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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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