JC Beall, Greg Restall
- Published in print:
- 2005
- Published Online:
- October 2011
- ISBN:
- 9780199288403
- eISBN:
- 9780191700491
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199288403.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of ...
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Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline of philosophy. This book presents and defends what it calls logical pluralism, arguing that the notion of logical consequence does not pin down one deductive consequence relation; it allows for many of them. In particular, the book argues that broadly classical, intuitionistic, and relevant accounts of deductive logic are genuine logical consequence relations; we should not search for one true logic, since there are many. The book's conclusions have profound implications for many linguists as well as for philosophers.
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Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline of philosophy. This book presents and defends what it calls logical pluralism, arguing that the notion of logical consequence does not pin down one deductive consequence relation; it allows for many of them. In particular, the book argues that broadly classical, intuitionistic, and relevant accounts of deductive logic are genuine logical consequence relations; we should not search for one true logic, since there are many. The book's conclusions have profound implications for many linguists as well as for philosophers.
Mary Leng
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199280797
- eISBN:
- 9780191723452
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199280797.001.0001
- Subject:
- Philosophy, Metaphysics/Epistemology, Logic/Philosophy of Mathematics
This book offers a defence of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to ...
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This book offers a defence of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focused on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. This book, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized.
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This book offers a defence of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focused on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. This book, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized.
Michael D. Resnik
- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250142
- eISBN:
- 9780191598296
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250142.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Mathematics is regarded as our most developed science, and yet philosophical troubles surface as soon as we inquire about its subject matter partly because mathematics itself says ...
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Mathematics is regarded as our most developed science, and yet philosophical troubles surface as soon as we inquire about its subject matter partly because mathematics itself says nothing about the metaphysical nature of its objects. Taking mathematics at face value seems to favour the Platonist view according to which mathematics concerns causally inert objects existing outside space‐time, but this view seems to preclude any account of how we acquire mathematical knowledge without using some mysterious intellectual intuition. In this book, I defend a version of mathematical realism, motivated by the indispensability of mathematics in science, according to which (1) mathematical objects exist independently of us and our constructions, (2) much of contemporary mathematics is true, and (3) mathematical truths obtain independently of our beliefs, theories, and proofs.
The ontological component of my realism is a form of structuralism according to which mathematical objects are featureless, abstract positions in structures, or patterns, and like geometric points, their identities are fixed only through their relationships to each other. Structuralism is also part of my epistemology in that material objects ‘fit’ simple patterns, and in doing so, they ‘fill’ the positions of simple mathematical structures. We may perceive the arrangements of objects but we cannot perceive their positions i.e. the abstract, non‐spatiotemporal mathematical objects, and the problem then consists in explaining how we can form beliefs about them.
Answering this question introduces a central notion of my epistemology, that of a posit: by representing and designing patterned objects our ancestors posited geometric objects as sui generis and started describing them by describing the patterns in which they are positions. Since positing mathematical objects, like positing new scientific entities, is an activity similar to making up a story, one might wonder how such an activity can lead to mathematical knowledge and truth, but I believe that our ancestors were justified in introducing mathematical objects and we are justified in retaining them, by pragmatic and global considerations: mathematics has proved immensely fruitful for science, technology, and practical life, and doing without it is now virtually impossible.
This account of justification introduces a further problem: if our justification for believing in mathematical truths is global and pragmatic, then it might turn out that one is not justified in accepting a mathematical claim unless it is accepted by science, and this is clearly at odds with the practice of mathematics where we hardly ever invoke such global considerations in order to justify a mathematical claim. In mathematics, we usually employ a local conception of evidence made up mainly of a priori proofs. However, arguing from the perspective of a Quinean epistemic holism, I claim that this feature of the practice should not make us conclude that mathematics is an a priori science, disconnected evidentially from both observation and natural science, for observation is relevant to mathematics, and technological and scientific success forms a vital part of our justification for believing in the truth of mathematics.
