# 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 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 Jan 3, 2016 at 18:58

That there is a threshold force required to move a body subject to friction is an interesting fact, but even cursory observation shows that this threshold is different for different bodies, and depends on their weight. Moreover, it is finite and noticeable, not infinitesimal. Nothing in Aristotle's quotes suggests that he thought otherwise. Some historians argue that Archimedes publicly demonstrated moving Syracusia (55 meter long ship) single-handedly using levers and pulleys specifically to refute Aristotle's "one man cannot move a ship". In any case, the story was well known in antiquity, so Aristotle's reasoning on the issue was contested.

The relationship between Aristotle and calculus is more complicated. Infinitesimals certainly would have been anathema to him as manifestations of actual infinite divisibility, existence of which he denied. They would have resurrected Zeno's paradoxes of motion exorcising which was one of his main goals. So calculus of Fermat and Leibniz is out of the question. Newton's calculus is a different matter. In mature works Newton replaced infinitesimals with quantities explicitly defined in terms of motion, which allowed him to talk about "first and last ratios" (limits) without any appeal to infinite divisibility or infinitesimals. Friedman discusses Newton's notion of quantity and approach to calculus in Kant's Theory of Geometry (pp.478-482). Newton's qualms about infinitesimals somewhat echo Aristotle's concerns about Zeno and coherence of motion in mathematics, and historians also point out some structural parallels between Aristotle's presentation of mechanics and Newton's in Principia.

However, these philosophical affinities are not really comparable to Archimedes's much more direct technical influence on Cavalieri's indivisibles, and later on integration. More in that spirit would be also Archimedes's On Spirals, which suggested kinematic method of finding tangents (using parallelogram of velocities), and is known to have directly influenced Newton's kinematic calculus of "fluxions". Toricelli revived the method in 1640s, and Newton's teacher Barrow was an admirer.

• Archimedes on hydraulics and spirals seems really important to me as well. In passing it may be worth looking at Serres' Birth of Geometry in this context which delves pretty deep into those materials. Jan 3, 2016 at 19:02