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Conclusion

Michael Potter

in Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap

Published in print:
2002
Published Online:
May 2007
ISBN:
9780199252619
eISBN:
9780191712647
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199252619.003.0013
Subject:
Philosophy, Logic/Philosophy of Mathematics

This concluding chapter summarizes the discussion in the preceding chapters. The book sought an answer to the question: can we give an account of arithmetic which does not make it depend for its ... More


The Contradiction (i): The Problem

David Bostock

in Russell's Logical Atomism

Published in print:
2012
Published Online:
September 2012
ISBN:
9780199651443
eISBN:
9780191741197
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199651443.003.0002
Subject:
Philosophy, History of Philosophy, Metaphysics/Epistemology

The chapter gives some background on the mathematical achievements of Cantor and Dedekind, and describes how Russell aimed to show that all mathematics can be derived from a purely logical starting ... More


THE LOGICISM OF FREGE, DEDEKIND, AND RUSSELL

William Demopoulos and Peter Clark

in The Oxford Handbook of Philosophy of Mathematics and Logic

Published in print:
2005
Published Online:
July 2005
ISBN:
9780195148770
eISBN:
9780199835560
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0195148770.003.0005
Subject:
Philosophy, Logic/Philosophy of Mathematics

The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our ... More


Cardinals

Michael Potter

in Set Theory and its Philosophy: A Critical Introduction

Published in print:
2004
Published Online:
September 2011
ISBN:
9780199269730
eISBN:
9780191699443
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199269730.003.0014
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter studies the concept of equinumerosity introduced in § 4.8 by means of the following definition: two sets are said to be equinumerous if there is a one-to-one correspondence between them. ... More


The Continuum

Mathieu Marion

in Wittgenstein, Finitism, and the Foundations of Mathematics

Published in print:
2008
Published Online:
September 2011
ISBN:
9780199550470
eISBN:
9780191701559
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199550470.003.0007
Subject:
Philosophy, History of Philosophy, Logic/Philosophy of Mathematics

Ludwig Wittgenstein wrote or spoke extensively on the nature of the continuum during the years 1929-–33. This is hardly surprising, since this topic was at the centre of the Grundlagenstreit: ... More


Mathematical Concepts: Fruitfulness and Naturalness

Jamie Tappenden

in The Philosophy of Mathematical Practice

Published in print:
2008
Published Online:
February 2010
ISBN:
9780199296453
eISBN:
9780191711961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296453.003.0011
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter addresses the question of when one concept or definition is properly regarded as more ‘natural’ than another with reference to the principle that a mark of good definitions is their ... More


How We Got Here

Stewart Shapiro

in Philosophy of Mathematics: Structure and Ontology

Published in print:
2000
Published Online:
November 2003
ISBN:
9780195139303
eISBN:
9780199833658
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0195139305.003.0006
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter sketches the historical development, within mathematics, of the idea that mathematics is the science of structure. We begin with the complex transition from geometry as the study of ... More


Reals by Abstraction

Bob Hale

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:
10.1093/0198236395.003.0016
Subject:
Philosophy, Logic/Philosophy of Mathematics

The neo‐Fregean argues that principles having a certain `abstractive’ form can play a distinctive foundational role in the philosophy of mathematics––the paradigm being the reduction of elementary ... More


The Development of Analysis: A Case Study

Philip Kitcher

in The Nature of Mathematical Knowledge

Published in print:
1985
Published Online:
November 2003
ISBN:
9780195035414
eISBN:
9780199833368
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0195035410.003.0011
Subject:
Philosophy, Logic/Philosophy of Mathematics

Concludes with a more sustained look at a single part of the history of mathematics, the development of analysis from the seventeenth century to the work of Dedekind, Cantor and Frege. The discussion ... More


The Basic Laws of Cardinal Number

Richard Kimberly Heck

in Essays on Frege's Basic Laws of Arithmetic

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198712084
eISBN:
9780191780240
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198712084.003.0001
Subject:
Philosophy, Logic/Philosophy of Mathematics

