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.

I'm looking for any authors who have discussed this issue specifically with respect to algebra and algebraic notation.

Answer 1

Commutative properties in algebra reflect fundamental truths about addition and multiplication. Addition's commutativity is an analytic judgment, while multiplication’s is a synthetic one. Identifying authors who combine ancient trade and algebraic notation requires specific historical mathematics research.

Step-by-step explanation:

The discussion of commutative properties in algebra and algebraic notation is rooted in the understanding of basic operations like addition and multiplication. Specifically, the concept of commutativity explains that in addition (A+B=B+A), the order of the terms does not affect the sum—you get the same result whether you add 30 pots of beer to 50 pots of beer or the other way around. This is considered an analytic judgment because it follows from the definition of what addition is.

On the other hand, the equality of AB=BA in multiplication is called a synthetic judgment, since it’s not immediately obvious from the definition of the operation. Rather, it is a surprising discovery that 5 lots of 3 equals 3 lots of 5. This property is a fundamental part of algebra that transcends cultural, historical, and practical contexts.

Authors who have delved into these algebraic principles include mathematicians and logicians who explore the foundational aspects of numbers and operations. However, the specific names of authors who address the historical trade context of ancient Sumer in combination with algebraic notation may not be as easily identified and could require extensive literature research within the history of mathematics.