Plato and Pythagoreanism

Phillip Sidney Horky 

 Oxford University Press | 2013 | 328 páginas | rar - pdf | 2,1 Mb

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Was Plato a Pythagorean? Plato's students and earliest critics thought so, but scholars since the nineteenth century have been more skeptical. With this probing study, Phillip Sidney Horky argues that a specific type of Pythagorean philosophy, called "mathematical" Pythagoreanism, exercised a decisive influence on fundamental aspects of Plato's philosophy. The progenitor of mathematical Pythagoreanism was the infamous Pythagorean heretic and political revolutionary Hippasus of Metapontum, a student of Pythagoras who is credited with experiments in harmonics that led to innovations in mathematics. The innovations of Hippasus and other mathematical Pythagoreans, including Empedocles of Agrigentum, Epicharmus of Syracuse, Philolaus of Croton, and Archytas of Tarentum, presented philosophers like Plato with novel ways to reconcile empirical knowledge with abstract mathematical theories. Plato and Pythagoreanism demonstrates how mathematical Pythagoreanism established many of the fundamental philosophical questions Plato dealt with in his central dialogues, including Cratylus, Phaedo, Republic, Timaeus, and Philebus. In the process, it also illuminates the historical significance of the mathematical Pythagoreans, a group whose influence on the development of philosophical and scientific methods has been obscured since late antiquity. The picture that results is one in which Plato inherits mathematical Pythagorean method only to transform it into a powerful philosophical argument about the essential relationships between the cosmos and the human being.

CONTENTS
Preface ix
Acknowledgments xv
Abbreviations xix
1. Aristotle on Mathematical Pythagoreanism in the Fourth Century bce 3
2. Hippasus of Metapontum and Mathematical Pythagoreanism 37
3 . Exoterism and the History of Pythagorean Politics 85
4 . Mathematical Pythagoreanism and Plato’s Cratylus 125
5 . What Is Wisest? Mathematical Pythagoreanism and Plato’s Phaedo 167
6 . Th e Method of the Gods: Mathematical Pythagoreanism and Discovery 201
Afterword 261
Bibliography 265
Index Locorum 281

General Index 295

Basic Math & Pre-Algebra Workbook

Mark Zegarelli 

For Dummies | 2014 -2ª edição | 315 Páginas | rar - PDF | 4,6 Mb

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1ª edição - 2012/2007

Basic Math and Pre-Algebra Workbook For Dummies, 2nd Edition helps take the guesswork out of solving math equations and will have you unraveling the mystery of FOIL in no time. Whether you need to brush up on the basics of addition, subtraction, multiplication, and division or you’re ready to tackle algebraic expressions and equations, this handy workbook will demystify math so you can get back to having fun in math class.

• Properly use negative numbers, units, inequalities, exponents, square roots, and absolute value

• Round numbers and estimate answers

• Solve problems with fractions, decimals, and percentages

• Navigate basic geometry

• Complete algebraic expressions and equations

• Understand statistics and sets

• Sample questions with step-by-step explanation

• Answers to practice problems so you can check your work

Let Basic Math and Pre-Algebra Workbook For Dummies, 2nd Edition take the guessing out of math and help you discover your problem solving potential.

