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.
José Ferreirós
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.003.0002
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter is a general introduction to the current trend of studies of mathematical practice, with particular emphasis on historical and philosophical work. It offers a preliminary explanation of ...
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This chapter is a general introduction to the current trend of studies of mathematical practice, with particular emphasis on historical and philosophical work. It offers a preliminary explanation of the notion of mathematical practice, first by considering the work of historians and philosophers on mathematical practices, from Archimedes and David Hilbert to Jens Høyrup, Penelope Maddy, Marcus Giaquinto, and Philip S. Kitcher. The chapter then characterizes the notion of mathematical practice by successively proposing several constraints. It argues that several different levels of practice and knowledge are coexistent and that their interrelationships are crucial to mathematical knowledge. It shows how the scheme of a web of interrelated practices—counting practices, measuring practices, technical practices, scientific practices—with their systematic links acting as constraint and guide, can be applied in the analysis of very different levels of mathematical activity.Less
This chapter is a general introduction to the current trend of studies of mathematical practice, with particular emphasis on historical and philosophical work. It offers a preliminary explanation of the notion of mathematical practice, first by considering the work of historians and philosophers on mathematical practices, from Archimedes and David Hilbert to Jens Høyrup, Penelope Maddy, Marcus Giaquinto, and Philip S. Kitcher. The chapter then characterizes the notion of mathematical practice by successively proposing several constraints. It argues that several different levels of practice and knowledge are coexistent and that their interrelationships are crucial to mathematical knowledge. It shows how the scheme of a web of interrelated practices—counting practices, measuring practices, technical practices, scientific practices—with their systematic links acting as constraint and guide, can be applied in the analysis of very different levels of mathematical activity.
José Ferreirós
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.003.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This book proposes a novel analysis of mathematical knowledge from a practice-oriented standpoint and within the context of the philosophy of mathematics. The approach it is advocating is a ...
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This book proposes a novel analysis of mathematical knowledge from a practice-oriented standpoint and within the context of the philosophy of mathematics. The approach it is advocating is a cognitive, pragmatist, historical one. It emphasizes a view of mathematics as knowledge produced by human agents, on the basis of their biological and cognitive abilities, the latter being mediated by culture. It also gives importance to the practical roots of mathematics—that is, its roots in everyday practices, technical practices, mathematical practices themselves, and scientific practices. Finally, the approach stresses the importance of analyzing mathematics' historical development, and of accepting the presence of hypothetical elements in advanced mathematics. The book's main thesis is that several different levels of knowledge and practice are coexistent, and that their links and interplay are crucial to mathematical knowledge. This chapter offers some remarks that may help readers locate the book's arguments within a general scheme.Less
This book proposes a novel analysis of mathematical knowledge from a practice-oriented standpoint and within the context of the philosophy of mathematics. The approach it is advocating is a cognitive, pragmatist, historical one. It emphasizes a view of mathematics as knowledge produced by human agents, on the basis of their biological and cognitive abilities, the latter being mediated by culture. It also gives importance to the practical roots of mathematics—that is, its roots in everyday practices, technical practices, mathematical practices themselves, and scientific practices. Finally, the approach stresses the importance of analyzing mathematics' historical development, and of accepting the presence of hypothetical elements in advanced mathematics. The book's main thesis is that several different levels of knowledge and practice are coexistent, and that their links and interplay are crucial to mathematical knowledge. This chapter offers some remarks that may help readers locate the book's arguments within a general scheme.
Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
We can gain empirical knowledge of elementary arithmetic and elementary geometry because the primitive core of these subjects consists of truths about manipulations of reality. Full arithmetic and ...
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We can gain empirical knowledge of elementary arithmetic and elementary geometry because the primitive core of these subjects consists of truths about manipulations of reality. Full arithmetic and geometry idealize these operations. Later mathematics attributes much more extensive powers to the ideal agent who performs mathematical operations.Less
We can gain empirical knowledge of elementary arithmetic and elementary geometry because the primitive core of these subjects consists of truths about manipulations of reality. Full arithmetic and geometry idealize these operations. Later mathematics attributes much more extensive powers to the ideal agent who performs mathematical operations.
