Duality in 19th and 20th Century Mathematical Thinking
Duality in 19th and 20th Century Mathematical Thinking

Author: Ralf Krömer

Publisher: Springer Nature

Published: 2024

Total Pages: 962

ISBN-13: 3031597974

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This volume brings together scholars across various domains of the history and philosophy of mathematics, investigating duality as a multi-faceted phenomenon. Encompassing both systematic analysis and historical examination, the book endeavors to elucidate the status, roles, and dynamics of duality within the realms of 19th and 20th-century mathematics. Eschewing a priori notions, the contributors embrace the diverse interpretations and manifestations of duality, thus presenting a nuanced and comprehensive perspective on this intricate subject. Spanning a broad spectrum of mathematical topics and historical periods, the book uses detailed case studies to investigate the different forms in which duality appeared and still appears in mathematics, to study their respective histories, and to analyze interactions between the different forms of duality. The chapters inquire into questions such as the contextual occurrences of duality in mathematics, the influence of chosen forms of representation, the impact of investigations of duality on mathematical practices, and the historical interconnections among various instances of duality. Together, they aim to answer a core question: Is there such a thing as duality in mathematics, or are there just several things called by the same name and similar in some respect? What emerges is that duality can be considered as a basic structure of mathematical thinking, thereby opening new horizons for the research on the history and the philosophy of mathematics and the reflection on mathematics in general. The volume will appeal not only to experts in the discipline but also to advanced students of mathematics, history, and philosophy intrigued by the complexities of this captivating subject matter.

Research in History and Philosophy of Mathematics
Research in History and Philosophy of Mathematics

Author: Maria Zack

Publisher: Springer Nature

Published: 2024-01-18

Total Pages: 190

ISBN-13: 3031461932

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This volume contains 8 papers that have been collected by the Canadian Society for History and Philosophy of Mathematics. It showcases rigorously reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics.Some of the topics explored include: A way to rethink how logic is taught to philosophy students by using a rejuvenated version of the Aristotelian idea of an argument schema A quantitative approach using data from Wikipedia to study collaboration between nineteenth-century British mathematicians The depiction and perception of Émilie Du Châtelet’s scientific contributions as viewed through the frontispieces designed for books written by or connected to her A study of the Cambridge Women’s Research Club, a place where British women were able to participate in scholarly scientific discourse in the middle of the twentieth century An examination of the research and writing process of mathematicians by looking at their drafts and other preparatory notes A global history of al-Khwārāzmī’s Kitāb al-jabr wa-l-muqābala as obtained by tracing its reception through numerous translations and commentaries Written by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics.

Monoidal Category Theory
Monoidal Category Theory

Author: Noson S. Yanofsky

Publisher: MIT Press

Published: 2024-11-05

Total Pages: 669

ISBN-13: 026238079X

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A comprehensive, cutting-edge, and highly readable textbook that makes category theory and monoidal category theory accessible to students across the sciences. Category theory is a powerful framework that began in mathematics but has since expanded to encompass several areas of computing and science, with broad applications in many fields. In this comprehensive text, Noson Yanofsky makes category theory accessible to those without a background in advanced mathematics. Monoidal Category Theorydemonstrates the expansive uses of categories, and in particular monoidal categories, throughout the sciences. The textbook starts from the basics of category theory and progresses to cutting edge research. Each idea is defined in simple terms and then brought alive by many real-world examples before progressing to theorems and uncomplicated proofs. Richly guided exercises ground readers in concrete computation and application. The result is a highly readable and engaging textbook that will open the world of category theory to many. Makes category theory accessible to non-math majors Uses easy-to-understand language and emphasizes diagrams over equations Incremental, iterative approach eases students into advanced concepts A series of embedded mini-courses cover such popular topics as quantum computing, categorical logic, self-referential paradoxes, databases and scheduling, and knot theory Extensive exercises and examples demonstrate the broad range of applications of categorical structures Modular structure allows instructors to fit text to the needs of different courses Instructor resources include slides

