24. Relevant logics; 25. Fuzzy logics; Postscript: a methodological coda.
Hardback (ISBN-13: 9780521493543)
Modular forms are functions with an enormous amount of symmetry that play a central role in number theory, connecting it with analysis and geometry. They have played a prominent role in mathematics since the 19th century and their study continues to flourish today. Modular forms formed the inspiration for Langlands' conjectures and play an important role in the description of the cohomology of varieties defined over number fields. This collection of up-to-date articles originated from the conference 'Modular Forms' held on the Island of Schiermonnikoog in the Netherlands. A broad range of topics is covered including Hilbert and Siegel modular forms, Weil representations, Tannakian categories and Torelli's theorem. This book is a good source for all researchers and graduate students working on modular forms or related areas of number theory and algebraic geometry.
* Collection of articles by leaders in the field; presents the state of
the art in modular forms * Topics covered include Siegel modular forms,
Hecke eigenvalues of Hilbert modular forms, Weil representations, Tannakian
categories and Torelli’s theorem * Ideal for academic researchers and
graduate students in number theory and algebraic geometry; string theorists
will also find the collection of interest
Preface; Contributors; 1. Modular forms Bas Edixhoven, Gerard van der Geer and Ben Moonen; 2. On the basis problem for Siegel modular forms with level Siegfried Bocherer, Hidenori Katsurada and Rainer Shulze-Pillot; 3. Mock theta functions, weak Maass forms, and applications Kathrin Bringmann; 4. Sign changes of coefficients of half integral weight modular forms Jan Hendrik Bruinier and Winfried Kohnen; 5. Gauss map on the theta divisor and Green’s functions Robin de Jong; 6. A control theorem for the images of Galois actions on certain infinite families of modular forms Luis Dieulefait; 7. Galois realizations of families of Projective Linear Groups via cusp forms Luis Dieulefait; 8. A strong symmetry property of Eisenstein series Bernhard Heim; 9. A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences Tomoyoshi Ibukiyama; 10. Petersson's trace formula and the Hecke eigenvalues of Hilbert modular forms Andrew Knightly and Charles Li; 11. Modular shadows and the Levy-Mellin ∞-adic transform Yuri I. Manin and Matilde Marcolli; 12. Jacobi forms of critical weight and Weil representations Nils-Peter Skoruppa; 13. Tannakian categories attached to abelian varieties Rainer Weissauer; 14. Torelli's theorem from the topological point of view Rainer Weissauer; 15. Existence of Whittaker models related to four dimensional symplectic Galois representations Rainer Weissauer; 16. Multiplying modular forms Martin H. Weissman; 17. On projective linear groups over finite fields Gabor Wiese.
Hardback (ISBN-13: 9780521895460)
Paperback (ISBN-13: 9780521719780)
* Encourages a questioning and active nature rather than a passive one,
leading the reader to develop a deeper understanding of mathematics * Emphasises
the use of examples and counterexamples to illuminate theorems * Essential
for any starting undergraduates in mathematics; also of benefit to engineers
and physicists who need high-level mathematics, and students that require
logic such as computer scientists, philosophers, and linguists
0. Preface; Part I. Study Skills For Mathematicians: 1. Sets and functions;
2. Reading mathematics; 3. Writing mathematics I; 4. Writing mathematics
II; 5. How to solve problems; Part II. How To Think Logically: 6. Making
a statement; 7. Implications; 8. Finer points concerning implications;
9. Converse and equivalence; 10. Quantifiers * For all and There exists;
11. Complexity and negation of quantifiers; 12. Examples and counterexamples;
13. Summary of logic; Part III. Definitions, Theorems and Proofs: 14. Definitions,
theorems and proofs; 15. How to read a definition; 16. How to read a theorem;
17. Proof; 18. How to read a proof; 19. A study of Pythagoras’ Theorem;
Part IV. Techniques of Proof: 20. Techniques of proof I: direct method;
21. Some common mistakes; 22. Techniques of proof II: proof by cases; 23.
Techniques of proof III: Contradiction; 24. Techniques of proof IV: Induction;
25. More sophisticated induction techniques; 26. Techniques of proof V:
contrapositive method; Part V. Mathematics That All Good Mathematicians
Need: 27. Divisors; 28. The Euclidean Algorithm; 29. Modular arithmetic;
30. Injective, surjective, bijective * and a bit about infinity; 31. Equivalence
relations; Part VI. Closing Remarks: 32. Putting it all together; 33. Generalization
and specialization; 34. True understanding; 35. The biggest secret; Appendices:
A. Greek alphabet; B. Commonly used symbols and notation; C. How to prove
that …; Index.
