Charming Proofs A Journey into Elegant Mathematics

Claudi Alsina e Roger B. Nelsen

Dolciani mathematical expositions, nº 42

The Mathematical Association of America | 2011 | 320 páginas | rar - pdf | 2,35 Mb

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Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy'. Charming Proofs presents a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, and to develop the ability to create proofs themselves. The authors consider proofs from topics such as geometry, number theory, inequalities, plane tilings, origami and polyhedra. Secondary school and university teachers can use this book to introduce their students to mathematical elegance. More than 130 exercises for the reader (with solutions) are also included.

Contents
A garden of integers
Distinguished numbers
Points in the plane
The polygonal playground
A treasury of triangle theorems
The enchantment of the equilateral triangle
The quadrilaterals' corner
Squares everywhere
Curves ahead
Adventures in tiling and coloring
Geometry in three dimensions
Additional theorems, problems, and proofs.

Proof and Knowledge in Mathematics

Michael Detlefsen
Routledge | 1992 | 256 páginas | 

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These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is a priori or a posteriori in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification.

CONTENTS
1 PROOF AS A SOURCE OF TRUTH
Michael D.Resnik
2 REFLECTIONS ON THE CONCEPT OF A PRIORI TRUTH AND ITS CORRUPTION BY KANT
William W.Tait
3 LOGICISM
Steven J.Wagner
4 EMPIRICAL INQUIRY AND PROOF
Shelley Stillwell
5 ON THE CONCEPT OF PROOF IN ELEMENTARY GEOMETRY
Pirmin Stekeler-Weithofer
6 MATHEMATICAL RIGOR IN PHYSICS
Mark Steiner
7 FOUNDATIONALISM AND FOUNDATIONS OF MATHEMATICS
Stewart Shapiro
8 BROUWERIAN INTUITIONISM
Michael Detlefsen

Bridge to Abstract Mathematics

(Mathematical Association of America Textbooks)

Ralph W. Oberste-Vorth, Aristides Mouzakitis e Bonita A. Lawrence

Mathematical Association of America | 2012 | 253 páginas | pdf | 3,6 Mb

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Mathematics is a science that concerns theorems that must be proved within a system of axioms and definitions. With this book, the mathematical novice will learn how to prove theorems and explore the universe of abstract mathematics. The introductory chapters familiarise the reader with some fundamental ideas, including the axiomatic method, symbolic logic and mathematical language. This leads to a discussion of the nature of proof, along with various methods for proving statements. The subsequent chapters present some foundational topics in pure mathematics, including detailed introductions to set theory, number systems and calculus. Through these fascinating topics, supplemented by plenty of examples and exercises, the reader will hone their proof skills. This complete guide to proof is ideal preparation for a university course in pure mathematics, and a valuable resource for educators.

• A complete guide to constructing proofs
• Introduces students to the world of abstract mathematics
• Prepares students for further study in linear algebra, calculus and topology

Table of Contents
Some notes on notation
To the students
For the professors
Part I. The Axiomatic Method:
1. Introduction
2. Statements in mathematics
3. Proofs in mathematics
Part II. Set Theory:
4. Basic set operations
5. Functions
6. Relations on a set
7. Cardinality
Part III. Number Systems:
8. Algebra of number systems
9. The natural numbers
10. The integers
11. The rational numbers
12. The real numbers
13. Cantor's reals
14. The complex numbers
Part IV. Time Scales:
15. Time scales
16. The Delta Derivative
Part V. Hints:
17. Hints for (and comments on) the exercises
Index.

Theorems in School: From History, Epistemology and Cognition to Classroom Practice

(New Directions in Mathematics and Science Education)

