Peter Hylton
- Published in print:
- 2005
- Published Online:
- May 2010
- ISBN:
- 9780199286355
- eISBN:
- 9780191713309
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199286355.001.0001
- Subject:
- Philosophy, History of Philosophy, Logic/Philosophy of Mathematics
The work of Bertrand Russell had a decisive influence on the emergence of analytic philosophy, and on its subsequent development. The essays collected in this volume, by one of the authorities on ...
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The work of Bertrand Russell had a decisive influence on the emergence of analytic philosophy, and on its subsequent development. The essays collected in this volume, by one of the authorities on Russell's philosophy, all aim at recapturing and articulating aspects of Russell's philosophical vision during his most influential and important period, the two decades following his break with Idealism in 1899. One theme of the collection concerns Russell's views about propositions and their analysis, and the relation of those ideas to his rejection of Idealism. Another theme is the development of Russell's logicism, culminating in Whitehead's and Russell's Principia Mathematica, and the author offers a revealing view of the conception of logic that underlies it. Here again there is an emphasis on Russell's argument against Idealism, on the idea that his logicism was a crucial part of that argument. A further focus of the volume is Russell's views about functions and propositional functions. This theme is part of a contrast that the author draws between Russell's general philosophical position and that of Frege; in particular, there is a close parallel with the quite different views that the two philosophers held about the nature of philosophical analysis. The author also sheds light on the much-disputed idea of an operation, which Wittgenstein advances in the Tractatus Logico-Philosophicus.Less
The work of Bertrand Russell had a decisive influence on the emergence of analytic philosophy, and on its subsequent development. The essays collected in this volume, by one of the authorities on Russell's philosophy, all aim at recapturing and articulating aspects of Russell's philosophical vision during his most influential and important period, the two decades following his break with Idealism in 1899. One theme of the collection concerns Russell's views about propositions and their analysis, and the relation of those ideas to his rejection of Idealism. Another theme is the development of Russell's logicism, culminating in Whitehead's and Russell's Principia Mathematica, and the author offers a revealing view of the conception of logic that underlies it. Here again there is an emphasis on Russell's argument against Idealism, on the idea that his logicism was a crucial part of that argument. A further focus of the volume is Russell's views about functions and propositional functions. This theme is part of a contrast that the author draws between Russell's general philosophical position and that of Frege; in particular, there is a close parallel with the quite different views that the two philosophers held about the nature of philosophical analysis. The author also sheds light on the much-disputed idea of an operation, which Wittgenstein advances in the Tractatus Logico-Philosophicus.
Alan Weir
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199541492
- eISBN:
- 9780191594915
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199541492.003.0010
- Subject:
- Philosophy, Philosophy of Language, Logic/Philosophy of Mathematics
This chapter summaries the key thesis of the book: that, given the Sense/Circumstance/World framework in the philosophy of language, we can show that mathematics expresses objective truths, but not ...
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This chapter summaries the key thesis of the book: that, given the Sense/Circumstance/World framework in the philosophy of language, we can show that mathematics expresses objective truths, but not by dint of delineating a mind-independent reality. Neo-formalism is held to incorporate sound aspects of rival views, such as classic formalism; neo-logicism, in the idea that stipulation of axioms grounds mathematical truth; constructivism, insofar as the constructivist links truth with provability; strict finitism to the extent that the strict finitist emphasizes that the ontology of mathematics can include only a finite corpus of concrete tokens. Neo-formalism even incorporates elements of platonism, insofar as it upholds the objectivity of mathematical truth. However neo-formalism, in rejecting the platonistic realm of an external mathematical ontology as a mythological projection of human activity, avoids the crippling metaphysical and epistemological problems of platonism.Less
This chapter summaries the key thesis of the book: that, given the Sense/Circumstance/World framework in the philosophy of language, we can show that mathematics expresses objective truths, but not by dint of delineating a mind-independent reality. Neo-formalism is held to incorporate sound aspects of rival views, such as classic formalism; neo-logicism, in the idea that stipulation of axioms grounds mathematical truth; constructivism, insofar as the constructivist links truth with provability; strict finitism to the extent that the strict finitist emphasizes that the ontology of mathematics can include only a finite corpus of concrete tokens. Neo-formalism even incorporates elements of platonism, insofar as it upholds the objectivity of mathematical truth. However neo-formalism, in rejecting the platonistic realm of an external mathematical ontology as a mythological projection of human activity, avoids the crippling metaphysical and epistemological problems of platonism.
