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• Introduction : Mind over data
• The ladder of causation
• From buccaneers to guinea pigs : the genesis of causal inference
• From evidence to causes : Reverend Bayes meets Mr. Holmes
• Confounding and deconfounding : or, slaying the lurking variable
• The smoke-filled debate : clearing the air
• Paradoxes galore!
• Beyond adjustment : the conquest of Mount Intervention
• Counterfactuals : mining worlds that could have been
• Mediation : the search for a mechanism
• Big data, artificial intelligence, and the big questions.

A Turing Award-winning computer scientist and statistician shows how understanding causality has revolutionized science and will revolutionize artificial intelligence "Correlation is not causation." This mantra, chanted by scientists for more than a century, has led to a virtual prohibition on causal talk. Today, that taboo is dead. The causal revolution, instigated by Judea Pearl and his colleagues, has cut through a century of confusion and established causality--the study of cause and effect--on a firm scientific basis. His work explains how we can know easy things, like whether it was rain or a sprinkler that made a sidewalk wet; and how to answer hard questions, like whether a drug cured an illness. Pearl's work enables us to know not just whether one thing causes another: it lets us explore the world that is and the worlds that could have been. It shows us the essence of human thought and key to artificial intelligence. Anyone who wants to understand either needs The Book of Why.
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• Preface ixAcknowledgments xi1 Measurement 12 Judgment 223 Psychology 484 Frequency 625 Mathematics 796 Inverse Inference 1007 Unification 1228 Algorithmic Randomness 1459 Physical Chance 16510 Induction 190Appendix: Probability Tutorial 209Notes 225Annotated Select Bibliography 239Image Credits 247Index 249.
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A fascinating account of the breakthrough ideas that transformed probability and statistics In the sixteenth and seventeenth centuries, gamblers and mathematicians transformed the idea of chance from a mystery into the discipline of probability, setting the stage for a series of breakthroughs that enabled or transformed innumerable fields, from gambling, mathematics, statistics, economics, and finance to physics and computer science. This book tells the story of ten great ideas about chance and the thinkers who developed them, tracing the philosophical implications of these ideas as well as their mathematical impact. Persi Diaconis and Brian Skyrms begin with Girolamo Cardano, a sixteenth-century physician, mathematician, and professional gambler who helped develop the idea that chance actually can be measured. They describe how later thinkers showed how the judgment of chance also can be measured, how frequency is related to chance, and how chance, judgment, and frequency could be unified. Diaconis and Skyrms explain how Thomas Bayes laid the foundation of modern statistics, and they explore David Hume's problem of induction, Andrey Kolmogorov's general mathematical framework for probability, the application of computability to chance, and why chance is essential to modern physics. A final idea--that we are psychologically predisposed to error when judging chance--is taken up through the work of Daniel Kahneman and Amos Tversky. Complete with a brief probability refresher, Ten Great Ideas about Chance is certain to be a hit with anyone who wants to understand the secrets of probability and how they were discovered.
