Steffen L. Lauritzen
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509721
- eISBN:
- 9780191709197
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509721.001.0001
- Subject:
- Mathematics, Probability / Statistics
Thorvald Nicolai Thiele was a brilliant Danish researcher of the 19th century. He was a professor of Astronomy at the University of Copenhagen and the founder of Hafnia, the first Danish private ...
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Thorvald Nicolai Thiele was a brilliant Danish researcher of the 19th century. He was a professor of Astronomy at the University of Copenhagen and the founder of Hafnia, the first Danish private insurance company. Thiele worked in astronomy, mathematics, actuarial science, and statistics, his most spectacular contributions were in the latter two areas, where his published work was far ahead of his time. This book is concerned with his statistical work. It evolves around his three main statistical masterpieces, which are now translated into English for the first time: 1) his article from 1880 where he derives the Kalman filter; 2) his book from 1889, where he lays out the subject of statistics in a highly original way, derives the half-invariants (today known as cumulants), the notion of likelihood in the case of binomial experiments, the canonical form of the linear normal model, and develops model criticism via analysis of residuals; and 3) an article from 1899 where he completes the theory of the half-invariants. This book also contains three chapters, written by A. Hald and S. L. Lauritzen, which describe Thiele's statistical work in modern terms and puts it into an historical perspective.Less
Thorvald Nicolai Thiele was a brilliant Danish researcher of the 19th century. He was a professor of Astronomy at the University of Copenhagen and the founder of Hafnia, the first Danish private insurance company. Thiele worked in astronomy, mathematics, actuarial science, and statistics, his most spectacular contributions were in the latter two areas, where his published work was far ahead of his time. This book is concerned with his statistical work. It evolves around his three main statistical masterpieces, which are now translated into English for the first time: 1) his article from 1880 where he derives the Kalman filter; 2) his book from 1889, where he lays out the subject of statistics in a highly original way, derives the half-invariants (today known as cumulants), the notion of likelihood in the case of binomial experiments, the canonical form of the linear normal model, and develops model criticism via analysis of residuals; and 3) an article from 1899 where he completes the theory of the half-invariants. This book also contains three chapters, written by A. Hald and S. L. Lauritzen, which describe Thiele's statistical work in modern terms and puts it into an historical perspective.
Michael Potter
- Published in print:
- 2002
- Published Online:
- May 2007
- ISBN:
- 9780199252619
- eISBN:
- 9780191712647
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199252619.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant ...
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This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as the understanding of mathematics. The book argues that through the problem of arithmetic participates in the larger puzzle of the relationship between thought, language, experience, and the world, we can distinguish accounts that look to each of these to supply the content we require: those that involve the structure of our experience of the world; those that explicitly involve our grasp of a ‘third realm’ of abstract objects distinct from the concrete objects of the empirical world and the ideas of the author's private Gedankenwelt; those that appeal to something non-physical that is nevertheless an aspect of reality in harmony with which the physical aspect of the world is configured; and finally those that involve only our grasp of language.Less
This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as the understanding of mathematics. The book argues that through the problem of arithmetic participates in the larger puzzle of the relationship between thought, language, experience, and the world, we can distinguish accounts that look to each of these to supply the content we require: those that involve the structure of our experience of the world; those that explicitly involve our grasp of a ‘third realm’ of abstract objects distinct from the concrete objects of the empirical world and the ideas of the author's private Gedankenwelt; those that appeal to something non-physical that is nevertheless an aspect of reality in harmony with which the physical aspect of the world is configured; and finally those that involve only our grasp of language.
Kurt Smith
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199583652
- eISBN:
- 9780191723155
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199583652.001.0001
- Subject:
- Philosophy, History of Philosophy, Metaphysics/Epistemology
Why is there a material world? Why is it fundamentally mathematical? This book explores a seventeenth‐century answer to these questions as it emerged from the works of Descartes and Leibniz. What we ...
