Not so long ago the scientific revolution was interpreted in terms of competing metaphysically loaded worldviews. For example, Koyré argued, to simplify, that Huygens and Leibniz rejected Newton’s physics because they (correctly) thought it incompatible with the mechanical philosophy, which Newton, in turn, rejected as wedded to hypotheses (of the wrong sort) (see, e.g., Koyré 1950: 262). And those that objected to this somewhat Hegelian narrative argued, in the spirit of Cassirer, that from this conflict a more modest and simultaneously more general science of law-like quantitative relations was born during the eighteenth century.
While not strictly false this picture is misleading because the debate between Huygens and Newton was also centered on empirical arguments (as George Smith and I showed). In fact, by building on Cohen’s idea of a ‘Newtonian style,’ and by focusing on Newton’s evidential arguments, George Smith (see, especially, his "Closing the Loop") and Bill Harper revolutionized Newton studies. One casualty of their research (see also this influential paper by Howard Stein), was the idea that Newton embraced hypothetico-deductive reasoning in his 'deduction' of the gravitational law.* This is a fascinating topic, but one may well wonder why this matters to philosophy in a broader sense.
But I think the strength of Newton’s evidential arguments are intrinsic to the significance of Newton to the history of philosophy in four ways. (I leave aside here the would be emulators of Newton in philosophy!) First, because in virtue of the these he was capable of transforming many questions central to philosophy into empirical questions or into significant, distinct empirical research projects (as Katherine Brading argues). Second, in so doing he contributed to the emerging split between philosophy and science. In particular, and to speak anachronistically, within science questions that increasingly came to be associated with ‘philosophy’ could be forestalled or postponed indefinitely (or so I argue here; and with Zvi Biener here.) Feynman's “Shut up and calculate” is an extreme, but historically well-trodden version of this. (To be clear: I am not claiming this is always true in science, or that Newton intended this outcome.)
In addition, Newton helped make prominent -- and this he did intend -- a style of argument such that the authority of science could be used to settle debate within metaphysics and philosophy more generally. I have called this latter move ‘Newton’s Challenge’ to philosophy (including a number of distinct versions of Newton’s Challenge). And while plenty of self-described philosophical naturalists and synthetic philosophers would defend this stance, it has put many forms of philosophical activity on the proverbial back-foot.
Finally, the stunning success of Newton's evidential arguments helped stimulate a certain style of argument in philosophy that relied on a decisive open-ended progressive conception of science. I think this is pretty clearly happening in Clarke's use of Cicero/Tully and Newton in his Demonstration on the Attributes and Being of God. And it is implied by Halley's use of tropes from Seneca, although Halley seems to think Newton may have completed science. That is to say, rather than promoting a cyclical or degenerative meta-philosophical horizon, Newton helps inaugurate an age of open-ended progress.
Notice that all these four points are, in a sense, orthogonal to the content of Newton's own metaphysics and ethics.But it is possible that Newton's metaphysics, his relentless conceptual tinkering and innovations, made these four effects possible. If that is so, then understanding Newton's metaphysics is a way to understand much larger forces that shape our philosophical scene. In particular, but that is a promissory note, it is possible that the content of Newton's metaphysics has generated path dependencies that are otherwise invisible to us because they operate, as it were, behind the scenes of our conventional philosophical self-understanding.
*Stein himself used his work on Newton to shape the philosophy of space-time in the second half of the twentieth century, along the way recasting the debate between substantivalism and relationism
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