Philosophy of mathematics Books

Book SynopsisPhilosophy of Mathematics: An Introduction provides a critical analysis of the major philosophical issues and viewpoints in the concepts and methods of mathematics - from antiquity to the modern era. Offers beginning readers a critical appraisal of philosophical viewpoints throughout history Gives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicism Provides readers with a non-partisan discussion until the final chapter, which gives the author''s personal opinion on where the truth lies Designed to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals Trade Review“Given this caveat, Bostock’s new book is highly recommendable as a text for undergraduate seminars in the philosophy of mathematics and also for individual study. It covers all the essentials and more. It should appeal not only to students who have already developed a preference for the general approach and style of contemporary analytic philosophy, but also to a broader audience of students and to people with a non-professional interest in philosophy and mathematics.” (Erkenn, 2011) "This is a concise as well as comprehensive presentation of core topics in the philosophy of mathematics, written in a clear and engaged manner, hence well readable." (Zentralblatt MATH, 2011) "This book is an undergraduate introduction to the basic ideas on the nature of mathematics that have played a significant role in the development of philosophy from Antiquity to contemporary debates . . . throughout the book the emphasis is on the basic ideas as well as their current variations, leading up to recent debates between realists and nominalists." (Mathematical Reviews, 2011) Table of ContentsIntroduction. Part I: Plato versus Aristotle:. A. Plato. 1. The Socratic Background. 2. The Theory of Recollection. 3. Platonism in Mathematics. 4. Retractions: the Divided Line in Republic VI (509d−511e). B. Aristotle. 5. The Overall Position. 6. Idealizations. 7. Complications. 8. Problems with Infinity. C. Prospects. Part II: From Aristotle to Kant:. 1. Medieval Times. 2. Descartes. 3. Locke, Berkeley, Hume. 4. A Remark on Conceptualism. 5. Kant: the Problem. 6. Kant: the Solution. Part III: Reactions to Kant:. 1. Mill on Geometry. 2. Mill versus Frege on Arithmetic. 3. Analytic Truths. 4. Concluding Remarks. Part IV: Mathematics and its Foundations:. 1. Geometry. 2. Different Kinds of Number. 3. The Calculus. 4. Return to Foundations. 5. Infinite Numbers. 6. Foundations Again. Part V: Logicism:. 1. Frege. 2. Russell. 3. Borkowski/Bostock. 4. Set Theory. 5. Logic. 6. Definition. Part VI: Formalism:. 1. Hilbert. 2. Gödel. 3. Pure Formalism. 4. Structuralism. 5. Some Comments. Part VII: Intuitionism:. 1. Brouwer. 2. Intuitionist Logic. 3. The Irrelevance of Ontology. 4. The Attack on Classical Logic. Part VIII: Predicativism:. 1. Russell and the VCP. 2. Russell’s Ramified Theory and the Axiom of Reducibility. 3. Predicative Theories after Russell. 4. Concluding Remarks. Part IX: Realism versus Nominalism:. A. Realism. 1. Gödel. 2. Neo-Fregeans. 3. Quine and Putnam. B. Nominalism. 4. Reductive Nominalism. 5. Fictionalism. 6. Concluding Remarks. References. Index

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Book SynopsisPhilosophy of Mathematics: An Introduction provides a critical analysis of the major philosophical issues and viewpoints in the concepts and methods of mathematics - from antiquity to the modern era. Offers beginning readers a critical appraisal of philosophical viewpoints throughout history Gives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicism Provides readers with a non-partisan discussion until the final chapter, which gives the author''s personal opinion on where the truth lies Designed to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals Trade Review“Given this caveat, Bostock’s new book is highly recommendable as a text for undergraduate seminars in the philosophy of mathematics and also for individual study. It covers all the essentials and more. It should appeal not only to students who have already developed a preference for the general approach and style of contemporary analytic philosophy, but also to a broader audience of students and to people with a non-professional interest in philosophy and mathematics.” (Erkenn, 2011) "This is a concise as well as comprehensive presentation of core topics in the philosophy of mathematics, written in a clear and engaged manner, hence well readable." (Zentralblatt MATH, 2011) Table of ContentsIntroduction. Part I: Plato versus Aristotle:. A. Plato. 1. The Socratic Background. 2. The Theory of Recollection. 3. Platonism in Mathematics. 4. Retractions: the Divided Line in Republic VI (509d−511e). B. Aristotle. 5. The Overall Position. 6. Idealizations. 7. Complications. 8. Problems with Infinity. C. Prospects. Part II: From Aristotle to Kant:. 1. Medieval Times. 2. Descartes. 3. Locke, Berkeley, Hume. 4. A Remark on Conceptualism. 5. Kant: the Problem. 6. Kant: the Solution. Part III: Reactions to Kant:. 1. Mill on Geometry. 2. Mill versus Frege on Arithmetic. 3. Analytic Truths. 4. Concluding Remarks. Part IV: Mathematics and its Foundations:. 1. Geometry. 2. Different Kinds of Number. 3. The Calculus. 4. Return to Foundations. 5. Infinite Numbers. 6. Foundations Again. Part V: Logicism:. 1. Frege. 2. Russell. 3. Borkowski/Bostock. 4. Set Theory. 5. Logic. 6. Definition. Part VI: Formalism:. 1. Hilbert. 2. Gödel. 3. Pure Formalism. 4. Structuralism. 5. Some Comments. Part VII: Intuitionism:. 1. Brouwer. 2. Intuitionist Logic. 3. The Irrelevance of Ontology. 4. The Attack on Classical Logic. Part VIII: Predicativism:. 1. Russell and the VCP. 2. Russell’s Ramified Theory and the Axiom of Reducibility. 3. Predicative Theories after Russell. 4. Concluding Remarks. Part IX: Realism versus Nominalism:. A. Realism. 1. Gödel. 2. Neo-Fregeans. 3. Quine and Putnam. B. Nominalism. 4. Reductive Nominalism. 5. Fictionalism. 6. Concluding Remarks. References. Index