 P. Boero

Sense Publishers | 2007 | 335 páginas | pdf | 3 Mb

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During the last decade, a revaluation of proof and proving within mathematics curricula was recommended; great emphasis was put on the need of developing proof-related skills since the beginning of primary school. This book, addressing mathematics educators, teacher-trainers and teachers, is published as a contribution to the endeavour of renewing the teaching of proof (and theorems) on the basis of historical-epistemological, cognitive and didactical considerations. Authors come from eight countries and different research traditions: this fact offers a broad scientific and cultural perspective. In this book, the historical and epistemological dimensions are dealt with by authors who look at specific research results in the history and epistemology of mathematics with an eye to crucial issues related to educational choices. Two papers deal with the relationships between curriculum choices concerning proof (and the related implicit or explicit epistemological assumptions and historical traditions) in two different school systems, and the teaching and learning of proof there. The cognitive dimension is important in order to avoid that the didactical choices do not fit the needs and the potentialities of learners. Our choice was to firstly deal with the features of reasoning related to proof, mainly concerning the relationships between argumentation and proof. The second part of this book concentrates on some crucial cognitive and didactical aspects of the development of proof from the early approach in primary school, to high school and university. We will show how suitable didactical proposals within appropriate educational contexts can match the great (yet, underestimated!) young students' potentialities in approaching theorems and theories.

CONTENTS
Preface
The ongoing value of proof
Gila Hanna 3
Introduction
Theorems in school: An introduction
Paolo Boero 19
Part I: The historical and epistemological dimension
1 Origin of mathematical proof: History and epistemology
Gilbert Arsac 27
2 The proof in the 20th century: From Hilbert to automatic theorem proving
Ferdinando Arzarello 43
3 Students’ proof schemes revisited
Guershon Harel 65
Part II: Curricular choices, historical traditions and learning of proof: Two national case studies
4 Curriculum change and geometrical reasoning
Celia Hoyles and Lulu Healy 81
5 The tradition and role of proof in mathematics education in Hungary
Julianna Szendrei-Radnai and Judit Török 117
Part III: Argumentation and proof
6 Cognitive functioning and the understanding ofmathematical processes of proof
Raymond Duval 137
7 Some remarks about argumentation and proof
Nadia Douek 163
Part IV: Didactical aspects
8 Making possible the discussion of “impossible in mathematics”
Greisy Winicki-Landman 185
9 The development of proof making by students
Carolyn A. Maher, Ethel M. Muter and Regina D. Kiczek 197
10 Approaching and developing the culture of geometry theorems in school: A theoretical framework
Marolina Bartolini Bussi, Paolo Boero, Franca Ferri, Rossella Garuti and Maria Alessandra Mariotti 211
11 Construction problems in primary school: A case from the geometry of circle
Maria G. Bartolini Bussi, Mara Boni and Franca Ferri 219
12 Approaching theorems in grade VIII: Some mental processes underlying producing and proving
conjectures, and conditions suitable to enhance them
Paolo Boero, Rossella Garuti and Enrica Lemut 249
13 From dynamic exploration to “theory” and “theorems” (from 6th to 8th grades)
Laura Parenti, Maria Teresa Barberis, Massima
Pastorino and Paola Viglienzone 265
14 Geometrical proof: The mediation of a microworld
Maria Alessandra Mariotti 285
15 The transition to formal proof in geometry
Ferdinando Arzarello, Federica Olivero, Domingo Paola and Ornella Robutti 305

The History of Mathematical Proof in Ancient Traditions

Karine Chemla, Centre National de la Recherche Scientifique (CNRS)

Cambridge University Press | 2012 | 614 páginas | PDF | 7,52 Mb

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'This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of 19th-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers, and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew how to prove the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof.' Jeremy Gray, Open University