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• * Introduction* The case of academician Luzin in the collective memory of the scientific community* Minutes of the meetings of the USSR Academy of Sciences Commission in the case of academician Luzin* Minutes of the meeting of the USSR Academy of Sciences Commission in the matter of academician Luzin: 7 July Minutes of the meeting of the USSR Academy of Sciences Commission on the matter of academician Luzin*9 July Minutes of the meeting of the USSR Academy of Sciences Commission on the matter of academician Luzin*11 July Minutes of the meeting of the USSR Academy of Sciences Commission in the matter of academician Luzin*13 July Minutes of the meeting of the USSR Academy of Sciences Commission in the matter of academician Luzin*15 July Commentaries on the minutes of the meetings of the USSR Academy of Sciences Commission in the case of academician Luzin* Commentaries on the minutes of the meetings of the USSR Academy of Sciences Commission in the case of academician Luzin Commentary on the minutes of the meeting of the USSR Academy of Sciences Commission in the matter of academician Luzin*7 July 1936 Commentary on the minutes of the USSR Academy of Sciences Commission in the matter of academician Luzin*9 July 1936 Commentary on the minutes of the meeting of the USSR Academy of Sciences Commission in the matter of academician Luzin*11 July 1936 Commentary on the minutes of the meeting of the USSR Academy of Sciences Commission in the matter of academician Luzin*13 July 1936 Commentary on the minutes of the meeting of the USSR Academy of Sciences Commission in the matter of academician Luzin*15 July 1936 Literature Appendices* Appendices introduction* A pleasant disillusionment* Reply to academician N. Luzin* Enemies wearing a Soviet mask* Letter from L. Z. Mekhlis, editor of $\textit{Pravda}$, to the Central Committee, 3 July 1936* Resolution concerning the articles "Response to academician Luzin" and "Enemies wearing a Soviet mask" in $\textit{Pravda}$* Draft of the proposal of the special session of the Presidium of the USSR Academy of Sciences, 4 July 1936* Letter from P. L. Kapitsa to Molotov, 6 July 1936* Excerpt from the minutes of the Presidum meeting of 7 July 1936* Letters from V. I. Vernadski iand N. V. Nasonov to the Academy of Sciences Division of Mathematical and Natural Sciences and to academicians A. E. Fersman and N. P. Gorbunov in support of academician Luzin* Letter from academician N. N. Luzin to the Central Committee of the Communist Party 7 July 1936* Traditions of servility* Resolution of the General Assembly of Scientists of the Department of Mechanics and Mathematics and Institute of Mathematics, Mechanics, and Astronomy at Moscow University* Letter from Luzin to an undetermined addressee, 11 July 1936* Enemies wearing a Soviet mask* The Leningrad scholars respond* Letter from L. Z. Mekhlis, Editor of Pravda, to Stalin and Molotov, 14 July 1936* The enemy exposed Luzin's statement to the Presidium of the Academy of Sciences, 14 July 1936* Academician Gubkin on so-called academician Luzin* The Belarus scholars on the exposed enemy Luzin* The scholarly community condemns enemies wearing a Soviet mask* Note accompanying the draft of the findings of the Presidium of the USSR Academy of Sciences regarding academician N. N. Luzin, 25 July 1936* Conclusion of the Commission On academician N. N. Luzin* Findings of the Presidium of the USSR Academy of Sciences, 5 August 1936* To rid academia of Luzinism* Glossary of Soviet terms and people* Subject index* Name index.
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The Soviet school, one of the glories of twentieth-century mathematics, faced a serious crisis in the summer of 1936. It was suffering from internal strains due to generational conflicts between the young talents and the old establishment. At the same time, Soviet leaders (including Stalin himself) were bent on "Sovietizing" all of science in the USSR by requiring scholars to publish their works in Russian in the Soviet Union, ending the nearly universal practice of publishing in the West. A campaign to "Sovietize" mathematics in the USSR was launched with an attack on Nikolai Nikolaevich Luzin, the leader of the Soviet school of mathematics, in Pravda. Luzin was fortunate in that only a few of the most ardent ideologues wanted to destroy him utterly. As a result, Luzin, though humiliated and frightened, was allowed to make a statement of public repentance and then let off with a relatively mild reprimand. A major factor in his narrow escape was the very abstractness of his research area (descriptive set theory), which was difficult to incorporate into a propaganda campaign aimed at the broader public. The present book contains the transcripts of five meetings of the Academy of Sciences commission charged with investigating the accusations against Luzin, meetings held in July of 1936. Ancillary material from the Soviet press of the time is included to place these meetings in context.
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• Meetings across the tree of life. Two meetings and a departure ; Outlines
• A history of animals. Beginnings ; Living together ; Neurons and nervous systems ; The garden ; Senses ; The fork
• Mischief and craft. In a sponge garden ; Evolution of the cephalopods ; Puzzles of octopus intelligence ; Visiting Octopolis ; Nervous evolution ; Body and control ; Convergence and divergence
• From white noise to consciousness. What it's like ; Evolution of experience ; Latecomer versus transformation ; The case of the octopus
• Making colors. The giant cuttlefish ; Making colors ; Seeing colors ; Being seen ; Baboon and squid ; Symphony
• Our minds and others ; From Hume to Vygotsky ; Word made flesh ; Conscious experience ; Full circle
• Experience compressed. Decline ; Life and death ; A swarm of motorcycles ; Long and short lives ; Ghosts
• Octopolis. An armful of octopuses ; Origins of Octopolis ; Parallel lines ; The oceans.

