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## Pure Mathematics

*E. Brian Davies*

### in Science in the Looking Glass: What Do Scientists Really Know?

- Published in print:
- 2007
- Published Online:
- September 2008
- ISBN:
- 9780199219186
- eISBN:
- 9780191711695
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219186.003.0005
- Subject:
- Physics, History of Physics

This chapter seeks to demolish the myth that mathematics is uniquely free of controversy, and therefore a guaranteed source of objective and eternal knowledge. Topics discussed include the origins of ... More

## Tolerance and Analyticity in Carnap's Philosophy of Mathematics

*Michael Friedman*

### in Future Pasts: The Analytic Tradition in Twentieth Century Philosophy

- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195139167
- eISBN:
- 9780199833214
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019513916X.003.0011
- Subject:
- Philosophy, History of Philosophy

This essay is an interpretation of Carnap’s principle of tolerance in his 1934 Logical Syntax of Language, as expressed in his philosophy of mathematics. Friedman argues that Carnap’s deepest ... More

## Frege on Knowing the Third Realm (1992)

*TYLER BURGE*

### in Truth, Thought, Reason: Essays on Frege

- Published in print:
- 2005
- Published Online:
- May 2010
- ISBN:
- 9780199278534
- eISBN:
- 9780191706943
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199278534.003.0009
- Subject:
- Philosophy, History of Philosophy, Philosophy of Language

Frege regarded the foundations of mathematics as self-evident. He maintained that reason could enable one to know of the existence and nature of both mind-independent abstract objects, such as the ... More

## Wittgenstein and Intuitionism

*Robert J. Fogelin*

### in Philosophical Interpretations

- Published in print:
- 1992
- Published Online:
- November 2003
- ISBN:
- 9780195071627
- eISBN:
- 9780199833221
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019507162X.003.0015
- Subject:
- Philosophy, General

This essay explores one important strain that runs through Wittgenstein's Remarks on the Foundations of Mathematics: his commitment to themes characteristic of the intuitionist movement in logic and ... More

## Set Theory and its Philosophy: A Critical Introduction

*Michael Potter*

- Published in print:
- 2004
- Published Online:
- September 2011
- ISBN:
- 9780199269730
- eISBN:
- 9780191699443
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269730.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an ... More

## Mathematics Based on Fuzzy Logic

*Radim Bělohlávek, Joseph W. Dauben, and George J. Klir*

### in Fuzzy Logic and Mathematics: A Historical Perspective

- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780190200015
- eISBN:
- 9780190200039
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190200015.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, General

Mathematical reasoning is governed by the laws of classical logic, based on the principle of bivalence. With the acceptance of intermediate truth degrees, the situation changed substantially. This ... More

## Parameterisation

*Graham Priest*

### in Beyond the Limits of Thought

- Published in print:
- 2002
- Published Online:
- October 2011
- ISBN:
- 9780199254057
- eISBN:
- 9780191698194
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199254057.003.0011
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

This chapter examines the relevance of mathematician Frank Ramsey's essay The Foundations of Mathematics to the contradictions in the limits of thought. Ramsey has provided the most acute early ... More

## Unfolding FOLDS: A Foundational Framework for Abstract Mathematical Concepts

*Jean-Pierre Marquis*

### in Categories for the Working Philosopher

- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198748991
- eISBN:
- 9780191811593
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198748991.003.0008
- Subject:
- Philosophy, History of Philosophy

FOLDS, first-order logic with dependent sorts, has been introduced by the logician Michael Makkai as a foundational framework to capture the abstract nature of contemporary mathematical concepts. In ... More

## The Logical Must: Wittgenstein on Logic

*Penelope Maddy*

- Published in print:
- 2014
- Published Online:
- August 2014
- ISBN:
- 9780199391752
- eISBN:
- 9780199391776
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199391752.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, General

This book is an examination of Wittgenstein’s philosophy of logic, early and late, undertaken from an austere naturalistic perspective called ‘‘Second Philosophy’’. On a generally metaphysical ... More

## Turing, Wittgenstein, and Emergence

*Juliet Floyd*

### in Philosophy of Emerging Media: Understanding, Appreciation, Application

- Published in print:
- 2015
- Published Online:
- March 2016
- ISBN:
- 9780190260743
- eISBN:
- 9780190277772
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780190260743.003.0014
- Subject:
- Philosophy, Philosophy of Language

Emerging media are a product of the digital world that Alan Turing fathered. In 1936 Turing transformed the ancient notion of algorithm—by the 1920s central to mathematical logic—into the idea of a ... More

## The mathematical practice of definitions by abstraction from Euclid to Frege (and beyond)

*Paolo Mancosu*

### in Abstraction and Infinity

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198746829
- eISBN:
- 9780191809095
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198746829.003.0002
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, History of Philosophy

The first chapter is devoted to showing that, contrary to what Frege intimates in the Grundlagender Arithmetik, abstraction principles were quite widespread in the mathematical practice that preceded ... More

## Rule-Following and Logic

*Penelope Maddy*

### in The Logical Must: Wittgenstein on Logic

- Published in print:
- 2014
- Published Online:
- August 2014
- ISBN:
- 9780199391752
- eISBN:
- 9780199391776
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199391752.003.0006
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, General

Like the Second Philosopher, the later Wittgenstein also rejects the priority of sense. In Philosophical Investigations and Remarks on the Foundations of Mathematics, the famous rule-following ... More

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