Charles S. Chihara
- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239758
- eISBN:
- 9780191597190
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239750.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
A continuation of the study of mathematical existence begun in Ontology and the Vicious‐Circle Principle (published in 1973); in the present work, Quine's indispensability argument is rebutted by the ...
More
A continuation of the study of mathematical existence begun in Ontology and the Vicious‐Circle Principle (published in 1973); in the present work, Quine's indispensability argument is rebutted by the development of a new nominalistic version of mathematics (the Constructibility Theory) that is specified as an axiomatized theory formalized in a many‐sorted first‐order language. What is new in the present work is its abandonment of the predicative restrictions of the earlier work and its much greater attention to the applications of mathematics in science and everyday life. The book also contains detailed discussions of rival views (Mathematical Structuralism, Field's Instrumentalism, Burgess's Moderate Realism, Maddy's Set Theoretical Realism, and Kitcher's Ideal Agent account of mathematics), in which many comparisons with the Constructibility Theory are made.Less
A continuation of the study of mathematical existence begun in Ontology and the Vicious‐Circle Principle (published in 1973); in the present work, Quine's indispensability argument is rebutted by the development of a new nominalistic version of mathematics (the Constructibility Theory) that is specified as an axiomatized theory formalized in a many‐sorted first‐order language. What is new in the present work is its abandonment of the predicative restrictions of the earlier work and its much greater attention to the applications of mathematics in science and everyday life. The book also contains detailed discussions of rival views (Mathematical Structuralism, Field's Instrumentalism, Burgess's Moderate Realism, Maddy's Set Theoretical Realism, and Kitcher's Ideal Agent account of mathematics), in which many comparisons with the Constructibility Theory are made.
Hartry Field
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780199242894
- eISBN:
- 9780191597381
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199242895.001.0001
- Subject:
- Philosophy, General
This is a collection of papers, written over many years, with substantial postscripts tying them together and giving an updated perspective on them. The first five are on the notions of truth and ...
More
This is a collection of papers, written over many years, with substantial postscripts tying them together and giving an updated perspective on them. The first five are on the notions of truth and truth‐conditions, and their role in a theory of meaning and of the content of our mental states. The next five deal with what I call ‘factually defective discourse’—discourse that gives rise to issues about which, it is tempting to say that, there is no fact of the matter as to the right answer; one particular kind of factually defective discourse is called ‘indeterminacy’, and it gets the bulk of the attention. The final bunch of papers deal with issues about objectivity, closely related to issues about factual defectiveness; two deal with the question of whether the axioms of mathematics are as objective as is often assumed, and one deals with the question of whether our epistemological methods are as objective as they are usually assumed to be.Less
This is a collection of papers, written over many years, with substantial postscripts tying them together and giving an updated perspective on them. The first five are on the notions of truth and truth‐conditions, and their role in a theory of meaning and of the content of our mental states. The next five deal with what I call ‘factually defective discourse’—discourse that gives rise to issues about which, it is tempting to say that, there is no fact of the matter as to the right answer; one particular kind of factually defective discourse is called ‘indeterminacy’, and it gets the bulk of the attention. The final bunch of papers deal with issues about objectivity, closely related to issues about factual defectiveness; two deal with the question of whether the axioms of mathematics are as objective as is often assumed, and one deals with the question of whether our epistemological methods are as objective as they are usually assumed to be.
Mary Leng
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199280797
- eISBN:
- 9780191723452
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199280797.003.0003
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology
This chapter discusses Hartry Field's attempt to respond to the indispensability argument by showing how to dispense with mathematics in formulating our best scientific theories. It is pointed out ...
