Question

Let's go back to the original meanings of addition and multiplication back in ancient Sumer when arithmetic was primarily used as a tool in the trade of sheep and beer. Addition meant something like this: Adamen sold me 30 pots of beer and Biluda sold me 50 pots of beer. So I'll use this new-fangled technology to save me the time of counting all of my pots of beer in my warehouse. Instead, I'll just calculate 30+50=80; I have 80 pots of beer.

Now consider how the law of commutation applies to this application. What is A+B=B+A saying? It's not saying that if I count A pots first and then B pots, I'll get the same as if I count B pots first and then A pots, because I never contemplated counting the pots in any particular order. I have a warehouse full of beer and all I contemplated was counting all of the pots in the warehouse. There's no reason to think that I would count all of the Adamen pots together and all of the Biluda pots together; the pots could be counted in any convenient order, so the notation A+B doesn't imply any order, that's just an artifact of the notation. A+B means exactly the same things as B+A. It's an analytic judgement.

By contrast AB=BA is not at all obvious. It means something like this: Adamen sells beer in lots of 3, and I bought 5 lots. Biluda sells beer in lots of 5 and I bough 3 lots. How does my number of Adamen pots compare to my number of Biluda pots? Well, 5 lots of 3 is the same as 3 lots of 5. Really? That's surprising! When I said 5 lots of 3, I didn't at all mean the same as when I said 3 lots of 5. The knowledge that these two numbers are equal is not analytic, but synthetic.

Answer 1

The commutative property in arithmetic, both for addition (A+B=B+A) and multiplication (AB=BA), indicates that the order of numbers does not impact the sum or product. While this is self-evident or analytic for addition, it is less obvious or synthetic for multiplication; nevertheless, the result remains the same regardless of order.

Step-by-step explanation:

The question essentially asks for clarification on the commutative property of addition and multiplication in arithmetic. The commutative property states that the order in which numbers are added or multiplied does not affect the result. With the example of counting pots of beer, A+B=B+A means that whether you add 30 pots from Adamen to 50 pots from Biluda or vice versa, the result is the same: 80 pots. The order of adding does not matter. This property is analytic because its truth is self-evident by the definition of addition.

Regarding multiplication, the question notes that realizing AB=BA (such as calculating the total pots if Adamen sells in lots of 3 and you buy 5 lots, compared to Biluda selling in lots of 5, but you buy 3 lots) is not immediately obvious and is synthetic. It's surprising because the process of buying a certain number of lots seems inherently ordered, yet the totals come out the same (5 lots of 3 is 15 pots, and 3 lots of 5 is also 15 pots). This illustrates the commutative property for multiplication, which says that the product of two numbers is unaffected by the order in which they are multiplied.