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## Mathematics Developed: The Case of the Reals

*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/9780691167510.003.0008
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter considers two crucial shifts in mathematical knowledge: the natural numbers ℕ and the real number system ℝ. ℝ has proved to serve together with the natural numbers ℕ as one of the two ... More

## Representing Numbers Using Fibonacci Variants

*Stephen K. Lucas*

### in The Mathematics of Various Entertaining Subjects: Research in Recreational Math

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164038
- eISBN:
- 9781400881338
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164038.003.0017
- Subject:
- Mathematics, History of Mathematics

This chapter introduces the Zeckendorf representation of a Fibonacci sequence, a form of a natural number which can be easily found using a greedy algorithm: given a number, subtract the largest ... More

## Where Integers Come From

*Alan M. Leslie, C. R. Gallistel, and Rochel Gelman*

### in The Innate Mind, Volume 3: Foundations and the Future

- Published in print:
- 2008
- Published Online:
- January 2008
- ISBN:
- 9780195332834
- eISBN:
- 9780199868117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195332834.003.0007
- Subject:
- Philosophy, Philosophy of Mind

This chapter examines the innate basis of our concepts of the positive integers. In practice, real valued variables are never exactly equal; nor is it easy to specify an algorithm for establishing ... More

## Classical Arithmetization

*John Stillwell*

### in Reverse Mathematics: Proofs from the Inside Out

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691196411
- eISBN:
- 9781400889037
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691196411.003.0002
- Subject:
- Mathematics, History of Mathematics

This chapter describes how one proceeds from natural to rational numbers, then to real and complex numbers, and to continuous functions—thus arithmetizing the foundations of analysis and geometry. ... More

## Frege's Account of Classes

*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/9780199252619.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

In the Grundlagen, Frege sketched a proof of the basic properties of natural numbers from the numerical equivalence and showed how that equivalence could be proved if numbers were explicitly defined ... More

## ‘The Form of a Relation’: Peirce and Mathematical Structuralism

*Christopher Hookway*

### in The Pragmatic Maxim: Essays on Peirce and pragmatism

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199588381
- eISBN:
- 9780191745089
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199588381.003.0007
- Subject:
- Philosophy, History of Philosophy, Metaphysics/Epistemology

Mathematics raises a number of problems for pragmatist philosophers: how can pragmatists tolerate concepts such as numbers?; how can we apply the pragmatic maxim to clarify mathematical concepts?; of ... More

## The Natural Numbers and Analysis

*Geoffrey Hellman*

### in Mathematics without Numbers: Towards a Modal-Structural Interpretation

- Published in print:
- 1993
- Published Online:
- November 2003
- ISBN:
- 9780198240341
- eISBN:
- 9780191597664
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198240341.003.0002
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Motivation and details of a systematic translation of number‐theoretic sentences into a suitable modal‐logical language are provided, eliminating reference to numbers as objects and expressing the ... More

## Arithmetic

*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/9780199269730.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

It has to be shown that the axioms of ZU guarantee the existence of structures with the familiar properties the natural, rational, and real numbers are expected to have. This chapter makes a start on ... More

## Counting

*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/9780199269730.003.0009
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter studies those properties of numbers that depend not on the algebraic operations but on the order in which they are arranged. It starts by setting up the terminology for talking about and ... More

## Gödel Numbering and Gödel’s Theorem

*J. R. LUCAS*

### in The Freedom of the Will

- Published in print:
- 1970
- Published Online:
- October 2011
- ISBN:
- 9780198243434
- eISBN:
- 9780191680687
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198243434.003.0024
- Subject:
- Philosophy, Metaphysics/Epistemology, Moral Philosophy

Gödel devised a method whereby the formulae of a given formal language could be put in a 1-1 correspondence with a subset of the natural numbers: that is to say, to every number there is correlated ... More

## Arithmetical Comprehension

*John Stillwell*

### in Reverse Mathematics: Proofs from the Inside Out

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691196411
- eISBN:
- 9781400889037
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691196411.003.0006
- Subject:
- Mathematics, History of Mathematics

This chapter focuses on arithmetical comprehension. Arithmetical comprehension is the most obvious set existence axiom to use when developing analysis in a system based on Peano arithmetic (PA) with ... More

## Ordinal arithmetic

*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/9780199269730.003.0017
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter defines operations of addition, multiplication, and exponentiation for ordinals. It takes as a model the recursive definitions of the corresponding operations for natural numbers (§ ... 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

## Definitions of Numbers and Their Applications

*Bob Hale*

### in Abstractionism: Essays in Philosophy of Mathematics

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780199645268
- eISBN:
- 9780191755330
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199645268.003.0017
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Everyone agrees that the applicability of mathematics is of enormous importance, and at the very least demands explanation. But should the very possibility of application be somehow built into ... More

## On Frege’s Applications Constraint

*Paul McCallion*

### in Abstractionism: Essays in Philosophy of Mathematics

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780199645268
- eISBN:
- 9780191755330
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199645268.003.0015
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

As Benacerraf famously observed, the natural numbers may be reduced to sets in many different ways and these various set-theoretic reductions seem to be equally appealing. There is often more than ... More

## Ontology and Metaphysics

*Thomas Hofweber*

### in Ontology and the Ambitions of Metaphysics

- Published in print:
- 2016
- Published Online:
- September 2016
- ISBN:
- 9780198769835
- eISBN:
- 9780191822650
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198769835.003.0001
- Subject:
- Philosophy, Metaphysics/Epistemology, Logic/Philosophy of Mathematics

Many important metaphysical questions are closely tied to ontological questions. But ontology, the part of metaphysics that deals with what the world is made from or what exists, gives rise to a ... More

## Radical Contingentism, or; Why Not Even Numbers Exist Necessarily

*Peter Simons*

### in Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198792161
- eISBN:
- 9780191866876
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198792161.003.0005
- Subject:
- Philosophy, Moral Philosophy

Bob Hale championed the view that some objects exist of necessity, most prominently, mathematical objects like numbers. In contrast, this chapter upholds radical contingentism, the view that no ... More

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