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"Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research.The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis.Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of ‘infinity’ and its permissible uses in mathematics, number systems, and evolving views of the role of axioms.Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity.Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject."

Mathematics --- Inventions --- History. --- Mathematical models. --- Creative ability in technology --- Prior art (Patent law) --- Research, Industrial --- Math --- Science --- mathematical ideas --- development of numbers --- development number systems --- negative numbers --- irrationals numbers --- complex numbers

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In line with the emerging field of philosophy of mathematical practice, this book pushes the philosophy of mathematics away from questions about the reality and truth of mathematical entities and statements and toward a focus on what mathematicians actually do--and how that evolves and changes over time. How do new mathematical entities come to be? What internal, natural, cognitive, and social constraints shape mathematical cultures? How do mathematical signs form and reform their meanings? How can we model the cognitive processes at play in mathematical evolution? And how does mathematics tie together ideas, reality, and applications?Roi Wagner uniquely combines philosophical, historical, and cognitive studies to paint a fully rounded image of mathematics not as an absolute ideal but as a human endeavor that takes shape in specific social and institutional contexts. The book builds on ancient, medieval, and modern case studies to confront philosophical reconstructions and cutting-edge cognitive theories. It focuses on the contingent semiotic and interpretive dimensions of mathematical practice, rather than on mathematics' claim to universal or fundamental truths, in order to explore not only what mathematics is, but also what it could be. Along the way, Wagner challenges conventional views that mathematical signs represent fixed, ideal entities; that mathematical cognition is a rigid transfer of inferences between formal domains; and that mathematics' exceptional consensus is due to the subject's underlying reality. The result is a revisionist account of mathematical philosophy that will interest mathematicians, philosophers, and historians of science alike.

Mathematics --- Benedetto. --- Black-Scholes formula. --- Eugene Wigner. --- Friedrich W.J. Schelling. --- George Lakoff. --- Gilles Deleuze. --- Hermann Cohen. --- Hilary Putnam. --- Johann G. Fichte. --- Logic of Sensation. --- Mark Steiner. --- Rafael Nez. --- Stanislas Dehaene. --- Vincent Walsh. --- Water J. Freeman III. --- abbaco. --- algebra. --- arithmetic. --- authority. --- cognitive theory. --- combinatorics. --- conceptual freedom. --- constraints. --- economy. --- gender role stereotypes. --- generating functions. --- geometry. --- inferences. --- infinities. --- infinity. --- mathematical cognition. --- mathematical concepts. --- mathematical cultures. --- mathematical domains. --- mathematical entities. --- mathematical evolution. --- mathematical interpretation. --- mathematical language. --- mathematical metaphor. --- mathematical norms. --- mathematical objects. --- mathematical practice. --- mathematical signs. --- mathematical standards. --- mathematical statements. --- mathematics. --- natural order. --- natural sciences. --- nature. --- negative numbers. --- number sense. --- option pricing. --- philosophy of mathematics. --- reality. --- reason. --- relevance. --- semiosis. --- sexuality. --- stable marriage problem. --- Philosophy --- History.

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The legendary Renaissance math duel that ushered in the modern age of algebraThe Secret Formula tells the story of two Renaissance mathematicians whose jealousies, intrigues, and contentious debates led to the discovery of a formula for the solution of the cubic equation. Niccolò Tartaglia was a talented and ambitious teacher who possessed a secret formula—the key to unlocking a seemingly unsolvable, two-thousand-year-old mathematical problem. He wrote it down in the form of a poem to prevent other mathematicians from stealing it. Gerolamo Cardano was a physician, gifted scholar, and notorious gambler who would not hesitate to use flattery and even trickery to learn Tartaglia's secret.Set against the backdrop of sixteenth-century Italy, The Secret Formula provides new and compelling insights into the peculiarities of Renaissance mathematics while bringing a turbulent and culturally vibrant age to life. It was an era when mathematicians challenged each other in intellectual duels held outdoors before enthusiastic crowds. Success not only enhanced the winner's reputation, but could result in prize money and professional acclaim. After hearing of Tartaglia's spectacular victory in one such contest in Venice, Cardano invited him to Milan, determined to obtain his secret by whatever means necessary. Cardano's intrigues paid off. In 1545, he was the first to publish a general solution of the cubic equation. Tartaglia, eager to take his revenge by establishing his superiority as the most brilliant mathematician of the age, challenged Cardano to the ultimate mathematical duel.A lively and compelling account of genius, betrayal, and all-too-human failings, The Secret Formula reveals the epic rivalry behind one of the fundamental ideas of modern algebra.

