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24142 results for Mathematics

Page 1 of 1208

Title match

  • Chapter 11 - Mathematics

  • from Part III - Metaphysics
  • Jamin Asay, The University of Hong Kong
  • Book:
    A Theory of Truthmaking
    Published online:
    18 April 2020
    Print publication:
    30 April 2020, pp 223-246

  • Summary

    This chapter demonstrates how thinking about truthmakers can bring some clarity to the ongoing debate concerning the ontological status of mathematical entities, and advances one position on the topic. It begins with what is nowadays the most familiar argument in the ontology of mathematics: the indispensability argument. Close inspection of it reveals that the notion of truthmaking is indeed playing a role, and that by thinking about indispensability and truthmaking in tandem, both ideas emerge with greater clarity. Focusing on indispensability will shed light on what the ontological stakes are in the overall debate, so the second section presents what is seen to be the most tenable ontological positions. It then offers a personal (rather deflationary) contribution to the truthmaking question about mathematics, and concludes by comparing and contrasting that view with a recent brand of mathematical trivialism.

Title match

  • Mathematics

  • from ENTRIES
  • Edited by Lawrence Nolan, California State University, Long Beach
  • Book:
    The Cambridge Descartes Lexicon
    Published online:
    05 January 2016
    Print publication:
    01 January 2015, pp 470-474

  • Summary

    According to many of its past and present readers, and maybe the author himself, the Geometry forms the core of Descartes’ mathematical practice. From the point of view of its posterity, this judgment seems uncontroversial. By identifying the “geometrical” and the “algebraic,” the treatise from 1637 opened a new era in mathematics. It is well known that other (“infinitesimal”) techniques were also used by the philosopher in his correspondence when dealing with the quadrature of the cycloid or “Debeaune's problem” (Houzel 1997). But even when one acknowledges these other techniques (dealt with only outside of the published treatise), the comparison with the Geometry seems the proper background to assess Descartes’ practice. Yet this comparison raises the following questions: Is the Geometry the ending point of a continuous and self-conscious evolution? Is it part of a homogeneous and coherent practice characterizing Descartes’ mathematics as a whole?

    These questions are of interest not only for historians of mathematics. The Discourse on Method presents in effect the study of mathematics as a privileged field where the famous “method” is developed. Just after the statement of the four “rules,” Descartes mentions as an origin for his methodology “the long chains of reasonings, every one simple and easy, which geometers habitually employ to reach their most difficult proofs” (AT VI 19, CSM I 120). Symmetrically, it is with mathematics – the most simple and certain science – that one has to start the inquiry. However, he then makes clear that one need not learn the whole of mathematics but only what forms its core: the study of “ratios and proportions,” which he proposes to represent by lines, which in turn are represented by symbols (AT VI 20, CSM I 121). The program exposed by Descartes in Discourse II seems to have been achieved: starting from the three disciplines he learned when he was young (logic, geometrical analysis, and algebra), Descartes finally succeeds in finding a method that “retained the advantages of all three but was free from their defects” (AT VI 17, CSM I 19).

Title match

  • Mathematics

  • Edited by Marla Parker, SunSoft
  • Book:
    She Does Math!
    Published by:
    Mathematical Association of America
    Published online:
    05 June 2012
    Print publication:
    01 June 1995, pp 23-25

  • Summary

    When I was growing up, women were not expected to choose a career in mathematics. Since I was always a rather hard-headed, stubborn individual, I rebelled against this notion and decided to do it anyway. And now, as a mathematician, I derive great satisfaction at the surprise in people's faces when they find out what I do for a living.

    Mathematics is great. I can sit and think, deduce and reason, be creative all day long—and get paid for it. And what's really interesting is that most of the work I do has no immediate practical value, because mathematicians don't usually attack a problem with an application in mind. We leave it to the scientists to take our results and find some use for them.