More
This chapter discusses Hartry Field's attempt to respond to the indispensability argument by showing how to dispense with mathematics in formulating our best scientific theories. It is pointed out that Field does not want to stop us from using mathematics in our scientific theorizing, but rather, that he wishes to explain the use of mathematics as a ‘theoretical juice extractor’, which allows us to draw out the consequences of our non‐mathematical assumptions. Objections to Field's programme are considered, including objections to the logical assumptions made by his account of applications, and objections to his claim that mathematics can always be dispensed with. While these objections are not conclusive, it is noted that mathematical assumptions may be valuable enough to remain present in even our best formulations of our scientific theories. Hence the book's project, to consider the case for anti‐platonism on the assumption that mathematics is indispensable to empirical science.Less
This chapter discusses Hartry Field's attempt to respond to the indispensability argument by showing how to dispense with mathematics in formulating our best scientific theories. It is pointed out that Field does not want to stop us from using mathematics in our scientific theorizing, but rather, that he wishes to explain the use of mathematics as a ‘theoretical juice extractor’, which allows us to draw out the consequences of our non‐mathematical assumptions. Objections to Field's programme are considered, including objections to the logical assumptions made by his account of applications, and objections to his claim that mathematics can always be dispensed with. While these objections are not conclusive, it is noted that mathematical assumptions may be valuable enough to remain present in even our best formulations of our scientific theories. Hence the book's project, to consider the case for anti‐platonism on the assumption that mathematics is indispensable to empirical science.
Mark Colyvan
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195137545
- eISBN:
- 9780199833139
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019513754X.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Hartry Field has attacked the indispensability argument by arguing that mathematics is in fact dispensable to science. He does this by showing a way to dispense with quantification over mathematical ...
More
Hartry Field has attacked the indispensability argument by arguing that mathematics is in fact dispensable to science. He does this by showing a way to dispense with quantification over mathematical entities in Newtonian gravitation theory. He, thus, advances a kind of fictionalism about mathematics. In this chapter, Field's program is outlined and the prospects for its success are evaluated. An important part of the task of assessing what is required of Field's program involves understanding how we should read the crucial term “indispensable” in Quine's argument.Less
Hartry Field has attacked the indispensability argument by arguing that mathematics is in fact dispensable to science. He does this by showing a way to dispense with quantification over mathematical entities in Newtonian gravitation theory. He, thus, advances a kind of fictionalism about mathematics. In this chapter, Field's program is outlined and the prospects for its success are evaluated. An important part of the task of assessing what is required of Field's program involves understanding how we should read the crucial term “indispensable” in Quine's argument.
Charles S. Chihara
- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239758
- eISBN:
- 9780191597190
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239750.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Focuses on Hartry Field's Instrumentalism. The ‘Conservation Theorems’, upon which Field bases so much of his form of Instrumentalism, are examined in detail, as is Field's attempt to ‘nominalize’ ...
More
Focuses on Hartry Field's Instrumentalism. The ‘Conservation Theorems’, upon which Field bases so much of his form of Instrumentalism, are examined in detail, as is Field's attempt to ‘nominalize’ physics. Doubts are raised about the adequacy of Field's views of mathematics and physics, and a detailed comparison with the Constructibility Theory is presented.Less
Focuses on Hartry Field's Instrumentalism. The ‘Conservation Theorems’, upon which Field bases so much of his form of Instrumentalism, are examined in detail, as is Field's attempt to ‘nominalize’ physics. Doubts are raised about the adequacy of Field's views of mathematics and physics, and a detailed comparison with the Constructibility Theory is presented.
Hartry Field
- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780199241279
- eISBN:
- 9780191597107
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199241279.003.0006
- Subject:
- Philosophy, Metaphysics/Epistemology
Hartry Field argues that meaning‐based approaches to explaining the apriority of certain propositions fail to succeed in their endeavour. He suggests that adopting what one calls a ‘non‐factualist’ ...
More
Hartry Field argues that meaning‐based approaches to explaining the apriority of certain propositions fail to succeed in their endeavour. He suggests that adopting what one calls a ‘non‐factualist’ view of justification itself removes the mystery of the apriority of such propositions, and sketches what such a view of justification involves.Less
Hartry Field argues that meaning‐based approaches to explaining the apriority of certain propositions fail to succeed in their endeavour. He suggests that adopting what one calls a ‘non‐factualist’ view of justification itself removes the mystery of the apriority of such propositions, and sketches what such a view of justification involves.
Bob Hale
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780198236399
- eISBN:
- 9780191597565
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198236395.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This paper argues that the epistemological challenge against a Platonist conception of mathematics can be met. The challenge is expounded in a general version and does not rely on the specific ...
