Notions of Infinitesimals — Large Values of 0?
It might seem obvious that there should be a distinction, but what actual reasons are there to treat infinitesimals (think: reciprocals of infinities) as distinct from 0? Consider the notion of coverage “almost nowhere” in measure theory or an event with probability 0 happening “almost never”. These sure seem like infinitesimals to me!
I know that dual numbers have ε^2 = 0 definitionally, but this is often considered problematic and is why they're mainly of interest in engineering contexts as a "hack" that allows computer implementations of automatic differentiation. And anyway, if you interpret ε = 0 without distinguishing 0 from infinitesimals, it actually kind of makes dual numbers better behaved (albeit more confusing), not worse. I know less about hyperreal numbers and nonstandard analysis, but the main thing I've seen is that 0's lack of a multiplicative inverse is preserved in accordance with the transfer principle, whereas infinitesimals have infinite reciprocals. So…is that somehow not a problem in these other contexts like probability? I guess by calling infinitesimals "0", we simply dodge the issue there?
Maybe I'm missing something huge tiny…or nothing at all. 😛