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Inconsistent Arithmetic

Graham Priest

in In Contradiction

Published in print:
2006
Published Online:
May 2007
ISBN:
9780199263301
eISBN:
9780191718823
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199263301.003.0018
Subject:
Philosophy, Logic/Philosophy of Mathematics

The study of formal inconsistent arithmetics has already occasioned a number of interesting technical results, as well as philosophical spin-offs. This chapter looks at some of these. It shows how ... More


Hilbert's Programme

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/9780199252619.003.0010
Subject:
Philosophy, Logic/Philosophy of Mathematics

By the time Hilbert's first series of published research on the foundations of mathematics came to an end around 1904, he had formulated but not solved the problem of finding — for a formal system of ... More


Arithmetic Certainty

José Ferreirós

in Mathematical Knowledge and the Interplay of Practices

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691167510
eISBN:
9781400874002
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691167510.003.0007
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter considers the idea that we have certainty in our basic arithmetic knowledge. The claim that arithmetical knowledge enjoys certainty cannot be extended to a similar claim about number ... More


Elementary Logic

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/9780199651443.003.0001
Subject:
Philosophy, History of Philosophy, Metaphysics/Epistemology

Frege, Peano, and Russell all shared what we now call elementary logic. Nowadays we think of this as given mainly by its criterion for validity, i.e. truth in all interpretations. But they were more ... More


Logic

Alan Weir

in Truth Through Proof: A Formalist Foundation for Mathematics

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199541492
eISBN:
9780191594915
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199541492.003.0009
Subject:
Philosophy, Philosophy of Language, Logic/Philosophy of Mathematics

Gappy semantics is used to model indeterminacy with ‘upwards falsity-preservation’ emphasized as much as downwards truth-preservation. This two-way idea is generalized to multiple premiss/multiple ... More


Arithmetic

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/9780199219186.003.0003
Subject:
Physics, History of Physics

This chapter explains why our concept of number is not nearly as simple as most people think. It argues that a long historical process has resulted in the creation of a powerful numerical structure ... More


The Emergence of Foundational Study

Jan von Plato

in The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691174174
eISBN:
9781400885039
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691174174.003.0003
Subject:
History, History of Science, Technology, and Medicine

This chapter looks at how modern foundational study has twofold mathematical roots. One is the discovery of non-Euclidean geometries, especially the proof of independence of the parallel postulate by ... More


Frege’s Discovery of Formal Reasoning

Jan von Plato

in The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691174174
eISBN:
9781400885039
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691174174.003.0005
Subject:
History, History of Science, Technology, and Medicine

This chapter considers Gottlob Frege as the founder of contemporary logic through his little book Begriffsschrift that came out in 1879. The name stands for something like “writing for concepts,” in ... More


Geometry on my Mind

David D. Nolte

in Galileo Unbound: A Path Across Life, the Universe and Everything

Published in print:
2018
Published Online:
August 2018
ISBN:
9780198805847
eISBN:
9780191843808
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805847.003.0005
Subject:
Physics, History of Physics

This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who ... More


The logical and philosophical reflection on definitions by abstraction: From Frege to the Peano school and Russell

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/9780198746829.003.0003
Subject:
Philosophy, Logic/Philosophy of Mathematics, History of Philosophy

The second chapter is divided into two parts, the first of which analyzes Frege’s discussion of abstraction principles in section 64 of the Grundlagen der Arithmetik. In his investigations on Frege, ... More


Contemporary Reviews of 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/9780198712084.003.0022
Subject:
Philosophy, Logic/Philosophy of Mathematics

Frege’s Grundgesetze der Arithmetik received four short reviews during his lifetime (by H. Courbe, R. Hoppe, C. Th. Michaëlis, and C. Faerber) and a longer critical discussion by G. Peano. We present ... 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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