Customer Reviews

Berkeley's Philosophy of Mathematics (Science and Its Conceptual Foundations series)

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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.