Anyway, I prefer to see 2+2=4 as deriving from set theory, rather than arithmetic. Set theory has its formal rules, and some version of 2+2=4 is one of them.
The question is, do we find things in our world that can be usefully modelled by set theory? We can. For a start, there seem to be many objects that have persistence—cups, trees and planets don’t generally split or multiply in the course of a conversation. Also, we can usefully group objects together by their properties, and it is often useful to ignore their differences. Two similar looking trees are likely to do similar kinds of things in similar situations (ie induction over classes of objects is not completely useless).
So two cups of water plus two cups of sugar makes four cups—as long as we’re interested in cups, not in the difference between sugar and water. Two elms and two oaks make four trees—as long as we’re interest in trees in general, and as long as we aren’t thinking on the scale of decades, and as long there isn’t a guy with a chainsaw and an itchy trigger finger.
In fact, set theory is so useful about so many things in the world, that we abstract it to a universal truth − 2+2=4 in so many different circumstances, that simply 2+2=4.
So we can’t make two apples plus two apples equal five apples (at least not in a few seconds) because apples, in the way that we use the term, and on the time scales that we use the term, are objects that obey the axioms of set theory.
Anyway, I prefer to see 2+2=4 as deriving from set theory, rather than arithmetic. Set theory has its formal rules, and some version of 2+2=4 is one of them.
The question is, do we find things in our world that can be usefully modelled by set theory? We can. For a start, there seem to be many objects that have persistence—cups, trees and planets don’t generally split or multiply in the course of a conversation. Also, we can usefully group objects together by their properties, and it is often useful to ignore their differences. Two similar looking trees are likely to do similar kinds of things in similar situations (ie induction over classes of objects is not completely useless).
So two cups of water plus two cups of sugar makes four cups—as long as we’re interested in cups, not in the difference between sugar and water. Two elms and two oaks make four trees—as long as we’re interest in trees in general, and as long as we aren’t thinking on the scale of decades, and as long there isn’t a guy with a chainsaw and an itchy trigger finger.
In fact, set theory is so useful about so many things in the world, that we abstract it to a universal truth − 2+2=4 in so many different circumstances, that simply 2+2=4.
So we can’t make two apples plus two apples equal five apples (at least not in a few seconds) because apples, in the way that we use the term, and on the time scales that we use the term, are objects that obey the axioms of set theory.