Contents at a Glance

Introduction...... 1

Part I: Getting Started with Basic Math and Pre-Algebra.... 5

Chapter 1: We’ve Got Your Numbers......... 7

Chapter 2: Smooth Operators: Working with the Big Four Operations.. 23

Chapter 3: Getting Down with Negative Numbers..... 37

Chapter 4: It’s Just an Expression...... 49

Chapter 5: Dividing Attention: Divisibility, Factors, and Multiples... 69

PartII: Slicing Things Up: Fractions, Decimals, and Percents...... 89

Chapter 6: Fractions Are a Piece of Cake........ 91

Chapter 7: Fractions and the Big Four...... 109

Chapter 8: Getting to the Point with Decimals...... 143

Chapter 9: Playing the Percentages.... 165

Part III: A Giant Step Forward: Intermediate Topics...... 177

Chapter 10: Seeking a Higher Power through Scientific Notation... 179

Chapter 11: Weighty Questions on Weights and Measures. 189

Chapter 12: Shaping Up with Geometry.. 203

Chapter 13: Getting Graphic: Xy-Graphs...... 223

Part IV: The X Factor: Introducing Algebra..... 235

Chapter 14: Expressing Yourself with Algebraic Expressions... 237

Chapter 15: Finding the Right Balance: Solving Algebraic Equations.....257

Part V: The Part of Tens...... 273

Chapter 16: Ten Alternative Numeral and Number Systems.... 275

Chapter 17: Ten Curious Types of Numbers..... 283

Index....... 289

The beginnings of Greek mathematics

A. Szabó e A.M. Ungar

Springer | 1978 | 363 páginas | pdf | 16 Mb

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Table of contents : 
Title......Page 1
Contents......Page 2
Chronological Table......Page 4
Introduction......Page 5
1.1 Current views of the theory's development......Page 14
1.2 The concept of 'Dynamis'......Page 16
1.3 The mathematic part of the 'Theaetetus'......Page 18
1.4 The usage and chronology of 'Dynamis'......Page 20
1.5 'Tetragonismos'......Page 21
1.6 The mean proportional......Page 22
1.7 The mathematics lecture delivered by Theodorus......Page 25
1.8. The mathematical discoveries of Plato's Theaethetus......Page 28
1.9 The 'independence' of Theaetetus......Page 31
1.10 A glance at some rival theories......Page 33
1.11 The so called 'Theaetetus problem'......Page 35
1.12 The discovery of incommensurability......Page 40
1.13 The problem of doubling the square......Page 43
1.14 Doubling the square and the mean proportional......Page 46
2.1 Introduction......Page 47
2.2 A survey of the most important terms......Page 49
2.3 Consonances and intervals......Page 52
B. Diastema as interval......Page 53
2.4 The diastema between two numbers......Page 55
2.5 A digression on the theory of music......Page 57
2.6 End points and intervals pictured as straight lines......Page 60
2.7 'Diplasion', 'Hemiolion', 'Epitriton'......Page 62
2.8 The euclidean algorithm......Page 65
2.9 The canon......Page 66
2.10 Arithmetical operations on the canon......Page 68
2.12 'Aναλογια as 'geometric proportion'......Page 70
2.13 'Aνάλογον......Page 72
2.14 The preposition άνά......Page 73
2.15 The elliptic expression άνά λόγον......Page 75
2.16 The subsequent history of ανάλογον as a mathematical term......Page 76
2.17 Out of the canon and musical means......Page 78
2.18 The creation of the mathematical concept of λόγος......Page 81
2.19 A digression on the history of the word λόγος......Page 82
2.20 The appication of the theory of proportions to arithmetic and geometry..Page 83
2.21 The mean proportional in music, arithmetic and geometry......Page 85
2.22 The construction of the mean proportional......Page 86
2.23 Conclusion......Page 88
3.1 'Proof' in Greek mathematics......Page 90
3.2 The proof of incommensurability......Page 97
3.3 The origin of anti-empiricism and indirect proof......Page 106
3.4 Euclid's foundations......Page 108
3.5 Aristotle and foundations of mathematics......Page 111
3.6 'Hypotheseis'......Page 114
3.7 The 'assumptions' in dialectic......Page 116
3.8 How 'hypotheseis' were used......Page 117
3.9 'Hypotheseis' and the method of indirect proof......Page 119
3.10 A question of priority......Page 120
3.11 Zeno, the inventor of dialectic......Page 122
3.12 Plato and the Eleatics......Page 123
3.13 'Hypotheseis' and the foundations of mathematics......Page 124
3.14 The definition of 'Unit'......Page 126
3.15 Arithmetic and the teaching of the Eleatics......Page 128
3.16 The divisibility of numbers......Page 130
3.17 The problem of the 'aithemata'......Page 132
3.18 Euclid's postulates......Page 133
3.19 The construction of oenopides......Page 134
3.20 The first three postulates in the 'Elements'......Page 136
3.21 The 'Koinai Ennoiai'......Page 138
3.22 The word άξιωμα......Page 139
3.23 Plato's όμολογήματα and Euclid's άξιώματα......Page 141
3.24 "The whole is greater than the part"......Page 143
3.25 A complex of axioms......Page 147
3.26 The difference between postulates and axioms......Page 149
3.27 Arithmetic and geometry......Page 150
3.28 The science of space......Page 151
3.29 The foundations of geometry......Page 154
3.30 A reconsideration of some problems relating to early greek mathematics......Page 156
Postscript......Page 163
1. The prevailing view......Page 164
3. Elements of a pythagorean theory about the areas of parallelograms......Page 165
4. How to find a square with the same area as a given rectangle......Page 171
5. Conclusion......Page 174