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Why is there a material world? Why is it fundamentally mathematical? This book explores a seventeenth‐century answer to these questions as it emerged from the works of Descartes and Leibniz. What we learn is the sense in which these philosophers held that an analysis of the material world must inevitably lead to mathematics, and that mathematics must inevitably take matter as its object. Here the connection between matter and mathematics was cast in terms of the conditions of intelligibility—matter is what underwrote the very intelligibility of mathematics. Thus, in every world in which mathematics in intelligible, matter exists, and vice versa. On this view, then, matter is not seen as a cosmic anomaly or divine afterthought, but as an essential constituent of the universe. As the title of the book asserts: matter matters.Less
Why is there a material world? Why is it fundamentally mathematical? This book explores a seventeenth‐century answer to these questions as it emerged from the works of Descartes and Leibniz. What we learn is the sense in which these philosophers held that an analysis of the material world must inevitably lead to mathematics, and that mathematics must inevitably take matter as its object. Here the connection between matter and mathematics was cast in terms of the conditions of intelligibility—matter is what underwrote the very intelligibility of mathematics. Thus, in every world in which mathematics in intelligible, matter exists, and vice versa. On this view, then, matter is not seen as a cosmic anomaly or divine afterthought, but as an essential constituent of the universe. As the title of the book asserts: matter matters.
Mark Colyvan
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195137545
- eISBN:
- 9780199833139
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019513754X.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Looks at the Quine–Putnam indispensability argument in the philosophy of mathematics. This argument urges us to place mathematical entities on the same ontological footing as other theoretical ...
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Looks at the Quine–Putnam indispensability argument in the philosophy of mathematics. This argument urges us to place mathematical entities on the same ontological footing as other theoretical entities indispensable to our best scientific theories. The indispensability argument has come under serious scrutiny in recent times, with many influential philosophers unconvinced of its cogency. This book outlines the indispensability argument in considerable detail, before defending it against various challenges.Although the focus is squarely on the indispensability argument, in order to appreciate the argument's full force, it is necessary to consider many other interesting and related topics. These include questions about ontological commitments and the applications of mathematics to physical theories. Of particular interest here is the Quinean backdrop from which the indispensability argument emerges. This backdrop consists of the doctrines of holism and naturalism. The latter is crucial to the whole indispensability debate, so a considerable portion of this work is spent discussing naturalism.Less
Looks at the Quine–Putnam indispensability argument in the philosophy of mathematics. This argument urges us to place mathematical entities on the same ontological footing as other theoretical entities indispensable to our best scientific theories. The indispensability argument has come under serious scrutiny in recent times, with many influential philosophers unconvinced of its cogency. This book outlines the indispensability argument in considerable detail, before defending it against various challenges.
Although the focus is squarely on the indispensability argument, in order to appreciate the argument's full force, it is necessary to consider many other interesting and related topics. These include questions about ontological commitments and the applications of mathematics to physical theories. Of particular interest here is the Quinean backdrop from which the indispensability argument emerges. This backdrop consists of the doctrines of holism and naturalism. The latter is crucial to the whole indispensability debate, so a considerable portion of this work is spent discussing naturalism.
E. Brian Davies
- Published in print:
- 2007
- Published Online:
- September 2008
- ISBN:
- 9780199219186
- eISBN:
- 9780191711695
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219186.001.0001
- Subject:
- Physics, History of Physics
How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book ...
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How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book discusses the basis for scientists' claims to knowledge about the world. It looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. The book rejects the claim that all scientific knowledge is provisional, by citing examples from chemistry, biology, and geology. A major feature of the book is its defence of the view that mathematics was invented rather than discovered. While experience has shown that disentangling knowledge from opinion and aspiration is a hard task, this book provides a clear guide to the difficulties. Including many examples and quotations, and with a scope ranging from psychology and evolution to quantum theory and mathematics, this book aims to bring alive issues at the heart of all science.Less
How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book discusses the basis for scientists' claims to knowledge about the world. It looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. The book rejects the claim that all scientific knowledge is provisional, by citing examples from chemistry, biology, and geology. A major feature of the book is its defence of the view that mathematics was invented rather than discovered. While experience has shown that disentangling knowledge from opinion and aspiration is a hard task, this book provides a clear guide to the difficulties. Including many examples and quotations, and with a scope ranging from psychology and evolution to quantum theory and mathematics, this book aims to bring alive issues at the heart of all science.