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Mathematics is regarded as our most developed science, and yet philosophical troubles surface as soon as we inquire about its subject matter partly because mathematics itself says nothing about the metaphysical nature of its objects. Taking mathematics at face value seems to favour the Platonist view according to which mathematics concerns causally inert objects existing outside space‐time, but this view seems to preclude any account of how we acquire mathematical knowledge without using some mysterious intellectual intuition. In this book, I defend a version of mathematical realism, motivated by the indispensability of mathematics in science, according to which (1) mathematical objects exist independently of us and our constructions, (2) much of contemporary mathematics is true, and (3) mathematical truths obtain independently of our beliefs, theories, and proofs.
The ontological component of my realism is a form of structuralism according to which mathematical objects are featureless, abstract positions in structures, or patterns, and like geometric points, their identities are fixed only through their relationships to each other. Structuralism is also part of my epistemology in that material objects ‘fit’ simple patterns, and in doing so, they ‘fill’ the positions of simple mathematical structures. We may perceive the arrangements of objects but we cannot perceive their positions i.e. the abstract, non‐spatiotemporal mathematical objects, and the problem then consists in explaining how we can form beliefs about them.
Answering this question introduces a central notion of my epistemology, that of a posit: by representing and designing patterned objects our ancestors posited geometric objects as sui generis and started describing them by describing the patterns in which they are positions. Since positing mathematical objects, like positing new scientific entities, is an activity similar to making up a story, one might wonder how such an activity can lead to mathematical knowledge and truth, but I believe that our ancestors were justified in introducing mathematical objects and we are justified in retaining them, by pragmatic and global considerations: mathematics has proved immensely fruitful for science, technology, and practical life, and doing without it is now virtually impossible.
This account of justification introduces a further problem: if our justification for believing in mathematical truths is global and pragmatic, then it might turn out that one is not justified in accepting a mathematical claim unless it is accepted by science, and this is clearly at odds with the practice of mathematics where we hardly ever invoke such global considerations in order to justify a mathematical claim. In mathematics, we usually employ a local conception of evidence made up mainly of a priori proofs. However, arguing from the perspective of a Quinean epistemic holism, I claim that this feature of the practice should not make us conclude that mathematics is an a priori science, disconnected evidentially from both observation and natural science, for observation is relevant to mathematics, and technological and scientific success forms a vital part of our justification for believing in the truth of mathematics.
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 ...
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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.
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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.
Timothy Williamson
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199552078
- eISBN:
- 9780191752506
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199552078.001.0001
- Subject:
- Philosophy, Metaphysics/Epistemology, Logic/Philosophy of Mathematics
Are there such things as merely possible people, who would have lived if our ancestors had acted differently? Are there future people, who have not yet been conceived? The book argues for positive ...
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Are there such things as merely possible people, who would have lived if our ancestors had acted differently? Are there future people, who have not yet been conceived? The book argues for positive answers to these and similar questions about contingency and change, applying the technical resources of modal logic to provide structural cores for metaphysical theories. The search for a metaphysically neutral logic is rejected as futile. There is detailed historical discussion of how the metaphysical issues emerged in the twentieth-century development of quantified modal logic, through the work of such figures as Rudolf Carnap, Ruth Barcan Marcus, Arthur Prior, Saul Kripke, David Lewis, Robert Stalnaker, Alvin Plantinga, and Kit Fine. Higher-order modal logic is proposed as a new setting in which to resolve such metaphysical questions scientifically, by the construction of systematic logical theories embodying rival answers and their comparison by normal scientific standards, more specifically by abduction (inference to the best explanation). The book provides a rigorous introduction to the technical background—for example, concerning possible worlds semantics for modal logic and the interpretation of higher-order quantifers—needed to understand metaphysical questions in modal logic, as well as an extended argument for specific answers to them. A view is reached on which there is no contingency or change in what there is, only in how it is.Less
Are there such things as merely possible people, who would have lived if our ancestors had acted differently? Are there future people, who have not yet been conceived? The book argues for positive answers to these and similar questions about contingency and change, applying the technical resources of modal logic to provide structural cores for metaphysical theories. The search for a metaphysically neutral logic is rejected as futile. There is detailed historical discussion of how the metaphysical issues emerged in the twentieth-century development of quantified modal logic, through the work of such figures as Rudolf Carnap, Ruth Barcan Marcus, Arthur Prior, Saul Kripke, David Lewis, Robert Stalnaker, Alvin Plantinga, and Kit Fine. Higher-order modal logic is proposed as a new setting in which to resolve such metaphysical questions scientifically, by the construction of systematic logical theories embodying rival answers and their comparison by normal scientific standards, more specifically by abduction (inference to the best explanation). The book provides a rigorous introduction to the technical background—for example, concerning possible worlds semantics for modal logic and the interpretation of higher-order quantifers—needed to understand metaphysical questions in modal logic, as well as an extended argument for specific answers to them. A view is reached on which there is no contingency or change in what there is, only in how it is.
