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12 of 13 people found the following review helpful
4.0 out of 5 stars "As in mathematics, so in natural philosophy", May 24, 2012
This review is from: Isaac Newton: Philosophical Writings (Cambridge Texts in the History of Philosophy) (Paperback)
This is a perfectly fine volume of Newton's philosophical writings. I contend, however, that the editor, and many with him, do not sufficiently appreciate the importance of fact that Newton was primarily a mathematician---"a mathematician to his toe-tips," as Whiteside once put it.

I shall argue in this review that Newton's philosophy of mathematics is the key to his views on the metaphysics of gravity, space and time, etc. Unfortunately this commonsensical hypothesis is never considered by the editor and his colleagues in philosophy and history departments, who are ignorant of mathematics and prefer to focus on only the "soft" parts of Newton's philosophy, as if these could be understood in isolation.

In a revealing phrase, the editor warns that Newton should not be thought of "solely as a 'scientist'" (p. x). If Newton was "solely" anything it was not a scientist but a mathematician. And yet the editor's introduction contains no trace of the notion that Newton held views on the philosophy of mathematics, despite Newton's explicit dictum "as in mathematics, so in natural philosophy" (p. 139, from the Queries to the Opticks).

The editor's anti-mathematical blinders intervene illustratively in Newton's 1693 correspondence with Leibniz, where the editor omits by ellipsis their discussion of a particular mathematical problem (pp. 106, 108), leaving only the mushier discussion on the cause of gravity. In my opinion the omitted material is a crucial clue to understanding the clash between Newton's views and that of the continentals both in mathematics and in physics. Needless to say, the editor and his colleagues who zone out at the first sight of mathematics will be forever blind to the illumination this can bring.

In the omitted passages Leibniz asks for, in his words, "something big" regarding the construction of curves, namely a general method for reducing quadratures to rectifications. It was a consensus opinion among continental mathematicians that such a reduction was an important foundational desideratum, as "a line is simpler than an area." In reply (in the other omitted passage), Newton offers with considerable indifference the solution "which you seem to want," taking no interest whatever in the foundational aspects of this problem.

I claim that the striking difference in opinion here stems from two different ways of interpreting the role of constructions in ancient Greek geometry. The continental tradition is associated with the idea that "that which is known is that which is constructed," thus making constructions the bedrock of geometrical knowledge. Newton, on the other hand, takes construction postulates as a licence for ignorance, stipulating what falls outside the purview of geometry proper. From this point of view the problem of the construction of transcendental curves, so fundamental in the Leibnizian tradition, becomes a non-problem, or a non-geometrical one at any rate.

This is what Newton is referring to in the preface to the Principia, where he writes: "Geometry does not teach how to describe ... straight lines and circles, but postulates such a description. ... To describe straight lines and to describe circles are problems, but not problems in geometry." (p. 40)

In Newton's view, then, the essence of geometry is to reduce everything back to simple first principles, the justification of which is not the business of geometry. On the continental view, by contrast, one must start with intuitively acceptable first principles and build everything up constructively from this foundation.

This exact dichotomy underlies their differences on the cause of gravity, and that in almost the same words. "With the cause of gravity [I] meddle not" (p. 116), says Newton, admitting that "I have so little fancy to things of this nature" (p. 11), mirroring his indifference towards Leibniz's "big" geometrical problem. Leibniz on the other hand complains that to say that gravity is "performed without any mechanism by a simple primitive quality, or by a law of God, who produces that effect without using any intelligible means, it is an unreasonable occult quality, and so very occult, that it is impossible it should ever be clear, though an angel, or God himself, should undertake to explain it" (p. 112). This entire quotation applies verbatim to Newton's views on the construction of curves. Indeed, Newton explicitly stated the opposite view that "any plane figures executed by God ... are measured by geometry" (MP.VII.289, not in the present volume).

Again in the discussion of absolute versus relative space one finds the same pattern: Newton is happy to postulate absolute space (p. 64ff.), whereas the continentals insist on a relative notion of space derived constructively from the intuitively given relations of bodies.

In all of these cases---curves in geometry, the cause of gravity, the philosophy of space---Newton's strategy is to reduce the phenomena to simple first principles which are to be postulated without explanation, whereas the continental strategy is to start with intuitively acceptable first principles and construct the rest from there. Both strategies, in my opinion, are modelled on classical Euclidean geometry. Newton said it himself: "As in mathematics, so in natural philosophy." It is a pity that no one takes his words more seriously.
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Isaac Newton: Philosophical Writings (Cambridge Texts in the History of Philosophy)
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