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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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