    As for my history: I majored in mathematics in college and obtained my BA from the University of Florida. Next, I joined the Peace Corps and went to Liberia, West Africa, where I had my first experience as a teacher. I taught mathematics, and my students taught me about their culture and what it meant to be an African. After my tour in Africa, I went to Scotland to teach math, and found that mathematics is important in all cultures.

    Upon returning to the United States, I went back to school, earning a BA in art at the University of California, Santa Cruz (UCSC).

Title match

  • Appendix A - Mathematics

  • from Part VIII - Fundaments
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 381-404

  • Summary

    The fundaments of mathematics include:

  • • A.1: a brief introduction to the algebra of vectors and tensors in three dimensions;

  • • A.2: a summary of the calculus of three-dimensional vectors;

  • • A.3: an introduction to curvilinear coordinate systems;

  • • A.4: an introduction to Taylor series;

  • • A.5: an introduction to Fourier series and integrals;

  • • A.6: classification of several simple types of linear, second-order partial differential equations;

  • • A.7: a listing of the Greek symbols that are commonly used in applied mathematics; and

  • • A.8: an introduction to scalar and vector potentials.

    • These mathematical fundaments will employ the orthogonal reference coordinates system introduced in § 2.1 (and visualized in Figure 2.1) and will use extensively the associated unit vectors 11, 12 and 13. These vectors are arranged in a right-hand configuration such that 11 × 12 = 13. Any or all of these will be designated by 1 i, with i = 1, 2 and/or 3, as the situation requires. These vectors satisfy the orthogonality relation 1 i •1 j = δij, where δij is the Kronecker delta.

    Vectors and Tensors

    This fundament is not a general summary, but instead focuses on vectors and tensors in three-dimensional space, specifically vectors having three elements (or components) and tensors of rank two having nine elements.

    Definitions, Notation and Representation

    A tensor is an ordered set of nk numbers, where k and n are non-negative integers, that obeys certain tensor transformation rules. The integer k is the rank of the tensor. The A tensor of rank

    A tensor of rank 0 is a scalar: a number.

    A tensor of rank 1 is a vector: an ordered set of n numbers. Each number is a component of the vector. (Typically n = 3)

    Tensors may be thought of as square arrays in k dimensions. A matrix is a related two-dimensional array of size m by n, where m and n are positive integers. A matrix is square if m = n. A tensor of rank two behaves algebraically the same as a square matrix, and the tensor elements may be written out and manipulated in matrix-display form.

    In this document, a scalar is denoted by a lower case letter in italics (e.g., x).

Title match

  • 21 - Mathematics

  • David C. Lindberg, Michael H. Shank
  • Book:
    The Cambridge History of Science
    Published online:
    05 September 2013
    Print publication:
    07 October 2013, pp 512-531

  • Summary

    Since Roman times, the word mathematici had applied especially to mathematicians, and in the thirteenth century Roger Bacon still strove at length to convince his audience of the difference between false mathematics and true mathematics. Boethius produced an influential Latin work on arithmetic, which was closely based on the Greek Introduction to Arithmetic of the neo-Pythagorean Nicomachus of Gerasa and also a substantial treatise on music, a science that was then regarded as intimately dependent upon arithmetic. Geometrical figures themselves could perform important semiotic functions, as when in the fourteenth century Nicole Oresme used graphic-like diagrams to symbolize variations of speeds and qualities. In standard histories of mathematics, the name of Jordanus de Nemore is linked with that of Leonardo of Pisa as another medieval mathematician of undoubted originality. A striking feature of medieval mathematics, especially in its theoretical genre, is the amount of discussion that was conducted in terms of ratios and their equalities or proportions.

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Title match

  • Mathematics

  • Edited by Marla Parker, SunSoft
  • Book:
    She Does Math!
    Published by:
    Mathematical Association of America
    Published online:
    05 June 2012
    Print publication:
    01 June 1995, pp 87-91

  • Summary

    My Nutley High School yearbook caption says that I “would like to teach college math,” probably because I had taken five year-long mathematics courses, plus the usual college prep curriculum. But why did I choose—and remain with—math? No doubt for a variety of reasons: the influence of my father's engineering background; my mother's competence in basic consumer mathematics; the American commitment to mathematics and science in the aftermath of the Soviet Union launching the spacecraft, Sputnik; and the fact that I liked mathematics—and find it a continuing challenge.