Peter Godfrey-Smith is a leading philosopher of science. He is also an accomplished scuba diver whose underwater videos of warring octopuses have attracted wide notice. In this book, he brings his parallel careers together to tell a bold new story of how nature became aware of itself. Mammals and birds are widely seen as the smartest creatures on earth. But one other branch of the tree of life has also sprouted higher intelligence: the cephalopods, consisting of the squid, the cuttlefish, and above all the octopus. New research shows that these marvelous creatures display remarkable gifts, with each of their tentacles even capable of thinking for itself. What does it mean that higher intelligence on earth has evolved not once but twice? And that the mind of the octopus is nonetheless so different from our own? Combining science and philosophy with first hand accounts of his cephalopod encounters, Godfrey-Smith shows how primitive organisms bobbing in the ocean began sending signals to each other and how these early forms of communication gave rise to the advanced nervous systems that permit cephalopods to change colours and human beings to speak. By tracing the problem of consciousness back to its roots and comparing the human brain to its most alien and perhaps most remarkable animal relative, Godfrey- Smith's Other Minds sheds new light on one of our most abiding mysteries.
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• An introduction to logics of knowledge and belief
• [pt.] I. Informational attitudes
• Only knowing
• Awareness
• Epistemic probabilistic logic
• [pt.] II. Dynamics of informational attitudes
• Knowledge and time
• Dynamic epistemic logic
• Dynamic logics of belief change
• [pt.] III. Applications
• Model checking temporal epistemic logic
• Epistemic foundations of game theory
• BDI logics
• Knowledge and ability
• Knowledge and security.

• Preface
• Acknowledgments
• 1. Rigor and Rigorization
• 2. Rigor and Foundations
• 3. Structure and Structuralism
• 4. Structure and Foundations
• Bibliography.
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While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.
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• MATHEMATICAL FOUNDATIONS.
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Topology is the mathematical study of the most basic geometrical structure of a space. Mathematical physics uses topological spaces as the formal means for describing physical space and time. This book proposes a completely new mathematical structure for describing geometrical notions such as continuity, connectedness, boundaries of sets, and so on, in order to provide a better mathematical tool for understanding space-time. This is the initial volume in a two-volume set, the first of which develops the mathematical structure and the second of which applies it to classical and Relativistic physics. The book begins with a brief historical review of the development of mathematics as it relates to geometry, and an overview of standard topology. The new theory, the Theory of Linear Structures, is presented and compared to standard topology. The Theory of Linear Structures replaces the foundational notion of standard topology, the open set, with the notion of a continuous line. Axioms for the Theory of Linear Structures are laid down, and definitions of other geometrical notions developed in those terms. Various novel geometrical properties, such as a space being intrinsically directed, are defined using these resources. Applications of the theory to discrete spaces (where the standard theory of open sets gets little purchase) are particularly noted. The mathematics is developed up through homotopy theory and compactness, along with ways to represent both affine (straight line) and metrical structure.
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• Preface ix CHAPTER ONE Philosophy and Biology 1 CHAPTER TWO Laws, Mechanisms, and Models 11 CHAPTER THREE Evolution and Natural Selection 28 CHAPTER FOUR Adaptation, Construction, Function 50 CHAPTER FIVE Individuals 66 CHAPTER SIX Genes 81 CHAPTER SEVEN Species and the Tree of Life 100 CHAPTER EIGHT Evolution and Social Behavior 120 CHAPTER NINE Information 144 References 159 Index 179.