Patricia A. Blanchette
- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199891610
- eISBN:
- 9780199933211
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199891610.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Frege’s Coneption of Logic explores the relationship between Frege’s understanding of conceptual analysis and his understanding of logic. It is argued that the fruitfulness of Frege’s ...
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Frege’s Coneption of Logic explores the relationship between Frege’s understanding of conceptual analysis and his understanding of logic. It is argued that the fruitfulness of Frege’s conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation. The first part of the book locates the role of conceptual analysis in Frege’s logicist project. It is argued that, despite a number of difficulties, Frege’s use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege’s intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. The second part of the book explores the resulting conception of logic itself, and some of the straightforward ways in which Frege’s conception differs from its now-familiar descendants. In particular, it’s argued that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege’s position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.Less
Frege’s Coneption of Logic explores the relationship between Frege’s understanding of conceptual analysis and his understanding of logic. It is argued that the fruitfulness of Frege’s conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation. The first part of the book locates the role of conceptual analysis in Frege’s logicist project. It is argued that, despite a number of difficulties, Frege’s use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege’s intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. The second part of the book explores the resulting conception of logic itself, and some of the straightforward ways in which Frege’s conception differs from its now-familiar descendants. In particular, it’s argued that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege’s position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.
Patricia A. Blanchette
- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199891610
- eISBN:
- 9780199933211
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199891610.003.0000
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter lays out the central theme of the book, the interaction between Frege’s understanding of conceptual analysis and his understanding of logic. It also sketches the contents of the ...
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This chapter lays out the central theme of the book, the interaction between Frege’s understanding of conceptual analysis and his understanding of logic. It also sketches the contents of the succeeding chapters.Less
This chapter lays out the central theme of the book, the interaction between Frege’s understanding of conceptual analysis and his understanding of logic. It also sketches the contents of the succeeding chapters.
Denis McManus
- Published in print:
- 2006
- Published Online:
- January 2007
- ISBN:
- 9780199288021
- eISBN:
- 9780191713446
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199288021.003.0002
- Subject:
- Philosophy, History of Philosophy
Frege and Russell were particularly important influences on the early Wittgenstein, and this chapter presents a brief sketch of some themes in their work. Two themes in particular are emphasized: (1) ...
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Frege and Russell were particularly important influences on the early Wittgenstein, and this chapter presents a brief sketch of some themes in their work. Two themes in particular are emphasized: (1) the notion that philosophical problems can be exposed as pseudo-problems through logical analysis of propositions; and (2) the question of the nature of logical truth. Later chapters in the book argue that Wittgenstein develops in radical ways the thought of Frege and Russell on these themes.Less
Frege and Russell were particularly important influences on the early Wittgenstein, and this chapter presents a brief sketch of some themes in their work. Two themes in particular are emphasized: (1) the notion that philosophical problems can be exposed as pseudo-problems through logical analysis of propositions; and (2) the question of the nature of logical truth. Later chapters in the book argue that Wittgenstein develops in radical ways the thought of Frege and Russell on these themes.
Michael Potter
- Published in print:
- 2002
- Published Online:
- May 2007
- ISBN:
- 9780199252619
- eISBN:
- 9780191712647
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199252619.003.0003
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
If our ability to count depends on the structure of space and time, this may explain its applicability to the things we intuit in space and time, but it will also limit its applicability to just ...