Louis A. Girifalco
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228966
- eISBN:
- 9780191711183
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228966.003.0004
- Subject:
- Physics, History of Physics
Descartes believed he was meant to create the ultimate natural philosophy. He relied primarily on reason and philosophic principles, so most of the physics he developed was wrong. But he was a great ...
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Descartes believed he was meant to create the ultimate natural philosophy. He relied primarily on reason and philosophic principles, so most of the physics he developed was wrong. But he was a great mathematician and created analytic geometry, which was a major step in developing modern mathematics. Mathematics had always primarily meant geometry, which was regarded as being the only absolute truth. Galileo, and even Newton, presented their results in geometric form. Descartes showed that there was a close connection between geometry and algebra. This ultimately led to modern powerful analytic tools. His contributions to philosophy were important because they stressed the need for rigorous logic and for making as few assumptions as possible.Less
Descartes believed he was meant to create the ultimate natural philosophy. He relied primarily on reason and philosophic principles, so most of the physics he developed was wrong. But he was a great mathematician and created analytic geometry, which was a major step in developing modern mathematics. Mathematics had always primarily meant geometry, which was regarded as being the only absolute truth. Galileo, and even Newton, presented their results in geometric form. Descartes showed that there was a close connection between geometry and algebra. This ultimately led to modern powerful analytic tools. His contributions to philosophy were important because they stressed the need for rigorous logic and for making as few assumptions as possible.
David M. Armstrong
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199590612
- eISBN:
- 9780191723391
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199590612.003.0012
- Subject:
- Philosophy, Philosophy of Mind, Metaphysics/Epistemology
Logical and mathematical truths differ from the empirical sciences in being necessary; they can be discovered a priori and in general can be proved (contra Quine). How is this possible? This problem ...
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Logical and mathematical truths differ from the empirical sciences in being necessary; they can be discovered a priori and in general can be proved (contra Quine). How is this possible? This problem is partly met by recognizing that the rational sciences are sciences of the possible. Only the mathematical structures that are instantiated in space‐time are existents. Furthermore, using the Entailment Principle, it is seen that only the logico‐mathematical axioms require truthmakers. We should recognize laws in these sciences, but laws that are necessary. Such laws will be truthmakers for truths about uninstantiated structures, for instance large infinite numbers. What is the source of these necessary laws? Perhaps it is a necessity in the nature of things.Less
Logical and mathematical truths differ from the empirical sciences in being necessary; they can be discovered a priori and in general can be proved (contra Quine). How is this possible? This problem is partly met by recognizing that the rational sciences are sciences of the possible. Only the mathematical structures that are instantiated in space‐time are existents. Furthermore, using the Entailment Principle, it is seen that only the logico‐mathematical axioms require truthmakers. We should recognize laws in these sciences, but laws that are necessary. Such laws will be truthmakers for truths about uninstantiated structures, for instance large infinite numbers. What is the source of these necessary laws? Perhaps it is a necessity in the nature of things.
Henry E. Kyburg
- Published in print:
- 1991
- Published Online:
- October 2011
- ISBN:
- 9780195062533
- eISBN:
- 9780199853038
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195062533.001.0001
- Subject:
- Philosophy, Philosophy of Science
This book presents views on a wide range of philosophical problems associated with the study and practice of science and mathematics. The main structure of the book consists of a presentation of ...
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This book presents views on a wide range of philosophical problems associated with the study and practice of science and mathematics. The main structure of the book consists of a presentation of notions of epistemic probability and its use in the scientific enterprise i.e., the effort to modify previously adopted beliefs in the light of experience. Intended for cognitive scientists and people in artificial intelligence, as well as for technically oriented philosophers, the book also provides a general overview of the philosophy of science for the non-philosopher.Less
This book presents views on a wide range of philosophical problems associated with the study and practice of science and mathematics. The main structure of the book consists of a presentation of notions of epistemic probability and its use in the scientific enterprise i.e., the effort to modify previously adopted beliefs in the light of experience. Intended for cognitive scientists and people in artificial intelligence, as well as for technically oriented philosophers, the book also provides a general overview of the philosophy of science for the non-philosopher.