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Book SynopsisBerto''s highly readable and lucid guide introduces students and the interested reader to Gödel''s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel''s arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel''s theories Written in an accessible, non-technical style Trade Review"There's Something about G¨odel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical." (Philosophia Mathematica, 2011) “There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: ‘It's all over.’ Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto. Berto's clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading ... .Berto is lucid and witty in exposing mistaken applications of Gödel's results ... [and] has provided a thoroughly recommendable guide to Gödel's theorems and their current status within, and outside, mathematical logic.” (Times Higher Education Supplement, February 2010)Table of ContentsPrologue. Acknowledgments. Part I: The Gödelian Symphony. 1 Foundations and Paradoxes. 1 "This sentence is false". 2 The Liar and Gödel. 3 Language and metalanguage. 4 The axiomatic method, or how to get the non-obvious out of the obvious. 5 Peano's axioms … . 6 … and the unsatisfied logicists, Frege and Russell. 7 Bits of set theory. 8 The Abstraction Principle. 9 Bytes of set theory. 10 Properties, relations, functions, that is, sets again. 11 Calculating, computing, enumerating, that is, the notion of algorithm. 12 Taking numbers as sets of sets. 13 It's raining paradoxes. 14 Cantor's diagonal argument. 15 Self-reference and paradoxes. 2 Hilbert. 1 Strings of symbols. 2 "… in mathematics there is no ignorabimus". 3 Gödel on stage. 4 Our first encounter with the Incompleteness Theorem … . 5 … and some provisos. 3 Gödelization, or Say It with Numbers! 1 TNT. 2 The arithmetical axioms of TNT and the "standard model" N. 3 The Fundamental Property of formal systems. 4 The Gödel numbering … . 5 … and the arithmetization of syntax. 4 Bits of Recursive Arithmetic … . 1 Making algorithms precise. 2 Bits of recursion theory. 3 Church's Thesis. 4 The recursiveness of predicates, sets, properties, and relations. 5 … And How It Is Represented in Typographical Number Theory. 1 Introspection and representation. 2 The representability of properties, relations, and functions … . 3 … and the Gödelian loop. 6 "I Am Not Provable". 1 Proof pairs. 2 The property of being a theorem of TNT (is not recursive!) 3 Arithmetizing substitution. 4 How can a TNT sentence refer to itself? 5 γ 6 Fixed point. 7 Consistency and omega-consistency. 8 Proving G1. 9 Rosser's proof. 7 The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2. 1 G2. 2 Technical interlude. 3 "Immediate consequences" of G1 and G2. 4 Undecidable1 and undecidable2. 5 Essential incompleteness, or the syndicate of mathematicians. 6 Robinson Arithmetic. 7 How general are Gödel's results? 8 Bits of Turing machine. 9 G1 and G2 in general. 10 Unexpected fish in the formal net. 11 Supernatural numbers. 12 The culpability of the induction scheme. 13 Bits of truth (not too much of it, though). Part II: The World after Gödel. 8 Bourgeois Mathematicians! The Postmodern Interpretations. 1 What is postmodernism? 2 From Gödel to Lenin. 3 Is "Biblical proof" decidable? 4 Speaking of the totality. 5 Bourgeois teachers! 6 (Un)interesting bifurcations. 9 A Footnote to Plato. 1 Explorers in the realm of numbers. 2 The essence of a life. 3 "The philosophical prejudices of our times". 4 From Gödel to Tarski. 5 Human, too human. 10 Mathematical Faith. 1 "I'm not crazy!" 2 Qualified doubts. 3 From Gentzen to the Dialectica interpretation. 4 Mathematicians are people of faith. 11 Mind versus Computer: Gödel and Artificial Intelligence. 1 Is mind (just) a program? 2 "Seeing the truth" and "going outside the system". 3 The basic mistake. 4 In the haze of the transfinite. 5 "Know thyself": Socrates and the inexhaustibility of mathematics. 12 Gödel versus Wittgenstein and the Paraconsistent Interpretation. 1 When geniuses meet … . 2 The implausible Wittgenstein. 3 "There is no metamathematics". 4 Proof and prose. 5 The single argument. 6 But how can arithmetic be inconsistent? 7 The costs and benefits of making Wittgenstein plausible. Epilogue. References. Index.

