Customer Review

September 19, 2010
This is a useful book with detailed contributions to many specific issues of Newtonian scholarship. Its conclusions, however, seem to me to be rather hasty at times. Consider for example Guicciardini's characterisation of "Newton's Interpretation" (p. 129) of his work on cubic curves:

"The lesson that Newton learned from his projective classification of cubic curves is again at odds with Descartes' defence of algebra as problematic analysis. First, contrary to what Descartes had stated, the curves defined by third-degree algebraic equations are far from simple: they show bewilderingly complex shapes." (p. 129)

No evidence whatsoever is offered to support the claim that Newton drew this conclusion from his work on cubics. Such a conclusion would be strange since what Descartes really claimed is that cubics are more complicated than conics and less complicated than higher-degree curves, and Newton's projective classification certainly gives no reason to reject this order; on the contrary it fits Descartes' scheme perfectly in that the projective classification of conics is trivial, that of cubics feasible, and that of higher-degree curves beyond reach. Furthermore, contrary to calling cubics "bewilderingly complex," Newton, as one might expect, emphasised his complete understanding of them. Indeed, Guicciardini's own chapter epigraph quotes Newton boasting that "it's plain to me by ye fountain I draw it from" (p. 109).

Ironically, while Guicciardini sanctions this fictional critique of Descartes, he dismisses Newton's actual critique, which is much more sound. In effect, Newton points out that Descartes' dictum that the simplicity of a curve is determined by the degree of its equation is absurd and that instead simplicity is determined by the means of describing and generating the curve. Guicciardini is insensitive in his evaluation of this stance:

"The weakness of Newton's position is that the concepts of simplicity of tracing or of elegance, to which he continually referred, are qualitative and subjective. No compelling reason is given in support of Newton's evaluations of the simplicity of his preferred constructions; his criteria are largely aesthetic." (p. 65; cf. also p. 74)

Why should reliance on qualitative or subjective or aesthetic criteria necessarily constitute a "weakness"? The assumption that they do is nothing but sheer prejudice on the part of Guicciardini. Not a shred of evidence is offered that such concerns existed at the time.

In his conclusion Guicciardini goes even further, claiming that Newton "relied on aesthetic criteria of exactness and simplicity that were arbitrary and unjustified" (p. 386). Now it is one thing to criticise the appeal to aesthetics in mathematics---however biased and anachronistic such as critique may be---but to claim that Newton's aesthetic criteria were "arbitrary"---that, I'm afraid, is not only biased and anachronistic but plainly false.

Guicciardini's prejudiced conclusions are at odds with the laudable goal stated on the first page of the preface:

"The basic questions that motivate this book are What was mathematics for Newton? and What did being a mathematician mean to him?" (p. xiii)

I say: As long as we keep dismissing Newton's aesthetic conception of mathematics as "arbitrary and unjustified" we will never be able to answer these questions truthfully.
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