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Philip Beeley - One of the best experts on this subject based on the ideXlab platform.

  • practical mathematicians and Mathematical Practice in later seventeenth century london
    The British Journal for the History of Science, 2019
    Co-Authors: Philip Beeley
    Abstract:

    Mathematical practitioners in seventeenth-century London formed a cohesive knowledge community that intersected closely with instrument-makers, printers and booksellers. Many wrote books for an increasingly numerate metropolitan market on topics covering a wide range of Mathematical disciplines, ranging from algebra to arithmetic, from merchants' accounts to the art of surveying. They were also teachers of mathematics like John Kersey or Euclid Speidell who would use their own rooms or the premises of instrument-makers for instruction. There was a high degree of interdependency even beyond their immediate milieu. Authors would cite not only each other, but also practitioners of other professions, especially those artisans with whom they collaborated closely. Practical Mathematical books effectively served as an advertising medium for the increasingly self-conscious members of a new emerging professional class. Contemporaries would talk explicitly of 'the London mathematicians' in distinction to their academic counterparts at Oxford or Cambridge. The article takes a closer look at this metropolitan knowledge culture during the second half of the century, considering its locations, its meeting places and the Mathematical clubs which helped forge the identity of its practitioners. It discusses their backgrounds, teaching Practices and relations to the London book trade, which supplied inexpensive practical Mathematical books to a seemingly insatiable public.

Sagüillo Fernández-vega, José Miguel - One of the best experts on this subject based on the ideXlab platform.

  • Hilary Putnam on the philosophy of logic and mathematics
    Servicio Editorial de la Universidad del País Vasco Euskal Herriko Unibertsitatearen Argitalpen Zerbitzua, 2018
    Co-Authors: Sagüillo Fernández-vega, José Miguel
    Abstract:

    This paper focuses on Putnam’s conception of logical truth as grounded in his picture of Mathematical Practice and ontology. Putnam’s 1971 book Philosophy of Logic came one year later than Quine’s homonymous volume. In the first section, I compare these two Philosophies of Logic which exemplify realist-nominalist viewpoints in a most conspicuous way. The next section examines Putnam’s views on modality, moving from the modal qualification of his intuitive conception to his official generalized non-modal second-order set-theoretic concept of logical truth. In the third section, I emphasize how Putnam´s “mathematics as modal logic” departs from Quine’s “reluctant Platonism”. I also suggest a complementary view of Platonism and modalism showing them perhaps interchangeable but underlying different stages of research processes that make up a rich and dynamic Mathematical Practice. The final, more speculative section, argues for the pervasive platonistic conception enhancing the aims of inquiry in the Practice of the working mathematician; Este artículo estudia la concepción de Putnam de verdad lógica que emana de su visión de la práctica de la matemática y de su ontología. Philosophy of Logic, el libro de 1971 de Putnam surge un año más tarde que el homónimo de Quine. En la primera sección, se comparan estas dos Filosofías de la Lógica que ejemplifican los puntos de vista del realismo y del nominalismo de modo conspicuo. La siguiente sección examina el enfoque de la modalidad de Putnam, que va desde la cualificación modal de su caracterización intuitiva de validez lógica a su concepción oficial generalizada no-modal conjuntista de segundo orden. La tercera sección subraya el modo en que "la matemática como lógica modal" de Putnam se distancia del "Platonism a regañadientes" de Quine. Aquí se sugiere una visión complementaria del Platonism y del modalismo, los cuales, aunque quizás intercambiables, se muestran subyaciendo a los diferentes estadios del proceso de investigación de una práctica de la matemática rica y dinámica. La sección final, más especulativa, conjetura algunas razones de la persistente concepción platónica implícita en la práctica del matemático

  • Hilary Putnam on the philosophy of logic and mathematics
    Universidad del País Vasco Euskal Herriko Unibertsitatea, 2018
    Co-Authors: Sagüillo Fernández-vega, José Miguel
    Abstract:

    This paper focuses on Putnam’s conception of logical truth as grounded in his picture of Mathematical Practice and ontology. Putnam’s 1971 book Philosophy of Logic came one year later than Quine’s homonymous volume. In the first section, I compare these two Philosophies of Logic which exemplify realist-nominalist viewpoints in a most conspicuous way. The next section examines Putnam’s views on modality, moving from the modal qualification of his intuitive conception to his official generalized non-modal second-order set-theoretic concept of logical truth. In the third section, I emphasize how Putnam´s “mathematics as modal logic” departs from Quine’s “reluctant Platonism”. I also suggest a complementary view of Platonism and modalism showing them perhaps interchangeable but underlying different stages of research processes that make up a rich and dynamic Mathematical Practice. The final, more speculative section, argues for the pervasive platonistic conception enhancing the aims of inquiry in the Practice of the working mathematician.Este artículo estudia la concepción de Putnam de verdad lógica que emana de su visión de la práctica de la matemática y de su ontología. Philosophy of Logic, el libro de 1971 de Putnam surge un año más tarde que el homónimo de Quine. En la primera sección, se comparan estas dos Filosofías de la Lógica que ejemplifican los puntos de vista del realismo y del nominalismo de modo conspicuo. La siguiente sección examina el enfoque de la modalidad de Putnam, que va desde la cualificación modal de su caracterización intuitiva de validez lógica a su concepción oficial generalizada no-modal conjuntista de segundo orden. La tercera sección subraya el modo en que «la matemática como lógica modal» de Putnam se distancia del «Platonism a regañadientes» de Quine. Aquí se sugiere una visión complementaria del Platonism y del modalismo, los cuales, aunque quizás intercambiables, se muestran subyaciendo a los diferentes estadios del proceso de investigación de una práctica de la matemática rica y dinámica. La sección final, más especulativa, conjetura algunas razones de la persistente concepción platónica implícita en la práctica del matemáticoThe research for this paper was supported by the Spanish Ministry of Economy and Competitivity and FEDER via the research projects FFI 2013-41415-P and FFI2017-82534-PS