Lance J. Rips
- Published in print:
- 2011
- Published Online:
- January 2011
- ISBN:
- 9780195183054
- eISBN:
- 9780199865109
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195183054.003.0003
- Subject:
- Psychology, Cognitive Psychology
This chapter canvasses proposed connections between minds and numbers that might make knowledge of mathematics possible. The goal is to determine whether any promising leads are available in ...
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This chapter canvasses proposed connections between minds and numbers that might make knowledge of mathematics possible. The goal is to determine whether any promising leads are available in accounting for people's ability to represent math objects. It argues that we have certain primitive concepts (e.g., CAUSE in the case of physical objects, UNIQUENESS in the case of mathematical ones) and certain primitive operations (instantiation, recursion, and other procedures specialized for concept combination) that allow us to form schemas or theories for both physical and mathematical domains. We may then posit that the best of these theories are true—that they correctly describe the nature of our world—and that the objects they describe are elements of that world. Such a schema-based approach has advantages over most current theories of mathematical knowledge.Less
This chapter canvasses proposed connections between minds and numbers that might make knowledge of mathematics possible. The goal is to determine whether any promising leads are available in accounting for people's ability to represent math objects. It argues that we have certain primitive concepts (e.g., CAUSE in the case of physical objects, UNIQUENESS in the case of mathematical ones) and certain primitive operations (instantiation, recursion, and other procedures specialized for concept combination) that allow us to form schemas or theories for both physical and mathematical domains. We may then posit that the best of these theories are true—that they correctly describe the nature of our world—and that the objects they describe are elements of that world. Such a schema-based approach has advantages over most current theories of mathematical knowledge.
Michael D. Resnik
- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250142
- eISBN:
- 9780191598296
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250142.003.0009
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
If, as I grant, mathematical objects are abstract entities existing outside of space and time, and if the idea of supernaturally grasping abstract entities is scientifically unacceptable, then we ...
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If, as I grant, mathematical objects are abstract entities existing outside of space and time, and if the idea of supernaturally grasping abstract entities is scientifically unacceptable, then we need to explain how we can attain mathematical knowledge using our ordinary faculties. I try to meet this challenge through a postulational account of the genesis of our mathematical knowledge, according to which our ancestors introduced mathematical objects by first positing geometric ideals and then postulating abstract mathematical entities. Since positing involves simply introducing a discourse about objects and affirming their existence, positing mathematical objects involves nothing more serious than writing fiction. For this reason, postulational approaches seem better suited for conventionalists; so in the second part of this chapter, I explain how positing in mathematics is different from positing in fiction, and how we can gain knowledge from the former. Finally, I try to make sense of the idea that mathematical postulates are about an independent mathematical reality and that we can refer to that reality through them, by giving an immanent and disquotational account of reference and contrasting it with a transcendent/causal account.Less
If, as I grant, mathematical objects are abstract entities existing outside of space and time, and if the idea of supernaturally grasping abstract entities is scientifically unacceptable, then we need to explain how we can attain mathematical knowledge using our ordinary faculties. I try to meet this challenge through a postulational account of the genesis of our mathematical knowledge, according to which our ancestors introduced mathematical objects by first positing geometric ideals and then postulating abstract mathematical entities. Since positing involves simply introducing a discourse about objects and affirming their existence, positing mathematical objects involves nothing more serious than writing fiction. For this reason, postulational approaches seem better suited for conventionalists; so in the second part of this chapter, I explain how positing in mathematics is different from positing in fiction, and how we can gain knowledge from the former. Finally, I try to make sense of the idea that mathematical postulates are about an independent mathematical reality and that we can refer to that reality through them, by giving an immanent and disquotational account of reference and contrasting it with a transcendent/causal account.
Philip Kitcher
- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780199241279
- eISBN:
- 9780191597107
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199241279.003.0004
- Subject:
- Philosophy, Metaphysics/Epistemology
Philip Kitcher defends his rejection of the thesis that mathematical knowledge is possible if such knowledge is a priori against a number of objections. His main line of argument is to suggest a ...