    I fulfilled my career goal by entering Douglass College, the women's college of Rutgers University in New Jersey, majoring in mathematics with a minor in French. Then, I went directly into Rutgers' graduate school under a fellowship, and completed my PhD in mathematics six years later. As a full-time faculty member for three of those six years, I had to make every minute count to meet all the responsibilities of both a teacher and a student. I put many miles (and even more kilometers!) on my car, consulting with my dissertation advisor who was taking a year's sabbatical in a distant city.

    Fortunately, I have always been a well-organized person so I was able to juggle both roles, but it did mean sacrificing a good deal of personal time.

    My field of research is in real analysis and special function theory, part of pure mathematics. Currently, my research interests focus on improving the teaching and learning of mathematics; engaging those who are under-represented in mathematics; curing math avoidance and anxiety; and strategic planning in higher education.

Title match

  • 10 - Mathematics

  • from Part II
  • Stuart Brock, Victoria University of Wellington, New Zealand
  • Book:
    Realism and Anti-Realism
    Published by:
    Acumen Publishing
    Published online:
    05 February 2013
    Print publication:
    30 March 2007, pp 150-173

  • Summary

    Introduction

    This chapter is slightly more technical than the others. This is required by the subject matter. We try as much as possible either to avoid technical notions or to restrict discussion of them to footnotes. The material presented here is appropriate for readers who have studied a little logic or even those who remember at least some of the mathematics that they learnt in school.

    A mathematical statement is a statement such as “2 + 2 = 4” or “the square root of two is not a fraction”. What makes these statements true? And what makes other mathematical statements false, such as “2 + 2 = 5” and “the square root of two is a fraction”? Realism about mathematics claims that there are mathematical objects, such as numbers, that have mind-independent properties that make these statements true or false. Thus, for example, realists claim that there is something that is the number two that has the property that it is half of the number four. One might think that it is obvious that there are numbers: that we produce numbers by writing them. But realists think that in addition to numerals that we write there are numbers that are not created by human beings.

    A mathematical theory is a set of statements about a putative class of mathematical objects. Thus, for example, arithmetic is a theory about the natural numbers, 0, 1, 2, 3, …. One way of viewing the realist debate in mathematics is to think in terms of mathematical theories.

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  • 6 - Mathematics

  • Alfred W. Crosby, University of Texas, Austin
  • Book:
    The Measure of Reality
    Published online:
    05 October 2013
    Print publication:
    28 November 1996, pp 109-126

  • Summary

    Wherefore in all great works are Clerks so much desired? Wherefore are Auditors so well fed? What causeth Geometricians so highly to be enhaunsed? Why are Astronomers so greatly advanced? Because that by number such things they finde, which else would farre excell mans minde.

    Robert Recorde (1540)

    Certain Western Europeans of the late Middle Ages and Renaissance began tentatively to consider the possibilities of absolute time and space. The advantages were that absolute properties, by definition, were permanent and universal, which meant that it was worth the effort to measure them and to analyze and manipulate the measurements in various ways. Measurement is numbers, and the manipulation of numbers is mathematics. Thomas Bradwardine, Schoolman and archbishop of Canterbury in the fourteenth century, said, “Whoever then has the effrontery to study physics while neglecting mathematics, should know from the start that he will never make his entry through the portals of wisdom.”

    Roger Bacon, John Buridan, Theodoric of Freiberg, Nicole Oresme, and others of a like mind prefigured Kepler and Galileo with their glorification of geometry and, particularly in the case of Oresme, with a conviction that numbers could be imposed where they had previously been thought inappropriate. Oresme (who spent much of his life in Paris and must have heard Charles V's authoritarian clock many times) wrote in a treatise entitled The Geometry of Qualities and Motion that for the measurement of things of “continuous quantity” – for instance, motion or heat – “it is necessary that points, lines, and surfaces, or their properties be imagined….