Bob Hale, Aviv Hoffmann (eds)
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199565818
- eISBN:
- 9780191722004
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199565818.001.0001
- Subject:
- Philosophy, Metaphysics/Epistemology, Logic/Philosophy of Mathematics
The philosophy of modality investigates necessity and possibility, and related notions — are they objective features of mind-independent reality? If so, are they irreducible, or can ...
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The philosophy of modality investigates necessity and possibility, and related notions — are they objective features of mind-independent reality? If so, are they irreducible, or can modal facts be explained in other terms? This book presents new work on modality by established leaders in the field and by up-and-coming philosophers. Between them, the chapters address fundamental questions concerning realism and anti-realism about modality, the nature and basis of facts about what is possible and what is necessary, the nature of modal knowledge, modal logic and its relations to necessary existence and to counterfactual reasoning. The general introduction locates the individual contributions in the wider context of the contemporary discussion of the metaphysics and epistemology of modality.
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The philosophy of modality investigates necessity and possibility, and related notions — are they objective features of mind-independent reality? If so, are they irreducible, or can modal facts be explained in other terms? This book presents new work on modality by established leaders in the field and by up-and-coming philosophers. Between them, the chapters address fundamental questions concerning realism and anti-realism about modality, the nature and basis of facts about what is possible and what is necessary, the nature of modal knowledge, modal logic and its relations to necessary existence and to counterfactual reasoning. The general introduction locates the individual contributions in the wider context of the contemporary discussion of the metaphysics and epistemology of modality.
Penelope Maddy
- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780198250753
- eISBN:
- 9780191597961
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250754.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
It is well known that certain natural statements of set theory, like Cantor's continuum hypothesis (CH), cannot be proved or disproved on the basis of the standard axioms ...
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It is well known that certain natural statements of set theory, like Cantor's continuum hypothesis (CH), cannot be proved or disproved on the basis of the standard axioms (Zermelo–Fraenkel with Choice or ZFC). Some philosophers take this to be the end of the story on these questions, but set theorists continue to look for answers by investigating candidates for new axioms. One way to understand this work, an approach pioneered by Gödel, is to embrace some brand of realism (sometimes called ‘Platonism’) about sets: there is an objective world of sets in which the CH is either true or false; ZFC does not completely describe this world; our job is to find new true axioms that will give a fuller description detailed enough to decide the CH. Recent versions of realism in the philosophy of mathematics have rested on Quine's indispensability arguments: the world posited by our best scientific theories includes mathematical entities. In this book, I argue that attention to the actual details of scientific methodology substantially undermines Quine's argument, leaving realism without its best support. As an alternative, I develop a naturalistic approach (drawing on other themes from Quine, Gödel, and Wittgenstein) that finds the justification for mathematical methods in mathematics rather than extra‐mathematical philosophy, and I apply this naturalism to the test case of a particular new axiom candidate (Gödel's Axiom of Constructibility).