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Philip Kitcher defends his rejection of the thesis that mathematical knowledge is possible if such knowledge is a priori against a number of objections. His main line of argument is to suggest a property that follows deductively from apriority, and then to argue that propositions cannot possess such a property.Less
Philip Kitcher defends his rejection of the thesis that mathematical knowledge is possible if such knowledge is a priori against a number of objections. His main line of argument is to suggest a property that follows deductively from apriority, and then to argue that propositions cannot possess such a property.
José Ferreirós
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.003.0009
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new ...
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This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new hypotheses are embedded in the web of mathematical practices, they become systematically linked with previous strata of mathematical knowledge, and this forces upon us agents (for example, research mathematicians or students of math) certain results, be they principles or conclusions. The chapter first considers a simple case that illustrates objective features in the introduction of basic mathematical hypotheses. It then discusses Georg Cantor's “purely arithmetical” proofs of his set-theoretic results, along with the notion of arbitrary set in relation to the Axiom of Choice that has strong roots in the theory of real numbers. It also explores Cantor's ordinal numbers and the Continuum Hypothesis.Less
This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new hypotheses are embedded in the web of mathematical practices, they become systematically linked with previous strata of mathematical knowledge, and this forces upon us agents (for example, research mathematicians or students of math) certain results, be they principles or conclusions. The chapter first considers a simple case that illustrates objective features in the introduction of basic mathematical hypotheses. It then discusses Georg Cantor's “purely arithmetical” proofs of his set-theoretic results, along with the notion of arbitrary set in relation to the Axiom of Choice that has strong roots in the theory of real numbers. It also explores Cantor's ordinal numbers and the Continuum Hypothesis.
Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
If we are to obtain a priori mathematical knowledge by following proofs, then we have to be able to have a priori knowledge of the axioms. This chapter (along with Chapter 4) examines the major ...
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If we are to obtain a priori mathematical knowledge by following proofs, then we have to be able to have a priori knowledge of the axioms. This chapter (along with Chapter 4) examines the major accounts of how such knowledge might be gained. It is argued that all these accounts fail.Less
If we are to obtain a priori mathematical knowledge by following proofs, then we have to be able to have a priori knowledge of the axioms. This chapter (along with Chapter 4) examines the major accounts of how such knowledge might be gained. It is argued that all these accounts fail.
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.
Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.003.0006
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
The next stage is to set up the main question for the rest of the book: How is empirical mathematical knowledge possible? The outline of an answer is given: relatively simple experiences provide ...
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The next stage is to set up the main question for the rest of the book: How is empirical mathematical knowledge possible? The outline of an answer is given: relatively simple experiences provide knowledge of elementary mathematics; the historical process extends the basic mathematical knowledge in extraordinary ways.Less
The next stage is to set up the main question for the rest of the book: How is empirical mathematical knowledge possible? The outline of an answer is given: relatively simple experiences provide knowledge of elementary mathematics; the historical process extends the basic mathematical knowledge in extraordinary ways.
C. A. J. Coady
- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198235514
- eISBN:
- 9780191597220
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198235518.003.0014
- Subject:
- Philosophy, Metaphysics/Epistemology
There is a common view that mathematical truths can be known only a priori, by intuition or grasping a proof, but the possibility of transmitting mathematical knowledge by testimony cuts against this ...
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There is a common view that mathematical truths can be known only a priori, by intuition or grasping a proof, but the possibility of transmitting mathematical knowledge by testimony cuts against this view. This chapter discusses the tension between the common view and the realities of testimonial knowledge. It examines the attempts to show that mathematical knowledge cannot be conveyed by telling, and argues that they are unsuccessful.Less
There is a common view that mathematical truths can be known only a priori, by intuition or grasping a proof, but the possibility of transmitting mathematical knowledge by testimony cuts against this view. This chapter discusses the tension between the common view and the realities of testimonial knowledge. It examines the attempts to show that mathematical knowledge cannot be conveyed by telling, and argues that they are unsuccessful.
José Ferreirós
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.003.0004
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter presents a thesis about what it regards as a crucial complementarity of elements in mathematical practice and in the configuration of mathematical knowledge: both conceptual and ...
