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Even if we believe that calculus satisfactorily solves Zeno's paradoxes, we can't earnestly say that conceptual clarity was achieved before Cauchy's definition of the limit in the 19th century.

For Descartes, there was only some very non-rigorous proto-calculus (as we know, he himself made important contributions).

How was this acceptable for Descartes?

He himself didn't like infinitesimals - in differentiation.

Unsurprisingly, since the concept behind dx/dt is extremely confused and vague (not talking about modern rigorous reformulations). Far removed from being "clear and distinct".

But this just means that there was no solution available to him. Just doing differentiation algebraically by unproven rules is even worse and reduces the mathematical treatment of motion to guess-work.

Did Descartes ever defend his choice to accept something as muddled as motion in his philosophy? Something which contemporary mathematicians had just an inchoate understanding of, and handled in a mysterious, "occult" conceptual framework?

Zeno's paradoxes, at the very least, show how much humans struggle with the concept. He couldn't have claimed that we simply possess a clear and distinct idea of motion.

  • "differentiation" and "concept behind dx/dt" in Descartes ? – Mauro ALLEGRANZA Jul 29 at 5:56
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    Motion was quite "clear" to Descartes: "Descartes’ Principles of Philosophy also presents his most extensive discussion of the phenomena of motion, which is defined as “the transfer of one piece of matter or of one body, from the neighborhood of those bodies immediately contiguous to it and considered at rest, into the neighborhood of others” (Pr II 25)." – Mauro ALLEGRANZA Jul 29 at 5:58
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    Why do you assume that the calculus solution could be the only one available to Descartes? Zeno's paradoxes were considered solved by Aristotle, and his solution was not questioned until 19th century, Why is Aristotle's objection not considered a resolution to Zeno's paradox? That aside, Euclid and Archimedes "clearly and distinctly" expounded the method of exhaustion, which can solve Zeno's paradoxes in the same fashion that calculus supposedly does. Btw, Descartes did not use infinitesimals, but did use Archimedean indivisibles. – Conifold Jul 29 at 9:03
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    He did not need to "believe". Archimedes did not "believe" in atoms either, but he knew how to convert arguments with them into double reductios. To Descartes, like others after Aristotle, infinite divisibility was only potential and Zeno's arguments could not get off the ground. – Conifold Aug 1 at 4:46
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You can read this reference

Newton's critique of Descartes's Theory of Motion Carmical, Alex W. Purdue University, ProQuest Dissertations Publishing, 2010.

https://docs.lib.purdue.edu/dissertations/AAI3413777/

and

The Descartes-Newton paradox: Clashing theories of planetary motion at the turn of the eighteenth century Jean-Sébastien Spratt Vassar College

https://digitalwindow.vassar.edu/cgi/viewcontent.cgi?article=1617&context=senior_capstone

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Rereading a few pages from Chap. 7 of The World makes the above questioning really puzzling.

Did Descartes ever defend his choice to accept something as muddled as motion in his philosophy? Something which contemporary mathematicians had just an inchoate understanding of, and handled in a mysterious, "occult" conceptual framework?

Descartes wrote

The |Philosophers| themselves avow that the nature of their motion is very little known. To render it in some way intelligible, they have still not been able to explain it more clearly than in these terms: motus est actus entis in potentia, prout in potentia est, which terms are for me so obscure that I am constrained to leave them here in their language, because I cannot interpret them. (And, in fact, the words, "motion is the act of a being in potency, insofar as it is in potency," are no clearer for being in [English].) On the contrary, the nature of the motion of which I mean to speak here is so easy to know that mathematicians themselves, who among all men studied most to conceive very distinctly the things they were considering, judged it simpler and more intelligible than their surfaces and their lines. So it appears from the fact that they explained the line by the motion of a point, and the surface by that of a line.

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  • I don't think the question is "really puzzling". The quoted paragraph also shows this. It does e.g. not apply to Fermat, who successfully used different (more advanced) methods to tackle motion. It's easy for Descartes to claim that mathematicians are men who clearly and distinctly understand motion, and they understand it like him ... if he usurps the role as supreme arbiter of who is a "true" mathematician. So thanks, the quote shows he did defend his choice: motion is just to be understood as a position function (x(t), y(t), z(t)). But his defense is very bad. – viuser Jul 31 at 20:12
  • With the concept of kinetic energy we can stomach the statement that being at point x a body has speed v and we understand all the rhetoric involved ; analyzing motion in terms of actuality and potentiality generates trouble just as analyzing it as position and speed. – sand1 Aug 1 at 12:07

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