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This is a concise, comprehensive, and accessible introduction to the philosophy of biology written by a leading authority on the subject. Geared to philosophers, biologists, and students of both, the book provides sophisticated and innovative coverage of the central topics and many of the latest developments in the field. Emphasizing connections between biological theories and other areas of philosophy, and carefully explaining both philosophical and biological terms, Peter Godfrey-Smith discusses the relation between philosophy and science; examines the role of laws, mechanistic explanation, and idealized models in biological theories; describes evolution by natural selection; and assesses attempts to extend Darwin's mechanism to explain changes in ideas, culture, and other phenomena. Further topics include functions and teleology, individuality and organisms, species, the tree of life, and human nature. The book closes with detailed, cutting-edge treatments of the evolution of cooperation, of information in biology, and of the role of communication in living systems at all scales. Authoritative and up-to-date, this is an essential guide for anyone interested in the important philosophical issues raised by the biological sciences.
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• Foreword--
• 1. A Cartesian introduction--
• 2. What makes mathematics mathematics?--
• 3. Why is there philosophy of mathematics?--
• 4. Proofs--
• 5. Applications--
• 6. In Plato's name--
• 7. Counter-Platonisms-- Disclosures.
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This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.
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• Part I: How Do We Compute? What Can We Prove? 1.Alan Mathison Turing 2.On Computable Numbers, with an Application to the Entscheidungsproblem 3.On Computable Numbers, with an Application to the Entscheidungsproblem - correction 4.Review of Turing 1936-7 5.Computability and -definability 6.The p-function in -K-conversion 7.Systems of Logic based on Ordinals 8.A Formal Theorem in Church's Theory of Types 9.The Use of Dots as Brackets in Church's System 10.Practical Forms of Type Theory 11.The Reform of Mathematical Notation Part II: Hiding and Unhiding Information: Cryptology, Complexity and Number Theory. 1.On the Gaussian Error Function 2.A Method for the Calculation of the Zeta-function 3.Some Calculations of the Riemann Zeta-function 4.On a Theorem of Littlewood 5.The Word Problem in Semi-groups with Cancellation 6.Solvable and Unsolvable Problems 7.The Word Problem in Compact Groups 8.On Permutation Groups 9.Rounding-off Errors in Matrix Processes 10.A Note on Normal Numbers 11.Turing's treatise on the Enigma (Prof's Book)-- Report by Turing on U. S. Navy cryptanalytic work and their machinery, November 1942-- Speech System 'Delilah' - report on progress, 6 June 1944-- Checking a Large Routine-- An early program proof by Alan Turing-- Programmers' Handbook for the Manchester electronic computer-- Local Programming Methods and Conventions Part III: Building a Brain: Intelligent Machines, Practice and Theory. 1.Lecture to the London Mathematical Society 2.Intelligent Machinery 3.Computing Machinery and Intelligence 4.Chess-- Solvable and Unsolvable Problems 5.Intelligent Machinery: A heretical theory-- Can digital computers think?-- Can automatic calculating machines be said to think? 6.Some Remarks on the Undecidability Results Part IV: The Mathematics of Emergence: The Mysteries of Morphogenesis. 1.The Chemical Basis of Morphogenesis 2.A Diffusion Reaction Theory of Morphogenesis in Plants 3.Morphogen Theory of Phyllotaxis-- Geometrical and Descriptive Phyllotaxis-- Chemical Theory of Morphogenesis-- A Solution of the Morphogenetical Equations for the Case of Spherical Symmetry 4.Outline of the Development of the Daisy.
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In this 2013 winner of the prestigious R.R. Hawkins Award from the Association of American Publishers, as well as the 2013 PROSE Awards for Mathematics and Best in Physical Sciences & Mathematics, also from the AAP, readers will find many of the most significant contributions from the four-volume set of the Collected Works of A. M. Turing. These contributions, together with commentaries from current experts in a wide spectrum of fields and backgrounds, provide insight on the significance and contemporary impact of Alan Turing's work. Offering a more modern perspective than anything currently available, Alan Turing: His Work and Impact gives wide coverage of the many ways in which Turing's scientific endeavors have impacted current research and understanding of the world. His pivotal writings on subjects including computing, artificial intelligence, cryptography, morphogenesis, and more display continued relevance and insight into today's scientific and technological landscape. This collection provides a great service to researchers, but is also an approachable entry point for readers with limited training in the science, but an urge to learn more about the details of Turing's work.
