[gallopingcomp]

2+2=4 is (roughly) the same kind of truth as “two groups each consisting of two elves have a total of four elves”, or “if you travel a distance of 2cm twice, you’ve traveled by 4cm”, except it isn’t tied to the real or fictional existence of centimeters or elves.

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.

[GoblinSlayer]

That's a tautology, you can similarly say "imagine a world where mathematics isn't tied to the real or fictional existence, in this world mathematics isn't tied to the real or fictional existence".

>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.

[gallopingcomp]

1. I would object to the “similarly”, because they are not similar types of statements. And yes, the tautology aspect is the whole point of the axiomatic method (which has limitations that cannot be directly blamed on that premise).

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 ;)

[fallingknife]

You were off with your first assumption that 0/0 = 1. In fact, you proved that it isn't.

[cnity]

> or all the symbols mean completely different things

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".

[gallopingcomp]

Yup. You run into that all the time in abstract algebra. Although people usually don’t like to touch the equality sign; the usual practice (based on my limited exposure) is to invent equivalent operator notations.

[GoblinSlayer]

Mathematics detached from context can't be used in practice, so it doesn't look like it's truly detached.

[gallopingcomp]

My program depends on glibc. Does glibc necessarily depend on my program?

[GoblinSlayer]

The program puts glibc in a context to implement business logic, detached from this context glibc can't be used for anything.

[gallopingcomp]

Exactly - it’s still there (and still has its own semantics), but it’s also inert.

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.

[pixl97]

Weird way to say it, instead of program say PC and see if the answer changes.

[BlueTemplar]

Math outside of ZFC is still math, even if not necessarily «useful» math.

Mandatory gesturing towards Gödel.

[gallopingcomp]

I actually agree with you - “the symbols mean something different now” isn’t a bug, it’s a feature. But I was trying to point out (what I saw as) a big ambiguity in parent’s comment.