And, to address your point directly, of course mathematics detached from context can be used in practice. One can certainly create-slash-discover an abstract algebra, derives theorems about it detached from outside context, and then later on discover a context in which the abstract structure is applicable, and apply the pre-derived theorems.
It's actually not entirely unproductive to consider this line of thought, whereby in this formulation equality actually means something like "arrivable via some number of zero divisions". I'm sure you could find all sorts of curiosities with this mathematical "toy".
>And on the second point, by assuming 0/0=1, either you have left the realm of natural numbers (or real numbers), or you have to break the distributive law of addition, or all the symbols mean completely different things.
I didn't do such things.
> Otherwise, you are essentially declaring both 1!=2 and 1=2, which is not math.
It's derived from initial assumptions, which is how all math works.
2. You didn’t do the first two. But the symbols now mean different things than their conventional interpretations in number theory.
3. > It's derived from initial assumptions, which is how all math works
It’s exactly how _logic_ works, and is how all math works, but that would only qualify it as (il)logic, and not inherently math. Necessary but not sufficient condition.
(Sorry ;)
Mandatory gesturing towards Gödel.
And loosely the same kind of truth as “imagine a world where elves live in Lorien… in this world, elves live in Lorien.”)
And on the second point, by assuming 0/0=1, either you have left the realm of natural numbers (or real numbers), or you have to break the distributive law of addition, or all the symbols mean completely different things. Otherwise, you are essentially declaring both 1!=2 and 1=2, which is not math.