Table of Contents

Prologue: historiography and history of mathematical proof: a research program Karine Chemla
Part I. Views on the Historiography of Mathematical Proof: 1. The Euclidean ideal of proof in The Elements and philological uncertainties of Heiberg's edition of the text Bernard Vitrac
2. Diagrams and arguments in ancient Greek mathematics: lessons drawn from comparisons of the manuscript diagrams with those in modern critical editions Ken Saito and Nathan Sidoli
3. The texture of Archimedes' arguments: through Heiberg's veil Reviel Netz
4. John Philoponus and the conformity of mathematical proofs to Aristotelian demonstrations Orna Harari
5. Contextualising Playfair and Colebrooke on proof and demonstration in the Indian mathematical tradition (1780–1820) Dhruv Raina
6. Overlooking mathematical justifications in the Sanskrit tradition: the nuanced case of G. F. Thibaut Agathe Keller
7. The logical Greek versus the imaginative Oriental: on the historiography of 'non-Western' mathematics during the period 1820–1920 François Charette
Part II. History of Mathematical Proof in Ancient Traditions: The Other Evidence: 8. The pluralism of Greek 'mathematics' Geoffrey Lloyd
9. Generalizing about polygonal numbers in ancient Greek mathematics Ian Mueller
10. Reasoning and symbolism in Diophantus: preliminary observations Reviel Netz
11. Mathematical justification as non-conceptualized practice: the Babylonian example Jens Høyrup
12. Interpretation of reverse algorithms in several Mesopotamian texts Christine Proust
13. Reading proofs in Chinese commentaries: algebraic proofs in an algorithmic context Karine Chemla
14. Dispelling mathematical doubts: assessing mathematical correctness of algorithms in Bhaskara's commentary on the mathematical chapter of the Aryabhatıya Agathe Keller
15. Argumentation for state examinations: demonstration in traditional Chinese and Vietnamese mathematics Alexei Volkov
16. A formal system of the Gougu method – a study on Li Rui's detailed outline of mathematical procedures for the right-angled triangle Tian Miao.

Outro livro de Karine Chemla , disponível no blog:

- K. Chemla (éd.) (2004). History of science, history of text, Springer, Collection « Boston studies in the philosophy of science 

Capítulos em livros de Karine Chemla , disponíveis no blog:

- Karine Chemla et Agathe Keller. (2002). The Sanskrit karanis, and the Chinese mian , Yvonne Dold-Samplonius, Joseph W. Dauben, Menso Folkerts, Benno van Dalen (éds.), From China to Paris: 2000 Years of Mathematical Transmission (Actes du Colloque de Bellagio, 5-2000) (pp. 87-132). Steiner Verlag, Stuttgart .

- K. Chemla (2005). The interplay between proof and algorithm in 3rd century China : The operation as prescription of computation and the operation as argument, in Paolo Mancosu, Klaus F. Jorgensen & Stig Andur Pedersen (éds.), Visualization, Explanation and Reasoning styles in mathematics,(pp. 123-145).  Synthese Library Series, volume 327, Springer, 

- K. Chemla (2009). Proof in the Wording : Two modalities from Ancient Chinese , in G. Hanna, H. N. Jahnke, H. Pulte (éds.), Explanation and Proof in Mathematics : Philosophical and Educational Perspectives, (pp. 253—285), Springer.

-  K. Chemla (2012). Using documents from ancient China to teach mathematical proof in G. Hanna and M. de Villiers (eds.), Proof and Proving in Mathematics Education (pp. 423 -). New ICMI Study Series 15

Capítulos em livros de Karine Chemla , disponíveis online (link externo ao blog):

- K. Chemla  (2010). A Chinese Canon in Mathematics and its two Layers of Commentaries : Reading a collection of texts as shaped by actors, in F. Bretelle-Establet (éd.), Looking at it from Asia : the processes that shaped the sources of history of science (pp 169—210), Springer, Boston Studies in the Philosophy of Science 265

Proof and Proving in Mathematics Education

The 19th ICMI Study

New ICMI Study Series, Vol. 15

Hanna, Gila; de Villiers, Michael (Eds.)

Springer | 2012 | 475 páginas | PDF | 7,23 Mb

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• First comprehensive book covering a wide range of aspects related to the teaching and learning of proof
• Provides illustrative examples of proof practices and activities in different contexts and levels
• Extensive survey of available research findings, and indicating directions for future research

One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels.

Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification.

This book, resulting from the 19^th ICMI Study, brings together a variety of viewpoints on issues such as:

• The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. 
• The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades.
   
• The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving.

The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.

