Babylonian algebra and abstraction

Reading the explanation of Babylonian algebra, I realized that our modern notation of algebra is mostly just a convenient shorthand. After all, when we first introduce the concepts, we use plain language to describe them - for example, x^2 is just 'any number multiplied by itself' - and the reason we use algebra over this language is that it would get far too cumbersome. But, as the Babylonians show us, it's certainly possible to formulate problems and solutions in this type of language. As concepts become more abstract, the amount of language we need to describe them grows, so having notation becomes more useful. For example, Fermat's Last Theorem for n=3 can be stated in English as "the sum of the volumes of two cubes with positive integer lengths is not the volume of any cube with a positive integer length", which is quite a mouthful. The theorem in its full generality would be even harder to express exactly.

Since my background is in pure math, it's tempting to say that math is all about abstraction and generalization, but I don't think that's quite true. Applied math, for example, uses those abstractions to solve specific questions using processes that are neither abstraction nor generalization, and I still consider that to be math. Also, recreational math sometimes uses tools of abstraction, but not exclusively. So while math is built around a framework of abstraction, it involves other types of thought as well.