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• Introduction-- Part I. Delimiting the Unconceived:
• 1. String theory--
• 2. The conceptual framework--
• 3. The assessment of scientific underdetermination in string theory-- Part II. A Wider Perspective:
• 4. The dynamics of high energy physics--
• 5. Scientific underdetermination in physics and beyond-- Part III. Physics and Truth:
• 6. Final theory claims--
• 7. An altered perspective on scientific realism-- References-- Index.
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String theory has played a highly influential role in theoretical physics for nearly three decades and has substantially altered our view of the elementary building principles of the Universe. However, the theory remains empirically unconfirmed, and is expected to remain so for the foreseeable future. So why do string theorists have such a strong belief in their theory? This book explores this question, offering a novel insight into the nature of theory assessment itself. Dawid approaches the topic from a unique position, having extensive experience in both philosophy and high-energy physics. He argues that string theory is just the most conspicuous example of a number of theories in high-energy physics where non-empirical theory assessment has an important part to play. Aimed at physicists and philosophers of science, the book does not use mathematical formalism and explains most technical terms.
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• Approaches to understanding human behavior
• Quantitative behavioral genetics
• Social-environmental approaches
• Molecular behavioral genetics
• Neurobiological approaches
• Integrative approaches
• Scope and limits of the approaches
• Epistemological, ontological, and social analysis
• What we could know
• Defining behavior
• The social life of behavioral science
• A brief conclusion.

In "Studying Human Behavior", Helen E. Longino enters into the complexities of human behavioral research, a domain still dominated by the age-old debate of "nature versus nurture." Rather than supporting one side or another or attempting to replace that dichotomy with a different framework for understanding behavior, Longino focuses on how scientists study it, specifically sexual behavior and aggression, and asks what can be known about human behavior through empirical investigation. She dissects five approaches to the study of behavior - quantitative behavioral genetics, molecular behavior genetics, developmental psychology, neurophysiology and anatomy, and social/environmental methods - highlighting the underlying assumptions of these disciplines, as well as the different questions and mechanisms each addresses. She also analyzes efforts to integrate different approaches. Longino concludes that there is no single "correct" approach but that each contributes to our overall understanding of human behavior. In addition, Longino reflects on the reception and transmission of this behavioral research in scientific, social, clinical, and political spheres. A highly significant and innovative study that bears on crucial scientific questions, "Studying Human Behavior" will be essential reading not only for scientists and philosophers but also for science journalists and anyone interested in the engrossing challenges of understanding human behavior.
(source: Nielsen Book Data)

In "Topics in the Foundations of General Relativity and Newtonian Gravitation Theory", David B. Malament presents the basic logical-mathematical structure of general relativity and considers a number of special topics concerning the foundations of general relativity and its relation to Newtonian gravitation theory. These special topics include the geometrized formulation of Newtonian theory (also known as Newton-Cartan theory), the concept of rotation in general relativity, and Godel spacetime. One of the highlights of the book is a no-go theorem that can be understood to show that there is no criterion of orbital rotation in general relativity that fully answers to our classical intuitions. "Topics" is intended for both students and researchers in mathematical physics and philosophy of science.
(source: Nielsen Book Data)

• Preface--
• 1. Logical dynamics, agency, and intelligent interaction--
• 2. Epistemic logic and semantic information--
• 3. Dynamic logic of public observation--
• 4. Multi-agent dynamic-epistemic logic--
• 5. Dynamics of inference and awareness--
• 6. Questions and issue management--
• 7. Soft information, correction, and belief change--
• 8. An encounter with probability--
• 9. Preference statics and dynamics--
• 10. Decisions, actions, and games--
• 11. Processes over time--
• 12. Epistemic group structure and collective agency--
• 13. Logical dynamics in philosophy--
• 14. Computation as conversation--
• 15. Rational dynamics in game theory--
• 16. Meeting cognitive realities--
• 17. Conclusion-- Bibliography.
