Counting – How Do We Know It?

Why even ask a question like that? Everyone knows that $3\times4=4\times3$ , just like everyone knows that $3+4=4+3$ . Mathematicians would say that multiplication or addition of two numbers is commutative, meaning that you get the same answer regardless of the order in which you do the multiplication or addition. But what does that really mean? When we say $3\times4$ , we literally mean “3, 4 times,” or 4 sets of 3 objects each, or $3+3+3+3$ objects. Likewise, when we say $4\times3$ , we mean “4, 3 times,” or 3 sets of 4 objects each, or $4+4+4$ objects. Most people would think it’s obvious that $3\times4=4\times3$ , but it’s equivalent to saying that $3+3+3+3=4+4+4$ , which seems quite a bit less obvious. So how do we know it? Is the fact that multiplication is commutative just something we assume about numbers, or is there a way to prove it?