I hope I don't come off as dense but suppose I constructed (informally) the decimal number system assuming the existence of symbols, 1-9 and defining the operation of addition on it with usual properties like closure, associativity et cetera.
Now suppose I have some quantity of real life object say sticks - |, ||, |||,...|||||||||, and I define addition to be the operation of clubbing or keeping group of sticks together.
Question is how is the second real life system equivalent to the first abstract one ? How do I 'see' or prove it?