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This book develops a view of logic as a theory of information-driven agency and intelligent interaction between many agents - with conversation, argumentation and games as guiding examples. It provides one uniform account of dynamic logics for acts of inference, observation, questions and communication, that can handle both update of knowledge and revision of beliefs. It then extends the dynamic style of analysis to include changing preferences and goals, temporal processes, group action and strategic interaction in games. Throughout, the book develops a mathematical theory unifying all these systems, and positioning them at the interface of logic, philosophy, computer science and game theory. A series of further chapters explores repercussions of the 'dynamic stance' for these areas, as well as cognitive science.
(source: Nielsen Book Data)

Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern 'effective' theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics. The book includes all the necessary background from (advanced) set theory, logic and recursion theory.
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• Preface--
• 1. Evidence--
• 2. Intelligent design--
• 3. Natural selection--
• 4. Common ancestry-- Conclusion-- References-- Index.
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How should the concept of evidence be understood? And how does the concept of evidence apply to the controversy about creationism as well as to work in evolutionary biology about natural selection and common ancestry? In this rich and wide-ranging book, Elliott Sober investigates general questions about probability and evidence and shows how the answers he develops to those questions apply to the specifics of evolutionary biology. Drawing on a set of fascinating examples, he analyzes whether claims about intelligent design are untestable; whether they are discredited by the fact that many adaptations are imperfect; how evidence bears on whether present species trace back to common ancestors; how hypotheses about natural selection can be tested, and many other issues. His book will interest all readers who want to understand philosophical questions about evidence and evolution, as they arise both in Darwin's work and in contemporary biological research.
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• Preface to the first edition-- Preface to the second edition-- Mathematical prolegomenon-- Part I. Propositional Logic:
• 1. Classical logic and the material conditional--
• 2. Basic modal logic--
• 3. Normal modal logics--
• 4. Non-normal modal logics-- strict conditionals--
• 5. Conditional logics--
• 6. Intuitionist logic--
• 7. Many-valued logics--
• 8. First degree entailment--
• 9. Logics with gaps, gluts, and worlds--
• 10. Relevant logics--
• 11. Fuzzy logics-- 11a. Appendix: many valued modal logics-- Postscript: an historical perspective on conditionals-- Part II. Qualification and Identity:
• 12. Classical logic--
• 13. Free logic--
• 14. Constant domain modal logics--
• 15. Variable domain modal logics--
• 16. Necessary identity in modal logic--
• 17. Contingent identity in modal logic--
• 18. Non-normal modal logics--
• 19. Conditional logics--
• 20. Intuitionist logic--
• 21. Many-valued logics--
• 22. First degree entailment--
• 23. Logics with gaps, gluts, and worlds--
• 24. Relevant logics--
• 25. Fuzzy logics-- Postscript: a methodological coda.
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This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area.
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• Contents Preface and Acknowledgements
• Chapter 1: Introduction: The Mathematical Image
• Chapter 2: Platonism
• Chapter 3: Picture-proofs and Platonism
• Chapter 4: What is Applied Mathematics?
• Chapter 5: Hilbert and Godel
• Chapter 6: Knots and Notation
• Chapter 7: What is a Definition?
• Chapter 8: Constructive Approaches
• Chapter 9: Proofs, Pictures and Procedures in Wittgenstein
• Chapter 10: Computation, Proof and Conjecture
• Chapter 11: How to Refute the Continuum Hypothesis
• Chapter 12: Calling the Bluff Notes Bibliography Index.
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In his long-awaited new edition of "Philosophy of Mathematics", James Robert Brown tackles important new as well as enduring questions in the mathematical sciences. Can pictures go beyond being merely suggestive and actually prove anything? Are mathematical results certain? Are experiments of any real value? This clear and engaging book takes a unique approach, encompassing non-standard topics such as the role of visual reasoning, the importance of notation, and the place of computers in mathematics, as well as traditional topics such as formalism, Platonism, and constructivism. The combination of topics and clarity of presentation make it suitable for beginners and experts alike. The revised and updated second edition of "Philosophy of Mathematics "contains more examples, suggestions for further reading, and expanded material on several topics including a novel approach to the continuum hypothesis.