An overview of what Frege accomplishes in Part II of Grundgesetze der Arithmetik, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, ... More


Axioms in Frege

Patricia A. Blanchette

in Essays on Frege's Basic Laws of Arithmetic

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198712084
eISBN:
9780191780240
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198712084.003.0002
Subject:
Philosophy, Logic/Philosophy of Mathematics

Frege’s conception of axioms is an old-fashioned one. According to it, each axiom is a determinate non-linguistic proposition, one with a fixed subject-matter, and with respect to which the notion of ... More


Frege’s Theorems on Simple Series

William Stirton

in Essays on Frege's Basic Laws of Arithmetic

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198712084
eISBN:
9780191780240
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198712084.003.0009
Subject:
Philosophy, Logic/Philosophy of Mathematics

The paper discusses theorems 207, 263, 327 and 348 and the proofs thereof. With fidelity to Frege’s own terminology, we can describe all four theorems as being about simple series. An attempt is made ... More


Frege’s Relation to Dedekind: Basic Laws and Beyond

Erich H. Reck

in Essays on Frege's Basic Laws of Arithmetic

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198712084
eISBN:
9780191780240
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198712084.003.0011
Subject:
Philosophy, Logic/Philosophy of Mathematics

Among all of Frege’s contemporaries, Richard Dedekind is arguably the thinker closest to him in terms of their general backgrounds and core projects. This essay provides a reexamination of Frege’s ... More


Frege on Creation

Michael Hallett

in Essays on Frege's Basic Laws of Arithmetic

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198712084
eISBN:
9780191780240
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198712084.003.0012
Subject:
Philosophy, Logic/Philosophy of Mathematics

In §§138–47 of the Basic Laws of Arithmetic, Frege attacks the notion that mathematical objects can be ‘created’, criticising Stolz, Hankel, and Dedekind directly, and Cantor and Hilbert indirectly. ... More


Mathematical Creation in Frege’s Grundgesetze

Philip A. Ebert and Marcus Rossberg

in Essays on Frege's Basic Laws of Arithmetic

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198712084
eISBN:
9780191780240
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198712084.003.0013
Subject:
Philosophy, Logic/Philosophy of Mathematics

We discuss a passage from Grundgesetze der Arithmetik that raises doubts regarding Frege’s attitude towards platonism. First, we motivate a platonist interpretation of Frege’s mature philosophy of ... More


Introduction

Tim Maudlin

in New Foundations for Physical Geometry: The Theory of Linear Structures

Published in print:
2014
Published Online:
April 2014
ISBN:
9780198701309
eISBN:
9780191771613
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198701309.003.0001
Subject:
Philosophy, Logic/Philosophy of Mathematics

The Introduction provides a short history of the development of geometry, with special attention the recasting of geometrical descriptions into algebraic forms. The invention of negative and ... More


Categoricity and the natural numbers

Tim Button and Sean Walsh

in Philosophy and Model Theory

Published in print:
2018
Published Online:
May 2018
ISBN:
9780198790396
eISBN:
9780191863424
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198790396.003.0007
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the ... More


Mathematics Transformed, Again

Danielle Macbeth

in Realizing Reason: A Narrative of Truth and Knowing

Published in print:
2014
Published Online:
May 2014
ISBN:
9780198704751
eISBN:
9780191774232
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198704751.003.0006
Subject:
Philosophy, History of Philosophy

In the nineteenth century, a new form of mathematical practice emerged, led above all by Riemann in Germany but also exemplified in the work of Galois, Dedekind, Hilbert, and others. Turning their ... More


The Natural Numbers

Øystein Linnebo

in Thin Objects: An Abstractionist Account

Published in print:
2018
Published Online:
June 2018
ISBN:
9780199641314
eISBN:
9780191863806
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199641314.003.0010
Subject:
Philosophy, Logic/Philosophy of Mathematics

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, ... More


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