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If our ability to count depends on the structure of space and time, this may explain its applicability to the things we intuit in space and time, but it will also limit its applicability to just those things: we shall, in short, be left without the ability to count anything that is not made up of spatio-temporal elements. This evidently did not bother Kant, but it has seemed implausible to many subsequent thinkers. The most notable of these was Frege, who published Die Grundlagen der Arithmetik (The Foundations of Arithmetic), a book arguing against Kant's conclusion, in 1884. Frege was the first to put forward the thesis, now known as logicism, that logic is capable of grounding mathematical truths without thereby rendering them wholly trivial.Less
If our ability to count depends on the structure of space and time, this may explain its applicability to the things we intuit in space and time, but it will also limit its applicability to just those things: we shall, in short, be left without the ability to count anything that is not made up of spatio-temporal elements. This evidently did not bother Kant, but it has seemed implausible to many subsequent thinkers. The most notable of these was Frege, who published Die Grundlagen der Arithmetik (The Foundations of Arithmetic), a book arguing against Kant's conclusion, in 1884. Frege was the first to put forward the thesis, now known as logicism, that logic is capable of grounding mathematical truths without thereby rendering them wholly trivial.
Bob Hale and Crispin Wright
- Published in print:
- 2005
- Published Online:
- July 2005
- ISBN:
- 9780195148770
- eISBN:
- 9780199835560
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195148770.003.0006
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
According to Gottlob Frege, his logicism died when it was discovered that the underlying theory of extensions is inconsistent. The neo-logicist attempts to found mathematics on other abstraction ...
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According to Gottlob Frege, his logicism died when it was discovered that the underlying theory of extensions is inconsistent. The neo-logicist attempts to found mathematics on other abstraction principles, such as the so-called Hume’s principle that two concepts have the same number if and only if they are equinumerous. This chapter discusses the state of neo-logicism, responding to various objections that have been raised against it.Less
According to Gottlob Frege, his logicism died when it was discovered that the underlying theory of extensions is inconsistent. The neo-logicist attempts to found mathematics on other abstraction principles, such as the so-called Hume’s principle that two concepts have the same number if and only if they are equinumerous. This chapter discusses the state of neo-logicism, responding to various objections that have been raised against it.
Agustín Rayo
- Published in print:
- 2005
- Published Online:
- July 2005
- ISBN:
- 9780195148770
- eISBN:
- 9780199835560
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195148770.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Roughly, logicism is the view that mathematics is logic. This chapter identifies several distinct logicist theses, and shows that their truth-values can be established on minimal assumptions. There ...
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Roughly, logicism is the view that mathematics is logic. This chapter identifies several distinct logicist theses, and shows that their truth-values can be established on minimal assumptions. There is also a discussion of “Neo-Logicism.”Less
Roughly, logicism is the view that mathematics is logic. This chapter identifies several distinct logicist theses, and shows that their truth-values can be established on minimal assumptions. There is also a discussion of “Neo-Logicism.”
Michael Potter
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780199215836
- eISBN:
- 9780191721243
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199215836.003.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter addresses the question of what led Wittgenstein to study philosophy with Russell in Cambridge. Since the narrative of Wittgenstein's life before 1911 is well summarized in available ...
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This chapter addresses the question of what led Wittgenstein to study philosophy with Russell in Cambridge. Since the narrative of Wittgenstein's life before 1911 is well summarized in available biographies, the focus will be on a few points including Wittgenstein's early life, his experiences as a research student at Manchester, his interest in the philosophy of mathematics, logicism, and Russell's paradox.Less
This chapter addresses the question of what led Wittgenstein to study philosophy with Russell in Cambridge. Since the narrative of Wittgenstein's life before 1911 is well summarized in available biographies, the focus will be on a few points including Wittgenstein's early life, his experiences as a research student at Manchester, his interest in the philosophy of mathematics, logicism, and Russell's paradox.
Scott Soames
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160726
- eISBN:
- 9781400850464
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160726.001.0001
- Subject:
- Philosophy, American Philosophy
This collection of recent and unpublished essays traces milestones in the field of analytic philosophy from its beginnings in Britain and Germany in the late nineteenth and early twentieth centuries, ...