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• Preface--
• 1. What Godel's Theorems say--
• 2. Decidability and enumerability--
• 3. Axiomatized formal theories--
• 4. Capturing numerical properties--
• 5. The truths of arithmetic--
• 6. Sufficiently strong arithmetics--
• 7. Interlude: taking stock--
• 8. Two formalized arithmetics--
• 9. What Q can prove--
• 10. First-order Peano Arithmetic--
• 11. Primitive recursive functions--
• 12. Capturing funtions--
• 13. Q is p.r. adequate--
• 14. Interlude: a very little about Principia--
• 15. The arithmetization of syntax--
• 16. PA is incomplete--
• 17. Godel's First Theorem--
• 18. Interlude: about the First Theorem--
• 19. Strengthening the First Theorem--
• 20. The Diagonalization Lemma--
• 21. Using the Diagonalization Lemma--
• 22. Second-order arithmetics--
• 23. Interlude: incompleteness and Isaacson's conjecture--
• 24. Godel's Second Theorem for PA--
• 25. The derivability conditions--
• 26. Deriving the derivability conditions--
• 27. Reflections--
• 28. Interlude: about the Second Theorem--
• 29. Recursive functions--
• 30. Undecidability and incompleteness--
• 31. Turing machines--
• 32. Turing machines and recursiveness--
• 33. Halting problems--
• 34. The Church-Turing Thesis--
• 35. Proving the Thesis?--
• 36. Looking back.
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In 1931, the young Kurt Godel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Godel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.
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• Introduction by Reuben Hersh.- A. Renyi: Socratic Dialogue.- C. Celluci: Filosofia e Matematica, introduction.- W. Thurston: On Proof and Progress in Mathematics.- A. Aberdein: The Informal Logic of Mathematical Proof.- Y. Rav: Philosophical Problems of Mathematics in Light of Evolutionary Epistemology.- B. Rotman: Towards a Semiotics of Mathematics.- D. Mackenzie: Computers and the Sociology of Mathematical Proof.- T. Stanway: From G.H.H. and Littlewood to XML and Maple: Changing Needs and Expectations in Mathematical Knowledge Management.- R. Nunez: Do Numbers Really Move?- T. Gowers: Does Mathematics Need a Philosophy?- J. Azzouni: How and Why Mathematics is a Social Practice.- G.C. Rota: The Pernicious Influence of Mathematics Upon Philosophy.- J. Schwartz: The Pernicious Influence of Mathematics on Science.- Alfonso Avila del Palacio: What is Philosophy of Mathematics Looking For?.- A. Pickering: Concepts and the Mangle of Practice: Constructing Quaternions.- E. Glas: Mathematics as Objective Knowledge and as Human Practice.- L. White: The Locus of Mathematical Reality: An Anthropological Footnote.- R. Hersh: Inner Vision, Outer Truth.
• (source: Nielsen Book Data)

This book collects some of the most interesting recent writings that are tackling, from various points of view, the problem of giving an accounting of the nature, purpose, and justification of real mathematical practice - mathematics as actually done by real live mathematicians. What is the nature of the objects being studied? What determines the directions and styles in which mathematics progresses (or, perhaps, degenerates)? What certifies its claim to certainty, or to a priori status, to independence of experience? Why is mathematics the same for all times and places, or is it really the same, or in what senses is it the same and in what senses different? Many of these writings were read at conferences in Europe and America under the heading of "history" or "cultural studies" as well as "philosophy." It is the editor's hope to help foster healthy interdisciplinary mutual aid in this young and fertile area.'I was pleasantly surprised to find that this book does not treat mathematics as dessicated formal logic but as a living organism, immediately recognizable to any working mathematician' - Sir Michael Atiyah, University of Edinburgh. 'A wonderful collection of essays on the philosophy of mathematics, some by mathematicians, others by philosophers, and all having significant things to say. Most readers will be informed, some will be infuriated, but all will be stimulated' - John H. Conway, John von Neumann Distinguished Professor of Mathematics, Princeton University.
(source: Nielsen Book Data)