History of Babylonian Word Problems

Image source: https://www.open.edu/openlearn/sites/www.open.edu.openlearn/files/imported/14056/cuneiform_catland.jpg

It's interesting to see that Babylonians had both math application questions and pure math ideas presented in application question form. While the applied math questions were for practical purposes, the "pure" math versions were just because they could. I think that ties into how humans have an urge to test boundaries. I was in an E.A. in a Family Studies course once and the teacher explained how the problem with constantly keeping children (or anyone, really) busy is that they don't have an opportunity to be bored. Boredom, she argued, is essential because it facilitates creativity. I'm sure after drilling an abundance of similar practical math exercises, someone eventually became bored enough to wonder how they could push their boundaries and find a new way to engage with math. 

As to practicality and abstraction with relation to contemporary algebra, I partially agree with Jacob Klein's notion that only modern algebraic notation qualifies as "purity of form"(p. 8). I think the clearest way to make generalizations in abstract math is through modern algebraic notation but I also recognize that it's the convention that I'm most familiar with and that there may be other representations out there that I am not aware of. In the case of the Babylonians, I can see how their pure mathematics wasn't deemed to have a pure form because the proposed problems were based off of real world applications, but I also believe that it doesn't make that math any less valuable. In my opinion, purity of form is a technicality and the substance of a pure math problem is more useful. I believe venturing into the realm of pure math as the Babylonians did was important progress in the field of math. It set the stage to continue pushing the boundaries of the use of math and created the foundation for what came afterwards.