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This collection of recent and unpublished essays traces milestones in the field of analytic philosophy from its beginnings in Britain and Germany in the late nineteenth and early twentieth centuries, through its subsequent growth in the United States, up to its present as the world’s most vigorous philosophical tradition. The central chapter chronicles how analytic philosophy developed in the United States out of American pragmatism, the impact of European visitors and immigrants, the mid-century transformation of the Harvard philosophy department, and the rapid spread of the analytic approach that followed. Another chapter explains the methodology guiding analytic philosophy, from the logicism of Frege and Russell through Wittgenstein’s linguistic turn and Carnap’s vision of replacing metaphysics with philosophy of science. Further chapters review advances in logic and the philosophy of mathematics that laid the foundation for a rigorous, scientific study of language, meaning, and information. Other chapters discuss W. V. O. Quine, David K. Lewis, Saul Kripke, the Frege–Russell analysis of quantification, Russell’s attempt to eliminate sets with his “no class theory,” and the Quine–Carnap dispute over meaning and ontology. The book then turns to topics at the frontier of philosophy of language. The final chapters, combining philosophy of language and law, advance a sophisticated originalist theory of interpretation and apply it to U.S. constitutional rulings about due process.Less
This collection of recent and unpublished essays traces milestones in the field of analytic philosophy from its beginnings in Britain and Germany in the late nineteenth and early twentieth centuries, through its subsequent growth in the United States, up to its present as the world’s most vigorous philosophical tradition. The central chapter chronicles how analytic philosophy developed in the United States out of American pragmatism, the impact of European visitors and immigrants, the mid-century transformation of the Harvard philosophy department, and the rapid spread of the analytic approach that followed. Another chapter explains the methodology guiding analytic philosophy, from the logicism of Frege and Russell through Wittgenstein’s linguistic turn and Carnap’s vision of replacing metaphysics with philosophy of science. Further chapters review advances in logic and the philosophy of mathematics that laid the foundation for a rigorous, scientific study of language, meaning, and information. Other chapters discuss W. V. O. Quine, David K. Lewis, Saul Kripke, the Frege–Russell analysis of quantification, Russell’s attempt to eliminate sets with his “no class theory,” and the Quine–Carnap dispute over meaning and ontology. The book then turns to topics at the frontier of philosophy of language. The final chapters, combining philosophy of language and law, advance a sophisticated originalist theory of interpretation and apply it to U.S. constitutional rulings about due process.
David Bostock
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199651443
- eISBN:
- 9780191741197
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199651443.003.0002
- Subject:
- Philosophy, History of Philosophy, Metaphysics/Epistemology
The chapter gives some background on the mathematical achievements of Cantor and Dedekind, and describes how Russell aimed to show that all mathematics can be derived from a purely logical starting ...
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The chapter gives some background on the mathematical achievements of Cantor and Dedekind, and describes how Russell aimed to show that all mathematics can be derived from a purely logical starting point (logicism). But it also describes Russell’s discovery of what he always called ‘the contradiction’, which we call ‘Russell’s paradox’, which presented a serious obstacle to that derivation. It gives Russell’s first reactions, namely the early theory of types sketched in 1903, the ‘zigzag’ theory, and the theory of limitation of size. The last is compared with the set theory of Zermelo-Fraenkel that is common nowadays.Less
The chapter gives some background on the mathematical achievements of Cantor and Dedekind, and describes how Russell aimed to show that all mathematics can be derived from a purely logical starting point (logicism). But it also describes Russell’s discovery of what he always called ‘the contradiction’, which we call ‘Russell’s paradox’, which presented a serious obstacle to that derivation. It gives Russell’s first reactions, namely the early theory of types sketched in 1903, the ‘zigzag’ theory, and the theory of limitation of size. The last is compared with the set theory of Zermelo-Fraenkel that is common nowadays.
Alan Weir
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199541492
- eISBN:
- 9780191594915
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199541492.003.0005
- Subject:
- Philosophy, Philosophy of Language, Logic/Philosophy of Mathematics
Relativism is distinguished from pluralism; mathematical theses which are true in some systems but not others are held to express different propositions which different truth values in each. ...
