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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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