Comment by LegionMammal978

> there's still a definite integer that it corresponds to.

The formalism says that there's still a definite integer that it corresponds to. The ultrafinitist would deny that the formalism keeps capturing truth past where we've verified it to be true, or some unknown distance farther.

> I hypothesized a universe in which BB(748) could actually exist, but you can equally hypothesize ones in which not only can it exist, it exists comfortably and is considered a small number by its denizens.

Sure, but the ultrafinitist would argue, "All this is still just a shallow hypothesis: you've said the words, but that's not enough to breathe much 'life' into the concept. It is but the simplest of approximations that can fit into our heads, and such large things (if they could exist) would likely have an entirely different nature that is incomprehensible to us."

> We can't conceive of such a thing but there's no particular a priori reason to suppose it couldn't exist.

That's why I wouldn't call myself an ultrafinitist, but would prefer an agnostic approach. There may be no great a priori reason to suppose it cannot exist, but I similarly do not see any such reason it must necessarily exist. We empirically notice that our formalism works for numbers small enough to work with, and we pragmatically round it off to "this formalism is true", but one could argue that surprising claims about huge numbers need stronger support than mere pragmatism.