# What is the relation between calculus and Aristotle's view of infinite divisibility?

According to an article by Rowan, Aristotle

very practically, pointed out that there was a threshold to get something moving when there is resistance to friction: 'one man cannot move a ship' as he put it.

This sounds plausible given our own experience, but looking at the passage in question it appears, at least to me, that Aristotle is supposing something entirely else - and this from his analysis of the paradoxes of Zeno; he writes in Physics VII.5:

After all, the fact that a given power as a whole has moved an object such-and-such a distance does not mean that half the power will move it any distance in any time. If, it did, one man could move a ship, since the power of the haulers and the distance which they all moved the ship together are divisible by the number of haulers.

He means there is a physical limit to how small a power there is, ontologically speaking, that can cause motion; a quanta or atom of power. He connects this to a paradox of Zeno that is little known - at least I haven't come across it before.

That is why Zeno is wrong in arguing that the tiniest fragment of millet makes a sound; there is no reason why the fragment should be able to move in any amount of time the air which the whole bushel moved as it fell.

And this analysis appears to be ratified by what he writes in the last passage of the book:

however, the fact that agent of alteration ... causes such-and-such an amount of alteration ... does not make it inevitable that it will alter ... an object half the size in half the amount of time...; no, it may well be it will cause no alteration or increase at all...

That this analysis in the limit of the small (not the large) follows immediately that of power (force), and motion; suggests that it was understood by Aristotle that just above the limit of the small, these concepts are related linearly - hence his introduction of ratios; and this is the notion that is quantified much later by the infinitesimal calculus of Newton and Liebniz; and like Newton, perhaps not at all coincidentally - it was discovered, after all by the analysis of motion.

Question 1: am I correct in thinking that this is at least one origin of the infinitesimal calculus, in the same way one takes the integration of areas in taking the limit of inscribed polygons by Archimedes?

Question 2: is the above analysis correct in suggesting that Aristotle is theorising the possibility of a quanta of power?

Question 3: what exactly is the relation between the calculus of Newton and Liebniz and the qualitative notions of Aristotle; and to what extent can this relationship be properly assessed - and has been?

• There is an interesting book that touches on several of the things you ask about. "Spooky Action At A Distance" by George Musser Commented Nov 20, 2015 at 15:44
• One question per question :) I'm picking part of the third as the headline but ofc feel free to refactor as you see fit Commented Jan 3, 2016 at 18:58