Title match

  • Mathematics

  • Edited by Marla Parker, SunSoft
  • Book:
    She Does Math!
    Published by:
    Mathematical Association of America
    Published online:
    05 June 2012
    Print publication:
    01 June 1995, pp 156-160

  • Summary

    I didn't know I liked math until I took the Kuder Personal Preference Test and a course in algebra. A typical question in the Kuder Test is: would you rather grow a flower, sell a flower, or develop a new breed of flower? The test told me that my interest in computation was at the 96th percentile. Discovery of my interest in math (according to the test) surprised me because I had thought that math was boring and also because no one else in my family was interested in math or science. Then I discovered algebra. It gave me a powerful tool to solve the kinds of puzzles I enjoyed doing anyway, so I was convinced that math could really be interesting and useful.

    I took algebra in the 9th grade, geometry in the 10th, algebra II in the 11th, and two math courses in the first semester of my senior year—trigonometry and solid geometry—and calculus the second semester (a rigorous, epsilon-and-delta course!). I always got A's in math. The only sciences I took were chemistry and physics. I was nervous about taking physics, because I thought it was a “boy's subject,” but my counselor convinced me to try it and I became the best student in the class. I also took two semesters of drafting in high school, one of them in summer school; that remains one of my proudest accomplishments, since I went from F's on all my early work up through D's, C's, B's, and finally A's.

Title match

  • 1 - Mathematics

  • Andreu Mas-Colell, Harvard University, Massachusetts
  • Book:
    The Theory of General Economic Equilibrium
    Published online:
    05 January 2013
    Print publication:
    30 August 1985, pp 6-56

  • Summary

    This chapter gathers a number of mathematical definitions and theorems that, to different extents, will be needed for the economic theory of the following seven chapters. It is not meant to be read systematically before the rest of the book. The chapter is divided into twelve sections. The ordering of the sections is one of convenience and not of intrinsic importance for later developments. This being a book on differentiable techniques, it stands to reason that the central sections are Sections B on linear algebra, C on differential calculus, D on optimization, H on differentiable manifolds, I on transversality theory, and J on degrees of functions and indices of zeros of vector fields.

    The economic theory of this book is presented in Chapters 2 through 8 with a fairly strict adherence to the axiomatic method and without presupposing much. No similar claim is made, however, for this chapter. A quick reading of the headings of the different sections will convince the reader that their content cannot be a systematic, complete, or rigorous exposition. It serves to fix terminology and to facilitate reference, but if one wishes to go deeper into the purely mathematical aspects, this chapter is no substitute for the study of the pertinent mathematical sources cited at the end of every section. It should suffice to say that the chapter does not contain any proofs.

Title match

  • 11 - Mathematics

  • from Part II - Pedagogy in Interaction
  • Edited by Amelia Church, University of Melbourne, Amanda Bateman, Swansea University
  • Book:
    Talking with Children
    Published online:
    16 June 2022
    Print publication:
    30 June 2022, pp 227-246

  • Summary

    Early childhood educators are faced daily with supporting young children’s learning in educational environments that include digital technologies. This chapter first discusses what is currently known about young children’s use of digital technologies from health and educational perspectives. Specifically, this chapter shows how young children’s digital technology use in early childhood classrooms is influenced by a range of factors, such as access to digital technology and educator beliefs about the benefits of technology, and how to integrate digital technology in ways that align with early childhood pedagogy. Next, this chapter shows how educators and children engage with digital technologies through employing interactional practices that support inquiry-based learning, problem solving, and conceptual engagement, including digital and critical literacy skills. In this way, it highlights how teacher pedagogy-in-use makes possible children’s rich engagement with digital technologies. Finally, this chapter shows how these understandings are be applied by educators in their own practices.