Equations --- Algebra --- Mathematics --- History. --- Ars Magna. --- Arturo Sangalli. --- Blaise Pascal. --- Cardano-Tartaglia formula. --- Carl Friedrich Gauss. --- Euler's Pioneering Equation. --- Ian Stewart. --- Isaac Newton. --- Italian Renaissance. --- Joseph-Louis Lagrange. --- Leibniz. --- Lodovico Ferrari. --- Maria Gaetana Agnesi. --- Omar Khayyám. --- Pierre de Fermat. --- Rafael Bombelli. --- Renaissance men. --- Robin Wilson. --- Scipione del Ferro. --- Significant Figures. --- algebra. --- binomial coefficients. --- binomial theorem. --- coefficients. --- imaginary numbers. --- negative numbers. --- roots of the cubic function. --- Tartaglia, Niccolò, --- Cardano, Girolamo, --- Cardano, Girolamo --- Tartaglia, Niccolò --- 1500-1599 --- Tartaglia, Nicolaus --- Fontana, Niccolò --- Tartalea, Niccolò --- Tartaglia, Nicolas --- Tartalea, Nicolo --- Tartaglia, Nicolo --- Tartalea --- Mathematiker --- Rechenmeister --- Übersetzer --- Ingenieur --- Lehrer --- Prof. --- Brescia --- Venedig --- Verona --- 1500-1557 --- Cardano, Geronimo --- Cardano, Hieronimo --- Cardano, Hieronymus --- Cardano --- Cardani, Hieronymus --- Cardanus, Girolamo --- Cardanus, Hieronimus Castellioneus --- Cardanus, Hieronymus Castellioneus --- Cardanus, Hieronimus C. --- Cardanus, Hier. --- Cardanus --- Cardanus, H. --- Cardini, Hieronymus --- Giralmo --- Hieronymus --- Kardano, Dzhirolamo --- Cardano, Gerolamo --- Cardano, Giralmo --- Cardan, Jérôme --- Cardan, Hierosme --- Cardanus, Hieronymus C. --- Cardanus, Hieronymus --- Cardanus, Hierosme --- Castellioneus, Hieronymus --- Geronimo --- Girolamo --- Jérome --- Cardan, Girolamo --- Cardano, Hieronimus --- Jérôme --- Cardan --- Cardan, H. --- Naturwissenschaftler --- Astronom --- Philosoph --- Arzt --- Pavia --- Rom --- Padua --- Mailand --- Bologna --- Arnaldus --- Benvenutus Rambaldi --- Caesius, Georg --- Cardano, Giovanni Battista --- Cigalini, Francesco --- Dasypodius, Konrad --- Frölich, Huldrich --- Majoragio, Marcantonio --- Petrarca, Francesco --- Ursinus, Adam --- 24.09.1501-20.09.1576 --- 24.11.1501-01.09.1575 --- 1501-1576 --- 1501-1575 --- Fontana, Nicolò, --- Tartaglia, Nicholas, --- Tartaglia, Nicolas, --- Tartaglia, Nicolò, --- Tartaia, Nicolò, --- Tartalea, Nicolai, --- Cardan, H., --- Cardan, Jérôme, --- Cardano, Gerolamo, --- Cardano, Hieronimo, --- Cardanus, Hierome, --- Cardanus, Hieronymus, --- Cardanus, Hierosme, --- Kardano, Dzhirolamo, --- Cardan, Jerome --- Cardan, Jerome,