Maybe you should be dubious about Graham's number. It seems like you should always be able to add an apple to a pile of apples and get a pile with more apples in, but that's not actually the case; at some point, well before Graham's number, your pile would collapse into a black hole.
If we start defining numbers by physical quantities ('what really exists') we'll have a pretty hard to work with numerical system. Because MAX_APPLES is of course smaller than MAX_DUST_MITES, so they define two different number systems. And I can probably pretty easily subdivide until maybe a hundredth of an apple, but a hundredth of a dust mite is much less physically plausible (at human scales).
Not to mention that you could say: well sure, I can imagine a trillion trillion apples, but they don't really exist, they're just a fantasy. In reality we only have MAX_REAL_APPLE apples, and damn it, Johnny just ate one so the largest 'really existing' number just went down.
Overall, we can play this game a lot, but the reality is that pi is as real as Aleph0 which is as real as 1 and 2 and 0.0000...1 and +Inf etc.
> If we start defining numbers by physical quantities ('what really exists') we'll have a pretty hard to work with numerical system. Because MAX_APPLES is of course smaller than MAX_DUST_MITES, so they define two different number systems. And I can probably pretty easily subdivide until maybe a hundredth of an apple, but a hundredth of a dust mite is much less physically plausible (at human scales).
Sure. So a simplified model is probably the right tradeoff to make. But we should not forget that it is a tradeoff; by working in a simplified world of notation we run the risk of forming constructions that can't actually be carried back to the real world of counting apples or dust mites. The map is not the territory.
> Overall, we can play this game a lot, but the reality is that pi is as real as Aleph0 which is as real as 1 and 2 and 0.0000...1 and +Inf etc.
"All models are wrong, but some are useful" - but not all models are equally wrong, and not all are equally useful. Including pi gives us a model that's easier to work with, but introduces a certain risk of forming a construction that doesn't translate back to reality. Including infinities brings a substantially bigger version of that risk. It may still be the right choice, but it's a choice that should be made carefully and deliberately; blindly assuming that all mathematical models are appropriate to a given situation is quite unwise.