Stewart Shapiro
- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780195139303
- eISBN:
- 9780199833658
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195139305.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
The philosophy of mathematics articulated and defended in this book goes by the name of “structuralism”, and its slogan is that mathematics is the science of structure. The subject matter of ...
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The philosophy of mathematics articulated and defended in this book goes by the name of “structuralism”, and its slogan is that mathematics is the science of structure. The subject matter of arithmetic, for example, is the natural number structure, the pattern common to any countably infinite system of objects with a distinguished initial object and a successor relation that satisfies the induction principle. The essence of each natural number is its relation to the other natural numbers. One way to understand structuralism is to reify structures as ante rem universals. This would be a platonism concerning mathematical objects, which are the places within such structures. Alternatively, one can take an eliminative, in re approach, and understand talk of structures as shorthand for talk of systems of objects or, invoking modality, talk of possible systems of objects. Shapiro argues that although the realist, ante rem approach is the most perspicuous, in a sense, the various accounts are equivalent. Along the way, the ontological and epistemological aspects of the structuralist philosophies are assessed. One key aspect is to show how each philosophy deals with reference to mathematical objects. The view is tentatively extended to objects generally: to science and ordinary discourse.Less
The philosophy of mathematics articulated and defended in this book goes by the name of “structuralism”, and its slogan is that mathematics is the science of structure. The subject matter of arithmetic, for example, is the natural number structure, the pattern common to any countably infinite system of objects with a distinguished initial object and a successor relation that satisfies the induction principle. The essence of each natural number is its relation to the other natural numbers. One way to understand structuralism is to reify structures as ante rem universals. This would be a platonism concerning mathematical objects, which are the places within such structures. Alternatively, one can take an eliminative, in re approach, and understand talk of structures as shorthand for talk of systems of objects or, invoking modality, talk of possible systems of objects. Shapiro argues that although the realist, ante rem approach is the most perspicuous, in a sense, the various accounts are equivalent. Along the way, the ontological and epistemological aspects of the structuralist philosophies are assessed. One key aspect is to show how each philosophy deals with reference to mathematical objects. The view is tentatively extended to objects generally: to science and ordinary discourse.
Geoffrey Hellman
- Published in print:
- 1993
- Published Online:
- November 2003
- ISBN:
- 9780198240341
- eISBN:
- 9780191597664
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198240341.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Develops a structuralist understanding of mathematics, as an alternative to set‐ or type‐theoretic foundations, that respects classical mathematical truth while minimizing Platonist commitments to ...
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Develops a structuralist understanding of mathematics, as an alternative to set‐ or type‐theoretic foundations, that respects classical mathematical truth while minimizing Platonist commitments to abstract entities. Modal logic is combined with notions of part/whole (mereology) enabling a systematic interpretation of ordinary mathematical statements as asserting what would be the case in any (suitable) structure there (logically) might be, e.g. for number theory, functional analysis, algebra, pure geometry, etc. Structures are understood as comprising objects, whatever their nature, standing in suitable relations as given by axioms or defining conditions in mathematics proper. The characterization of structures is aided by the addition of plural quantifiers, e.g. ‘Any objects of sort F’ corresponding to arbitrary collections of Fs, achieving the expressive power of second‐order logic, hence a full logic of relations. (See the author's ‘Structuralism without Structures’, Philosophia Mathematica 4 (1996): 100–123.) Claims of absolute existence of structures are replaced by claims of (logical) possibility of enough structurally interrelated objects (modal‐existence postulates). The vast bulk of ordinary mathematics, and scientific applications, can thus be recovered on the basis of the possibility of a countable infinity of atoms. As applied to set theory itself, these ideas lead to a ‘many worlds’—– as opposed to the standard ‘fixed universe’—view, inspired by Zermelo (1930), respecting the unrestricted, indefinite extendability of models of the Zermelo–Fraenkel axioms. Natural motivation for (‘small’) large cardinal axioms is thus provided. In sum, the vast bulk of abstract mathematics is respected as objective, while literal reference to abstracta and related problems with Platonism are eliminated.Less
