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E-grāmata: X Marks the Spot: The Lost Inheritance of Mathematics [Taylor & Francis e-book]

, (Oakland University, USA)
  • Taylor & Francis e-book
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"X Marks the Spot is written from the point of view of the users of mathematics. Since the beginning, mathematical concepts and techniques (such as arithmetic and geometry) were created as tools with a particular purpose like counting sheep and measuringland areas. Understanding those purposes leads to a greater understanding of why mathematics developed as it did. Later mathematical concepts came from a process of abstracting and generalizing earlier mathematics. This process of abstraction is very powerful, but often comes at the price of intuition and understanding. This book strives to give a guided tour of the development of various branches of mathematics (and what they're used for) that will give the reader this intuitive understanding. Features:Treats mathematical techniques as tools, and areas of mathematics as the result of abstracting and generalizing earlier mathematical tools; Written in a relaxed conversational and occasionally humorous style making it easy to follow even when discussing esoterica; Unravels how mathematicians think, demystifying math and connecting it to the ways non-mathematicians think and connecting math to people's lives; Discusses how math education can be improved in order to prevent future generations from being turned off by math"--

X Marks the Spot is written from the point of view of the users of mathematics. Since the beginning, mathematical concepts and techniques (such as arithmetic and geometry) were created as tools with a particular purpose like counting sheep and measuring land areas.

Understanding those purposes leads to a greater understanding of why mathematics developed as it did. Later mathematical concepts came from a process of abstracting and generalizing earlier mathematics. This process of abstraction is very powerful, but often comes at the price of intuition and understanding. This book strives to give a guided tour of the development of various branches of mathematics (and what they’re used for) that will give the reader this intuitive understanding.