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Relativism is distinguished from pluralism; mathematical theses which are true in some systems but not others are held to express different propositions which different truth values in each. Comparisons are drawn with if-thenism, postulationism, and deductivism. A number of objections are tackled: that proofs need not be formal, that mathematicians believe theses without having proofs, that axiom systems can be inadequate and incomplete, that any consistent sentence counts as a mathematical truth, since it is provable from some system. In response, further comparisons are drawn with neo-logicism, and the role of abstraction principles such as Hume's principle discussed. The superiority of neo-formalism is urged, in the treatment for example of the ‘Julius Caesar problem’ and the problem of pairwise inconsistent abstraction principles. Finally Gödel's incompleteness theorem and the problem of true, but unprovable, Gödel sentences, is raised. A quick response by the neo-formalist is rejected, and the difficulty set aside until the last chapters.Less
Relativism is distinguished from pluralism; mathematical theses which are true in some systems but not others are held to express different propositions which different truth values in each. Comparisons are drawn with if-thenism, postulationism, and deductivism. A number of objections are tackled: that proofs need not be formal, that mathematicians believe theses without having proofs, that axiom systems can be inadequate and incomplete, that any consistent sentence counts as a mathematical truth, since it is provable from some system. In response, further comparisons are drawn with neo-logicism, and the role of abstraction principles such as Hume's principle discussed. The superiority of neo-formalism is urged, in the treatment for example of the ‘Julius Caesar problem’ and the problem of pairwise inconsistent abstraction principles. Finally Gödel's incompleteness theorem and the problem of true, but unprovable, Gödel sentences, is raised. A quick response by the neo-formalist is rejected, and the difficulty set aside until the last chapters.
Peter Hylton
- Published in print:
- 2005
- Published Online:
- May 2010
- ISBN:
- 9780199286355
- eISBN:
- 9780191713309
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199286355.003.0004
- Subject:
- Philosophy, History of Philosophy, Logic/Philosophy of Mathematics
This chapter articulates a major theme in Russell's thought: his conception of logic and of the philosophy of logic. It begins by raising the question of the philosophical significance that logicism, ...
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This chapter articulates a major theme in Russell's thought: his conception of logic and of the philosophy of logic. It begins by raising the question of the philosophical significance that logicism, the reduction of mathematics to logic, had for Russell when he first developed that doctrine. The answer is that it was part of a complex argument against Kant and post-Kantian Idealism. For this argument to work, logic must be thought of as made up of absolute and unconditioned truths. A certain conception of logic is thus implicit in the philosophical use that Russell makes of logicism. The chapter articulates this conception and contrasts it with a widely held modern conception according to which the central notion is truth in an interpretation, rather than truth tout court; the notion of an interpretation is alien to Russell's thought. It is argued that given his general conception of logic, it is natural, perhaps inevitable, that logic will be higher-order logic, equivalent to set theory. Russell's use of logicism, however, is cast in doubt by the need to accommodate the paradox that bears his name. The theory of types undermines his conception of logic as consisting of universal and unconditioned truths. The infinitude of objects can no longer be proved, but is taken as an explicit assumption when needed; this threatens the idea that it is indeed mathematics which is being reduced to logic. The magnificent intellectual achievement of Principia Mathematica is thus, cut off from the philosophical motivations that lay behind Russell's initial formulation of logicism.Less
This chapter articulates a major theme in Russell's thought: his conception of logic and of the philosophy of logic. It begins by raising the question of the philosophical significance that logicism, the reduction of mathematics to logic, had for Russell when he first developed that doctrine. The answer is that it was part of a complex argument against Kant and post-Kantian Idealism. For this argument to work, logic must be thought of as made up of absolute and unconditioned truths. A certain conception of logic is thus implicit in the philosophical use that Russell makes of logicism. The chapter articulates this conception and contrasts it with a widely held modern conception according to which the central notion is truth in an interpretation, rather than truth tout court; the notion of an interpretation is alien to Russell's thought. It is argued that given his general conception of logic, it is natural, perhaps inevitable, that logic will be higher-order logic, equivalent to set theory. Russell's use of logicism, however, is cast in doubt by the need to accommodate the paradox that bears his name. The theory of types undermines his conception of logic as consisting of universal and unconditioned truths. The infinitude of objects can no longer be proved, but is taken as an explicit assumption when needed; this threatens the idea that it is indeed mathematics which is being reduced to logic. The magnificent intellectual achievement of Principia Mathematica is thus, cut off from the philosophical motivations that lay behind Russell's initial formulation of logicism.