Develops a structuralist understanding of mathematics, as an alternative to set‐ or type‐theoretic foundations, that respects classical mathematical truth while minimizing Platonist commitments to abstract entities. Modal logic is combined with notions of part/whole (mereology) enabling a systematic interpretation of ordinary mathematical statements as asserting what would be the case in any (suitable) structure there (logically) might be, e.g. for number theory, functional analysis, algebra, pure geometry, etc. Structures are understood as comprising objects, whatever their nature, standing in suitable relations as given by axioms or defining conditions in mathematics proper. The characterization of structures is aided by the addition of plural quantifiers, e.g. ‘Any objects of sort F’ corresponding to arbitrary collections of Fs, achieving the expressive power of second‐order logic, hence a full logic of relations. (See the author's ‘Structuralism without Structures’, Philosophia Mathematica 4 (1996): 100–123.) Claims of absolute existence of structures are replaced by claims of (logical) possibility of enough structurally interrelated objects (modal‐existence postulates). The vast bulk of ordinary mathematics, and scientific applications, can thus be recovered on the basis of the possibility of a countable infinity of atoms. As applied to set theory itself, these ideas lead to a ‘many worlds’—– as opposed to the standard ‘fixed universe’—view, inspired by Zermelo (1930), respecting the unrestricted, indefinite extendability of models of the Zermelo–Fraenkel axioms. Natural motivation for (‘small’) large cardinal axioms is thus provided. In sum, the vast bulk of abstract mathematics is respected as objective, while literal reference to abstracta and related problems with Platonism are eliminated.
Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
The Nature of Mathematical Knowledge develops and defends an empiricist approach to mathematical knowledge. After offering an account of a priori knowledge, it argues that none of the ...
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The Nature of Mathematical Knowledge develops and defends an empiricist approach to mathematical knowledge. After offering an account of a priori knowledge, it argues that none of the available accounts of a priori mathematical knowledge is viable. It then constructs an approach to the content of mathematical statements, viewing mathematics as grounded in our manipulations of physical reality. From these crude beginnings, mathematics unfolds through the successive modifications of mathematical practice, spurred by the presence of unsolved problems. This process of unfolding is considered in general, and illustrated by considering the historical development of analysis from the seventeenth century to the end of the nineteenth.Less
The Nature of Mathematical Knowledge develops and defends an empiricist approach to mathematical knowledge. After offering an account of a priori knowledge, it argues that none of the available accounts of a priori mathematical knowledge is viable. It then constructs an approach to the content of mathematical statements, viewing mathematics as grounded in our manipulations of physical reality. From these crude beginnings, mathematics unfolds through the successive modifications of mathematical practice, spurred by the presence of unsolved problems. This process of unfolding is considered in general, and illustrated by considering the historical development of analysis from the seventeenth century to the end of the nineteenth.
Michael Potter
- Published in print:
- 2004
- Published Online:
- September 2011
- ISBN:
- 9780199269730
- eISBN:
- 9780191699443
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269730.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an ...
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This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. The book offers a simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. The book interweaves a presentation of the technical material with a philosophical critique. The book does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.Less
This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. The book offers a simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. The book interweaves a presentation of the technical material with a philosophical critique. The book does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.
Richard Robinson
- Published in print:
- 1963
- Published Online:
- October 2011
- ISBN:
- 9780198241607
- eISBN:
- 9780191680397
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198241607.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Philosophy of Language
Definition has been practised and discussed for nearly two and a half millenniums. It has been more widely adopted, and less often reviled, than any other part of the original theory of logic drawn ...
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Definition has been practised and discussed for nearly two and a half millenniums. It has been more widely adopted, and less often reviled, than any other part of the original theory of logic drawn up by Aristotle. Today it is probably the best known idea in the field of logic, except for the idea of inference. The word ‘definition’ is more often used by the general public than any other peculiarly logical term except the word ‘logic’ itself. The purpose of this book, as a whole, is to clarify our conception of definition and to improve our defining activities. Topics covered include disagreements about definition, word-thing definition, lexical definition, stipulative definition, methods of word-thing definition, real definition, and definition in mathematics.Less
Definition has been practised and discussed for nearly two and a half millenniums. It has been more widely adopted, and less often reviled, than any other part of the original theory of logic drawn up by Aristotle. Today it is probably the best known idea in the field of logic, except for the idea of inference. The word ‘definition’ is more often used by the general public than any other peculiarly logical term except the word ‘logic’ itself. The purpose of this book, as a whole, is to clarify our conception of definition and to improve our defining activities. Topics covered include disagreements about definition, word-thing definition, lexical definition, stipulative definition, methods of word-thing definition, real definition, and definition in mathematics.