Table of contents

1. Aspects of proof in mathematics education: Gila Hanna and Michael de Villiers.- Part I: Proof and cognition.- 2. Cognitive development of proof: David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva, and Ying-Hao Cheng .- 3. Theorems as constructive visions: Giuseppe Longo.- Part II: Experimentation: Challenges and opportunities.- 4. Exploratory experimentation: Digitally-assisted discovery and proof: Jonathan M. Borwein.- 5. Experimental approaches to theoretical thinking: Artefacts and proofs.- Ferdinando Arzarello, Maria Giuseppina Bartolini Bussi, Allen Leung, Maria Alessandra Mariotti, and Ian Stevenson (With response by J. Borwein and J. Osborn).- Part III: Historical and educational perspectives of proof.- 6. Why proof?  A historian’s perspective: Judith V. Grabiner.- 7. Conceptions of proof – in research and in teaching: Richard Cabassut, AnnaMarie Conner, Filyet Asli Ersoz, Fulvia Furinghetti, Hans Niels Jahnke, and Francesca Morselli.- 8. Forms of proof and proving in the classroom: Tommy Dreyfus, Elena Nardi, and Roza Leikin.- 9. The need for proof and proving: mathematical and pedagogical perspectives: Orit Zaslavsky, Susan D. Nickerson, Andreas Stylianides, Ivy Kidron, and Greisy Winicki.- 10. Contemporary proofs for mathematics education: Frank Quinn.- Part IV: Proof in the school curriculum.- 11. Proof, Proving, and teacher-student interaction: Theories and contexts: Keith Jones and Patricio Herbst.- 12. From exploration to proof production: Feng-Jui Hsieh, Wang-Shian Horng, and Haw-Yaw Shy.- 13. Principles of task design for conjecturing and proving: Fou-Lai Lin, Kyeong-Hwa Lee, Kai-Lin Yang, Michal Tabach, and Gabriel Stylianides.- 14. Teachers’ professional learning of teaching proof and proving: Fou-Lai Lin, Kai-Lin Yang, Jane-Jane Lo, Pessia Tsamir, Dina Tirosh, and Gabriel Stylianides.- Part V: Argumentation and transition to tertiary level.- 15. Argumentation and proof in the mathematics classroom: Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna Epp, and Denis Tanguay.- 16. Examining the role of logic in teaching proof: Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna Epp, and Denis Tanguay.- 17. Transitions and proof and proving at tertiary level: Annie Selden.- Part VI: Lessons from the Eastern cultural traditions.- 18. Using documents from ancient China to teach mathematical proof: Karine Chemla .- 19. Proof in the Western and Eastern traditions: Implications for mathematics education: Man Keung Siu.- Acknowledgements.- Appendix 1: Discussion Document.- Appendix 2: Conference Proceedings: Table of contents.- Author Index.- Subject Index.

Actas do Encontro

Fou-Lai Lin, Feng-Jui Hsieh
Gila Hanna, Michael de Villiers

The Department of Mathematics, National Taiwan Normal University | 2009 | PDF

vol 1

140.122.140.1 (link direto)

vol 2

140.122.140.1 (link direto)

Proofs Without Words II: More Exercises in Visual Thinking

Roger B. Nelsen

(Classroom Resource Materials) (v. 2)

The Mathematical Association of America | 2001 | 142 páginas | djvu | 10,52 Mb

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Like its predecessor, Proofs without Words, this book is a collection of pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. The proofs in this collection are arranged by topic into five chapters: geometry and algebra; trigonometry, calculus and analytic geometry; inequalities; integer sums; and sequences and series. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.

Experimental Mathematics in Action

David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke, Victor Moll

A K Pеters | 2007 | 200 páginas | PDF | 3,60 Mb

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The last twenty years have been witness to a fundamental shift in the way mathematics is practiced. With the continued advance of computing power and accessibility, the view that real mathematicians don't compute no longer has any traction for a newer generation of mathematicians that can really take advantage of computer-aided research, especially given the scope and availability of modern computational packages such as Maple, Mathematica, and MATLAB. The authors provide a coherent variety of accessible examples of modern mathematics subjects in which intelligent computing plays a significant role.

Roads to Infinity: The Mathematics of Truth and Proof

John Stillwell

A K Peters | 2010 | 250 páginas | PDF | 1,3 Mb

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This popular account of set theory and mathematical logic introduces the reader to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics.The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Goedel.