Stephen Yablo
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199266487
- eISBN:
- 9780191594274
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199266487.003.0009
- Subject:
- Philosophy, Metaphysics/Epistemology, Logic/Philosophy of Mathematics
Numbers have many puzzling features. Their properties are mostly essential to them, but they exist in all possible worlds. Number theory seems a priori, yet it makes existence claims and existence ...
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Numbers have many puzzling features. Their properties are mostly essential to them, but they exist in all possible worlds. Number theory seems a priori, yet it makes existence claims and existence (setting aside the Cogito) is not supposed to be a priori knowable. If-thenism can perhaps explain the felt a priority, but it makes numerical truth relative where it seems absolute. A figuralist solution is proposed: ‘2 + 3 = 5’ seems necessary, a priori, and absolute because it has a logical truth as its assertive content. A rule that associates logical truths with each arithmetical truth is given, and also a rule that associates a logical truth (modulo concrete combinatorics) with each truth about hereditarily finite impure sets. The view that emerges takes something from Frege and something from Kant; one might call it Kantian logicism. The view is Kantian because it sees mathematics as arising out of our representations. Numbers and sets are ‘there’ because they are inscribed on the spectacles through which we see other things. It is logicist because the facts seen through our numerical spectacles are facts of first-order logic.Less
Numbers have many puzzling features. Their properties are mostly essential to them, but they exist in all possible worlds. Number theory seems a priori, yet it makes existence claims and existence (setting aside the Cogito) is not supposed to be a priori knowable. If-thenism can perhaps explain the felt a priority, but it makes numerical truth relative where it seems absolute. A figuralist solution is proposed: ‘2 + 3 = 5’ seems necessary, a priori, and absolute because it has a logical truth as its assertive content. A rule that associates logical truths with each arithmetical truth is given, and also a rule that associates a logical truth (modulo concrete combinatorics) with each truth about hereditarily finite impure sets. The view that emerges takes something from Frege and something from Kant; one might call it Kantian logicism. The view is Kantian because it sees mathematics as arising out of our representations. Numbers and sets are ‘there’ because they are inscribed on the spectacles through which we see other things. It is logicist because the facts seen through our numerical spectacles are facts of first-order logic.
Stewart Shapiro (ed.)
- Published in print:
- 2005
- Published Online:
- July 2005
- ISBN:
- 9780195148770
- eISBN:
- 9780199835560
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195148770.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This book provides comprehensive and accessible coverage of the disciplines of philosophy of mathematics and philosophy of logic. After an introduction, the book begins with a historical section, ...
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This book provides comprehensive and accessible coverage of the disciplines of philosophy of mathematics and philosophy of logic. After an introduction, the book begins with a historical section, consisting of a chapter on the modern period, Kant and his intellectual predecessors, a chapter on later empiricism, including Mill and logical positivism, and a chapter on Wittgenstein. The next section of the volume consists of seven chapters on the views that dominated the philosophy and foundations of mathematics in the early decades of the 20th century: logicism, formalism, and intuitionism. They approach their subjects from a variety of historical and philosophical perspectives. The next section of the volume deals with views that dominated in the later twentieth century and beyond: Quine and indispensability, naturalism, nominalism, and structuralism. The next chapter in the volume is a detailed and sympathetic treatment of a predicative approach to both the philosophy and the foundations of mathematics, which is followed by an extensive treatment of the application of mathematics to the sciences. The last six chapters focus on logical matters: two chapters are devoted to the central notion of logical consequence, one on model theory and the other on proof theory; two chapters deal with the so-called paradoxes of relevance, one pro and one contra; and the final two chapters concern second-order logic (or higher-order logic), again one pro and one contra.Less
This book provides comprehensive and accessible coverage of the disciplines of philosophy of mathematics and philosophy of logic. After an introduction, the book begins with a historical section, consisting of a chapter on the modern period, Kant and his intellectual predecessors, a chapter on later empiricism, including Mill and logical positivism, and a chapter on Wittgenstein. The next section of the volume consists of seven chapters on the views that dominated the philosophy and foundations of mathematics in the early decades of the 20th century: logicism, formalism, and intuitionism. They approach their subjects from a variety of historical and philosophical perspectives. The next section of the volume deals with views that dominated in the later twentieth century and beyond: Quine and indispensability, naturalism, nominalism, and structuralism. The next chapter in the volume is a detailed and sympathetic treatment of a predicative approach to both the philosophy and the foundations of mathematics, which is followed by an extensive treatment of the application of mathematics to the sciences. The last six chapters focus on logical matters: two chapters are devoted to the central notion of logical consequence, one on model theory and the other on proof theory; two chapters deal with the so-called paradoxes of relevance, one pro and one contra; and the final two chapters concern second-order logic (or higher-order logic), again one pro and one contra.