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This chapter presents a thesis about what it regards as a crucial complementarity of elements in mathematical practice and in the configuration of mathematical knowledge: both conceptual and formalistic ingredients enter the game, both necessary, and neither reducible to the other. The approach advocates a rather natural way in which semantic considerations emerge from the interplay between agents and frameworks, in the context of mathematical knowledge and practices. As a result, the difficulties involved in postulating abstract semantic entities are avoided. The chapter first considers form and content in mathematics before discussing formal systems and intended models used in mathematics. It then explains the proposed approach, which emphasizes the coexistence of a multiplicity of practices and the centrality of their interplay, and concludes by taking a look at the case of complex numbers in order to illustrate the idea of the complementarity of symbols and thought.Less
This chapter presents a thesis about what it regards as a crucial complementarity of elements in mathematical practice and in the configuration of mathematical knowledge: both conceptual and formalistic ingredients enter the game, both necessary, and neither reducible to the other. The approach advocates a rather natural way in which semantic considerations emerge from the interplay between agents and frameworks, in the context of mathematical knowledge and practices. As a result, the difficulties involved in postulating abstract semantic entities are avoided. The chapter first considers form and content in mathematics before discussing formal systems and intended models used in mathematics. It then explains the proposed approach, which emphasizes the coexistence of a multiplicity of practices and the centrality of their interplay, and concludes by taking a look at the case of complex numbers in order to illustrate the idea of the complementarity of symbols and thought.
Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.003.0010
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Describes some types of inference and principles of theory choice that are involved in the growth of mathematics.
Describes some types of inference and principles of theory choice that are involved in the growth of mathematics.
Philip Kitcher
- Published in print:
- 1985
- Published Online:
- November 2003
- ISBN:
- 9780195035414
- eISBN:
- 9780199833368
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195035410.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Chapters 7–9 offer a general account of the growth of mathematics. Introduce the notion of a mathematical practice, a multidimensional entity consisting of a language, accepted statements, accepted ...
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Chapters 7–9 offer a general account of the growth of mathematics. Introduce the notion of a mathematical practice, a multidimensional entity consisting of a language, accepted statements, accepted questions, accepted means of inference, and methodological maxims. Mathematics grows by modifying one or more components in response to the problems posed by others. So new language, language that is not initially well understood, may be introduced in order to answer questions taken to be important but resisting solution by available methods; in consequence, there may be a new task of clarifying the language or taming the methods that the extended language allows. The chapters attempt to show how such processes have occurred in the history of mathematics, and how they link the rich state of present mathematics to the crude beginnings of the subject. In Chapter 7, in particular, Kitcher compares mathematical change with scientific change, attempting to show that the growth of mathematical knowledge is far more similar to the growth of scientific knowledge than is usually appreciated.Less
Chapters 7–9 offer a general account of the growth of mathematics. Introduce the notion of a mathematical practice, a multidimensional entity consisting of a language, accepted statements, accepted questions, accepted means of inference, and methodological maxims. Mathematics grows by modifying one or more components in response to the problems posed by others. So new language, language that is not initially well understood, may be introduced in order to answer questions taken to be important but resisting solution by available methods; in consequence, there may be a new task of clarifying the language or taming the methods that the extended language allows. The chapters attempt to show how such processes have occurred in the history of mathematics, and how they link the rich state of present mathematics to the crude beginnings of the subject. In Chapter 7, in particular, Kitcher compares mathematical change with scientific change, attempting to show that the growth of mathematical knowledge is far more similar to the growth of scientific knowledge than is usually appreciated.
Michael D. Resnik
- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250142
- eISBN:
- 9780191598296
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250142.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
One of the strongest motivations for being an anti‐realist with regard to mathematics is the difficulty in formulating a plausible realist epistemology, given that there seems to be a lack of ties ...
