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If we believe that calculus satisfactorily solves Zeno’s paradoxes of motion, conceptual clarity about real analysis was not achieved before Cauchy's definition of the limit (in “Cours d'Analyse”, 1821).

But in Descartes’ time there was only some sort of proto-calculus, not even calculus on the level of Newton or Leibniz.

How was this acceptable for Descartes?

Descartes himself didn't accept infinitesimals. He judged the concept behind dx/dt to be confused and vague and not reaching his standard of a “clear and distinct” idea. He even got into a quarrel with Pierre de Fermat on this issue.

This means that there was no solution available to him. Just doing differentiation algebraically by “unproven” rules (like for polynomials) is no solution.

Did Descartes ever defend his choice to accept something as vague as motion in his philosophy? Something of which contemporary mathematicians only had an inchoate understanding of, and which they handled in a “mysterious” conceptual framework?

Zeno's paradoxes, at least, show that the human mind struggles with the concept of motion.

Descartes must have known Zeno, so he couldn't have simply claimed that we possess an innate, clear and distinct idea of motion.

viuser
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    "differentiation" and "concept behind dx/dt" in Descartes ? – Mauro ALLEGRANZA Jul 29 '20 at 05:56
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    *Motion* was quite "clear" to [Descartes](https://plato.stanford.edu/entries/descartes-physics/#SpacBodyMoti): "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 '20 at 05:58
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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?](https://philosophy.stackexchange.com/q/26441/9148) 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 '20 at 09:03
  • @Conifold method of exhaustion seems different: based on proof by contradiction. For Zeno we need *something* like the completeness axiom - there's no way to get a 2nd sequence involved in Achilles & tortoise. Archimedes was able to calculate special cases of series, like geometric series (the one for Achilles & tortoise), but he did this by an entirely geometric method: Equating one sequence of areas with another tractable seq. of areas, both intuitively real. Much more innocent, and much more restricted, than abstracting from this all and doing it with Cauchy's definition of the limit. – viuser Jul 31 '20 at 22:21
  • @Conifold so did Descartes believe in time and space being composed of zero-duration/zero-length instants/positions or not? – viuser Jul 31 '20 at 22:40
  • @MauroALLEGRANZA sure, not Descartes nor anybody at his time would've known what dx/dt means. But some proto-differentiation or hints of infinitesimals were present, at least in Fermat. – viuser Jul 31 '20 at 23:56
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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 01 '20 at 04:46
  • @Conifold no need for the scare quotes. It was more about the Aristotelian counterargument. I don't 100% know if Descartes believed (= was part of his metaphysics, etc) in zero duration instants; it was a genuine question. It seems to me **he did** ("... time of my life may be divided into an infinity of parts ..." Med III). And that's also the majority view, isn't it? So if time consists of 0 duration temporal atoms in Cartesian metaphysics, Aristotle's solution shouldn't be available to him. Can you show that Descartes thought time was not composed of temporal atoms, or ones of duration > 0? – viuser Aug 03 '20 at 01:37
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    [Schmaltz, The Metaphysics of Surfaces in Suárez and Descartes](https://quod.lib.umich.edu/cgi/p/pod/dod-idx/metaphysics-of-surfaces-in-suarez-and-descartes.pdf?c=phimp;idno=3521354.0019.008;format=pdf) pp. 4 and 13. – Conifold Aug 03 '20 at 05:12

2 Answers2

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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.

sand1
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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 '20 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 01 '20 at 12:07