José Ferreirós
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction ...
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This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, the book uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, the book shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. It argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. It demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, the book challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.Less
This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, the book uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, the book shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. It argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. It demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, the book challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.
Ian Hinckfuss
- Published in print:
- 1975
- Published Online:
- October 2011
- ISBN:
- 9780198245193
- eISBN:
- 9780191680854
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198245193.001.0001
- Subject:
- Philosophy, Metaphysics/Epistemology
This book is intended as an introduction to the philosophical problems of space and time, suitable for any reader who has an interest in the nature of the universe and who has a secondary-school ...
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This book is intended as an introduction to the philosophical problems of space and time, suitable for any reader who has an interest in the nature of the universe and who has a secondary-school knowledge of physics and mathematics. In particular, it is hoped that the book may find a use in philosophy departments and physics departments within universities and other tertiary institutions. The attempt is always to introduce the problems from a twentieth-century point of view. It is preferable to introduce the history of the topic if and when that history becomes relevant to the development and solution of the problems, rather than to introduce a problem that was of importance in some previous age and to trace the development of it down the years.Less
This book is intended as an introduction to the philosophical problems of space and time, suitable for any reader who has an interest in the nature of the universe and who has a secondary-school knowledge of physics and mathematics. In particular, it is hoped that the book may find a use in philosophy departments and physics departments within universities and other tertiary institutions. The attempt is always to introduce the problems from a twentieth-century point of view. It is preferable to introduce the history of the topic if and when that history becomes relevant to the development and solution of the problems, rather than to introduce a problem that was of importance in some previous age and to trace the development of it down the years.
Emily R. Grosholz
- Published in print:
- 1991
- Published Online:
- October 2011
- ISBN:
- 9780198242505
- eISBN:
- 9780191680502
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198242505.001.0001
- Subject:
- Philosophy, History of Philosophy, Logic/Philosophy of Mathematics
Cartesian method, construed as a way of organizing domains of knowledge according to the ‘order of reason’, was a powerful reductive tool. Descartes produced important ...
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Cartesian method, construed as a way of organizing domains of knowledge according to the ‘order of reason’, was a powerful reductive tool. Descartes produced important results in mathematics, physics, and metaphysics by relating certain complex items and problems back to simpler elements that serve as starting points for his inquiries. However, his reductive method also impoverished these domains in important ways, for it tended to restrict geometry to the study of straight line segments, physics to the study of ambiguously constituted bits of matter in motion, and metaphysics to the study of the isolated, incorporeal knower. This book examines in detail the impact, negative and positive, of Descartes's method on his scientific and philosophical enterprises, exemplified by the Geometry, the Principles of Philosophy, the Treatise of Man, and the Meditations on First Philosophy.Less
Cartesian method, construed as a way of organizing domains of knowledge according to the ‘order of reason’, was a powerful reductive tool. Descartes produced important results in mathematics, physics, and metaphysics by relating certain complex items and problems back to simpler elements that serve as starting points for his inquiries. However, his reductive method also impoverished these domains in important ways, for it tended to restrict geometry to the study of straight line segments, physics to the study of ambiguously constituted bits of matter in motion, and metaphysics to the study of the isolated, incorporeal knower. This book examines in detail the impact, negative and positive, of Descartes's method on his scientific and philosophical enterprises, exemplified by the Geometry, the Principles of Philosophy, the Treatise of Man, and the Meditations on First Philosophy.