Title match

  • 9 - Mathematics

  • Tim J. Stevens, MRC Laboratory of Molecular Biology, Cambridge, Wayne Boucher, University of Cambridge
  • Book:
    Python Programming for Biology
    Published online:
    05 February 2015
    Print publication:
    12 February 2015, pp 137-159

  • Summary

    Using Python for mathematics

    Given that Python is an interpreted programming language, rather than a fast compiled language, many people do not consider it for writing programs that involve extensive numerical work. While Python programs are certainly slower to execute than the equivalent written in something like C or FORTRAN, mathematical functionality certainly exists in Python and has the inherent advantages of the language; it is easy for people to use and conveniently links to other helpful data structures. Of course speed of calculation may not be so important, for a scientific investigation it may not matter if something takes 1 second or 0.1 second to run. Fortunately, computers get faster and the Python interpreter becomes improved, so you can do quite a bit of numerical work without concern. However, if calculation speed really is important in a given situation then there are a few things you can do to make things faster while still keeping the convenience of Python. For example, you can write code in C, a very efficient numerical language, and use it from within Python (this is called a C extension), effectively extending the vocabulary of the interpreted language with speedy subroutines. More recently the language Cython has helped make C extensions very easy to write. Cython is a Python-like language, and virtually all Python programs can be interpreted by it, without alteration, but the language ultimately generates C code that can be compiled. Cython can be used to call fast library code written in pure C, and can incorporate a mixture of Python and C data structures in the same code; although less flexible, the C data structures are very efficient. Writing C extensions and Cython modules is discussed in Chapter 27.

    Python includes standard arithmetic operations as part of the core functionality: add, multiply etc. There is an additional module, math, which always comes packaged with Python and which provides further numerical functionality: logarithms, trigonometry etc. For numerical calculations that are not especially intensive, the core functionality and the math module will often suffice. There has been a history of trying to provide modules for quick numerical algorithms in Python. The first attempt, begun in 1995, was called Numeric, and the second attempt was called Numarray. These two are now deemed to be obsolete, but the third attempt, begun in 2005, is called NumPy (http://numpy.scipy.org/), incorporates elements from the earlier attempts and will hopefully last longer.

Title match

  • 1 - Mathematics

  • Franco M. Capaldi, Merrimack College, Massachusetts
  • Book:
    Continuum Mechanics
    Published online:
    05 August 2012
    Print publication:
    18 June 2012, pp 1-34

  • Summary

    As scientists and engineers, we make sense of the world around us through observation and experimentation. Using mathematics, we attempt to describe our observations andmake useful predictions based on these observations. For example, a simple experimental observation that the distance traversed by an object traveling at a constant velocity is linearly related to both the velocity and the time can be formalized using the relation, d = vt, where d is the distance vector, v is the velocity vector, and t is the time. The distance, velocity, and time are physical quantities that can be measured or controlled. Physical quantities such as distance, velocity, and time are represented mathematically as tensors. A scalar, for example, is a zeroth-order tensor. Only amagnitude is required to specify the value of a zeroth-order tensor. In our previous example, time is such a quantity. If you are told that the duration of an event was 3 seconds, you need no other information to fully characterize this physical quantity. Velocity, on the other hand, requires both a magnitude and a direction to specify its meaning. The velocity would be represented using a first-order tensor, also known as a vector. The internal stress in a material is a second-order tensor, which requires a magnitude and two directions to specify its value. You may recognize that the two required directions are the normal of the surface on which the stress acts and the direction of the traction vector on this surface.

Title match

Title match

  • Mathematics

  • from SCIENCE
  • Edited by Ray Lester
  • Book:
    The New Walford Guide to Reference Resources
    Published by:
    Facet
    Published online:
    08 June 2018
    Print publication:
    18 May 2004, pp 60-83

  • Summary

    Introductions to the subject

    1089 and all that: a journey into mathematics

    D. Acheson Oxford University Press, 2002, 184pp. £12.99. ISBN 0198516231.

    http://home.jesus.ox.ac.uk/∼dacheson [COMPANION]

    A rewarding and enjoyable introduction.

