Berkeley’s Philosophy of Mathematics (Science and Its Conceptual Foundations series) 1st Edition by Douglas M. Jesseph (PDF)

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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 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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is overall a pretty useful book. However, I maintain that Jesseph overestimates the merit of Berkeley’s critique of the foundations of the calculus. I wish to illustrate this by considering what Jesseph calls “the most incontestable mathematical thesis Berkeley advances in the Analyst,” namely “his objection to Newton’s proof, in the Principia, of the product rule.” “Newton’s procedure here is clearly inadmissible,” according to Jesseph (pp. 226-227).The issue essentially concerns whether the dy associated with a given dx is (1) dy = y(x+dx/2)-y(x-dx/2) or (2) dy = y(x+dx)-y(x), both of which are obviously intuitively acceptable. Newton uses (1) because the algebra comes out more neatly that way. Thus to find the increment in the product AB when A increases by a and B by b, Newton computes (A+a/2)(B+b/2)-(A-a/2)(B-b/2). Berkeley and Jesseph have issues with this proof by insisting that one must use (2) rather than (1):”[Berkeley] rightly points out that the ‘direct and true’ method of finding the increment of the area is to compare the product AB to the product (A+a)(B+b). … Berkeley astutely reveals a fundamental flaw in Newton’s procedure. … Newton’s procedure here is utterly mysterious, since he actually takes the increment in the rectangle (A-a/2)(B-b/2).” (pp. 190-191)Clearly, the nonsense that Berkeley’s critique is “astute” and that Newton’s proof is “inadmissible” and “fundamentally flawed” and “utterly mysterious” all rests on the assumption that (2) is “direct and true” whereas (1) is “inadmissible.” But as far as I can see neither Berkeley nor Jesseph is “astute” enough to recognise that this assumption needs justification before their favoured conclusions follow.

⭐on one of the first critics of infinitesimal analysis and an eminent philosopher. Jesseph succeeds brillantly in describing Berkeley’s philosophy of maths and he gives deep insights into Berkeley’s writings.

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