# Berkeley’s Philosophy of Mathematics

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# Berkeley’s Philosophy of Mathematics

In this first modern, critical assessment of the place of mathematics in Berkeley’s philosophy and Berkeley’s place in the history of mathematics, Douglas M. Jesseph provides a bold reinterpretation of Berkeley’s work. Jesseph challenges the prevailing view that Berkeley’s mathematical writings are peripheral to his philosophy and argues that mathematics is in fact central to his thought, developing out of his critique of abstraction. Jesseph’s argument situates Berkeley’s ideas within the larger historical and intellectual context of the Scientific Revolution.

Jesseph begins with Berkeley’s radical opposition to the received view of mathematics in the philosophy of the late seventeenth and early eighteenth centuries, when mathematics was considered a "science of abstractions." Since this view seriously conflicted with Berkeley’s critique of abstract ideas, Jesseph contends that he was forced to come up with a nonabstract philosophy of mathematics. Jesseph examines Berkeley’s unique treatments of geometry and arithmetic and his famous critique of the calculus in

By putting Berkeley’s mathematical writings in the perspective of his larger philosophical project and examining their impact on eighteenth-century British mathematics, Jesseph makes a major contribution to philosophy and to the history and philosophy of science.

Jesseph begins with Berkeley’s radical opposition to the received view of mathematics in the philosophy of the late seventeenth and early eighteenth centuries, when mathematics was considered a "science of abstractions." Since this view seriously conflicted with Berkeley’s critique of abstract ideas, Jesseph contends that he was forced to come up with a nonabstract philosophy of mathematics. Jesseph examines Berkeley’s unique treatments of geometry and arithmetic and his famous critique of the calculus in

*The Analyst*.By putting Berkeley’s mathematical writings in the perspective of his larger philosophical project and examining their impact on eighteenth-century British mathematics, Jesseph makes a major contribution to philosophy and to the history and philosophy of science.

## Table of Contents

Works Frequently Cited

Introduction

1. Abstraction and the Berkeleyan Philosophy of Mathematics

Aristotelian and Scholastic Background

Seventeenth-Century Background

Berkeley’s Case against Abstract Ideas

Sources of Berkeley’s Antiabstractionism

2. Berkeley’s New Foundations for Geometry

The Early View

Abstraction and Geometry in the

*Principles*

Geometry in the

*New Theory of Vision*

Geometry and Abstraction in the Later Works

3. Berkeley’s New Foundations for Arithmetic

Geometry versus Arithmetic

Numbers as Creatures of the Mind

The Nonabstract Nature of Numbers

Berkeley’s Arithmetical Formalism

Algebra as an Extension of Arithmetic

The Primacy of Practice over Theory

Berkeley’s Formalism Evaluated

4. Berkeley and the Calculus: The Background

Classical Geometry and the Proof by Exhaustion

Infinitesimal Mathematics

The Method of Indivisibles

Leibniz and the Differential Calculus

The Newtonian Method of Fluxions

5. Berkeley and the Calculus: Writings before the

*Analyst*

The Calculus in the

*Philosophical Commentaries*

The Essay "Of Infinites"

The

*Principles*and Other Works

6. Berkeley and the Calculus: The

*Analyst*

The Object of the Calculus

The Principles and Demonstrations of the Calculus

The Compensation of Errors Thesis

Ghosts of Departed Quantities and Other Vain Abstractions

The

*Analyst*Evaluated

7. The Aftermath of the

*Analyst*

Berkeley’s Disputes with Jurin and Walton

Other Reponses to Berkeley

The Significance of the

*Analyst*

Conclusions

Bibliography

Index

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