More
This paper argues that the epistemological challenge against a Platonist conception of mathematics can be met. The challenge is expounded in a general version and does not rely on the specific adoption of a causal theory of knowledge, but rather it disputes the ability to explain our reliability of our beliefs concerning abstract objects. It is then argued that for a Platonist, the above challenge can be met by giving a satisfactory epistemology of necessary truths in general. That is to reduce the explanation of our reliability in mathematical beliefs to beliefs of necessary statements as such. In the following, the worry is raised that such a transition is illicit, as it might still be the case that albeit logical beliefs are necessary, our mathematical beliefs, even if they were true are never necessarily so.Less
This paper argues that the epistemological challenge against a Platonist conception of mathematics can be met. The challenge is expounded in a general version and does not rely on the specific adoption of a causal theory of knowledge, but rather it disputes the ability to explain our reliability of our beliefs concerning abstract objects. It is then argued that for a Platonist, the above challenge can be met by giving a satisfactory epistemology of necessary truths in general. That is to reduce the explanation of our reliability in mathematical beliefs to beliefs of necessary statements as such. In the following, the worry is raised that such a transition is illicit, as it might still be the case that albeit logical beliefs are necessary, our mathematical beliefs, even if they were true are never necessarily so.
Crispin Wright
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780198236399
- eISBN:
- 9780191597565
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198236395.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This paper starts by offering a brief reconstruction of the Neo‐Fregean approach as suggested in Frege's Conception of Numbers as Objects and distinguishes various challenges against the method of ...
More
This paper starts by offering a brief reconstruction of the Neo‐Fregean approach as suggested in Frege's Conception of Numbers as Objects and distinguishes various challenges against the method of Abstraction. It then focuses on one line of criticism––Rejectionism––which is endorsed by Field in his Review of the previously mentioned book. The thought is to grant that the method of abstraction provides singular terms, however questions its ability to produce true statements. Furthermore, Field draws an analogy between the stipulation of Hume's Principle, which commits one to the existence of numbers and the ontological argument, which commits one to the existence of God. It is then shown that this analogy is amiss and that there is no real point of affinity with the Fregean platonist's ontological strategy and the ontological arguments. A further objection concerning the tacit ontological commitments on the right hand side of Abstraction Principle is discussed. The paper concludes considering ‘the onus of proof’ – issue for Nominalism‐Platonism debates.Less
This paper starts by offering a brief reconstruction of the Neo‐Fregean approach as suggested in Frege's Conception of Numbers as Objects and distinguishes various challenges against the method of Abstraction. It then focuses on one line of criticism––Rejectionism––which is endorsed by Field in his Review of the previously mentioned book. The thought is to grant that the method of abstraction provides singular terms, however questions its ability to produce true statements. Furthermore, Field draws an analogy between the stipulation of Hume's Principle, which commits one to the existence of numbers and the ontological argument, which commits one to the existence of God. It is then shown that this analogy is amiss and that there is no real point of affinity with the Fregean platonist's ontological strategy and the ontological arguments. A further objection concerning the tacit ontological commitments on the right hand side of Abstraction Principle is discussed. The paper concludes considering ‘the onus of proof’ – issue for Nominalism‐Platonism debates.
Bob Hale and Crispin Wright
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780198236399
- eISBN:
- 9780191597565
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198236395.003.0006
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This paper discusses the idea, labelled as “the traditional connection” that implicit definitions aim to found a priori knowledge of logic and mathematics. In the first part, it discusses and rejects ...
