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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


Objectivity in Mathematical Knowledge

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/9780691167510.003.0009
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new ... More


Mathematical Preliminaries

Partha P. Mitra and Hemant Bokil

in Observed Brain Dynamics

Published in print:
2007
Published Online:
May 2009
ISBN:
9780195178081
eISBN:
9780199864829
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195178081.003.0005
Subject:
Neuroscience, Techniques, Molecular and Cellular Systems

This chapter reviews a broad range of mathematical topics relevant to the rest of the book. It begins with a brief discussion of real and complex numbers and elementary real and complex functions, ... More


Real numbers

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/9780199269730.003.0011
Subject:
Philosophy, Logic/Philosophy of Mathematics

It was noted in the Introduction to this part of the book that Weierstrass based his rigorous presentation of the calculus on the assumption that the real numbers form a complete ordered field — that ... 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


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


What Is Mathematics About ?

Michael Dummett

in The Seas of Language

Published in print:
1996
Published Online:
November 2003
ISBN:
9780198236214
eISBN:
9780191597350
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198236212.003.0018
Subject:
Philosophy, Philosophy of Language

While it is relatively clear what the subject matter of empirical sciences is, puzzles persist about the proper subject matter of mathematics. The logicists took mathematics to be concerned solely ... More


Conclusions: Reflections and Hopes

Kumaraswamy Velupillai

in Computable Economics: The Arne Ryde Memorial Lectures

Published in print:
2000
Published Online:
November 2003
ISBN:
9780198295273
eISBN:
9780191596988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198295278.003.0010
Subject:
Economics and Finance, Macro- and Monetary Economics

In the concluding chapter, the themes broached in the first chapter are reconsidered in the light of the results and framework developed in the book as computable economics. In particular, the idea ... More


Mathematical Knowledge and the Interplay of Practices

José Ferreirós

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691167510
eISBN:
9781400874002
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691167510.001.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction ... More


Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier

Laura Capuano, Peter Jossen, Christina Karolus, and Francesco Veneziano

in Arithmetic and Geometry: Ten Years in Alpbach (AMS-202)

Published in print:
2019
Published Online:
May 2020
ISBN:
9780691193779
eISBN:
9780691197548
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691193779.003.0003
Subject:
Mathematics, Geometry / Topology

This chapter details Umberto Zannier's minicourse on hyperelliptic continued fractions and generalized Jacobians. It begins by presenting the Pell equation, which was studied by Indian, and later by ... More


Frege on the Real Numbers

Eric Snyder and Stewart Shapiro

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/9780198712084.003.0014
Subject:
Philosophy, Logic/Philosophy of Mathematics

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece Grundgesetze der Arithmetik. It is perhaps unsurprising that Frege’s theory ... More


Classical Analysis

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/9780691196411.003.0003
Subject:
Mathematics, History of Mathematics

This chapter explores the basic concepts that arise when real numbers and continuous functions are studied, particularly the limit concept and its use in proving properties of continuous functions. ... More


Real Numbers on an Aristotelian Continuum

Geoffrey Hellman and Stewart Shapiro

in Varieties of Continua: From Regions to Points and Back

Published in print:
2018
Published Online:
March 2018
ISBN:
9780198712749
eISBN:
9780191781087
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198712749.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter is an attempt to recover “points” and (something like) real numbers in the more Aristotelian framework presented in Chapter 3. The techniques go beyond Aristotelian resources, borrowing ... 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


Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance

Hartry Field

in Science without Numbers

Published in print:
2016
Published Online:
December 2016
ISBN:
9780198777915
eISBN:
9780191823381
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198777915.003.0003
Subject:
Philosophy, Philosophy of Science, Logic/Philosophy of Mathematics

Shows that the use of real numbers in geometry can be accounted for by the conservativeness of mathematics, without assuming the truth of the theory of real numbers. Focuses on Hilbert’s ... More


Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Ehud Hrushovski and François Loeser

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691161686
eISBN:
9781400881222
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161686.001.0001
Subject:
Mathematics, Geometry / Topology

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model ... 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


Nominalism and the Structure of Physical Space

Hartry Field

in Science without Numbers

Published in print:
2016
Published Online:
December 2016
ISBN:
9780198777915
eISBN:
9780191823381
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198777915.003.0004
Subject:
Philosophy, Philosophy of Science, Logic/Philosophy of Mathematics

Argues that the Hilbert theory of the previous chapter not only dispenses with real numbers, but is (or can be made with a little rewriting) a genuinely nominalistic theory of the structure of ... More


Mathematics and Reality

Roger Penrose and Martin Gardner

in The Emperor's New Mind: Concerning Computers, Minds, and The Laws of Physics

Published in print:
1989
Published Online:
November 2020
ISBN:
9780198519737
eISBN:
9780191917080
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198519737.003.0011
Subject:
Computer Science, Artificial Intelligence, Machine Learning

Let us imagine that we have been travelling on a great journey to some far-off world. We shall call this world Tor’Bled-Nam. Our remote sensing device has picked up a ... More


The Peircean Continuum

Francisco Vargas and Matthew E. Moore

in The History of Continua: Philosophical and Mathematical Perspectives

Published in print:
2020
Published Online:
December 2020
ISBN:
9780198809647
eISBN:
9780191846915
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198809647.003.0014
Subject:
Philosophy, Metaphysics/Epistemology

Charles Sanders Peirce’s views on continuity, the concept he lionized as “the master-key which … unlocks the arcana of philosophy”, are of vital importance for students of his philosophy, but have ... More


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