Paul C. Gutjahr
- Published in print:
- 2011
- Published Online:
- May 2011
- ISBN:
- 9780199740420
- eISBN:
- 9780199894703
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199740420.003.0041
- Subject:
- Religion, Church History
Chapter forty-one deals with the years immediately following the death of so many of Hodge’s friends and family. He underwent intense bouts of grief and his physical health was not strong. He was ...
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Chapter forty-one deals with the years immediately following the death of so many of Hodge’s friends and family. He underwent intense bouts of grief and his physical health was not strong. He was also named to Princeton College’s Board of Trustees in 1850. He served on the Board until his death in 1878. While a Trustee, Hodge worked closely with Presidents Carnahan, Maclean and McCosh to keep religious instruction an important part of the school’s curriculum. He also stressed a broad-based liberal arts approach to the College’s curricular agenda.Less
Chapter forty-one deals with the years immediately following the death of so many of Hodge’s friends and family. He underwent intense bouts of grief and his physical health was not strong. He was also named to Princeton College’s Board of Trustees in 1850. He served on the Board until his death in 1878. While a Trustee, Hodge worked closely with Presidents Carnahan, Maclean and McCosh to keep religious instruction an important part of the school’s curriculum. He also stressed a broad-based liberal arts approach to the College’s curricular agenda.
Penelope Maddy
- Published in print:
- 1992
- Published Online:
- November 2003
- ISBN:
- 9780198240358
- eISBN:
- 9780191597978
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019824035X.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Many mathematicians understand their work as an effort to describe the denizens and features of an abstract mathematical world or worlds. Most philosophers of mathematics consider views of this sort ...
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Many mathematicians understand their work as an effort to describe the denizens and features of an abstract mathematical world or worlds. Most philosophers of mathematics consider views of this sort highly problematic, largely due to two stark difficulties laid out by Benacerraf: first, if mathematical things are abstract, and thus not to be found in space and time, how can we come to know anything about them? Second, how can mathematics be the study of certain particular things, when all that seems to matter mathematically are various structural features and relations? The goal of this book is to develop a philosophically defensible version of the mathematician's pre‐theoretic realism (sometimes called ‘Platonism’) about mathematical things. Beginning from an analysis of the strengths and weaknesses of Quine's and Gödel's versions of mathematical realism, I propose an alternative called ‘set theoretic realism’ and argue that it avoids both of Benacerraf's problems. In their place, I raise a new problem: given that some open questions of mathematics (like Cantor's Continuum Hypothesis) cannot be settled on the basis of the standard axioms, how are we rationally to evaluate new candidates for axiomatic status (such as Gödel's Axiom of Constructibility or various large cardinal axioms)? Set theoretic realism and its realistic cousins are not the only positions that face this important new challenge—various popular versions of nominalism and structuralism do as well—which suggests that it taps into a fundamental issue.Less
Many mathematicians understand their work as an effort to describe the denizens and features of an abstract mathematical world or worlds. Most philosophers of mathematics consider views of this sort highly problematic, largely due to two stark difficulties laid out by Benacerraf: first, if mathematical things are abstract, and thus not to be found in space and time, how can we come to know anything about them? Second, how can mathematics be the study of certain particular things, when all that seems to matter mathematically are various structural features and relations? The goal of this book is to develop a philosophically defensible version of the mathematician's pre‐theoretic realism (sometimes called ‘Platonism’) about mathematical things. Beginning from an analysis of the strengths and weaknesses of Quine's and Gödel's versions of mathematical realism, I propose an alternative called ‘set theoretic realism’ and argue that it avoids both of Benacerraf's problems. In their place, I raise a new problem: given that some open questions of mathematics (like Cantor's Continuum Hypothesis) cannot be settled on the basis of the standard axioms, how are we rationally to evaluate new candidates for axiomatic status (such as Gödel's Axiom of Constructibility or various large cardinal axioms)? Set theoretic realism and its realistic cousins are not the only positions that face this important new challenge—various popular versions of nominalism and structuralism do as well—which suggests that it taps into a fundamental issue.
Hartry Field
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780199242894
- eISBN:
- 9780191597381
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199242895.001.0001
- Subject:
- Philosophy, General
This is a collection of papers, written over many years, with substantial postscripts tying them together and giving an updated perspective on them. The first five are on the notions of truth and ...
