Varieties of Continua: From Regions to Points and Back 1st Edition Geoffrey Hellman

Varieties of Continua: From Regions to Points and Back 1st Edition Geoffrey Hellman – Ebook Instant Download/Delivery ISBN(s): 9780198712749,019871274X,9780191021350,0191021350

Product details:

• ISBN 10: 0191021350
• ISBN 13: 9780191021350
• Author: Geoffrey Hellman, Stewart Shapiro

Varieties of Continua explores the development of the idea of the continuous. Hellman and Shapiro begin with two historical episodes. The first is the remarkably rapid transition in the course of the nineteenth century from the ancient Aristotelian view, that a true continuum cannot be composed of points, to the now standard, point-based frameworks for analysis and geometry found in modern mainstream mathematics (stemming from the work of Bolzano, Cauchy, Weierstrass, Dedekind, Cantor, et al.). The second is the mid-tolate-twentieth century revival of pre-limit methods in analysis and geometry using infinitesimals including non-standard analysis (due to Abraham Robinson), and the more radical smooth infinitesimal analysis that uses intuitionistic logic. Hellman and Shapiro present a systematic comparison of these and related alternatives (including constructivist and predicative conceptions), weighing various trade-offs, helping articulate a modern pluralist perspective, and articulate a modern pluralist perspective on continuity. The main creative work of the book is the development of rigorous regions-based theories of classical continua, including Euclidean and non-Euclidean geometries, that are mathematically equivalent (inter-reducible) to the currently standard, point-based accounts in mainstream mathematics.

Table contents:

Part 1: The Old Orthodoxy (Aristotle) vs the New Orthodoxy (Dedekind–Cantor)

1.1 Common Ground

1.2 Points

1.3 Succession, Contiguity, and Continuity

1.4 Infinity

1.5 Plan

Part 2: The Classical Continuum without Points

2.1 Atomless Mereological Continuum

2.2 Recovering R

2.3 Topological Models

2.4 “Points” in G

2.5 Brief Philosophical Interlude

2.6 Comparisons with Some Other Constructions

Part 3: Aristotelian and Predicative Continua

3.1 Introduction

3.2 Recapitulation of the “Semi-Aristotelian” Continuum (with Emphasis on “Semi”)

3.3 Basic Aristotelian Theory

3.4 Going Modal

3.5 Looking Ahead

3.6 Predicative, Regions-based Continuum

Part 4: Real Numbers on an Aristotelian Continuum

4.1 Introduction

4.2 A “Semi-Aristotelian” Continuum: The Basics

4.3 Semi-Aristotelian Superstructure: Putting “Points” Back in

4.4 Basic Aristotelian Theory

4.5 Superstructure: Putting “Points” Back in (and Taking Potential Infinity Seriously)

4.6 Rational Numbers

4.7 The Matter of Logic—So Far

4.8 Cauchy Sequences

4.9 The Matter of Logic (More Serious this Time)

4.10 Continuity and Decomposability

Part 5: Regions-based Two-dimensional Continua: The Euclidean Case

5.1 Introduction

5.2 Regions-based Two-dimensional Continuum: Derivation of the Archimedean property (Restricted)

5.3 Generalization of the Archimedean Property and Recovery of the Parallels Postulate

5.4 Closing Reflections

Part 6: Non-Euclidean Extensions

6.1 Hyperbolic Space

6.2 Spherical Geometry: Letting Bigons Be Bigons

Part 7: The Matter of Points

7.1 Introduction

7.2 Connections

7.3 Mathematical and Scientific Limitations

7.4 Metaphysical Matters

7.5 Verbal Disputes

Part 8: Scorecard

8.1 The Accounts

8.2 The Intuitions

8.3 Ancient Atomism (Radical Version)

8.4 Aristotle

8.5 Dedekind–Cantor—The Triumvirate

8.6 Non-standard Analysis

8.7 Intuitionistic Analysis

8.8 Smooth Infinitesimal Analysis

8.9 Predicative Analysis

8.10 Point-free Geometry and Analysis

People also search:

varieties of corn

varieties of corn plants

continua vs continuum

varieties of covid

varieties of disturbance

a continual source of infection