In what way is math empirical? If one wants to prove that 1+1=2, one does not take one apple, add another apple, and observe that one now has two apples. One de

In what way is math empirical? If one wants to prove that 1+1=2, one does not take one apple, add another apple, and observe that one now has two apples. One defines: x + 0 = x x + successor(y) = successor(x) + y 1=successor(0) 2=successor(1) One then takes the true true statement 1+1=1+1 | Identity And finds: 1+successor(0)=1+1 | Definition of 1 successor(1) + 0 = 1+1 | Definition of + 2 + 0 = 1+1 | Definition of 2 2=1+1 | Definition of + I made no observation to prove 1+1=2. What I do observe is that if we map the counting numbers to the count of how many apples I have, and we map the addition operation to combing to groups of apples, then the mathematical result is consistent with reality. However, if we empirically discover that the mathematical result is not constitent with reality, than that does not mean that the math is wrong, it simply means that the reality does not behave in a way that corresponds to the math.