The Proof Is in the Pudding: The Changing Nature of Mathematical Proof

Steven G. Krantz

Springer-Verlag | 2010 | 240 Páginas | PDF | 3,9 Mb

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Versão draft - online: math.wustl.edu

Krantz takes the reader on a journey around the globe and through centuries of history, exploring the many transformations that mathematical proof has undergone from its inception at the time of Euclid and Pythagoras to its versatile, present-day use. The author elaborates on the beauty, challenges and metamorphisms of thought that have accompanied the search for truth through proof. The first two chapters examine the early beginnings of concept of proof and the creation of its elegant structure and language, touching on some of the logic and philosophy behind these developments. The history then unfolds as the author explains the changing face of proofs. The more well-known proofs , the mathematicians behind them, and the world that surrounded them are all highlighted. Each story has its own unique past; there was often a philosophical, sociological, technological or competitive edge that restricted or promoted progress. But the author's commentary and insights create a seamless thread throughout the many vignettes Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. This is shown in noting some of the more prominent discussions currently underway, such as Gorenstein's effort to classify finance groups, Thomas Hale's resolution of the Kepler sphere-packing problem, and other modern tales. Most of the proofs are discussed in detail with figures and some equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.

Theorems, Corollaries, Lemmas, and Methods of Proof

(Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

Richard J. Rossi

Wiley-Interscience | 2006 | 318 páginas | PDF | 8,35 Mb

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A hands-on introduction to the tools needed for rigorous and theoretical mathematical reasoning

Successfully addressing the frustration many students experience as they make the transition from computational mathematics to advanced calculus and algebraic structures, Theorems, Corollaries, Lemmas, and Methods of Proof equips students with the tools needed to succeed while providing a firm foundation in the axiomatic structure of modern mathematics.

This essential book:
* Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs
* Reinforces the foundations of calculus and algebra
* Explores how to use both a direct and indirect proof to prove a theorem
* Presents the basic properties of real numbers
* Discusses how to use mathematical induction to prove a theorem
* Identifies the different types of theorems
* Explains how to write a clear and understandable proof
* Covers the basic structure of modern mathematics and the key components of modern mathematics


A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. In addition, the author has supplied many clear and detailed algorithms that outline these proofs.

Theorems, Corollaries, Lemmas, and Methods of Proof uniquely introduces scratch work as an indispensable part of the proof process, encouraging students to use scratch work and creative thinking as the first steps in their attempt to prove a theorem. Once their scratch work successfully demonstrates the truth of the theorem, the proof can be written in a clear and concise fashion. The basic structure of modern mathematics is discussed, and each of the key components of modern mathematics is defined. Numerous exercises are included in each chapter, covering a wide range of topics with varied levels of difficulty.

Intended as a main text for mathematics courses such as Methods of Proof, Transitions to Advanced Mathematics, and Foundations of Mathematics, the book may also be used as a supplementary textbook in junior- and senior-level courses on advanced calculus, real analysis, and modern algebra.

Representation and Productive Ambiguity in Mathematics and the Sciences

Emily R. Grosholz

Oxford University Press, USA |  2007 |  250 páginas | PDF | 2,21 Mb

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Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry, from ancient Greece to the present day, but focusing particularly on the seventeenth and twentieth centuries. She argues that reductive methods are effective not because they diminish but because they multiply and juxtapose modes of representation. Such problem-solving is, she argues, best understood in terms of Leibnizian "analysis"--the search for conditions of intelligibility. Discovery and justification are then two aspects of one rational way of proceeding, which produces the mathematician's formal experience.

Grosholz defends the importance of iconic, as well as symbolic and indexical, signs in mathematical representation, and argues that pragmatic, as well as syntactic and semantic, considerations are indispensable fore mathematical reasoning. By taking a close look at the way results are presented on the page in mathematical (and biological, chemical, and mechanical) texts, she shows that when two or more traditions combine in the service of problem solving, notations and diagrams are subtly altered, multiplied, and juxtaposed, and surrounded by prose in natural language which explains the novel combination. Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productivebecause they are ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, will be of considerable interest to philosophers and historians of mathematics and science, and also have far-reaching consequences for epistemology and philosophy of language.

Math Proofs Demystified

Stan Gibilisco

McGraw-Hill | 290 páginas| PDF | 2,60 Mb

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Here's an entertaining way to learn to prove math theorems! You won't need formal training, unlimited time, or a genius IQ. In Math Proofs Demystified, best-selling math and science writer Stan Gibilisco provides an effective and painless way to overcome the intimidation most studentsfeel when venturing into math theory. By the time you finish this book, you'll be proving math theorems with confidence! You'll also understand the fundamentals of elementary logic.

