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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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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 --- Mathematics --- Philosophy --- History. --- History. --- 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.

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Logic is sometimes called the foundation of mathematics: the logician studies the kinds of reasoning used in the individual steps of a proof. Alonzo Church was a pioneer in the field of mathematical logic, whose contributions to number theory and the theories of algorithms and computability laid the theoretical foundations of computer science. His first Princeton book, The Calculi of Lambda-Conversion (1941), established an invaluable tool that computer scientists still use today. Even beyond the accomplishment of that book, however, his second Princeton book, Introduction to Mathematical Logic, defined its subject for a generation. Originally published in Princeton's Annals of Mathematics Studies series, this book was revised in 1956 and reprinted a third time, in 1996, in the Princeton Landmarks in Mathematics series. Although new results in mathematical logic have been developed and other textbooks have been published, it remains, sixty years later, a basic source for understanding formal logic. Church was one of the principal founders of the Association for Symbolic Logic; he founded the Journal of Symbolic Logic in 1936 and remained an editor until 1979 At his death in 1995, Church was still regarded as the greatest mathematical logician in the world.

Logic, Symbolic and mathematical. --- Abstract algebra. --- Acta Mathematica. --- Arithmetic. --- Axiom of choice. --- Axiom of infinity. --- Axiom of reducibility. --- Axiom schema. --- Axiom. --- Axiomatic system. --- Binary function. --- Boolean algebra (structure). --- Boolean ring. --- Calculus ratiocinator. --- Characterization (mathematics). --- Class (set theory). --- Classical mathematics. --- Commutative property. --- Commutative ring. --- Conditional disjunction. --- David Hilbert. --- Decision problem. --- Deduction theorem. --- Denotation. --- Disjunctive syllogism. --- Double negation. --- Duality (mathematics). --- Elementary algebra. --- Elementary arithmetic. --- English alphabet. --- Equation. --- Existential quantification. --- Expression (mathematics). --- Formation rule. --- Frege (programming language). --- Function (mathematics). --- Functional calculus. --- Fundamenta Mathematicae. --- Gödel numbering. --- Gödel's completeness theorem. --- Gödel's incompleteness theorems. --- Hilbert's program. --- Hypothetical syllogism. --- Imperative logic. --- Inference. --- Introduction to Mathematical Philosophy. --- Lambda calculus. --- Linear differential equation. --- Logic. --- Logical connective. --- Logical disjunction. --- Material implication (rule of inference). --- Mathematical analysis. --- Mathematical induction. --- Mathematical logic. --- Mathematical notation. --- Mathematical practice. --- Mathematical problem. --- Mathematical theory. --- Mathematics. --- Mathematische Zeitschrift. --- Metatheorem. --- Modal logic. --- Modus ponendo tollens. --- Natural number. --- Naturalness (physics). --- Negation. --- Notation. --- Number theory. --- Object language. --- Parity (mathematics). --- Predicate (mathematical logic). --- Prenex normal form. --- Principia Mathematica. --- Propositional calculus. --- Propositional function. --- Propositional variable. --- Quantifier (logic). --- Range (mathematics). --- Real number. --- Recursion (computer science). --- Restriction (mathematics). --- Riemann surface. --- Ring (mathematics). --- Rule of inference. --- Scientific notation. --- Second-order arithmetic. --- Series (mathematics). --- Sign (mathematics). --- Skolem normal form. --- Special case. --- Tautology (logic). --- Term logic. --- The Principles of Mathematics. --- Theorem. --- Three-dimensional space (mathematics). --- Transfinite number. --- Triviality (mathematics). --- Truth table. --- Variable (mathematics). --- Zermelo set theory.