Patricia A. Blanchette
- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199891610
- eISBN:
- 9780199933211
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199891610.003.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter explains the role of conceptual analysis in Frege’s logicist project. It is explained that Frege employs conceptual analyses in order to reduce arithmetical truths to complexes of ...
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This chapter explains the role of conceptual analysis in Frege’s logicist project. It is explained that Frege employs conceptual analyses in order to reduce arithmetical truths to complexes of relatively simple components in order to facilitate the proof of those truths. The importance of the adequacy of the conceptual analyses is emphasized, and Frege’s view of that adequacy is explored in a preliminary way by looking carefully at a selection of Frege’s analyses in Begriffsschrift, Grundlagen, and Grundgesetze. Some of the difficulties surrounding Frege’s conception of analysis are explained, and it is pointed out that a clear conception of Frege’s understanding of analysis requires a clear understanding of his notion of thought, the subject of Chapter 2.Less
This chapter explains the role of conceptual analysis in Frege’s logicist project. It is explained that Frege employs conceptual analyses in order to reduce arithmetical truths to complexes of relatively simple components in order to facilitate the proof of those truths. The importance of the adequacy of the conceptual analyses is emphasized, and Frege’s view of that adequacy is explored in a preliminary way by looking carefully at a selection of Frege’s analyses in Begriffsschrift, Grundlagen, and Grundgesetze. Some of the difficulties surrounding Frege’s conception of analysis are explained, and it is pointed out that a clear conception of Frege’s understanding of analysis requires a clear understanding of his notion of thought, the subject of Chapter 2.
Stewart Shapiro
- Published in print:
- 2005
- Published Online:
- July 2005
- ISBN:
- 9780195148770
- eISBN:
- 9780199835560
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195148770.003.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter provides a broad overview of the philosophy of mathematics and the philosophy of logic. It gives brief coverage to the various issues and positions, such as platonism or realism, ...
More
This chapter provides a broad overview of the philosophy of mathematics and the philosophy of logic. It gives brief coverage to the various issues and positions, such as platonism or realism, varieties of nominalism, or anti-realism, logicism, intuitionism, empiricism, and structuralism.Less
This chapter provides a broad overview of the philosophy of mathematics and the philosophy of logic. It gives brief coverage to the various issues and positions, such as platonism or realism, varieties of nominalism, or anti-realism, logicism, intuitionism, empiricism, and structuralism.
William Demopoulos and Peter Clark
- Published in print:
- 2005
- Published Online:
- July 2005
- ISBN:
- 9780195148770
- eISBN:
- 9780199835560
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195148770.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our ...
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The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our critical exposition of the view proceeds by tracing its answers to three fundamental questions: (1) What is the basis for our knowledge of the infinity of the numbers? (2) How is arithmetic applicable to reality? (3) Why is reasoning by induction justified?Less
The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our critical exposition of the view proceeds by tracing its answers to three fundamental questions: (1) What is the basis for our knowledge of the infinity of the numbers? (2) How is arithmetic applicable to reality? (3) Why is reasoning by induction justified?