With Math Proofs Demystified, you master the subject one simple step at a time -- at your own speed. This unique self-teaching guide offers problems at the end of each chapter and section to pinpoint weaknesses, and a 70-question final exam to reinforce the entire book.

If you want to build or refresh your math proof skills, here's a fast and entertaining self-teaching course that's specially designed to minimize anxiety. Get ready to:

• Explore the wonders of the mathematical world
• Learn all about logic, validity, paradoxes, fallacies, and forms of deception
• Investigate famous theorems
• Develop a plan of action for each proof
• Construct proofs for algebra, geometry, set theory, number theory, and other math fields
• See how theorems relate to graphs and functions
• Take a "final exam" and grade it yourself!

Simple enough for beginners but challenging enough for math-savvy readers, Math Proofs Demystified is a fun way to learn or brush up on the fundamentals of this fascinating subject.

Reading, Writing, and Proving: A Closer Look at Mathematics

2nd Edition (Undergraduate Texts in Mathematics)

Ulrich Daepp, Pamela Gorkin
2.ª edição

Springer | 2011 | 391 páginas | pdf | 5 Mb

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This book, which is based on Polya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics and ends with suggested projects for independent study. Students will follow Polya's four step approach: analyzing the problem, devising a plan to solve the problem, carrying out that plan, and then determining the implication of the result. In addition to the Polya approach to proofs, this book places special emphasis on reading proofs carefully and writing them well. The authors have included a wide variety of problems, examples, illustrations and exercises, some with hints and solutions, designed specifically to improve the student's ability to read and write proofs. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.

Contents
Chapter 1: The How, When, and Why of Mathematics
Solutions to Exercises
Spotlight: George Pólya
Chapter 2: Logically Speaking
Chapter 3: Introducing the Contrapositive and Converse
Chapter 4: Set Notation and Quantifiers
Chapter 5: Proof Techniques
Chapter 6: Sets
Spotlight: Paradoxes
Chapter 7: Operations on Sets
Chapter 8: More on Operations on Sets
Chapter 9: The Power Set and the Cartesian Product
Tips on Writing Mathematics
Chapter 10: Relations
Tips on Reading Mathematics
Chapter 11: Partitions
Tips on Putting It All Together
Chapter 12: Order in the Reals
Chapter 13: Consequences of the Completeness of R
Tips: You Solved It. Now What?
Chapter 14: Functions, Domain, and Range
Spotlight: The Definition of Function
Chapter 15: Functions, One-to-One, and Onto
Chapter 16: Inverses
Chapter 17: Images and Inverse Images
Spotlight: Minimum or Infimum?
Chapter 18: Mathematical Induction
Chapter 19: Sequences
Chapter 20: Convergence of Sequences of Real Numbers
Chapter 21: Equivalent Sets
Chapter 22: Finite Sets and an Infinite Set
Chapter 23:Countable and Uncountable Sets
Chapter 24: The Cantor–Schröder–Bernstein Theorem
Spotlight: The Continuum Hypothesis
Chapter 25:Metric Spaces
Chapter 26: Getting to Know Open and Closed Sets
Chapter 27: Modular Arithmetic
Chapter 28: Fermat’s Little Theorem
Spotlight: Public and Secret Research
Chapter 29: Projects
Tips on Talking about Mathematics
29.1 Picture Proofs
Guided Project
Open-Ended Project
Notes and Sources
29.2 The Best Number of All (and Some Other Pretty Good Ones)
29.3 Set Constructions
29.4 Rational and Irrational Numbers
29.5 Irrationality of e and p
29.6 A Complex Project
29.7 When Does f-1 = 1/f ?
29.8 Pascal’s Triangle
29.9 The Cantor Set
29.10 The Cauchy–Bunyakovsky–Schwarz Inequality
29.11 Algebraic Numbers
29.12 The Axiom of Choice
29.13 The RSA Code
Spotlight: Hilbert’s Seventh Problem
Appendix
Algebraic Properties of R
Order Properties of R
Axioms of Set Theory
Pólya’s List
References
Index

Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms

(Mathematics Education Library)
Carolyn A. Maher, Arthur B. Powell, Elizabeth B. Uptegrove

Springer | 2010 | 224 páginas |  PDF | 3,35 MB

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Descrição: Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms is based on the accomplishments of a cohort group of learners from first grade through high school and beyond, concentrating on their work on a set of combinatorics tasks. By studying these students, the editors gain insight into the foundations of proof building, the tools and environments necessary to make connections, activities to extend and generalize combinatoric learning, and even explore implications of this learning on the undergraduate level.