New Horizons in Geometry

(Dolciani Mathematical Expositions)

Tom Apostol e Mamikon Mnatsakania

The Mathematical Association of America | 2013 | 528 páginas | rar - pdf | 5 Mb

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New Horizons in Geometry represents the fruits of 15 years of work in geometry by a remarkable team of prize-winning authors Tom Apostol and Mamikon Mnatsakanian. It serves as a capstone to an amazing collaboration. Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems that are often surprising and sometimes astounding. It is mathematical exposition of the highest order. 
The hundreds of full color illustrations by Mamikon are visually enticing and provide great motivation to read further and savor the wonderful results. Lengths, areas, and volumes of curves, surfaces, and solids are explored from a visually captivating perspective. It is an understatement to say that Apostol and Mamikon have breathed new life into geometry.

CONTENTS
Preface . ix
Introduction . . xi
Foreword . . xiii
Chapter 1. Mamikon’s Sweeping Tangent Theorem  . . 1
Chapter 2. Cycloids and Trochoids . . 31
Chapter 3. Cyclogons and Trochogons  . . 65
Chapter 4. Circumgons and Circumsolids . . 101
Chapter 5. The Method of Punctured Containers. . 135
Chapter 6. Unwrapping Curves from Cylinders and Cones . . 169
Chapter 7. New Descriptions of Conics via Twisted Cylinders, Focal Disks, and Directors . 213
Chapter 8. Ellipse to Hyperbola: “With This String I Thee Wed” . . 243
Chapter 9. Trammels . . 267
Chapter 10. Isoperimetric and Isoparametric Problems . . 295
Chapter 11. Arclength and Tanvolutes  331
Chapter 12. Centroids . . 375
Chapter 13. New Balancing Principles with Applications . . 401
Chapter 14. Sums of Squares. . 443
Chapter 15. Appendix . . 473

Warren Page

The Mathematical Association of America | 2013 | 153 páginas |  pdf | 861 kb

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Applications of Mathematics in Economics presents an overview of the (qualitative and graphical) methods and perspectives of economists. Its objectives are not intended to teach economics, but rather to give mathematicians a sense of what mathematics is used at the undergraduate level in various parts of economics, and to provide students with the opportunities to apply their mathematics in relevant economics contexts.

The volume’s applications span a broad range of mathematical topics and levels of sophistication. Each article consists of self-contained, stand-alone, expository sections whose problems illustrate what mathematics is used, and how, in that subdiscipline of economics. The problems are intended to be richer and more informative about economics than the economics exercises in most mathematics texts. Since each section is self-contained, instructors can readily use the economics background and worked-out solutions to tailor (simplify or embellish) a section’s problems to their students’ needs. Overall, the volume’s 47 sections contain more than 100 multipart problems. Thus, instructors have ample material to select for classroom uses, homework assignments, and enrichment activities.

Contents

Preface ix

Notes on the Sections xi

Problems and Subject Areas xvii

1 Microeconomics 1

Mary H. Lesser and Warren Page

2 Scenarios Involving Marginal Analysis 17

Julie Glass, Lynn Paringer, Jane Lopus

3 Intermediate Macroeconomics Theory 27

Michael K. Salemi

4 Closed Linear Systems 45

Warren Page and Alan Parks

5 Mathematics in Behavioral Economics 55

Michael Murray

6 Econometrics 75

Rae Jean B. Goodman

7 The Portfolio Problem 91

Kevin J. Hastings

8 Topics in Modern Finance 99

Frank Wang

9 Maximizing Profit with Production Constraints 115

Jennifer Wilson

About the Editor 131

About the Authors 133

Developments in Mathematical Education: Proceedings of the Second International Congress on Mathematical Education

