Subtraction is just addition with negation on the second element and division is just multiplication with inverse on the second element. Then, multiplication follows addition from the distributive property of fields: $a(b+c)=ab+ac$. Real numbers with addition and multiplications form a field and one of the properties of fields is the distributive property. If you wanted to switch them around, you would have to change the operations fundamentally. –  fhyve Dec 14 '12 at 9:55