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Book SynopsisBerto''s highly readable and lucid guide introduces students and the interested reader to Gödel''s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel''s arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel''s theories Written in an accessible, non-technical style Trade Review"This is a beautifully clear and accurate presentation of the material, with no technical demands beyond what is required for accuracy, and filled with interesting philosophical suggestions." (John Woods, University of British Columbia) "There's Something about G¨odel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical." (Philosophia Mathematica, 2011) "There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: ‘It's all over.’ Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto. Berto's clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading ... .Berto is lucid and witty in exposing mistaken applications of Gödel's results ... [and] has provided a thoroughly recommendable guide to Gödel's theorems and their current status within, and outside, mathematical logic.” (Times Higher Education Supplement, February 2010)Table of ContentsPrologue. Acknowledgments. Part I: The Gödelian Symphony. 1 Foundations and Paradoxes. 1 "This sentence is false". 2 The Liar and Gödel. 3 Language and metalanguage. 4 The axiomatic method, or how to get the non-obvious out of the obvious. 5 Peano's axioms … . 6 … and the unsatisfied logicists, Frege and Russell. 7 Bits of set theory. 8 The Abstraction Principle. 9 Bytes of set theory. 10 Properties, relations, functions, that is, sets again. 11 Calculating, computing, enumerating, that is, the notion of algorithm. 12 Taking numbers as sets of sets. 13 It's raining paradoxes. 14 Cantor's diagonal argument. 15 Self-reference and paradoxes. 2 Hilbert. 1 Strings of symbols. 2 "… in mathematics there is no ignorabimus". 3 Gödel on stage. 4 Our first encounter with the Incompleteness Theorem … . 5 … and some provisos. 3 Gödelization, or Say It with Numbers! 1 TNT. 2 The arithmetical axioms of TNT and the "standard model" N. 3 The Fundamental Property of formal systems. 4 The Gödel numbering … . 5 … and the arithmetization of syntax. 4 Bits of Recursive Arithmetic … . 1 Making algorithms precise. 2 Bits of recursion theory. 3 Church's Thesis. 4 The recursiveness of predicates, sets, properties, and relations. 5 … And How It Is Represented in Typographical Number Theory. 1 Introspection and representation. 2 The representability of properties, relations, and functions … . 3 … and the Gödelian loop. 6 "I Am Not Provable". 1 Proof pairs. 2 The property of being a theorem of TNT (is not recursive!) 3 Arithmetizing substitution. 4 How can a TNT sentence refer to itself? 5 γ 6 Fixed point. 7 Consistency and omega-consistency. 8 Proving G1. 9 Rosser's proof. 7 The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2. 1 G2. 2 Technical interlude. 3 "Immediate consequences" of G1 and G2. 4 Undecidable1 and undecidable2. 5 Essential incompleteness, or the syndicate of mathematicians. 6 Robinson Arithmetic. 7 How general are Gödel's results? 8 Bits of Turing machine. 9 G1 and G2 in general. 10 Unexpected fish in the formal net. 11 Supernatural numbers. 12 The culpability of the induction scheme. 13 Bits of truth (not too much of it, though). Part II: The World after Gödel. 8 Bourgeois Mathematicians! The Postmodern Interpretations. 1 What is postmodernism? 2 From Gödel to Lenin. 3 Is "Biblical proof" decidable? 4 Speaking of the totality. 5 Bourgeois teachers! 6 (Un)interesting bifurcations. 9 A Footnote to Plato. 1 Explorers in the realm of numbers. 2 The essence of a life. 3 "The philosophical prejudices of our times". 4 From Gödel to Tarski. 5 Human, too human. 10 Mathematical Faith. 1 "I'm not crazy!" 2 Qualified doubts. 3 From Gentzen to the Dialectica interpretation. 4 Mathematicians are people of faith. 11 Mind versus Computer: Gödel and Artificial Intelligence. 1 Is mind (just) a program? 2 "Seeing the truth" and "going outside the system". 3 The basic mistake. 4 In the haze of the transfinite. 5 "Know thyself": Socrates and the inexhaustibility of mathematics. 12 Gödel versus Wittgenstein and the Paraconsistent Interpretation. 1 When geniuses meet … . 2 The implausible Wittgenstein. 3 "There is no metamathematics". 4 Proof and prose. 5 The single argument. 6 But how can arithmetic be inconsistent? 7 The costs and benefits of making Wittgenstein plausible. Epilogue. References. Index.

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Book SynopsisThis volume will provide invaluable assistance for mathematicians, historians of mathematics and users of mathematics in the retrieval of information about mathematicians and topics in mathematics and closely related fields. The major portion of the book is a classified and annotated bibliography of some 24,000 relevant publications udner about 3,750 headings. Preceeding the bibliography is a guide to personal information retrieval, storage, analysis and use. An appendix provides a useful list of over 3,000 mathematical and historical journals cited in the bibliography or otherwise known to the author, with pertinent publishing information.

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Book SynopsisWhy bother to praise mathematics when you claim, as Alain Badiou does, that philosophy is first and foremost a metaphysics of happiness, or else it’s not worth an hour of trouble? What possible relationship can there be between mathematics and happiness? That is precisely the issue at stake in this dialogue, which serves as a very accessible introduction to what mathematics is and an exploration of the crucial influence it has always exerted on the greatest philosophers. Far from the thankless, pointless exercises they are often thought to be, mathematics and logic are indispensable guides to ridding ourselves of dominant opinions and making possible an access to truths, or to a human experience of the utmost value. That is why mathematics may well be the shortest path to the true life, which, when it exists, is characterized by an incomparable happiness.Trade Review�Badiou allows not only those in the know, but also those ignorant of geometry to enter here into his enchanting defense of mathematics. Packed with a variety of pleasures, this brief text introduces readers to brilliantly quirky mathematicians; philosophical problems with mathematical underpinnings; tricks of the trade, such as how to use the false to snare the truth; the passion of form; and the exquisite joy of the QED.� Joan Copjec, Brown UniversityTable of Contents Contents I Mathematics Must be Saved II Philosophy and Mathematics, or the Story of an Old Couple III What is Mathematics About? IV An Attempt at a Mathematics-based Metaphysics V Does Mathematics Bring Happiness? Conclusion

