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Proof and Knowledge in Mathematics
by Michael DetlefsenThese questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is a priori or a posteriori in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification,
Proof!: How the World Became Geometrical
by Amir AlexanderA “lucid and convincingly argued” narrative of how ancient geometric principles continue to shape the contemporary world (Publishers Weekly).On a cloudy day in 1413, a balding young man stood at the entrance to the Cathedral of Florence, facing the ancient Baptistery across the piazza. As puzzled passers-by looked on, he raised a small painting to his face, then held a mirror in front of the painting. Few at the time understood what he was up to; even he barely had an inkling of what was at stake. But on that day, the master craftsman and engineer Filippo Brunelleschi would prove that the world and everything within it was governed by the ancient science of geometry.In Proof!, the award-winning historian Amir Alexander traces the path of the geometrical vision of the world as it coursed its way from the Renaissance to the present, shaping our societies, our politics, and our ideals. Geometry came to stand for a fixed and unchallengeable universal order, and kings, empire-builders, and even republican revolutionaries would rush to cast their rule as the apex of the geometrical universe. For who could doubt the right of a ruler or the legitimacy of a government that drew its power from the immutable principles of Euclidean geometry?From the elegant terraces of Versailles to the broad avenues of Washington, DC, and on to the boulevards of New Delhi and Manila, the geometrical vision was carved into the landscape of modernity. Euclid, Alexander shows, made the world as we know it possible.
Proof, Computation and Agency
by Johan Van Benthem Amitabha Gupta Rohit ParikhProof, Computation and Agency: Logic at the Crossroads provides an overview of modern logic and its relationship with other disciplines. As a highlight, several articles pursue an inspiring paradigm called 'social software', which studies patterns of social interaction using techniques from logic and computer science. The book also demonstrates how logic can join forces with game theory and social choice theory. A second main line is the logic-language-cognition connection, where the articles collected here bring several fresh perspectives. Finally, the book takes up Indian logic and its connections with epistemology and the philosophy of science, showing how these topics run naturally into each other.
Proof, Logic and Formalization
by Michael DetlefsenThe mathematical proof is the most important form of justification in mathematics. It is not, however, the only kind of justification for mathematical propositions. The existence of other forms, some of very significant strength, places a question mark over the prominence given to proof within mathematics. This collection of essays, by leading figures working within the philosophy of mathematics, is a response to the challenge of understanding the nature and role of the proof.
Proof: The Art and Science of Certainty
by Adam KucharskiAn award-winning mathematician—he "made me smile and made me feel clever" (Peter Frankopan)—shows how we prove what&’s true, and what to do when we can&’t How do we establish what we believe? And how can we be certain that what we believe is true? And how do we convince other people that it is true? For thousands of years, from the ancient Greeks to the Arabic golden age to the modern world, science has used different methods—logical, empirical, intuitive, and more—to separate fact from fiction. But it all had the same goal: find perfect evidence and be rewarded with universal truth. As mathematician Adam Kucharski shows, however, there is far more to proof than axioms, theories, and laws: when demonstrating that a new medical treatment works, persuading a jury of someone&’s guilt, or deciding whether you trust a self-driving car, the weighing up of evidence is far from simple. To discover proof, we must reach into a thicket of errors and biases and embrace uncertainty—and never more so than when existing methods fail. Spanning mathematics, science, politics, philosophy, and economics, this book offers the ultimate exploration of how we can find our way to proof—and, just as importantly, of how to go forward when supposed facts falter.
Proofiness
by Charles SeifeThe bestselling author of Zero shows how mathematical misinformation pervades-and shapes-our daily lives. According to MSNBC, having a child makes you stupid. You actually lose IQ points. Good Morning America has announced that natural blondes will be extinct within two hundred years. Pundits estimated that there were more than a million demonstrators at a tea party rally in Washington, D. C. , even though roughly sixty thousand were there. Numbers have peculiar powers-they can disarm skeptics, befuddle journalists, and hoodwink the public into believing almost anything. "Proofiness," as Charles Seife explains in this eye-opening book, is the art of using pure mathematics for impure ends, and he reminds readers that bad mathematics has a dark side. It is used to bring down beloved government officials and to appoint undeserving ones (both Democratic and Republican), to convict the innocent and acquit the guilty, to ruin our economy, and to fix the outcomes of future elections. This penetrating look at the intersection of math and society will appeal to readers of Freakonomics and the books of Malcolm Gladwell. .