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One of the strongest motivations for being an anti‐realist with regard to mathematics is the difficulty in formulating a plausible realist epistemology, given that there seems to be a lack of ties between the mathematical apparatus and observation. In this chapter, I discuss a few puzzles that the mathematical realist has to solve in order to formulate an acceptable epistemology, and I hint at the direction in which one might hope to find the solution to these puzzles. One of the puzzles, that was first clearly formulated by Paul Benacerraf, is that since mathematical objects are supposed to be causally inert entities existing outside of space and time, it is hard to see how we can ever get in contact with them, i.e. how we can ever gain knowledge of them or refer to them. A second puzzle concerns the incompleteness of mathematical entities that has it, that we have no conceivable evidence and so no answers to questions of whether the objects that one mathematical theory discusses are identical to those that another theory treats. Realists can reject the presuppositions of both these puzzles. The first presupposes that there must be some sort of direct or indirect interaction between us and the objects of our knowledge. The second assumes that there is always a fact of the matter as to whether the objects of one theory are the same or distinct, as those of another. In later chapters, I will try to undermine both of these presuppositions.Less
One of the strongest motivations for being an anti‐realist with regard to mathematics is the difficulty in formulating a plausible realist epistemology, given that there seems to be a lack of ties between the mathematical apparatus and observation. In this chapter, I discuss a few puzzles that the mathematical realist has to solve in order to formulate an acceptable epistemology, and I hint at the direction in which one might hope to find the solution to these puzzles. One of the puzzles, that was first clearly formulated by Paul Benacerraf, is that since mathematical objects are supposed to be causally inert entities existing outside of space and time, it is hard to see how we can ever get in contact with them, i.e. how we can ever gain knowledge of them or refer to them. A second puzzle concerns the incompleteness of mathematical entities that has it, that we have no conceivable evidence and so no answers to questions of whether the objects that one mathematical theory discusses are identical to those that another theory treats. Realists can reject the presuppositions of both these puzzles. The first presupposes that there must be some sort of direct or indirect interaction between us and the objects of our knowledge. The second assumes that there is always a fact of the matter as to whether the objects of one theory are the same or distinct, as those of another. In later chapters, I will try to undermine both of these presuppositions.
John P. Burgess and Gideon Rosen
- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250128
- eISBN:
- 9780191597138
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250126.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Numbers and other mathematical objects are exceptional in having no locations in space and time and no causes or effects in the physical world. This makes it difficult to account for the possibility ...
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Numbers and other mathematical objects are exceptional in having no locations in space and time and no causes or effects in the physical world. This makes it difficult to account for the possibility of mathematical knowledge, leading many philosophers to embrace nominalism, the doctrine that there are no abstract entities. It has also led some of them to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects, eliminating so‐called ontological commitment to numbers, sets, and the like. These projects differ considerably in the apparatus they employ, and the spirit in which they are put forward. Some employ synthetic geometry, others modal logic. Some are put forward as revolutionary replacements for existing mathematics and science, others hermeneutic hypotheses about what they have meant all along. We attempt to cut through technicalities that have obscured previous discussions of these projects, and to present concise accounts with minimal prerequisites of a dozen strategies for nominalistic interpretation of mathematics. We also examine critically the aims and claims of such interpretations, suggesting that what they really achieve is something quite different from what the authors of such projects usually assume.Less
Numbers and other mathematical objects are exceptional in having no locations in space and time and no causes or effects in the physical world. This makes it difficult to account for the possibility of mathematical knowledge, leading many philosophers to embrace nominalism, the doctrine that there are no abstract entities. It has also led some of them to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects, eliminating so‐called ontological commitment to numbers, sets, and the like. These projects differ considerably in the apparatus they employ, and the spirit in which they are put forward. Some employ synthetic geometry, others modal logic. Some are put forward as revolutionary replacements for existing mathematics and science, others hermeneutic hypotheses about what they have meant all along. We attempt to cut through technicalities that have obscured previous discussions of these projects, and to present concise accounts with minimal prerequisites of a dozen strategies for nominalistic interpretation of mathematics. We also examine critically the aims and claims of such interpretations, suggesting that what they really achieve is something quite different from what the authors of such projects usually assume.
Colin McGinn
- Published in print:
- 2002
- Published Online:
- November 2003
- ISBN:
- 9780199251582
- eISBN:
- 9780191598012
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199251584.003.0003
- Subject:
- Philosophy, Metaphysics/Epistemology
McGinn defends a causal criterion for distinguishing a priori from a posteriori knowledge. In the case of a posteriori knowledge, the subject matter of a knower's ground for believing a proposition ...