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Book SynopsisWhy bother to praise mathematics when you claim, as Alain Badiou does, that philosophy is first and foremost a metaphysics of happiness, or else it’s not worth an hour of trouble? What possible relationship can there be between mathematics and happiness? That is precisely the issue at stake in this dialogue, which serves as a very accessible introduction to what mathematics is and an exploration of the crucial influence it has always exerted on the greatest philosophers. Far from the thankless, pointless exercises they are often thought to be, mathematics and logic are indispensable guides to ridding ourselves of dominant opinions and making possible an access to truths, or to a human experience of the utmost value. That is why mathematics may well be the shortest path to the true life, which, when it exists, is characterized by an incomparable happiness.Trade Review�Badiou allows not only those in the know, but also those ignorant of geometry to enter here into his enchanting defense of mathematics. Packed with a variety of pleasures, this brief text introduces readers to brilliantly quirky mathematicians; philosophical problems with mathematical underpinnings; tricks of the trade, such as how to use the false to snare the truth; the passion of form; and the exquisite joy of the QED.� Joan Copjec, Brown UniversityTable of ContentsContentsI Mathematics Must be SavedII Philosophy and Mathematics, or the Story of an Old CoupleIII What is Mathematics About?IV An Attempt at a Mathematics-based MetaphysicsV Does Mathematics Bring Happiness?Conclusion

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Book SynopsisThe use of diagrams in logic and geometry has encountered resistence throughout the years. For a proof to be valid in geometry it must not rely on the graphical properties of a diagram. In logic the teaching of proofs depends on the sentenial representations, ideas formed as natural language sentences such as "if A is true and B is true...". No serious formal proof system is based on diagrams. This text explores the reasons why structured graphics have been ignored in modern formal theories of axiomatic systems. The effects of historical forces on the evolution of diagrammatically-based systems of inference in logic and geometry are explored, from antiquity to the early 20th-century work of David Hilbert. From this exploration emerges an understanding that the present negative attitudes towards the use of diagrams in logic and geometry owe more to implicit appeals to their history and philosophical background than to any technical incompatibility with modern theories of logical systems.

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Book SynopsisThis book, following the three published volumes of the book, provides the main purpose to collect research papers and review papers to provide an overview of the main issues, results, and open questions in the cutting-edge research on the fields of modeling, optimization, and dynamics and their applications to biology, economy, energy, industry, physics, psychology and finance. Assuming the scientific relevance of the presenting innovative applications as well as merging issues in these areas, the purpose of this book is to collect papers of the world experts in mathematics, economics, and other applied sciences that is seminal to the future research developments. The majority of the papers presented in this book is authored by the participants in The Joint Meeting 6th International Conference on Dynamics, Games, and Science – DGSVI – JOLATE and in the 21st ICABR Conference. The scientific scope of the conferences is focused on the fields of modeling, optimization, and dynamics and their applications to biology, economy, energy, industry, physics, psychology, and finance. Assuming the scientific relevance of the presenting innovative applications as well as merging issues in these areas, the purpose of the conference is to bring together some of the world experts in mathematics, economics, and other applied sciences that reinforce ongoing projects and establish future works and collaborations.Table of ContentsA. Afsar, F. Martins, Bruno M. P. M. Oliveira, and A. A. Pinto, Immune response model fitting to CD4+ T cell data in lymphocytic choriomeningitis virus LCMV infection.- U. Agyüz, V. Purutçuoglu, E. Purutçuoglu and Y. Ürün, Construction of a New Model to Investigate Breast Cancer Data.- I. Baltas, M. Szczepanski, L. Dopierala, K. Kolodziejczyk, G.-W. Weber and A. N. Yannacopoulos, Optimal Pension Fund Management Under Risk and Uncertainty: The Case Study of Poland.- M. Bujidos-Casado, J. Navío-Marco and B. Rodrigo-Moya, Collaborative Innovation of Spanish SMEs in the European context: A compared study.- G. G. de Castro, A. O. Lopes and G. Mantovani, Haar systems, KMS states on von Neumann algebras and C*-algebras on dynamically defined groupoids and Noncommutative Integration.- C. Çıtak, T. Aksu, Ö. Harputlu and Gerhard-Wilhelm Weber, Mixed Compression Air-Intake Design for High-Speed Transportation.- D. Czerkawski, J. Małecka, G. Wilhelm Weber and B. Kjamili, Social Entrepreneurship Business Models for Handicapped People - Polish & Turkish case study of sharing public goods by doing business.- H. H. Ferreira, A. O. Lopes and E. R. Oliveira, An iterative process for approximating subactions.- A. D. Garcia and M. A. Szybisz, "Beat the gun": The phenomenon of liquidity.- E. Gómez-Escalonilla and Laura Parte, Board Knowledge and Bank Risk-Taking. An International Analysis.- F. Jiménez-Delgado, M. Dolores Reina-Paz, Israel J ThuissardVasallo and David Sanz-Rosa, The shopping experience in virtual sales: A study of the influence of website atmosphere on purchase intention.- Kyung B. Kim and José M. Labeaga, European Mobile Phone Industry: Demand Estimation Using Discrete Random Coefficients Models.- A. O. Lopes and M. Sebastiani, On Bertelson-Gromov Dynamical Morse Entropy, Rogério Martins, Synchronisation of weakly coupled oscillators.- Z. Kamisli Ozturk, Y. Cetin, Y. Isik and Z. I. Erzurum Cicek, Demand Forecasting with Clustering and Artificial Neural Networks Methods: an Application for Stock Keeping Units.- O. Palanci, S.Z. Alparslan Gok and Gerhard-Wilhelm Weber, On the Grey Obligation Rules.- Juan Diego Paredes-Gázquez, Eva Pardo and José Miguel Rodríguez-Fernández, Robustness checks in composite indicators: A responsible approach.- Elena V. Ravve, Zeev Volkovich, Gerhard-Wilhelm Weber, A Logic-Based Approach to Incremental Reasoning on Multi-Agent Systems.

