## Berkeley's Philosophy of MathematicsIn 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 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. |

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### Contents

Abstraction and the Berkeleyan Philosophy of Mathematics | 9 |

SeventeenthCentury Background | 13 |

Berkeleys Case against Abstract Ideas | 20 |

Sources of Berkeleys Antiabstractionism | 38 |

Berkeleys New Foundations for Geometry | 44 |

The Early View | 45 |

Abstraction and Geometry in the Principles | 69 |

Geometry in the New Theory of Vision | 78 |

Leibniz and the Differential Calculus | 138 |

The Newtonian Method of Fluxions | 143 |

Berkeley and the Calculus Writings before the Analyst | 152 |

The Calculus in the Philosophical Commentaries | 153 |

The Essay Of Infinities | 162 |

The Principles and Other Works | 173 |

Berkeley and the Calculus The Analyst | 178 |

The Object of the Calculus | 183 |

Geometry and Abstraction in the Later Works | 83 |

Berkeleys New Foundations for Arithmetic | 88 |

Geometry versus Arithmetic | 89 |

Numbers as Creatures of the Mind | 95 |

The Nonabstract Nature of Numbers | 99 |

Berkeleys Arithmetical Formalism | 106 |

Algebra as an Extension of Arithmetic | 114 |

The Primacy of Practice over Theory | 117 |

Berkeleys Formalism Evaluated | 118 |

Berkeley and the Calculus The Background | 123 |

Classical Geometry and the Proof by Exhaustion | 124 |

Infinitesimal Mathematics | 129 |

The Method of Indivisibles | 132 |

The Principles and Demonstrations of the Calculus | 189 |

The Compensation of Errors Thesis | 199 |

Ghosts of Departed Quantities and Other Vain Abstractions | 215 |

The Analyst Evaluated | 226 |

The Aftermath of the Analyst | 231 |

Berkeleys Disputes with Jurin and Walton | 233 |

Other Responses to Berkeley | 259 |

The Significance of the Analyst | 292 |

Conclusions | 297 |

301 | |

317 | |

### Common terms and phrases

abstract general idea abstract ideas Alciphron algebra Analyst angle of contact argues argument from impossibility Arithmetica Barrow Berkeley Berkeley claims Berkeley's account Berkeley's critique Berkeley's New Foundations Berkeley's views Berkeleyan Philosophy calculus circle classical geometry Commentaries conceive conception consider curve declares demonstration difference differential calculus doctrine of abstract epistemology equal equation Euclidean geometry evanescent exist figures flowing quantity formalistic Foundations for Geometry idea of number indivisibles infinite divisibility infinite number infinitely small infinitesimal magnitudes Infinities insists Jurin Leibniz Leibnizian limit Locke's Maclaurin Malebranche mathe mathematicians Mathesis Universalis matics method of exhaustion Method of Fluxions metic mind minima minimum sensible motion Newton Newtonian object of geometry Paman philosophy of mathematics Principles problems procedure proof proportion Quadrature reason rectangle rejection represent rigor Robins signs Sir Isaac subtangent supposed supposition tangent theorem Theory of Vision thesis of infinite things tion treat Treatise triangle truth ultimate ratios vanish Wallis Walton