ICME-2    1972      Exeter (UK)

A. G. Howson

Cambridge University Press | 1973 | 328 páginas | rar - pdf | 15,7 Mb

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Contents
A message from H.R.H. The Prince Philip, Duke of Edinburgh (p. v )
Editor's Acknowledgements (p. ix) 

Part I, A Congress Survey (p. 4) 
This chapter contains a very detailed analysis, presumably from Howson, of the planning, choices, and content of the Congress. It is divides into four parts:

1. Preliminaries and planning
2. The programme
3. The congress in action
4. The work of congress
5. The congress in retrospect

Part II The invited papers 
GORGE PÒLYA, As I read them (p. 77) 
JEAN PIAGET, Comments on mathematical education (p. 79) 
SIR JAMES LIGHTHILL, The Presidential Address ( p. 88) 
HANS FREUDENTHAL, What groups mean in mathematics and what they should mean in mathematics education (p. 101) 
DAVID HAWKINS, Nature, man and mathematics (p. 115) 
EDMUND LEACH, Some anthropological observations on number, time and common-sense (p. 136) 
HUGH PHILP, Mathematical education in developing countries - some problems of teaching and learning(p. 154) 
S. L. SOBOLEV, Some questions of mathematical education in the USSR (p. 181) 
RENÉ THOM, Modern mathematics: does it exist? (p. 194)

Part III A selection of congress papers 
E. BIGGS, Investigation and problem-solving in mathematical education (p. 213) 
E. FISCHBEIN, Intuition, structure and heuristic methods in the teaching of mathematics (p. 222) 
A. J. MALPAS, Mathematics and science in the secondary school (p. 233) 
B. E. MESERVE, Geometry as a gateway to mathematics (p. 241) 
J. B. MORGAN, The International Baccalaureate (p. 254) 
T. SHIBATA, The role of axioms in contemporary mathematics and in mathematical education (p. 262) 
M. SIME, Implications of the work of Piaget in the training of students to teach primary mathematics (p. 272) 
H. WHITNEY, Are we off the track in teaching mathematical concepts? (p. 283) 

Appendices

1. The congress committees and officers (p. 299)
2. The working groups (p. 300)
3. ICMI and congress recommendations (p. 305)
4. Films and video tapes on mathematics and its teaching (p. 307)

Index (p. 315) 

Love and Math: The Heart of Hidden Reality

Edward Frenkel

Basic Books | 2013 | 306 páginas | epub | 9 Mb

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mobi - 10 Mb - link 

pdf - 28 mb - link

What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren’t even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry.

In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we’ve never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space.

Love and Math tells two intertwined stories: of the wonders of mathematics and of one young man’s journey learning and living it. Having braved a discriminatory educational system to become one of the twenty-first century’s leading mathematicians, Frenkel now works on one of the biggest ideas to come out of math in the last 50 years: the Langlands Program. Considered by many to be a Grand Unified Theory of mathematics, the Langlands Program enables researchers to translate findings from one field to another so that they can solve problems, such as Fermat’s last theorem, that had seemed intractable before.

At its core, Love and Math is a story about accessing a new way of thinking, which can enrich our lives and empower us to better understand the world and our place in it. It is an invitation to discover the magic hidden universe of mathematics.

Contents
Preface
A Guide for the Reader
1 A Mysterious Beast
2 The Essence of Symmetry
3 The Fifth Problem
4 Kerosinka
5 Threads of the Solution
6 Apprentice Mathematician
7 The Grand Unified Theory
8 Magic Numbers
9 Rosetta Stone
10 Being in the Loop
11Conquering the Summit
12Tree of Knowledge
13Harvard Calling
14Tying the Sheaves of Wisdom
15A Delicate Dance
16Quantum Duality
17Uncovering Hidden Connections
18Searching for the Formula of Love
Epilogue
Acknowledgments
Notes
Glossary of Terms
Index