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Book SynopsisThis book analyses the role of diagrammatic reasoning in Plato’s philosophy: the readers will realize that Plato, describing the stages of human cognitive development using a diagram, poses a logic problem to stimulate the general reasoning abilities of his readers. Following the examination of mental models in this book, the readers will reflect on what inferences can be useful to approach this kind of logic problem. Plato calls for a collaboration between writer and readers. In this book the readers will examine the connection between diagrams and discovery, realizing the important epistemic role of visualization. They will recognize the crucial role that diagrams play in problem solving. The logic problem elaborated by Plato is addressed considering the epistemic function of mental models. These models introduce to an advanced stage of cognitive development, in which reasoning uses in its investigations a higher-level of mathematical complexity, represented by structuralism.Table of ContentsCHAPTER ONE: Introduction,- CHAPTER TWO: The Collaboration between Writer and Reader,- CHAPTER THREE: Visual Thinking,- CHAPTER FOUR: Diagrammatic Reasoning,- CHAPTER FIVE: Mental Models,-Chapter 6. Theoretical Adulthood and Structuralism.

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Book SynopsisBand VII der Hausdorff Edition widmet sich dem philosophischen Werk F. Hausdorffs. Der Band enthält den Aphorismenband "Sant' Ilario. Gedanken aus der Landschaft Zarathustras", das erkenntniskritische Buch "Das Chaos in kosmischer Auslese" sowie drei bemerkenswerte Essays über Nietzsches Werke - alle unter dem Pseudonym Paul Mongré veröffentlicht. Die beiden Bücher werden sehr eingehend kommentiert. In einer historischen Einführung des Herausgebers wird Hausdorffs philosophisches Werk in die Geschichte des philosophischen Denkens eingeordnet.Table of ContentsEinleitung des Herausgebers. - Paul Mongré (Felix Hausdorff): Sant' Ilario. Gedanken aus der Landschaft Zarthustras. - Paul Mongré (Felix Hausdorff): Selbstanzeige von Sant' Ilario. - Kommentar zu Sant' Ilario. - Paul Mongré (Felix Hausdorff): Das Chaos in kosmischer Auslese. - Paul Mongré (Felix Hausdorff): Selbstanzeige von Das Chaos in kosmischer Auslese. - Kommentar zu Das Chaos in kosmischer Auslese. - Paul Mongré (Felix Hausdorff): Nietzsches Wiederkunft des Gleichen. - Paul Mongré (Felix Hausdorff): Nietzsches Lehre von der Wiederkunft des Gleichen. - Paul Mongré (Felix Hausdorff): Der Wille zur Macht. - Personenverzeichnis.

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Book SynopsisDer Berliner Mathematiker Karl Weierstraß (1815-1897) lieferte grundlegende Beiträge zu den mathematischen Fachgebieten der Funktionentheorie, Algebra und Variationsrechnung. Er gilt weltweit als Begründer der mathematisch strengen Beweisführung in der Analysis. Mit seinem Namen verbunden ist zum Beispiel die berühmte Epsilon-Delta-Definition des Begriffs der Stetigkeit reeller Funktionen. Weierstraß’ Vorlesungszyklus zur Analysis in Berlin wurde weithin gerühmt und er lehrte teilweise vor 250 Hörern aus ganz Europa; diese starke mathematische Schule prägt bis heute die Mathematik. Aus Anlass seines 200. Geburtstags am 31. Oktober 2015 haben internationale Experten der Mathematik und Mathematikgeschichte diesen Festband zusammengestellt, der einen Einblick in die Bedeutung von Weierstraß’ Werk bis zur heutigen Zeit gibt.Die Herausgeber des Buches sind leitende Wissenschaftler am Weierstraß-Institut für Angewandte Analysis und Stochastik in Berlin, die Autoren eminente Mathematikhistoriker.Table of ContentsDie prägenden Jahre im Leben von Karl Weierstraß (Jürgen Elstrodt).- Zur Biographie von Karl Weierstraß und zu einigen Aspekten seiner Mathematik (Reinhard Bölling).- Weierstraß und die Preußische Akademie der Wissenschaften (Eberhard Knobloch).- Karl Weierstraß and the theory of Abelian and elliptic functions (Peter Ullrich).- Building analytic function theory: Weierstraß's approach in lecture courses and papers (Umberto Bottazzini).- Monodromy and normal forms (Fabrizio Catanese).- Weierstraß' Approximation Theorem (1885) and his 1886 lecture course revisted (Reinhard Siegmund-Schultze).- Counterexamples in Weierstraß' Work (Tom Archibald).

