From the commutative property of the reals, we have that: $$ x+y=y+x $$ In other words, the order of items does not matter, just their overall quantity when determining the sum. Imagine I have \$5. Here, $ 1+1+1+1+1=5 $. I can re-arrange the $1's$ in many different ways, $P_{5}^{5}$ to be exact, which yield 5 as a sum. In that sense, the dollars are fully fungible, and there is no well defined ``sequence''.
On the other hand, say someone asked me my age. I was born in 1990. As such, I am 30 years old at this point. But, there is a well defined sequence from 1990. My first year was 1990, second 1991, third 1992 and so on. Yes, I could perform the addition in whichever way possible, (add 2010 to 1991 to 1992 etc), but each number has a meaning, which is not as interchangeable as the previous case. In that sense, order has a meaning. However, arithmetic operations are still defined in the same case as above. In other words, why are arithmetic operations independent of whether or not the underlying quantities are interchangeable? Is there any class of operations that does indeed change in this setting?
Thanks!
