Editing Infinite divisibility (section)

== In philosophy ==

The origin of the idea in the Western tradition can be traced to the 5th century BCE starting with the Ancient Greek pre-Socratic philosopher [[Democritus]] and his teacher [[Leucippus]], who theorized matter's divisibility beyond what can be perceived by the senses until ultimately ending at an indivisible atom. The Indian philosopher, Maharshi [[Kanada (philosopher)|Kanada]] also proposed an atomistic theory, however there is ambiguity around when this philosopher lived, ranging from sometime between the 6th century to 2nd century BCE. Around 500 BC, he postulated that if we go on dividing matter (''[[Padārtha|padarth]]''), we shall get smaller and smaller particles. Ultimately, a time will come when we shall come across the smallest particles beyond which further division will not be possible. He named these particles ''Parmanu''. Another Indian philosopher, [[Pakudha Kaccayana|Pakudha Katyayama]], elaborated this doctrine and said that these particles normally exist in a combined form which gives us various forms of matter.<ref>{{cite book|url=https://ncert.nic.in/ncerts/l/iesc103.pdf}}</ref>
<ref>{{cite book|url=https://books.google.com/books?id=9RAwDwAAQBAJ&q=maharshi+Kanad&pg=RA1-PA15|title=The Science Springboard 9th|isbn=9789332585164|last1=Education|first1=Pearson|year=2016|publisher=Pearson India }}</ref>
[[Atomism]] is explored in [[Plato]]'s [[Timaeus (dialogue)|dialogue Timaeus]]. [[Aristotle]] proves that both length and time are infinitely divisible, refuting atomism.<ref>[[Physics (Aristotle)|Physics]] VI.I-III (231a21-234b10)</ref> [[Andrew Pyle (philosopher)|Andrew Pyle]] gives a lucid account of infinite divisibility in the first few pages of his ''Atomism and its Critics''. There he shows how infinite divisibility involves the idea that there is some [[extended item]], such as an apple, which can be divided infinitely many times, where one never divides down to point, or to atoms of any sort. Many philosophers{{who|date=September 2012}} claim that infinite divisibility involves either a collection of ''an infinite number of items'' (since there are infinite divisions, there must be an infinite collection of objects), or (more rarely), ''point-sized items'', or both. Pyle states that the mathematics of infinitely divisible extensions involve neither of these — that there are infinite divisions, but only finite collections of objects and they never are divided down to point extension-less items.

In [[Zeno's paradoxes#Arrow paradox|Zeno's arrow paradox]], Zeno questioned how an arrow can move if at one moment it is here and motionless and at a later moment be somewhere else and motionless.
{{blockquote|Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.<ref>{{cite web |url=http://classics.mit.edu/Aristotle/physics.6.vi.html#752 |work=The Internet Classics Archive |title=Physics |author=Aristotle}}</ref>|Aristotle|''[[Physics (Aristotle)|Physics]]'' VI:9, 239b5}}
In reference to Zeno's paradox of the arrow in flight, [[Alfred North Whitehead]] writes that "an infinite number of acts of becoming may take place in a finite time if each subsequent act is smaller in a convergent series":<ref name="Ross1983" />
{{blockquote|The argument, so far as it is valid, elicits a contradiction from the two premises: (i) that in a becoming something (''res vera'') becomes, and (ii) that every act of becoming is divisible into earlier and later sections which are themselves acts of becoming. Consider, for example, an act of becoming during one second. The act is divisible into two acts, one during the earlier half of the second, the other during the later half of the second. Thus that which becomes during the whole second presupposes that which becomes during the first half-second. Analogously, that which becomes during the first half-second presupposes that which becomes during the first quarter-second, and so on indefinitely. Thus if we consider the process of becoming up to the beginning of the second in question, and ask what then becomes, no answer can be given. For, whatever creature we indicate presupposes an earlier creature which became after the beginning of the second and antecedently to the indicated creature. Therefore there is nothing which becomes, so as to effect a transition into the second in question.<ref name="Ross1983">{{cite book |first=S.D. |last=Ross |year=1983 |title=Perspective in Whitehead's Metaphysics |series=Suny Series in Systematic Philosophy |publisher=State University of New York Press |isbn=978-0-87395-658-1 |lccn=82008332 |url=https://archive.org/details/perspectiveinwhi0000ross |url-access=registration |pages=[https://archive.org/details/perspectiveinwhi0000ross/page/182 182]–183}}</ref>|A.N. Whitehead|''[[Process and Reality]]''}}