Paolo Mancosu - One of the best experts on this subject based on the ideXlab platform.

  • the philosophy of Mathematical Practice
    2008
    Co-Authors: Paolo Mancosu
    Abstract:

    Introduction 1. Visualization 2. Diagrammatic Reasoning and Representational Systems 3. Explanation 4. Purity of Methods 5. Mathematical concepts 6. Philosophical Relevance of Category Theory 7. Philosophical Relevance of Computers in Mathematics 8. Philosophical Relevance of the interaction between Mathematical physics and pure mathematics

  • philosophy of mathematics and Mathematical Practice in the seventeenth century
    1996
    Co-Authors: Paolo Mancosu
    Abstract:

    The seventeenth century saw dramatic advances in Mathematical theory and Practice. With the recovery of many of the classical Greek Mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between Mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new Mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

Sarah Kate Selling - One of the best experts on this subject based on the ideXlab platform.

  • truth isn t everything promoting aesthetically guided choice in Mathematical work
    The Journal of Mathematical Behavior, 2016
    Co-Authors: Nicholas Fiori, Sarah Kate Selling
    Abstract:

    Abstract Most educational and philosophical thought about mathematics focuses on the logical structure of the subject and considers mathematicians and students to be people whose primary Practices are verifying statements within this structure. We claim that acts of discernment – careful choices driven by aesthetic considerations – are as important as acts of verification in Mathematical work. This paper offers a conceptualization of this “aesthetically guided choice” that differentiates between three interrelated acts of discernment: nominating ideas, arranging ideas, and balancing ideas. We argue that aesthetically guided choice should be supported in school, and that such acts are notably lacking from the “Standards for Mathematical Practice” in the Common Core State Standards. This paper is the development of a theory at heart, built around rich descriptions of a mathematics class with middle school students, in which young thinkers were engaged in aesthetic choice. It includes analyses of the history of mathematics and studies of Mathematical work to support claims about the nature of mathematics and to interpret the work of young mathematics learners from the perspective of mathematics as a discipline.

Brendan Larvor - One of the best experts on this subject based on the ideXlab platform.

  • As Thurston says? On using quotations from famous mathematicians to make points about philosophy and education
    ZDM, 2020
    Co-Authors: Gila Hanna, Brendan Larvor
    Abstract:

    It is commonplace in the educational literature on Mathematical Practice to argue for a general conclusion from isolated quotations from famous mathematicians. In this paper, we supply a critique of this mode of inference. We review empirical results that show the diversity and instability of mathematicians’ opinions on Mathematical Practice. Next, we compare mathematicians’ diverse and conflicting testimony on the nature and purpose of proof. We lay especial emphasis on the diverse responses mathematicians give to the challenges that digital technologies present to older conceptions of Mathematical Practice. We examine the career of one much cited and anthologised paper, WP Thurston’s ‘On Proof and Progress in Mathematics’ (1994). This paper has been multiply anthologised and cited hundreds of times in educational and philosophical argument. We contrast this paper with the views of other, equally distinguished mathematicians whose use of digital technology in mathematics paints a very different picture of Mathematical Practice. The interesting question is not whether mathematicians disagree—they are human so of course they do. The question is how homogenous is their Mathematical Practice . If there are deep differences in Practice between mathematicians, then it makes little sense to use isolated quotations as indicators of how mathematics is uniformly or usually done. The paper ends with reflections on the usefulness of quotations from research mathematicians for Mathematical education.

  • what philosophy of Mathematical Practice can teach argumentation theory about diagrams and pictures
    2013
    Co-Authors: Brendan Larvor
    Abstract:

    There has been a rising tide of interest among argumentation theorists in visual reasoning. In the hands of the leaders of this development the effort has been to assimilate visual reasoning to verbal argumentation. At the same time, there is a more mature but still advancing literature on the use of diagrams in Mathematical reasoning. There have been efforts to bring the two together. In this paper, I wish to use the philosophy of Mathematical Practice to identify a severe limitation in the attempt to assimilate visual reasoning to verbal reasoning, and by extension to criticise the approach to reasoning that treats all reasoning as if it were verbal reasoning.