Proofiness
by Charles SeifeThe bestselling author of Zero shows how mathematical misinformation pervades-and shapes-our daily lives. According to MSNBC, having a child makes you stupid. You actually lose IQ points. Good Morning America has announced that natural blondes will be extinct within two hundred years. Pundits estimated that there were more than a million demonstrators at a tea party rally in Washington, D.C., even though roughly sixty thousand were there. Numbers have peculiar powers-they can disarm skeptics, befuddle journalists, and hoodwink the public into believing almost anything. "Proofiness," as Charles Seife explains in this eye-opening book, is the art of using pure mathematics for impure ends, and he reminds readers that bad mathematics has a dark side. It is used to bring down beloved government officials and to appoint undeserving ones (both Democratic and Republican), to convict the innocent and acquit the guilty, to ruin our economy, and to fix the outcomes of future elections. This penetrating look at the intersection of math and society will appeal to readers of Freakonomics and the books of Malcolm Gladwell.
Proofs 101: An Introduction to Formal Mathematics
by Joseph KirtlandProofs 101: An Introduction to Formal Mathematics serves as an introduction to proofs for mathematics majors who have completed the calculus sequence (at least Calculus I and II) and a first course in linear algebra. The book prepares students for the proofs they will need to analyze and write the axiomatic nature of mathematics and the rigors of upper-level mathematics courses. Basic number theory, relations, functions, cardinality, and set theory will provide the material for the proofs and lay the foundation for a deeper understanding of mathematics, which students will need to carry with them throughout their future studies. Features Designed to be teachable across a single semester Suitable as an undergraduate textbook for Introduction to Proofs or Transition to Advanced Mathematics courses Offers a balanced variety of easy, moderate, and difficult exercises
Proofs and Algorithms
by Gilles DowekLogic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation. Proofs and Algorithms: Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, Gödel's incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself. Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.
Proofs and Models in Philosophical Logic (Elements in Philosophy and Logic)
by Greg RestallThis Element is an introduction to recent work proofs and models in philosophical logic, with a focus on the semantic paradoxes the sorites paradox. It introduces and motivates different proof systems and different kinds of models for a range of logics, including classical logic, intuitionistic logic, a range of three-valued and four-valued logics, and substructural logics. It also compares and contrasts the different approaches to substructural treatments of the paradox, showing how the structural rules of contraction, cut and identity feature in paradoxical derivations. It then introduces model theoretic treatments of the paradoxes, including a simple fixed-point model construction which generates three-valued models for theories of truth, which can provide models for a range of different non-classical logics. The Element closes with a discussion of the relationship between proofs and models, arguing that both have their place in the philosophers' and logicians' toolkits.
Proofs and Refutations
by John Worrall Elie Zahar Imre Lakatos Lakatos Imre Zahar ElieImre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a specially commissioned Preface written by Paolo Mancosu, this book has been revived for a new generation of readers.
Proofs and Refutations
by John Worrall Elie ZaharProofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.
Proofs of the Cantor-Bernstein Theorem
by Arie HinkisThis book offers an excursion through the developmental area of research mathematics. It presents some 40 papers, published between the 1870s and the 1970s, on proofs of the Cantor-Bernstein theorem and the related Bernstein division theorem. While the emphasis is placed on providing accurate proofs, similar to the originals, the discussion is broadened to include aspects that pertain to the methodology of the development of mathematics and to the philosophy of mathematics. Works of prominent mathematicians and logicians are reviewed, including Cantor, Dedekind, Schröder, Bernstein, Borel, Zermelo, Poincaré, Russell, Peano, the Königs, Hausdorff, Sierpinski, Tarski, Banach, Brouwer and several others mainly of the Polish and the Dutch schools. In its attempt to present a diachronic narrative of one mathematical topic, the book resembles Lakatos' celebrated book Proofs and Refutations. Indeed, some of the observations made by Lakatos are corroborated herein. The analogy between the two books is clearly anything but superficial, as the present book also offers new theoretical insights into the methodology of the development of mathematics (proof-processing), with implications for the historiography of mathematics.
Propaganda and International Criminal Law: From Cognition to Criminality
by Predrag DojčinovićThis book addresses the conceptual and evidentiary issues relating to the treatment of propaganda in international criminal law. Bringing together an interdisciplinary range of scholars, researchers and legal practitioners from Africa, Australia, Europe and the United States, the book provides an in-depth analysis of the nature, position and role of the concept of propaganda in mass atrocity crimes trials. A sequel to the earlier Propaganda, War Crimes Trials and International Law: From Speakers’ Corner to War Crimes (Routledge, 2011) this book is the first to synthesize the knowledge, procedures and methods of international criminal law with the social cognitive sciences. Including a comprehensive overview of the most relevant case law, jurisprudence and scientific studies, the book also offers a series of practical insights and strategies for both academics and legal professionals. An invaluable resource for those working in the area of international criminal law, this book will also be of interest to academics, practitioners and students with relevant interests in legal theory, politics, linguistics and psychology.
