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Sniffnoy comments on Rationality quotes: June 2010 - Less Wrong
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That's not true. He had perfectly good reasons for atomism in his context.
The ontological arguments of Parmenides (and as exposited by Melissus) lead to extremely unpalatable, if not outright contradictory, conclusions, such as there being no time or change or different entities. The arguments seem valid, and most of their premises are reasonable, but one of his most important and questionable premises is that void cannot exist.
Reject that premise and you are left with matter and void. How are matter and void distributed? Well, either matter can be indefinitely chopped up (continuous) or it must halt and be discrete at some point. The Pluralists like Anaxagoras take the former approach, but continuousness leads to its own issues with regard to change.* So to avoid issues with infinity, you must have discrete matter with size/divison limits - atoms.
So, Democritus and Leucippus are led to Atomism as the one safe path through a thicket of paradoxes and problems. Describing it as wild conjecture is deeply unfair, and, I hope, ignorant.
* One argument, if I remember it from Sextus Empiricus's Against the Physicists correctly, is that if matter really is infinitely divisible, then you should be able to divide it again and again, with void composing ever more of the original mass you started with; if you do division infinitely, then you must end up with nothing at all! That is a problem. Cantor dust would not have been acceptable to the ancient Greeks.
Even if that argument is valid, it doesn't seem to rule out matter being "infinitely divisible" in the sense that it could be divided arbitrarily many times - which is what a rejection of atomism really means - just the stronger sense of being able to divide something infinitely. As an argument against atomism it seems to rely on an equivocation between these two, unless that case is somehow ruled out.
I don't really follow. How is arbitrarily divisible not the same as infinitely divisible? If there is some limit to the arbitrariness then it's just atomism discussed. If there is no limit, then it seems like infinity to me.
(Most of the arguments against continuity use this sort of induction; continuity causes problems at this level, and it logically causes problems on n+1 levels; therefore it causes problems on all levels.)
If atomism is false, then for any matter and any n, the matter can be divided into n parts (whatever that may mean); this is a finite division, so if we started with a nonzero amount of "stuff" (whatever that may mean), the parts we divided it into should also comprise nonzero amounts of "stuff" that total to the original amount of "stuff". No problem there.
The apparent problem (which I would say isn't one really, but that's not the point, because there's no way the Greeks could have been expected to come up with why not) appears if you allow it to be divided into infinitely many parts, and then argue, since there was only finitely much "stuff" originally, and we've divided it into infinitely many parts, each part must have zero "stuff", but any sum of zeroes is zero.
Weak induction will reach all natural numbers, but it won't reach infinity.