On Sunday, October 1, 2017 at 2:02:07 PM UTC-6, John Gabriel wrote:
> S = Lim S
> STOPS RIGHT HERE AND NOW. It is the Eulerian Blunder and has no place in > rational thought, never mind mathematics.
S = Lim S isn't a statement of fact. It's a definition. It means that (in certain appropriate cases) when we write an S that doesn't really have a legitimate meaning, but Lim S does have a value, we are simply using S as a shorter way of writing Lim S.
So we use this not as a way of saying that the bad, invalid, S is now a good thing, but as a way of *getting rid* of S (and stuff like infinitesimals) and _only_ using real things which do have values - such as Lim S.
You have just misunderstood what Euler was trying to do. He wasn't trying to introduce bad messy thinking into mathematics; instead, he found some there, and he was using this method to get _rid_ of all the messy stuff quickly without having to make mathematicians do a lot of extra work.
Euler was not your enemy. He wasn't saying the false thing that S is just as good as Lim S. He knew that S was bad, but Lim S was good, and he was therefore doing just what you are trying to do: fix mathematics so that it rests on the good stuff and doesn't try to use the bad stuff.