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Book SynopsisDieses Buch bietet einen historisch orientierten Einstieg in die Algorithmik, also die Lehre von den Algorithmen, in Mathematik, Informatik und darüber hinaus. Besondere Merkmale und Zielsetzungen sind: Elementarität und Anschaulichkeit, die Berücksichtigung der historischen Entwicklung, Motivation der Begriffe und Verfahren anhand konkreter, aussagekräftiger Beispiele unter Einbezug moderner Werkzeuge (Computeralgebrasysteme, Internet). Als Zusatzmedien werden computer- und internetspezifische Interaktions- und Visualisierungsmöglichkeiten (kostenlos) zur Verfügung gestellt. Das Werk wendet sich an Studierende und Lehrende an Schulen und Hochschulen sowie an Nichtspezialisten, die an den Themen "Computer/Algorithmen/Programmierung" einschließlich ihrer historischen und geisteswissenschaftlichen Dimension interessiert sind.Table of ContentsEinleitung.- Begriffsbestimmungen.- Historische Bezüge.- Fundamentale heuristische Strategien des algorithmischen Problemlösens.- Effizienz von Algorithmen.- Korrektheit von Algorithmen, Korrektheit von Computerergebnissen.- Grenzen der Algorithmisierbarkeit, Grenzen des Computers.- Programmierung.- Informationstheorie, Codierung und Kryptographie.- Evolutionäre Algorithmen und neuronale Netze.

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Book SynopsisIn diesem Band sind die nachgelassenen Schriften Moritz Schlicks zur Logik und Philosophie der Mathematik gesammelt, ediert und kommentiert. Keine der zu Lebzeiten veröffentlichten Schriften Schlicks war ausschließlich diesen Themen gewidmet. Man sollte daraus jedoch nicht den Schluss zu ziehen, diese Themen hatten an der Peripherie von Schlicks Interesse gelegen. Überlegungen zur Logik und Mathematik ziehen sich durch Schlicks gesamtes Werk, von der Habilitation, über sein Opus Magnum, die Allgemeine Erkenntnislehre, bis zu seinen letzten stark von Ludwig Wittgenstein geprägten Aufsätzen in den 1930er Jahren. Es ist vielmehr so, dass Schlick Fragen der Logik und Mathematikphilosophie stets im Zusammenhang mit anderen Problemen sah und sie deshalb nie einzeln für sich behandelte. Ausnahmen machte er vor allem für Lehrveranstaltungen und so wundert es nicht, dass fast alle Texte dieses Bandes im Umkreis von solchen entstanden sind.Table of ContentsVorwort des Herausgebers.- Einleitung.- Die philosophischen Grundlagen der Mathematik.- Logik.- Logik.- Wahrscheinlichkeit.- Logik und Erkenntnistheorie.- Anhang.

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Book SynopsisOriginaltext und historischer und mathematischer Kommentar von Klaus VolkertDavid Hilberts „Festschrift“ Grundlagen der Geometrie“ aus dem Jahre 1899 wurde zu einem der einflussreichsten Texte der Mathematikgeschichte. Wie kein anderes Werk prägte es die Mathematik des 20. Jahrhunderts und ist auch heute noch von größtem Interesse.Aus der Perspektive eines Mathematikhistorikers schildert der Herausgeber die Entwicklung einer Axiomatik der Geometrie, die spätestens mit Euklids „Elemente“ (ca. 300 v. u. Z.) begann und erst durch Hilbert zu einem vollständigen und handhabbaren System geführt wurde. Nach einer ausführlichen Erläuterung des Hilbertschen Textes wird seine Rezeption bis 1905 umfassend dargestellt und daran anschließend viele der von ihm ausgehenden weiteren direkten und indirekten Entwicklungen skizziert.Die Faszination des Textes ist auch dem heutigen Leser direkt zugänglich, da Hilbert´s axiomatischer Ansatz ohne mengentheoretische Argumente oder formale Logik auskommt.Trade Review“There is wealth of information of a historical nature that makes this a very valuable addition to the literature. The decision to stop at 1905 … will make the reader wanting to learn more about the longer-term influence of this classical work … .” Victor V. Pambuccian, Mathematical Reviews, October, 2018)​Table of ContentsVorwort.- Einleitung.- Eine kurze Geschichte der Axiomatik insbesondere der Geometrie.- HilbertsWeg zu den „Grundlagen der Geometrie“.- Text der „Festschrift“.- Präsentation des Textes.- Die Rezeption der Hilbertschen „Festschrift“.- Nach der „Festschrift“.- A Klassische Sätze in Hilberts „Festschrift“ und seinen Vorlesungen.- B Hilberts Modelle.- Personenverzeichnis.- Stichwortverzeichnis.- Literaturverzeichnis.