More
This paper discusses the idea, labelled as “the traditional connection” that implicit definitions aim to found a priori knowledge of logic and mathematics. In the first part, it discusses and rejects a specific understanding of certain constraints (existence, uniqueness, possession problem, and explanation problem) on the theory of implicit definitions, as suggested by Horwich, on the basis of it being committing to some robust Platonist version of meaning facts. In contrast, it motivates further new constraints on the success of implicit definitions, such as arrogance, conservativeness, Evan's generality constraint, and harmony. Then, the standard view of implicit definitions for scientific terms, which appeals to so called Carnap Conditionals is discussed and an alternative model, i.e. the inverse Carnap Conditional is proffered. Lastly, this latter model is then applied to Hume's Principle and the conditionalized version of Hume's Principle as offered by Field is rejected. Furthermore, the problem of the ontological commitments of Hume's Principle and its status as a meaning––conferring successful stipulation are further discussed.Less
This paper discusses the idea, labelled as “the traditional connection” that implicit definitions aim to found a priori knowledge of logic and mathematics. In the first part, it discusses and rejects a specific understanding of certain constraints (existence, uniqueness, possession problem, and explanation problem) on the theory of implicit definitions, as suggested by Horwich, on the basis of it being committing to some robust Platonist version of meaning facts. In contrast, it motivates further new constraints on the success of implicit definitions, such as arrogance, conservativeness, Evan's generality constraint, and harmony. Then, the standard view of implicit definitions for scientific terms, which appeals to so called Carnap Conditionals is discussed and an alternative model, i.e. the inverse Carnap Conditional is proffered. Lastly, this latter model is then applied to Hume's Principle and the conditionalized version of Hume's Principle as offered by Field is rejected. Furthermore, the problem of the ontological commitments of Hume's Principle and its status as a meaning––conferring successful stipulation are further discussed.
Michael Dummett
- Published in print:
- 1996
- Published Online:
- November 2003
- ISBN:
- 9780198236214
- eISBN:
- 9780191597350
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198236212.003.0018
- Subject:
- Philosophy, Philosophy of Language
While it is relatively clear what the subject matter of empirical sciences is, puzzles persist about the proper subject matter of mathematics. The logicists took mathematics to be concerned solely ...
More
While it is relatively clear what the subject matter of empirical sciences is, puzzles persist about the proper subject matter of mathematics. The logicists took mathematics to be concerned solely with deductive arguments. Their programme attempted to combine three incompatible claims: that mathematics is a body of truths, that it is non‐empirical, and that it employs proofs obeying the rules of classical logic. By giving up the third contention, it becomes possible to salvage the logicist programme and to explain better mathematical practice, including the fact of its application in empirical disciplines.Less
While it is relatively clear what the subject matter of empirical sciences is, puzzles persist about the proper subject matter of mathematics. The logicists took mathematics to be concerned solely with deductive arguments. Their programme attempted to combine three incompatible claims: that mathematics is a body of truths, that it is non‐empirical, and that it employs proofs obeying the rules of classical logic. By giving up the third contention, it becomes possible to salvage the logicist programme and to explain better mathematical practice, including the fact of its application in empirical disciplines.
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 ...
More
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.
Micheal Slote
- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199790821
- eISBN:
- 9780199919185
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199790821.003.0003
- Subject:
- Philosophy, Moral Philosophy
This chapter homes in on Aristotle and recent Aristotelianism, which have mainly assumed that the virtues are unified, that in order to have one virtue one must have them all. What can be said about ...
More
This chapter homes in on Aristotle and recent Aristotelianism, which have mainly assumed that the virtues are unified, that in order to have one virtue one must have them all. What can be said about partial values and the inevitability of imperfection stands diametrically opposed to the Aristotelian picture of human good and virtue, and our examples—together with the complexities and richness of modern life that they illustrate—offer us some reason to finally reject the simpler Aristotelian ethical picture. The notion of partial values on which so much in the more complex account depends also turns out to have interesting parallels in what Freud said about “partial instincts” and what Hartry Field has more recently said about “partial signification.”Less
This chapter homes in on Aristotle and recent Aristotelianism, which have mainly assumed that the virtues are unified, that in order to have one virtue one must have them all. What can be said about partial values and the inevitability of imperfection stands diametrically opposed to the Aristotelian picture of human good and virtue, and our examples—together with the complexities and richness of modern life that they illustrate—offer us some reason to finally reject the simpler Aristotelian ethical picture. The notion of partial values on which so much in the more complex account depends also turns out to have interesting parallels in what Freud said about “partial instincts” and what Hartry Field has more recently said about “partial signification.”
