## 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

## 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

## 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

## 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

## 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 signal which is now displayed ... 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

## The Classical World

*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/9780198519737.003.0013
- Subject:
- Computer Science, Artificial Intelligence, Machine Learning

What need we know of the workings of Nature in order to appreciate how consciousness may be part of it? Does it really matter what are the laws that govern the constituent elements of bodies and ... More