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Book SynopsisSpaß an der Mathematik haben? Ja, das geht wirklich, wie dieses Buch zeigt! Es erzählt wie ein Roman eine „mathematische Geschichte“. Man könnte behaupten, diese recht verworrene Geschichte drehe sich um eine umständliche Entwicklung einer Formel, mit deren Hilfe man die Kreiszahl Pi berechnen kann. Aber eigentlich geht es um etwas ganz anderes: Das Buch nimmt den Leser an der Hand, fordert ihn aber durch eingestreute Fragen immer wieder zum Innehalten und Mitdenken auf. Dank der behutsamen Heranführung an die Themen können diese Fragen von jedem, der die Herausforderung annimmt, mit Schulkenntnissen gemeistert werden. Man bekommt so einen Einblick in „echte“ Mathematik zwischen Geometrie, Algebra, Analysis und Zahlentheorie. Man sieht, wie man an mathematische Fragestellungen herangehen kann. Und man erfährt, warum Mathematik früher ganz anders als heute war und wie sie sich erst mühsam entwickeln musste. Anekdoten über die Menschen hinter der Mathematik gibt's auch, denn der Autor plaudert gerne, philosophiert auch ab und zu und liebt Abschweifungen. Und das Schönste ist: Am Ende wartet keine Prüfung – der Leser kann sich einfach auf die Freude am Forschen und Verstehen einlassen.Table of ContentsAb in den Dschungel.- Nicht von Pythagoras .- Was beweisen Beweise?.- Die Kreativen.- Menschenwerk.- Nichts.- Die Diva.- Gibt es Pi überhaupt?.- Der Plan.- Millimeterpapier.- Die Atome der Mathematik.- Der Gott aus der Maschine.- Reste.- Der Amateur und die Windmühlen.- Die Badeanstalt.- Der erste Algorithmus.- Komplexes Intermezzo.- Außerirdische Mathematik.- Einfaches Sudoku.- Der letzte Brief.- Der schmale Rand.- Einfach die Regeln ändern.- Fünfzehntausend Seiten.- Endlich Punkte zählen!.- Dominoeffekte.- Noch eine Hypothese.- Von Fröschen und Mäusen.- Butterkeks.- Offenes Ende.- Epilog.- Anmerkungen.- Inhalt.- Index.

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Book SynopsisThis publication includes an unabridged and annotated translation of two works by Johann Heinrich Lambert (1728–1777) written in the 1760s: Vorläufige Kenntnisse für die, so die Quadratur und Rectification des Circuls suchen and Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques. The translations are accompanied by a contextualised study of each of these works and provide an overview of Lambert’s contributions, showing both the background and the influence of his work. In addition, by adopting a biographical approach, it allows readers to better get to know the scientist himself. Lambert was a highly relevant scientist and polymath in his time, admired by the likes of Kant, who despite having made a wide variety of contributions to different branches of knowledge, later faded into an undeserved secondary place with respect to other scientists of the eighteenth century. In mathematics, in particular, he is famous for his research on non-Euclidean geometries, although he is likely best known for having been the first who proved the irrationality of pi. In his Mémoire, he conducted one of the first studies on hyperbolic functions, offered a surprisingly rigorous proof of the irrationality of pi, established for the first time the modern distinction between algebraic and transcendental numbers, and based on such distinction, he conjectured the transcendence of pi and therefore the impossibility of squaring the circle.Table of ContentsPart I: Antecedents.- Chapter 1. From Geometry to Analysis.- Chapter 2. The situation in the first half of the 18th century. Euler and continued fractions.- Part II: Johann Heinrich Lambert (1728—1777).- Chapter 3. A biographical approach to Johann Heinrich Lambert.- Chapter 4. Outline of Lambert's Mémoire (1761/1768).- Chapter 5. An anotated translation of Lambert's Mémoire (1761/1768).- Chapter 6. Outine of Lambert's Vorläufige Kenntnisse (1766/1770).- Chapter 6. An anotated translation of Lambert's Vorläufige Kenntnisse (1766/1770).- Part III: The influence of Lambert's work and the development of irrational numbers.- Chapter 8. The state of irrationals until the turn of the century.- Chapter 9. Title to be set up.

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Using a complete interpretation of Whitehead’s philosophical and mathematical writings, this book argues that Whitehead has never been properly understood. It applies Whitehead’s philosophy to problems in the interpretation of science, empirical knowledge, and nature, and develops a new account of philosophical naturalism.

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Book SynopsisThe Unconscious as Space explores the experience of being and the practice of psychoanalysis by thinking of the unconscious in mathematical terms.Anca Carrington introduces mathematical models of space, from dimension theory to algebraic topology and knot theory, and considers their immediate psychoanalytic relevance. The hypothesis that the unconscious is structured like a space marked by impossibility is then examined. Carrington considers the clinical implications, with particular focus on the interplay between language and the unconscious as related topological spaces in which movement takes place along knot-like pathways.The Unconscious as Space will be of appeal to psychotherapists, psychoanalysts and mental health professionals in practice and in training.