David Enoch
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199579969
- eISBN:
- 9780191729010
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199579969.003.0007
- Subject:
- Philosophy, Moral Philosophy
A common objection to realism (robust or otherwise) is that realists owe us — very roughly speaking — an account of how it is that we can have epistemic access to the normative truths about which ...
More
A common objection to realism (robust or otherwise) is that realists owe us — very roughly speaking — an account of how it is that we can have epistemic access to the normative truths about which they are realists. This chapter first distinguishes between many different ways of understanding this epistemological challenge to Robust Realism, then focusing on the strongest version of the challenge, namely, the need to explain the correlation between our normative beliefs and the independent normative truths (or else accept that there is no such correlation, and that skepticism about the normative is the way to do). After the challenge is clearly stated, a way of coping it is suggested. The way to explain the correlation is by resorting to a (godless) pre-established-harmony kind of explanation, one that utilizes some plausible evolutionary speculations. In a final section there is a preliminary discussion of the somewhat related problem of accommodating semantic access.Less
A common objection to realism (robust or otherwise) is that realists owe us — very roughly speaking — an account of how it is that we can have epistemic access to the normative truths about which they are realists. This chapter first distinguishes between many different ways of understanding this epistemological challenge to Robust Realism, then focusing on the strongest version of the challenge, namely, the need to explain the correlation between our normative beliefs and the independent normative truths (or else accept that there is no such correlation, and that skepticism about the normative is the way to do). After the challenge is clearly stated, a way of coping it is suggested. The way to explain the correlation is by resorting to a (godless) pre-established-harmony kind of explanation, one that utilizes some plausible evolutionary speculations. In a final section there is a preliminary discussion of the somewhat related problem of accommodating semantic access.
Ulrich Meyer
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199599332
- eISBN:
- 9780191760648
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199599332.003.0003
- Subject:
- Philosophy, Philosophy of Science
This chapter argues against the view that the time-series is made up of metaphysically basic time points. It shows that the only plausible version of such a temporal substantivalism is an ...
More
This chapter argues against the view that the time-series is made up of metaphysically basic time points. It shows that the only plausible version of such a temporal substantivalism is an ontologically costly variant of a modal account of time. Unlike metaphysically basic spatial points, metaphysically basic time points would also not perform any theoretical work in accounting for inertial forces or free fields.Less
This chapter argues against the view that the time-series is made up of metaphysically basic time points. It shows that the only plausible version of such a temporal substantivalism is an ontologically costly variant of a modal account of time. Unlike metaphysically basic spatial points, metaphysically basic time points would also not perform any theoretical work in accounting for inertial forces or free fields.
Alex Rosenberg
- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780190462758
- eISBN:
- 9780190462772
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190462758.003.0004
- Subject:
- Philosophy, Philosophy of Science, General
Scientism is expounded. Then its two major challenges are stated and responses to them sketched. The first challenge is to its epistemology of mathematics-how we know the necessary truths of ...
More
Scientism is expounded. Then its two major challenges are stated and responses to them sketched. The first challenge is to its epistemology of mathematics-how we know the necessary truths of mathematics. The second challenge is to the very coherence of its eliminativist account of cognition. The first of these problems is likely to be taken more seriously by philosophers than by other advocates of scientism. It is a problem that has absorbed philosophers since Plato and on which little progress has been made. The second is often unnoticed, even among those who endorse scientism, since they don’t recognize their own commitment to eliminativism and so do not appreciate the threat of incoherence it poses. It is important for scientism to acknowledge these challenges.Less
Scientism is expounded. Then its two major challenges are stated and responses to them sketched. The first challenge is to its epistemology of mathematics-how we know the necessary truths of mathematics. The second challenge is to the very coherence of its eliminativist account of cognition. The first of these problems is likely to be taken more seriously by philosophers than by other advocates of scientism. It is a problem that has absorbed philosophers since Plato and on which little progress has been made. The second is often unnoticed, even among those who endorse scientism, since they don’t recognize their own commitment to eliminativism and so do not appreciate the threat of incoherence it poses. It is important for scientism to acknowledge these challenges.