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McGinn defends a causal criterion for distinguishing a priori from a posteriori knowledge. In the case of a posteriori knowledge, the subject matter of a knower's ground for believing a proposition is the cause of that knower's coming to believe that proposition. In the case of a priori knowledge, it is not the case that the subject matter of the knower's ground for believing a proposition is the cause of that knower's coming to believe that proposition. In this essay's first section, McGinn argues that a causal condition for knowledge is no part of an analysis of knowledge tout court, but merely part of a naturalized account of a posteriori knowledge. McGinn then argues for the extensional adequacy of his account of a priori knowledge for the cases of mathematical, logical, and analytic knowledge. He concludes by considering the connections between the a priori and the a posteriori, on the one hand, and the modal notions of necessity and contingency, on the other.Less
McGinn defends a causal criterion for distinguishing a priori from a posteriori knowledge. In the case of a posteriori knowledge, the subject matter of a knower's ground for believing a proposition is the cause of that knower's coming to believe that proposition. In the case of a priori knowledge, it is not the case that the subject matter of the knower's ground for believing a proposition is the cause of that knower's coming to believe that proposition. In this essay's first section, McGinn argues that a causal condition for knowledge is no part of an analysis of knowledge tout court, but merely part of a naturalized account of a posteriori knowledge. McGinn then argues for the extensional adequacy of his account of a priori knowledge for the cases of mathematical, logical, and analytic knowledge. He concludes by considering the connections between the a priori and the a posteriori, on the one hand, and the modal notions of necessity and contingency, on the other.
Charles S. Chihara
- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239758
- eISBN:
- 9780191597190
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239750.003.0012
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Takes up Field's version of Logicism—a position that he calls ‘deflationism’. Unlike traditional Logicists, Field does not analyse mathematical propositions into purely logical ones, but he does ...
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Takes up Field's version of Logicism—a position that he calls ‘deflationism’. Unlike traditional Logicists, Field does not analyse mathematical propositions into purely logical ones, but he does analyse mathematical knowledge into logical knowledge. Several objections are raised to deflationism, revolving around Field's contention that mathematics consists mostly of falsehoods. Contends that, although mathematics, literally and platonically construed, is not true, it does convey genuine (true) information.Less
Takes up Field's version of Logicism—a position that he calls ‘deflationism’. Unlike traditional Logicists, Field does not analyse mathematical propositions into purely logical ones, but he does analyse mathematical knowledge into logical knowledge. Several objections are raised to deflationism, revolving around Field's contention that mathematics consists mostly of falsehoods. Contends that, although mathematics, literally and platonically construed, is not true, it does convey genuine (true) information.
Michael D. Resnik
- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250142
- eISBN:
- 9780191598296
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250142.003.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Mathematics has often been described as the ‘Queen of Sciences’, yet philosophical problems arise as soon as one tries to define its subject matter. Anti‐realism concerning mathematical objects has ...
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Mathematics has often been described as the ‘Queen of Sciences’, yet philosophical problems arise as soon as one tries to define its subject matter. Anti‐realism concerning mathematical objects has proven to be problematic, but, on the other hand, realism gives rise to well‐known epistemological problems. In this book, I offer a type of mathematical realism according to which mathematical objects exist independently of our constructions, and mathematical truths are obtained independently of our beliefs. Central to my account of mathematical objects is a sort of structuralism, according to which mathematical objects are featureless, abstract positions in structures. On the epistemological level, I reject the apriority of mathematical knowledge, and claim that our justification for believing in the truth of mathematics is derived from pragmatic and holistic considerations concerning science.Less
Mathematics has often been described as the ‘Queen of Sciences’, yet philosophical problems arise as soon as one tries to define its subject matter. Anti‐realism concerning mathematical objects has proven to be problematic, but, on the other hand, realism gives rise to well‐known epistemological problems. In this book, I offer a type of mathematical realism according to which mathematical objects exist independently of our constructions, and mathematical truths are obtained independently of our beliefs. Central to my account of mathematical objects is a sort of structuralism, according to which mathematical objects are featureless, abstract positions in structures. On the epistemological level, I reject the apriority of mathematical knowledge, and claim that our justification for believing in the truth of mathematics is derived from pragmatic and holistic considerations concerning science.