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Book SynopsisOriginally published in 1966. An introduction to current studies of kinds of inference in which validity cannot be determined by ordinary deductive models. In particular, inductive inference, predictive inference, statistical inference, and decision making are examined in some detail. The last chapter discusses the relationship of these forms of inference to philosophical notions of rationality. Special features of the monograph include a discussion of the legitimacy of various criteria for successful predictive inference, the development of an intuitive model which exhibits the difficulties of choosing probability measures over infinite sets, and a comparison of rival views on the foundations of probability in terms of the amount of information which the members of these schools believe suitable for fruitful formalization. The bibliographies include articles by statisticians accessible to students of symbolic logic. Table of Contents1. Inductive and Predictive Inference 2. Hypothesis and Predictive Inference 3. Probability and Predictive Inference 4. Statistical Inference 5. Bayesian Statistical Inference 6. Statistical Decision and Utility 7. Theories and Rationality 8. Bibliography

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Book SynopsisThe Equation of Knowledge: From Bayes'' Rule to a Unified Philosophy of Science introduces readers to the Bayesian approach to science: teasing out the link between probability and knowledge. The author strives to make this book accessible to a very broad audience, suitable for professionals, students, and academics, as well as the enthusiastic amateur scientist/mathematician. This book also shows how Bayesianism sheds new light on nearly all areas of knowledge, from philosophy to mathematics, science and engineering, but also law, politics and everyday decision-making.Bayesian thinking is an important topic for research, which has seen dramatic progress in the recent years, and has a significant role to play in the understanding and development of AI and Machine Learning, among many other things. This book seeks to act as a tool for proselytising the benefits and limits of Bayesianism to a wider public. Features Trade ReviewLê Nguyên Hoang takes us on a fascinating intellectual journey into Bayesianism, cutting across many social and natural sciences. The Equation of Knowledge: From Bayes' Rule to a Unified Philosophy of Science is a real page turner.—George Zaccour, HEC Montréal and co-author of Handbook of Dynamic Game Theory"Each chapter is opened with a fascinating epigraph quoting famous persons, and is completed by the most recent references. There are multiple illustrations, and the Bayes’ formulae are many times presented via various funny symbols of emoji kind. The book is addressed to a wide audience of students, professionals, and actually any reader interested to be better acquainted with modern ideas in various sciences and philosophy of science, and their Bayesian statistical description and interpretation."— Stan Lipovetsky, Technometrics (Volume 63, 2021 - Issue 1)"[. . . ] Trained in the hard school of online videos, Le Nguyen Hoang has found a new tone to talk about science, a tone that is both rigorous and narrative, where examples illuminate the most abstract questions."— From the Foreword by Gilles Dowek, Professor at École Polytechnique and researcher at the Laboratoire d'Informatique de l'École Polytechnique and the Institut National de Recherche en Informatique et en Automatique (INRIA).Lê Nguyên Hoang takes us on a fascinating intellectual journey into Bayesianism, cutting across many social and natural sciences. The Equation of Knowledge: From Bayes' Rule to a Unified Philosophy of Science is a real page turner.— George Zaccour, HEC Montréal and co-author of Handbook of Dynamic Game Theory"Making math accessible to everyone, showing its connections with dozens of different domains, narrating scientific discovery as a personal human adventure, and sharing impressive enthusiasm: there is definitely something of Greg Chaitin's Meta Math! in Lê Nguyên Hoang's book!" — Rémi Peyre, École des Mines de Nancy"A remarkable piece of work, broad and insightful at the same time. This book is unique in that it gives an accessible journey from subtle probabilistic puzzles to the most advanced concepts at the heart of the machine learning revolution; with unrivalled clarity, it exposes deep ideas that have remained very confidential outside of specialized circles, and that yet are becoming fundamental in the way we understand our world."— Clément Hongler, Associate Professor and Chair of Statistical Field Theory, EPFL "As someone who practices research and publishes academic papers, it is frustrating to note how little we, scientists, are trained in epistemology. ‘How do we know that we know?’ This question is often neglected or taken for granted. The recent controversies about reproducibility of scientific publishing might be one of the tips of a larger iceberg. This book will, in my opinion, be remembered as one of those that helped melt the iceberg." — El Mahdi El Mhamdi, École Polytechnique Fédérale de Lausanne."The book has a lively writing style, rather like you are listening to an inspiring lecturer. Indeed the author has a French YouTube channel and is clearly enthusiastic about exposition. It is overtly an account of what the author personally finds interesting. [. . .] In teaching a basic college course, focused on the mathematical setup and on the analysis of data, I often find there is one student who comes to office hours and is interested in seeing connections with broad scientific fields, or in conceptual issues of the philosophy of science. I could certainly recommend this book to such a student. Similarly, for the MAA community it could be an innovative basis for an undergraduate seminar course, in which students would choose a topic from the book and delve deeper into it."— David Aldous, Mathematical Association of AmericaTable of ContentsSction I. Pure Bayesianism. 1. On A Transformative Journey. 2. Bayes Theorem. 3. Logically Speaking... 4. Let’s Generalize! 5. All Hail Prejudices. 6. The Bayesian Prophets. 7. Solomonoff’s Demon. Section II. Applied Bayesianism. 8. Can You Keep A Secret? 9. Game, Set and Math. 10. Will Darwin Select Bayes? 11. Exponentially Counter-Intuitive. 12. Ockham Cuts to the Chase. 13. Facts Are Misleading. Section III. Pragmatic Bayesianism. 14. Quick And Not Too Dirty. 15. Wish Me Luck. 16. Down Memory Lane. 17. Let’s Sleep On It. 18. The Unreasonable Effectiveness of Abstraction. 19. The Bayesian Brain. Section IV. Beyond Bayesianism. 20. It’s All Fiction. 21. Exploring The Origins Of Beliefs. 22. Beyond Bayesianism.

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