This volume underscores the power of attending to basic ideas in building arguments; it shows the importance of providing opportunities for the co-construction of knowledge by groups of learners; and it demonstrates the value of careful construction of appropriate tasks. Moreover, it documents how reasoning that takes the form of proof evolves with young children and discusses the conditions for supporting student reasoning.

Conteúdos:
Introduction.- The Longitudinal Study.- Methodology.- Representations as a Tool for Building Arguments.- Building Towers: Justifications Leading to Proof Making.- Making Pizzas: Reasoning by Cases and recursion.- Responding to Ankur's Challenge: Co-construction of Argument Leading to Proof.- Co-construction of Proof.- The Case of Stephanie.- Representations and Connections.- Pizzas, Block Towers, and Binomials.- Generalizing from Pizzas, Towers, Binomial Expansion, and Pascal's Triangle.- Extending and Generalizing the Isomorphism: Towers, Pizzas, Pascal's Triangle, and the Taxicab Problem.- College Students Building Towers and Making Pizzas.- Comparing the Problem Solving of College Students with Longitudinal Study Students.- Conclusions and Suggestions for Practice.

Explanation and Proof in Mathematics: Philosophical and Educational Perspectives

Gila Hanna, Hans Niels Jahnke, Helmut Pulte

Springer | 2009 | 204 páginas |  pdf | 4,34 Mb

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Descrição: In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles). A sampling of the coverage: The conjoint origins of proof and theoretical physics in ancient Greece. Proof as bearers of mathematical knowledge.Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein. Relationships between mathematical proof, problem-solving, and explanation. Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.

Proofs without Words: Exercises in Visual Thinking

(Classroom Resource Materials)
Roger B. Nelsen

The Mathematical Association of America | 1997 | 160 páginas | djvu | 3,24 Mb

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Descrição: Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. The proofs in this collection are arranged by topic into five chapters: Geometry and algebra; Trigonometry, calculus and analytic geometry; Inequalities; Integer sums; and Sequences and series. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.

Proof Theory: History and Philosophical Significance

(Synthese Library)
Vincent F. Hendricks, Stig Andur Pedersen, Klaus Frovin Jørgensen

Springer | 2000 | 256 páginas | djvu | 3,3 Mb

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Descrição: The conference on Proof Theory: History and Philosophical Significance held in 1997 at the University of Roskilde, Denmark, tracked the history of proof theory and its role in the analysis of the philosophical foundations of mathematics since Hilbert's original program to its modern, highly articulated form. This volume is a collection of papers presented at the conference. It can be read with profit and pleasure by philosophers, mathematicians, computer scientists and scholars with no professional training in proof theory, provided they have a general knowledge of foundational issues.

Conjecture and Proof

(Classroom Resource Materials)
Miklós Laczkovich

The Mathematical Association of America | 2001 | 140 páginas | djvu | 2, 9 Mb

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Descrição: The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to English-speaking students. This book is an elaborate version of the course on 'Conjecture and Proof'. It gives miniature introductions to various areas of mathematics by presenting some interesting and important, but easily accessible results and methods. The text contains complete proofs of deep results such as the transcendence of e, the Banach-Tarski paradox and the existence of Borel sets of arbitrary (finite) class. One of the purposes is to demonstrate how far one can get from the first principles in just a couple of steps. Prerequisites are kept to a minimum, and any introductory calculus course provides the necessary background for understanding the book. Exercises are included for the benefit of students. However, this book should prove fascinating for any mathematically literate reader.

Projective Geometry: From Foundations to Applications

Albrecht Beutelspacher, Ute Rosenbaum

Cambridge University Press | 1998 | 268 páginas | djvu | 1,77 Mb

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Descrição: This book introduces the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader to understand and construct proofs and write clear mathematics. The authors achieve this by exploring set theory, combinatorics and number theory, which include many fundamental mathematical ideas. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all time great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.