Alan Weir
- Published in print:
- 2005
- Published Online:
- July 2005
- ISBN:
- 9780195148770
- eISBN:
- 9780199835560
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195148770.003.0014
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Mathematics poses a difficult problem for methodological naturalists, those who embrace scientific method, and also for ontological naturalists who eschew non-physical entities such as Cartesian ...
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Mathematics poses a difficult problem for methodological naturalists, those who embrace scientific method, and also for ontological naturalists who eschew non-physical entities such as Cartesian souls. Mathematics seems both essential to science but also committed to abstract non-physical entities while methodologically it seems to have no place for experiment or empirical confirmation. The chapter critically reviews a number of responses naturalists have made including logicism, Quinean radical empiricism, and Penelope Maddy’s variant thereof and suggests some further problems both for ontological and for methodological naturalists.Less
Mathematics poses a difficult problem for methodological naturalists, those who embrace scientific method, and also for ontological naturalists who eschew non-physical entities such as Cartesian souls. Mathematics seems both essential to science but also committed to abstract non-physical entities while methodologically it seems to have no place for experiment or empirical confirmation. The chapter critically reviews a number of responses naturalists have made including logicism, Quinean radical empiricism, and Penelope Maddy’s variant thereof and suggests some further problems both for ontological and for methodological naturalists.
Bob Hale and Crispin Wright
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780198236399
- eISBN:
- 9780191597565
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198236395.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This volume collects together 15 papers in which Hale and Wright develop their neo‐Fregean approach to the philosophy of mathematics. Two key aspects of Frege's philosophy are drawn out and defended: ...
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This volume collects together 15 papers in which Hale and Wright develop their neo‐Fregean approach to the philosophy of mathematics. Two key aspects of Frege's philosophy are drawn out and defended: (1) the doctrine that mathematics is a body of knowledge about independently existing objects––Frege's platonism (2) the doctrine that this knowledge is a species of logical knowledge, broadly construed––Frege's logicism. In Wright and Hale's programme, a kind of contextual explanation––an Abstraction Principle––suffices to ground the fundamental concepts of a mathematical domain. A key result has become known as Frege's theorem: the basic axioms of arithmetic can be derived from a single ’abstraction principle’. Though Frege himself considers and rejects this principle as a foundation for arithmetic, Neo‐Fregeans take a more optimistic view. In the papers included in this volume, Wright and Hale combine elements drawn from Frege's own work with contemporary thinking to develop their distinctive approach to the philosophy of mathematics. The ground covered includes articles exploring the metaphysics, epistemology, and philosophy of language that forms the backdrop to the neo‐Fregean project; responses to critics of their programme; detailed exploration and defence of the case for neo‐Fregean logicism about arithmetic; and proposals for extending the neo‐Fregean programme to real analysis and set theory. A substantial introduction gives the framework for the neo‐Fregean programme and outlines how the papers fit together, while a postscript outlines a series of challenges for further research.Less
This volume collects together 15 papers in which Hale and Wright develop their neo‐Fregean approach to the philosophy of mathematics. Two key aspects of Frege's philosophy are drawn out and defended: (1) the doctrine that mathematics is a body of knowledge about independently existing objects––Frege's platonism (2) the doctrine that this knowledge is a species of logical knowledge, broadly construed––Frege's logicism. In Wright and Hale's programme, a kind of contextual explanation––an Abstraction Principle––suffices to ground the fundamental concepts of a mathematical domain. A key result has become known as Frege's theorem: the basic axioms of arithmetic can be derived from a single ’abstraction principle’. Though Frege himself considers and rejects this principle as a foundation for arithmetic, Neo‐Fregeans take a more optimistic view. In the papers included in this volume, Wright and Hale combine elements drawn from Frege's own work with contemporary thinking to develop their distinctive approach to the philosophy of mathematics. The ground covered includes articles exploring the metaphysics, epistemology, and philosophy of language that forms the backdrop to the neo‐Fregean project; responses to critics of their programme; detailed exploration and defence of the case for neo‐Fregean logicism about arithmetic; and proposals for extending the neo‐Fregean programme to real analysis and set theory. A substantial introduction gives the framework for the neo‐Fregean programme and outlines how the papers fit together, while a postscript outlines a series of challenges for further research.