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This is a collection of papers, written over many years, with substantial postscripts tying them together and giving an updated perspective on them. The first five are on the notions of truth and truth‐conditions, and their role in a theory of meaning and of the content of our mental states. The next five deal with what I call ‘factually defective discourse’—discourse that gives rise to issues about which, it is tempting to say that, there is no fact of the matter as to the right answer; one particular kind of factually defective discourse is called ‘indeterminacy’, and it gets the bulk of the attention. The final bunch of papers deal with issues about objectivity, closely related to issues about factual defectiveness; two deal with the question of whether the axioms of mathematics are as objective as is often assumed, and one deals with the question of whether our epistemological methods are as objective as they are usually assumed to be.Less
This is a collection of papers, written over many years, with substantial postscripts tying them together and giving an updated perspective on them. The first five are on the notions of truth and truth‐conditions, and their role in a theory of meaning and of the content of our mental states. The next five deal with what I call ‘factually defective discourse’—discourse that gives rise to issues about which, it is tempting to say that, there is no fact of the matter as to the right answer; one particular kind of factually defective discourse is called ‘indeterminacy’, and it gets the bulk of the attention. The final bunch of papers deal with issues about objectivity, closely related to issues about factual defectiveness; two deal with the question of whether the axioms of mathematics are as objective as is often assumed, and one deals with the question of whether our epistemological methods are as objective as they are usually assumed to be.
Stewart Shapiro
- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780198250296
- eISBN:
- 9780191598388
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250290.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
A language is second‐order, or higher‐order, if it has bound variables that range over properties or sets of the items in the range of the ordinary, first‐order variables. This book presents a formal ...
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A language is second‐order, or higher‐order, if it has bound variables that range over properties or sets of the items in the range of the ordinary, first‐order variables. This book presents a formal development of second‐ and higher‐order logic and an extended argument that higher‐order systems have an important role to play in the philosophy and foundations of mathematics. The development includes the languages, deductive systems, and model‐theoretic semantics for higher‐order languages, and the basic and advanced results in its meta‐theory: completeness, compactness, and the Löwenheim–Skolem theorems for Henkin semantics, and the failure of those results for standard semantics. Argues that second‐order theories and formalizations, with standard semantics, provide better models of important aspects of mathematics than their first‐order counterparts. Despite the fact that Quine is the main opponent of second‐order logic (arguing that second‐order logic is set‐theory in disguise), the present argument is broadly Quinean, proposing that there is no sharp line dividing mathematics from logic, especially the logic of mathematics. Also surveys the historical development in logic, tracing the emergence of first‐order logic as the de facto standard among logicians and philosophers. The connection between formal deduction and reasoning is related to Wittgensteinian issues concerning rule‐following. The book closes with an examination of several alternatives to second‐order logic: first‐order set theory, infinitary languages, and systems that are, in a sense, intermediate between first order and second order.Less
A language is second‐order, or higher‐order, if it has bound variables that range over properties or sets of the items in the range of the ordinary, first‐order variables. This book presents a formal development of second‐ and higher‐order logic and an extended argument that higher‐order systems have an important role to play in the philosophy and foundations of mathematics. The development includes the languages, deductive systems, and model‐theoretic semantics for higher‐order languages, and the basic and advanced results in its meta‐theory: completeness, compactness, and the Löwenheim–Skolem theorems for Henkin semantics, and the failure of those results for standard semantics. Argues that second‐order theories and formalizations, with standard semantics, provide better models of important aspects of mathematics than their first‐order counterparts. Despite the fact that Quine is the main opponent of second‐order logic (arguing that second‐order logic is set‐theory in disguise), the present argument is broadly Quinean, proposing that there is no sharp line dividing mathematics from logic, especially the logic of mathematics. Also surveys the historical development in logic, tracing the emergence of first‐order logic as the de facto standard among logicians and philosophers. The connection between formal deduction and reasoning is related to Wittgensteinian issues concerning rule‐following. The book closes with an examination of several alternatives to second‐order logic: first‐order set theory, infinitary languages, and systems that are, in a sense, intermediate between first order and second order.
