The fact that if we put any two objects into the same (previously empty) basket as any other two object we will in this basket have four objects is true before we can make any definitions. But the statement 2 + 2 = 4 does not make any sense before we have invented: (a) the numerals 2 and 4, (b) the symbol for addition + and (c) the symbol for equality =. When we have invented meanings for these symbols (symbols as things we use in formal manipulations are quite different from words and were invented quite late, much later than we started to actually solve problems mathematically) we have to show that they correspond to our intuitive meaning of putting things into baskets and counting them, but if they do they will also satisfy, say the Peano axioms for the natural numbers, which are the axioms we tend to start from to prove statements like 2+2=4 or “there are infinitely many prime numbers”.
If we were to assume differently such that 2+2 =5, then our notion of + and = would not correspond to the notions of adding objects to a basket and counting them. This is because we could walk through our proof step by step (as described in this post) to find the first line where we write down something that is not true for our usual notion of adding apples, there we would have an assumption or a rule of inference that was assumed in this new theory but which does not correspond to apple comparison.
The fact that if we put any two objects into the same (previously empty) basket as any other two object we will in this basket have four objects is true before we can make any definitions. But the statement 2 + 2 = 4 does not make any sense before we have invented: (a) the numerals 2 and 4, (b) the symbol for addition + and (c) the symbol for equality =. When we have invented meanings for these symbols (symbols as things we use in formal manipulations are quite different from words and were invented quite late, much later than we started to actually solve problems mathematically) we have to show that they correspond to our intuitive meaning of putting things into baskets and counting them, but if they do they will also satisfy, say the Peano axioms for the natural numbers, which are the axioms we tend to start from to prove statements like 2+2=4 or “there are infinitely many prime numbers”.
If we were to assume differently such that 2+2 =5, then our notion of + and = would not correspond to the notions of adding objects to a basket and counting them. This is because we could walk through our proof step by step (as described in this post) to find the first line where we write down something that is not true for our usual notion of adding apples, there we would have an assumption or a rule of inference that was assumed in this new theory but which does not correspond to apple comparison.