    ‘Parts of this book are extremely funny … [It is] an ideal stocking filler … An ideal present for friends and relatives who are not mathematicians, but have enough curiosity to spend a gentle afternoon trying to find out what mathematics is about … Buy this book.’ (London Mathematical Society Newsletter) ‘There are a few mathematicians who succeed in writing popular accounts of their craft without being superficial or condescending. With this book Acheson has joined the best of them.’ (Times Higher Education Supplement)

  • ■ Mathematics: a very short introduction T. Gowers Oxford University Press, 2002, 143pp. £6.99. ISBN 0192853619. A pleasant inviting read.

    • The A to Z of mathematics: a basic guide

    T.H. Sidebotham Wiley, 2002, 474pp. $69.95. ISBN 0471150452.

    ‘Makes math simple without making it simplistic’. Good reliable introductory guide for ‘the millions of people who would love to understand math but are turned away by fear of its complexity’.

    Conversations with a mathematician: math, art, science and the limits of reason: a collection of his most wide-ranging and non-technical lectures and interviews

    G.J. Chaitin Springer, 2002, 158pp. £14.99. ISBN 1852335491.

    ‘G. J. Chaitin is at the IBM Thomas J. Watson Research Center in New York. He has shown that God plays dice not only in quantum mechanics, but even in the foundations of mathematics, where Chaitin discovered mathematical facts that are true for no reason, that are true by accident. This book collects his most wide-ranging and non-technical lectures and interviews, and it will be of interest to anyone concerned with the philosophy of mathematics, with the similarities and differences between physics and mathematics, or with the creative process and mathematics as an art.’

    ‘This book is wonderful in both senses of the word … superlatively good and full of wonder. Nonmathematicians could read it, too, but as I read it I felt glad (and proud) to be a mathematician!’ (The American Mathematical Monthly)

    Development of mathematics 1950–2000 J.-P. Pier, ed. Birkhäuser, 2000, 1372pp. £147.50. ISBN 3764362804.

Title match

  • 13 - Mathematics

  • from Part II - Disciplines
  • Edited by Roy Porter, Wellcome Institute for the History of Medicine, University College London
  • Book:
    The Cambridge History of Science
    Published online:
    28 March 2008
    Print publication:
    17 March 2003, pp 305-327

  • Summary

    Considered broadly, mathematical activity in the eighteenth century was characterized by a strong emphasis on analysis and mechanics. The great advances occurred in the development of calculus-related parts of mathematics and in the detailed elaboration of the program of inertial mechanics founded during the Scientific Revolution. There were other mathematical developments of note – in the theory of equations, number theory, probability and statistics, and geometry – but none of them reached anything like the depth and scope attained in analysis and mechanics.

    The close relationship between mathematics and mechanics had a basis that extended deep into Enlightenment thought. In the Preliminary Discourse to the famous French Encyclopédie, Jean d’ Alembert distinguished between “pure” mathematics (geometry, arithmetic, algebra, calculus) and “mixed” mathematics (mechanics, geometrical astronomy, optics, art of conjecturing). He classified mathematics more generally as a “science of nature” and separated it from logic, a “science of man.” An internalized and critical spirit of inquiry, associated with the invention of new mathematical structures (for example, non-commutative algebra, non-Euclidean geometry, logic, set theory), represents characteristics of modern mathematics that would emerge only in the next century.

    Although there were several notable British mathematicians of the period–Abraham De Moivre, James Stirling, Brook Taylor, and Colin Maclaurin among them – the major lines of mathematical production occurred on the Continent, a trend that intensified as the century developed. Leadership was provided by a relatively small number of energetic figures: Jakob, Johann, and Daniel Bernoulli, Jakob Hermann, Leonhard Euler, Alexis Clairaut, Jean d’Alembert, Johann Heinrich Lambert, Joseph Louis Lagrange, Adrien Marie Legendre, and Pierre Simon Laplace.

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