Series: Major Themes in Education
ISBN: 978-0-415-43874-2
Binding: Hardback
Published by: Routledge
Publication Date: 15/10/2009
Pages: 1600
Mathematics Education is one of the most publicized and contested fields of endeavour in the area of education more generally. The entrails of international comparative mathematics achievement surveys are pored over by the media, politicians and educators alike; and, while for the last fifty years at least it has been assumed by most everyone in modern societies that mathematics should be a compulsory subject in all schools, parents and scholars alike argue furiously about whether traditional teaching and rote practising of mathematical skills is better or worse for pupils than conceptual teaching based on children’s own constructed ideas. University mathematics professors tend either to deplore the dropping of standards in their students, and thus the dropping of standards in teachers, or heartily embrace the new learning techniques made possible through careful use of the new technologies.
As academic thinking about and around mathematics education continues to flourish and develop, this new title in the Routledge series, Major Themes in Education, meets the need for an authoritative reference work to make sense of the subject’s vast literature and the continuing explosion in research output. Edited by Alan Bishop, a prominent scholar in the field, this Routledge Major Work is a four-volume collection of foundational and cutting-edge contributions that cover all of the major themes in mathematics education.
The first of the four volumes (‘Mathematics and the Curriculum’) brings together key work on the goals of mathematics education, as well as vital material on the relationship of the curriculum with numeracy, assessment, technology, and the place of marginalized students. The second volume covers the central theories of ‘Mathematics Learning and Learners’. The third volume (‘Mathematics Teaching and Teachers’) gathers the most important thinking on topics such as pedagogical practices; student-focused teaching; teaching at a distance; teacher education; and teachers as researchers. The final volume in the collection (‘The Contexts of Mathematics Education’) gathers vital material from the rich body of literature that explores the social, cultural and political contexts in which mathematics education sits.
With comprehensive introductions to each volume, newly written by the editor,
which place the collected material in its historical and intellectual context,
this Routledge Major Work is an essential work of reference. It is destined
to be valued by specialists in mathematics education and scholars working
in related areas*as well as by educational policy-makers and professionals*as
a vital one-stop research tool.
Part 1: Mathematics and the Curriculum 1. Goals of Mathematics Education 2. Mathematics in the Elementary School 3. Mathematics in the Secondary School 4. Mathematics at the Tertiary Level 5. Numeracy and the Mathematics Curriculum 6. Assessment and the Mathematics Curriculum 7. Technology in the Mathematics Curriculum 8. The Curriculum and Marginalized Students Part 2: Mathematics Learning and Learners 1. Theories of Mathematics Learning 2. Language and Mathematics Learning 3. Visualisation and Mathematics Learning 4. Gifted Learners 5. Disadvantaged Learners 6. Adult Learners 7. Gender Issues 8. Motivation and Affective Aspects Part 3: Mathematics Teaching and Teachers 1. Pedagogical Practices 2. Student Focused Teaching 3. Classroom Interactions 4. Teaching with Texts 5. Manipulatives, and Structured Materials 6. Teaching at a Distance 7. Out-of-School Instruction 8. Preservice Teacher Education 9. Inservice Teacher Education 10. Teachers as Researchers Part 4: The Contexts of Mathematics Education 1. History of Mathematics Education 2. Mathematics Education and Policy 3. Mathematics and Social Justice Issues 4. Popularizing Mathematics 5. Ethnomathematics and Mathematics Education 6. International Comparisons of Mathematics Achievement 7. International Cooperation in Mathematics Education 8. Trends in Research Approaches 9. Mathematics Education and the Media.