*Paul F. A. Bartha*

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780195325539
- eISBN:
- 9780199776313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195325539.003.0006
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter develops the thesis that the goal of an analogical argument is to generalize a particular logical, causal or explanatory relationship. Three separate types of similarity prominent in ...
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This chapter develops the thesis that the goal of an analogical argument is to generalize a particular logical, causal or explanatory relationship. Three separate types of similarity prominent in scientific analogies are characterized: feature matching, formal similarity, and parametric similarity (or continuity). These types are linked to prominent forms of generalization: common kinds, common mathematical formalisms and invariant relations. Notably, the chapter considers—and rejects—Steiner's thesis that an inscrutable class of “Pythagorean” analogies played a fundamental role in advancing nineteenth‐ and twentieth‐century physics.Less

This chapter develops the thesis that the goal of an analogical argument is to generalize a particular logical, causal or explanatory relationship. Three separate types of similarity prominent in scientific analogies are characterized: feature matching, formal similarity, and parametric similarity (or continuity). These types are linked to prominent forms of generalization: common kinds, common mathematical formalisms and invariant relations. Notably, the chapter considers—and rejects—Steiner's thesis that an inscrutable class of “Pythagorean” analogies played a fundamental role in advancing nineteenth‐ and twentieth‐century physics.

*Michael D. Resnik*

- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250142
- eISBN:
- 9780191598296
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250142.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

The indispensability thesis maintains both that using mathematical terms and assertions is an indispensable part of scientific practice and that this practice commits science to mathematical objects ...
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The indispensability thesis maintains both that using mathematical terms and assertions is an indispensable part of scientific practice and that this practice commits science to mathematical objects and truths. Anti‐realists have used several methods for attacking this thesis: Hartry Field has tried to show how science can do without mathematics by showing that it is possible to replace analytic mathematical scientific theories with synthetic versions that make no reference to mathematical objects. Phillip Kitcher and Charles Chichara have tried, instead, to maintain the mathematical formalism in science without being committed to mathematical realism, by giving a non‐realist account of mathematical objects. Finally, Geoffrey Hellman devised a modal structuralism that uses modal operators to translate standard mathematical language into a ‘structuralist’ language. In this chapter I discuss these positions and claim that, on the one hand, they have failed to show that science can do without mathematical objects, and on the other, that these approaches to mathematics do not represent an ontic and epistemic gain over standard realism.Less

The indispensability thesis maintains both that using mathematical terms and assertions is an indispensable part of scientific practice and that this practice commits science to mathematical objects and truths. Anti‐realists have used several methods for attacking this thesis: Hartry Field has tried to show how science can do without mathematics by showing that it is possible to replace analytic mathematical scientific theories with synthetic versions that make no reference to mathematical objects. Phillip Kitcher and Charles Chichara have tried, instead, to maintain the mathematical formalism in science without being committed to mathematical realism, by giving a non‐realist account of mathematical objects. Finally, Geoffrey Hellman devised a modal structuralism that uses modal operators to translate standard mathematical language into a ‘structuralist’ language. In this chapter I discuss these positions and claim that, on the one hand, they have failed to show that science can do without mathematical objects, and on the other, that these approaches to mathematics do not represent an ontic and epistemic gain over standard realism.

*Menachem Fisch*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780226514482
- eISBN:
- 9780226514659
- Item type:
- chapter

- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226514659.003.0009
- Subject:
- Philosophy, Philosophy of Science

The chapter describes how four major figures responded to the Treatise: Augustus DeMorgan, William Rowan Hamilton, the young and short-lived D. F. Gregory, and Peacock's Trinity colleague, William ...
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The chapter describes how four major figures responded to the Treatise: Augustus DeMorgan, William Rowan Hamilton, the young and short-lived D. F. Gregory, and Peacock's Trinity colleague, William Whewell. All four, as we shall see, treated it as a major contribution to be seriously reckoned with, and all to some extent misread it in the light of their own perspectives. And all four responded differently in ways that combined to herald in a new and essentially formalistic era in British mathematics, much earlier than in German or France. The final section describes the transformation of Peacock's own thinking as a result, as is evident in the 2nd edition of his work published too late to be effective, 15 years later. It is left to others to assess the lasting worth of the framework transition set in motion by Peacock's Treatise, which is presented here as a case-study of the kind of creative indecision, its sources and possible impact that the book argues is the key to understanding how framework transitions can be rational.Less

The chapter describes how four major figures responded to the Treatise: Augustus DeMorgan, William Rowan Hamilton, the young and short-lived D. F. Gregory, and Peacock's Trinity colleague, William Whewell. All four, as we shall see, treated it as a major contribution to be seriously reckoned with, and all to some extent misread it in the light of their own perspectives. And all four responded differently in ways that combined to herald in a new and essentially formalistic era in British mathematics, much earlier than in German or France. The final section describes the transformation of Peacock's own thinking as a result, as is evident in the 2nd edition of his work published too late to be effective, 15 years later. It is left to others to assess the lasting worth of the framework transition set in motion by Peacock's Treatise, which is presented here as a case-study of the kind of creative indecision, its sources and possible impact that the book argues is the key to understanding how framework transitions can be rational.