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The Indispensability of Mathematics

Mary Leng

in Mathematics and Reality

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


More on Revision‐Theoretic Conditionals

Hartry Field

in Saving Truth From Paradox

Published in print:
2008
Published Online:
May 2008
ISBN:
9780199230747
eISBN:
9780191710933
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230747.003.0018
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter explores the ‘revisionist’ theory of the previous chapter more closely. It shows how to fit it into the semantic framework of Chapter 15, and gives an alternative modal semantics as ... More


Kripke's Theory of Truth (Strong Kleene Version)

Hartry Field

in Saving Truth From Paradox

Published in print:
2008
Published Online:
May 2008
ISBN:
9780199230747
eISBN:
9780191710933
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230747.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter is an exposition of the strong Kleene version of Kripke's fixed point semantics, and the theories of truth that can be obtained from it. Emphasis is put on the distinction between ... More


Implicit Definition and the a Priori

Bob Hale and Crispin Wright

in The Reason's Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics

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


On the Philosophical Significance of Frege's Theorem

Crispin Wright

in The Reason's Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics

Published in print:
2001
Published Online:
November 2003
ISBN:
9780198236399
eISBN:
9780191597565
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198236395.003.0013
Subject:
Philosophy, Logic/Philosophy of Mathematics

Frege's theorem is the result that elementary arithmetic may be derived from the second‐ order sentence often referred to as ’Hume's Principle’ (HP). In this essay, Wright considers whether ... More


Science without Numbers

Hartry Field

Published in print:
2016
Published Online:
December 2016
ISBN:
9780198777915
eISBN:
9780191823381
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198777915.001.0001
Subject:
Philosophy, Philosophy of Science, Logic/Philosophy of Mathematics

The book argues that there is no reason to believe in the existence of mathematical entities, or the literal truth of mathematics, and in particular that physical theory does not require this. The ... More


First Illustration of Why Mathematical Entities are Useful: Arithmetic

Hartry Field

in Science without Numbers

Published in print:
2016
Published Online:
December 2016
ISBN:
9780198777915
eISBN:
9780191823381
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198777915.003.0002
Subject:
Philosophy, Philosophy of Science, Logic/Philosophy of Mathematics

Provides an elementary illustration of the kind of application of mathematics that can be accounted for by the conservativeness of mathematics alone, without invoking the assumption that the ... More


Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance

Hartry Field

in Science without Numbers

Published in print:
2016
Published Online:
December 2016
ISBN:
9780198777915
eISBN:
9780191823381
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198777915.003.0003
Subject:
Philosophy, Philosophy of Science, Logic/Philosophy of Mathematics

Shows that the use of real numbers in geometry can be accounted for by the conservativeness of mathematics, without assuming the truth of the theory of real numbers. Focuses on Hilbert’s ... More


Conservativeness and Deflationism

Leon Horsten

in The Tarskian Turn: Deflationism and Axiomatic Truth

Published in print:
2011
Published Online:
August 2013
ISBN:
9780262015868
eISBN:
9780262298643
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262015868.003.0068
Subject:
Philosophy, General

This chapter focuses on the relationship between deflationism and the arithmetical nonconservativeness of the compositional theory of truth. Deflationists insist that truth is somehow an ... More


The Categoricity of Logic

Vann McGee

in Foundations of Logical Consequence

Published in print:
2015
Published Online:
June 2015
ISBN:
9780198715696
eISBN:
9780191783388
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198715696.003.0006
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter develops the thesis, loosely attributed to Gentzen, that the meanings of logical terms are given by the rules of inference. The author reverses the usual order of explanation, which has ... More


Conservativeness, Cardinality, and Bad Company

Roy T. Cook, Philip A. Ebert, and Marcus Rossberg

in Abstractionism: Essays in Philosophy of Mathematics

Published in print:
2016
Published Online:
January 2017
ISBN:
9780199645268
eISBN:
9780191755330
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199645268.003.0011
Subject:
Philosophy, Logic/Philosophy of Mathematics

This paper argues that abstraction principles must be conservative in a number of distinct senses if they are to be acceptable, and uses this insight to present a solution to the Bad Company Problem. ... More


Why the Utility of Mathematical Entities is Unlike the Utility of Theoretical Entities

Hartry Field

in Science without Numbers

Published in print:
2016
Published Online:
December 2016
ISBN:
9780198777915
eISBN:
9780191823381
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198777915.003.0001
Subject:
Philosophy, Philosophy of Science, Logic/Philosophy of Mathematics

Argues for the legitimacy of using mathematics to draw nominalistic conclusions (ones statable without reference to mathematical entities) from nominalistic premises, without assuming that the ... More


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