Features

  • Treats mathematical techniques as tools, and areas of mathematics as the result of abstracting and generalizing earlier mathematical tools
  • Written in a relaxed conversational and occasionally humorous style making it easy to follow even when discussing esoterica.
  • Unravels how mathematicians think, demystifying math and connecting it to the ways non-mathematicians think and connecting math to people’s lives
    • Discusses how math education can be improved in order to prevent future generations from being turned off by math.
  • List of Figures ix
    Preface xvii
    Authors xix
    1 Why This Book?
    1(8)
    Part I The Roots of Mathematics
    2 Sticks and Stones
    9(40)
    The Unnumbered World
    10(1)
    Stones: Counting
    11(6)
    Sticks: Measuring
    17(6)
    Multiplying the Stick
    23(4)
    Tricks with Sticks
    27(5)
    Rate of Change
    32(2)
    Units Multiply Like Rabbits
    34(3)
    The volume?
    35(2)
    Infinity
    37(10)
    The Numbered World
    47(2)
    3 Abstraction, Mistrust, and Laziness
    49(20)
    Abstraction
    49(7)
    Mistrust
    56(1)
    Premises
    57(2)
    Methods of Reasoning in Proofs
    59(3)
    Kinds of Proof
    62(1)
    Elegance
    63(1)
    Laziness
    64(3)
    Putting the Three Together
    67(2)
    4 Algebra, Geometry, Analysis: The Mathematical Mindsets
    69(64)
    Algebra
    72(20)
    Here it is: Value Abstraction. Let's bring him out again, x the unknown, Variable Ex traordinaire!
    75(17)
    Geometry
    92(3)
    Sticks Are Lines
    95(7)
    When Are Shapes the Same?
    102(2)
    Shapes as Tools
    104(11)
    Analysis
    115(4)
    Building Machines
    119(3)
    Function Behavior and Unknown Functions
    122(1)
    Logic
    123(2)
    Logic in Math
    125(3)
    Logic and Meaning
    128(1)
    Digging up the Roots
    128(5)
    Part II Theory in Practice
    5 Analytic Geometry
    133(50)
    Planes and Space, Numbers by Numbers by Numbers
    135(9)
    Function is Shape
    144(9)
    Parametric Graphing
    153(5)
    The Fault Is Not in Our Stars
    158(4)
    Curves and Surfaces
    162(7)
    Coordinate Systems
    169(7)
    More Dimensions than You Can Shake a Stick at
    176(7)
    6 Calculus: Motion and Size
    183(46)
    The Derivative
    185(2)
    Limits
    187(4)
    Derivative Redux (Reduced, That Is)
    191(9)
    When Can We Differentiate?
    200(2)
    Function Approximation and Taylor Series
    202(5)
    Vectors
    207(3)
    Integral Calculus
    210(7)
    The Fundamental Theorem of Calculus
    217(7)
    Logarithm and Exponential
    224(3)
    The Absence and Presence of Calculus
    227(2)
    7 The Language of Motion
    229(8)
    Linear Digression
    231(6)
    8 Sound, Notes, and Harmonics
    237(10)
    9 Probability and Statistics
    247(32)
    Cards and Randomness
    262(1)
    Combinatorics: Counting Without Counting
    263(1)
    Counting Cards
    264(2)
    Probability Distributions: Stick the Chances
    266(4)
    Stochastic Processes
    270(2)
    Statistics
    272(7)
    10 Other Geometries: Not So Straight, These Sticks
    279(36)
    The Universe Is Bent
    281(7)
    Shortest Distance Between Two Points
    288(3)
    Metric Spaces
    291(6)
    Nature Spiky in Coast and Leaf
    297(10)
    Why Are They Called Fractals?
    307(4)
    Iteration and Chaos
    311(4)
    11 Algebra and the Rise of Abstraction
    315(38)
    Construction and Its Limits
    326(7)
    Imaginary and Complex Numbers
    333(5)
    Algebraic Structures: Abstraction Spreads Wide Its Arms
    338(5)
    Morphisms: Preservatives Added and Multiplied
    343(2)
    Algebraic View of Geometry: Slouching Toward Topology
    345(3)
    Category Theory
    348(5)
    Part III Toolkit of the Theoretical Universe
    12 The Smith and the Knight
    353(12)
    How the User Sees the Tool
    354(2)
    The Maker's View of the Tool
    356(2)
    Beauty in the Use of Tools
    358(1)
    Beauty in the Making of Tools: Artists and Artisans
    359(1)
    Mathematics as Toolkit
    360(2)
    Using the Toolkit
    362(1)
    Expanding the Toolkit
    363(2)
    13 Building the Theoretical Universe
    365(14)
    Fluids
    365(6)
    Electricity and Magnetism
    371(8)
    14 Computers
    379(36)
    Logic Embodied
    381(5)
    Binary Arithmetic Embodied
    386(3)
    Computer Programming
    389(11)
    Modular Programming, Procedures, and Functions
    400(7)
    Speed, Memory, Bandwidth, Price, Size, and Efficiency
    407(1)
    Human Thought and Computer "Thought"
    408(3)
    Computer Use in Math and Science
    411(4)
    15 The Theoretical Universe of Modern Physics: Toolkit Included
    415(30)
    Relativity
    415(12)
    General Relativity
    427(5)
    Quantum Mechanics
    432(9)
    Quantum Field Theory
    441(4)
    16 Math Education and Math in Education
    445(10)
    Welcome to Math Problem World
    449(1)
    How to Teach It
    450(1)
    Math in Education
    451(2)
    Reclaiming the Lost
    453(1)
    A Stone's Throw
    454(1)
    Index 455
    David Garfinkle was born in 1958 and wanted to be a physicist ever since his first year of high school. He got a bachelor's degree from Princeton University in 1980 and a PhD from The University of Chicago in 1985. Since 1991 he has been a physics professor at Oakland University in Michigan.

    David is the author of over 100 articles in physics journals. His main areas of research are black holes, spacetime singularities, and gravitational radiation. He performs computer simulations of gravitational collapse to resolve questions about black holes and singularities.

    David was named a Fellow of the American Physical Society (APS) with the citation reading "for his numerous contributions to a wide variety of topics in relativity and semiclassical gravity."

    David and Richard have written Three Steps to the Universe (U. of Chicago Press, 2008) a book on black holes and dark matter.

    For the better part of his early life, Richard Garfinkle thought he wanted to be a mathematician. He went so far as to spend four years studying math at the University of Chicago before discovering that he really wanted to be a writer. He has had several science fiction and fantasy novels published. His first Celestial Matters (Tor 1996) won the Compton Crook award for best first novel. Richard and David have written Three Steps to the Universe (U. of Chicago Press, 2008) a book on black holes and dark matter. For his day job, he programs computers. Richard lives in Chicago with his wife and children.