Framing Global Mathematics
Framing Global Mathematics

Author: Norbert Schappacher

Publisher: Springer Nature

Published: 2022-06-15

Total Pages: 20

ISBN-13: 3030956830

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This open access book is about the shaping of international relations in mathematics over the last two hundred years. It focusses on institutions and organizations that were created to frame the international dimension of mathematical research. Today, striking evidence of globalized mathematics is provided by countless international meetings and the worldwide repository ArXiv. The text follows the sinuous path that was taken to reach this state, from the long nineteenth century, through the two wars, to the present day. International cooperation in mathematics was well established by 1900, centered in Europe. The first International Mathematical Union, IMU, founded in 1920 and disbanded in 1932, reflected above all the trauma of WW I. Since 1950 the current IMU has played an increasing role in defining mathematical excellence, as is shown both in the historical narrative and by analyzing data about the International Congresses of Mathematicians. For each of the three periods discussed, interactions are explored between world politics, the advancement of scientific infrastructures, and the inner evolution of mathematics. Readers will thus take a new look at the place of mathematics in world culture, and how international organizations can make a difference. Aimed at mathematicians, historians of science, scientists, and the scientifically inclined general public, the book will be valuable to anyone interested in the history of science on an international level.

Reader's Guide to the History of Science
Reader's Guide to the History of Science

Author: Arne Hessenbruch

Publisher: Routledge

Published: 2013-12-16

Total Pages: 986

ISBN-13: 1134263015

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The Reader's Guide to the History of Science looks at the literature of science in some 550 entries on individuals (Einstein), institutions and disciplines (Mathematics), general themes (Romantic Science) and central concepts (Paradigm and Fact). The history of science is construed widely to include the history of medicine and technology as is reflected in the range of disciplines from which the international team of 200 contributors are drawn.

Visualization, Explanation and Reasoning Styles in Mathematics
Visualization, Explanation and Reasoning Styles in Mathematics

Author: P. Mancosu

Publisher: Springer Science & Business Media

Published: 2006-03-30

Total Pages: 315

ISBN-13: 1402033354

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In the 20th century philosophy of mathematics has to a great extent been dominated by views developed during the so-called foundational crisis in the beginning of that century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justi?cation and consistency of mathematics. Paradigmatic in this - spect is Hilbert’s program which inherits from Frege and Russell the project to formalize all areas of ordinary mathematics and then adds the requi- ment of a proof, by epistemically privileged means (?nitistic reasoning), of the consistency of such formalized theories. While interest in modi?ed v- sions of the original foundational programs is still thriving, in the second part of the twentieth century several philosophers and historians of mat- matics have questioned whether such foundational programs could exhaust the realm of important philosophical problems to be raised about the nature of mathematics. Some have done so in open confrontation (and hostility) to the logically based analysis of mathematics which characterized the cl- sical foundational programs, while others (and many of the contributors to this book belong to this tradition) have only called for an extension of the range of questions and problems that should be raised in connection with an understanding of mathematics. The focus has turned thus to a consideration of what mathematicians are actually doing when they produce mathematics. Questions concerning concept-formation, understanding, heuristics, changes instyle of reasoning, the role of analogies and diagrams etc.

How Mathematicians Think
How Mathematicians Think

Author: William Byers

Publisher: Princeton University Press

Published: 2010-05-02

Total Pages: 424

ISBN-13: 0691145997

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To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.

Collected Papers of Salomon Bochner, Part 4
Collected Papers of Salomon Bochner, Part 4

Author: Salomon Bochner

Publisher: American Mathematical Soc.

Published: 1992-01-01

Total Pages: 480

ISBN-13: 9780821801772

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During his long and productive career, Salomon Bochner worked in a variety of different areas of mathematics. This four-part set brings together his collected papers, illustrating the range and depth of his mathematical interests. The books are available either individually or as a set.