Propaganda and Nation Building: Selling the Irish Free State (Routledge New Directions in PR & Communication Research)
by Kevin HoraThis book examines the origins of Ireland in its first independent incarnation, the Irish Free State (1922-1937). It explores how contemporary public relations and propaganda techniques were used to construct an identity for this new state – a state which after enduring seven years of insurrection and civil war, became one of the most stable democracies in Europe. This stability, the book argues, was constructed not solely through policies enacted by governments, but through the construction of a Gaelic, Catholic and Celtic national identity. By shifting the perspective to how nation building was communicated, it weaves an interdisciplinary narrative that initiates a new understanding of nation building - providing insights of increasing relevance in current world events. Avoiding a simplistic cause and effect history of public relations, the book examines the uses and effects of early public relations from a political and societal perspective and suggests that while governments were only modestly successful in their varied propaganda efforts, cumulatively they facilitated a transition from violence to peace. This will be of interest to researchers and advanced students with an interest in public relations, propaganda studies, nation building and Irish studies.
Properties (Elements in Metaphysics)
by Anna-Sofia MaurinAlthough the subject matter of this Element is properties, do not expect in-depth introductions to the various views on properties 'on the market'. Instead, here that subject matter is treated meta-philosophically. Rather than ask and try to answer a question like do properties exist? this Element asks what reasons one might have for thinking that properties exist (what problem properties, if they exist, are there to solve), what counts as solving that (or those) problems (including what counts as 'a property'), as well as how we ought to proceed when trying to find out if properties exist (by which method this ought to be decided). As it turns out, these questions and their answers are all intricately intertwined. Theory comparison and theory evaluation is in other words (and perhaps not that surprisingly) tricky. Do properties exist? After reading this Element all we can say is therefore this: that depends.
Properties and Propositions: The Metaphysics of Higher-Order Logic
by Robert TruemanThis book articulates and defends Fregean realism, a theory of properties based on Frege's insight that properties are not objects, but rather the satisfaction conditions of predicates. Robert Trueman argues that this approach is the key not only to dissolving a host of longstanding metaphysical puzzles, such as Bradley's Regress and the Problem of Universals, but also to understanding the relationship between states of affairs, propositions, and the truth conditions of sentences. Fregean realism, Trueman suggests, ultimately leads to a version of the identity theory of truth, the theory that true propositions are identical to obtaining states of affairs. In other words, the identity theory collapses the gap between mind and world. This book will be of interest to anyone working in logic, metaphysics, the philosophy of language or the philosophy of mind.
Properties of Life: Toward a Theory of Organismic Biology (Vienna Series in Theoretical Biology)
by Bernd RosslenbroichA coherent and comprehensive theory of life that synthesizes the specific properties of living organisms.Despite continued advances, science has until now struggled to describe the specific properties that define a living being. By synthesizing several aspects of organismic biology and contemporary science, Properties of Life by Bernd Rosslenbroich generates a coherent concept of the singular quality of being alive—a concept that provides a crucial foundation for scientists, farmers, and medical practitioners and helps explain how we all interact with the world around us and within ourselves.Is an organism an aggregate of parts or an integrated system with agency? Is it a passive stimulus-response machine or a being equipped with subjectivity and consciousness? Rosslenbroich argues that the way people in different fields understand life determines their assumptions about organic function and behavior. In medicine, this extends to the human organism, which influences prevention, diagnosis, and treatment. Drawing attention to a long-standing but underappreciated line of thought in organismic biology, Rosslenbroich&’s original idea emphasizes the autonomy of living processes, their network characteristics, and their self-determined organization in time and structure. A timely and revelatory book, Properties of Life formulates an integrated, unified theory that remains flexible enough to accommodate future developments and resilient enough to withstand the challenges of different theoretical and disciplinary backgrounds.
Properties, Powers and Structures: Issues in the Metaphysics of Realism (Routledge Studies in Metaphysics)
by Brian Ellis Alexander Bird Howard SankeyWhile the phrase "metaphysics of science" has been used from time to time, it has only recently begun to denote a specific research area where metaphysics meets philosophy of science—and the sciences themselves. The essays in this volume demonstrate that metaphysics of science is an innovative field of research in its own right. The principle areas covered are: The modal metaphysics of properties: What is the essential nature of natural properties? Are all properties essentially categorical? Are they all essentially dispositions, or are some categorical and others dispositional? Realism in mathematics and its relation to science: What does a naturalistic commitment of scientific realism tell us about our commitments to mathematical entities? Can this question be framed in something other than a Quinean philosophy? Dispositions and their relation to causation: Can we generate an account of causation that takes dispositionality as fundamental? And if we take dispositions as fundamental (and hence not having a categorical causal basis), what is the ontological ground of dispositions? Pandispositionalism: Could all properties be dispositional in nature? Natural kinds: Are there natural kinds, and if so what account of their nature should we give? For example, do they have essences? Here we consider how these issues may be illuminated by considering examples from reals science, in particular biochemistry and neurobiology.
