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.
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.