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It is well known that certain natural statements of set theory, like Cantor's continuum hypothesis (CH), cannot be proved or disproved on the basis of the standard axioms (Zermelo–Fraenkel with Choice or ZFC). Some philosophers take this to be the end of the story on these questions, but set theorists continue to look for answers by investigating candidates for new axioms. One way to understand this work, an approach pioneered by Gödel, is to embrace some brand of realism (sometimes called ‘Platonism’) about sets: there is an objective world of sets in which the CH is either true or false; ZFC does not completely describe this world; our job is to find new true axioms that will give a fuller description detailed enough to decide the CH. Recent versions of realism in the philosophy of mathematics have rested on Quine's indispensability arguments: the world posited by our best scientific theories includes mathematical entities. In this book, I argue that attention to the actual details of scientific methodology substantially undermines Quine's argument, leaving realism without its best support. As an alternative, I develop a naturalistic approach (drawing on other themes from Quine, Gödel, and Wittgenstein) that finds the justification for mathematical methods in mathematics rather than extra‐mathematical philosophy, and I apply this naturalism to the test case of a particular new axiom candidate (Gödel's Axiom of Constructibility).
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 ...
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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.
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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.
Douglas Patterson (ed.)
- Published in print:
- 2008
- Published Online:
- September 2010
- ISBN:
- 9780199296309
- eISBN:
- 9780191712272
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296309.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This book shows the way to a proper understanding of the philosophical legacy of the great logician, mathematician, and philosopher Alfred Tarski (1902–983). The contributors are an ...
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This book shows the way to a proper understanding of the philosophical legacy of the great logician, mathematician, and philosopher Alfred Tarski (1902–983). The contributors are an international group of scholars, some expert in the historical background and context of Tarski's work, others specializing in aspects of his philosophical development, others more interested in understanding Tarski in the light of contemporary thought. The chapters can be seen as addressing Tarski's seminal treatment of four basic questions about logical consequence. (1) How are we to understand truth, one of the notions in terms of which logical consequence is explained? What is it that is preserved in valid inference, or that such inference allows us to discover new claims to have on the basis of old? (2) Among what kinds of things does the relation of logical consequence hold? (3) Given answers to the first two questions, what is involved in the consequence relationship itself? What is the preservation at work in ‘truth preservation’? (4) Finally, what do truth and consequence so construed have to do with meaning?
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This book shows the way to a proper understanding of the philosophical legacy of the great logician, mathematician, and philosopher Alfred Tarski (1902–983). The contributors are an international group of scholars, some expert in the historical background and context of Tarski's work, others specializing in aspects of his philosophical development, others more interested in understanding Tarski in the light of contemporary thought. The chapters can be seen as addressing Tarski's seminal treatment of four basic questions about logical consequence. (1) How are we to understand truth, one of the notions in terms of which logical consequence is explained? What is it that is preserved in valid inference, or that such inference allows us to discover new claims to have on the basis of old? (2) Among what kinds of things does the relation of logical consequence hold? (3) Given answers to the first two questions, what is involved in the consequence relationship itself? What is the preservation at work in ‘truth preservation’? (4) Finally, what do truth and consequence so construed have to do with meaning?
A. N. Prior
P. T. Geach, A. J. P. Kenny (eds)
- Published in print:
- 1971
- Published Online:
- October 2011
- ISBN:
- 9780198243540
- eISBN:
- 9780191680694
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198243540.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Philosophy of Language
This book is divided into two parts. The first concentrates on the logical properties of propositions, their relation to facts and sentences, and the parallel objects of commands and ...
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This book is divided into two parts. The first concentrates on the logical properties of propositions, their relation to facts and sentences, and the parallel objects of commands and questions. The second part examines theories of intentionality and discusses the relationship between different theories of naming and different accounts of belief.
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This book is divided into two parts. The first concentrates on the logical properties of propositions, their relation to facts and sentences, and the parallel objects of commands and questions. The second part examines theories of intentionality and discusses the relationship between different theories of naming and different accounts of belief.
Reasons as Defaults
Changes of Mind
Wittgenstein's Tractatus
Categories of Being
Frege’s Conception of Logic
Episteme, etc.