Property (Key Concepts in Political Theory #9)
by Robert LambFew political ideas are as divisive and controversial for some – and yet taken for granted by others – as the ownership of private property. For its defenders, private ownership is a fundamental right that protects individual freedom and ensures wider economic benefits for the community; for its critics, by contrast, property is institutionalised theft, responsible for lamentable levels of inequality and poverty. In this book, Robert Lamb explores philosophical arguments deployed to conceptualise, justify, and criticise private property ownership. He introduces the radical case against property advanced by anarchist and socialist writers, before analysing some of the most important and influential arguments in its favour. Lamb explains and assesses the various defences of property rights advanced by Locke, Hume, Hegel, J. S. Mill, and Nozick. He then shows how theorists such as John Rawls and his followers encourage us to rethink the very nature of ownership in a democratic society. This engaging synthesis of historical and contemporary theories of property will be essential reading for students and scholars of political philosophy.
Property Is Theft!
by Pierre-Joseph Proudhon Iain Mckay"An indispensable source book for anyone interested in Proudhon's ideas and the origins of the socialist and anarchist movements in nineteenth-century Europe."-Robert Graham, editor of Anarchism: A Documentary History of Libertarian Ideas"Iain McKay's introduction offers a sure-footed guide through the misconceptions surrounding Proudhon's thought."-Mark Leier, author of Bakunin: The Creative PassionMore influential than Karl Marx during his lifetime, Pierre-Joseph Proudon's work has long been out of print or unavailable in English. Iain McKay's comprehensive collection is a much-needed and timely historical corrective.Iain McKay is the editor of An Anarchist FAQ.
Property Rights: Philosophic Foundations (Routledge Revivals)
by Lawrence C. BeckerProperty Rights: Philosophic Foundations, first published in 1977, comprehensively examines the general justifications for systems of private property rights, and discusses with great clarity the major arguments as to the rights and responsibilities of property ownership. In particular, the arguments that hold that there are natural rights derived from first occupancy, labour, utility, liberty and virtue are considered, as are the standard anti-property arguments based on disutility, virtue and inequality, and the belief that justice in distribution must take precedence over private ownership. Lawrence Becker goes on to contend that there are four sound lines of argument for private property that, together with what is sound in the anti-property arguments, must be co-ordinated to form the foundations of a new theory. He therefore expounds a concise but sophisticated theory of property that is relevant to the modern world, and concludes by indicating some of the implications of his theory.
Property Rights: The Argument for Privatization (Palgrave Studies in Classical Liberalism)
by Walter E. BlockIn this timely book, Walter E. Block uses classical liberal theory to defend private property rights. Looking at how free enterprise, capitalism and libertarianism are cornerstones of economically prosperous civilizations, Block highlights why private property rights are crucial. Discussing philosophy, libertarian property rights theory, reparations and other property rights issues, this volume is of interest to academics, students, journalists and all those interested in this integral aspect of political economic philosophy.
Property and Justice: A Liberal Theory of Natural Rights (Political Philosophy for the Real World)
by Billy ChristmasThis book gives an account of a full spectrum of property rights and their relationship to individual liberty. It shows that a purely deontological approach to justice can deal with the most complex questions regarding the property system. Moreover, the author considers the economic, ecological, and technological complexities of our real-world property systems. The result is a more conceptually sound account of natural rights and the property system they demand. If we think that liberty should be at the centre of justice, what does that mean for the property system? Economists and lawyers widely agree that a property system must be composed of many different types of property: the kind of private ownership one has over one’s person and immediate possessions, as well as the kinds of common ownership we each have in our local streets, as well as many more. However, theories of property and justice have not given anything approaching an adequate account of the relationship between liberty and any other form of property other than private ownership. It is often thought that a basic commitment to liberty cannot really tell us how to arrange the major complexities of the property system, which diverge from simple private ownership. Property and Justice demonstrates how philosophical rigour coupled with interdisciplinary engagement enables us to think clearly about how to deal with real-world problems. It will be of interest to political philosophers, political theorists, and legal theorists working on property rights and justice.
Property and Justice: A Trend Towards Marxist Political Philosophy (China Perspectives)
by Zhang WenxiThis book attempts to answer the question “what on earth is the Marxist political philosophy?” The author discusses Karl Marx's vision of the good life and the good society, focusing in particular on the interrelationship between property and justice. Strictly speaking, the subject of the present book should not be placed within an a priori conceptual framework, which essentially focuses on academic notions of the good society, good government and justice. However, for the purposes of this book, Marxist political philosophy, namely communism, is already embodied in two aspects. This book argues that it is the noisy and confusing modern political philosophy that obscures the actualisation of human nature unfolded by the Marxist political philosophy before us. The book will be essential reading for students and scholars of philosophy, Marxism, Chinese studies and Chinese Marxist philosophy.