Babylonian Algebra and Mesopotamian Scribes - Thoughts

Before the development of notation, it is clear that math was still being done rhetorically, in the fashion of Babylonian algebra using words such as breadth and length to explicitly write out what we now abbreviate to symbols like ‘l’ and ‘w.’ When I think about how this can apply to more principle generalizations, I feel this may actually ease the transition from the specific to the general, since the principles of math themselves are often written in words to describe the key concept. For example, we may know the formula for the area of a rectangle using the symbolic shorthand for the sake of utility, but in stating a general principle on how we ought to view the area of a rectangle and how we can find it, we would certainly state this in terms of things like the rectangle's width and length and their relationship, rather than saying the rectangle's "l and w."

This could tie into how students learn in our classrooms today, since we may now see why it is actually rather ineffective or confusing for students to first introduce this short hand or symbolic methods of algebra and expect them to conclude from here the general principles which are free of the symbols and instead speak to the greater concepts and relationships that actually drive why things work. Perhaps this should inform how we teach, in that students likely could develop firmer/deeper understandings of math, rather than just the ability to apply memorized symbolic formulas, if we moved them from rhetoric, to syncopated, and only lastly on to symbolic. This way, they learn algebra in plain words which they already know, then begin minor substituting or mixing of words and symbols, until they are fully comfortable moving to symbols entirely, as it is now simply a representative shorthand for a broader idea they have already learnt.

I find this also speaks to why mathematics is not all about generalization and abstraction, even if these categories help to cover or sum up much of the broad concepts of math. Generalizations and abstractions often have the key benefit of being highly transferable within the broad topic and they help us make connections between different areas as well. However, mathematical application is an important part of math that lies in clear contrast to abstraction. Additionally, highly specific proofs can be very important for understanding or 'proving' areas of math, even though they are far from being general or transferable topics. For other areas of mathematical knowledge, like geometry and number theory, I therefore think it is still possible to state general or abstract relationships without algebra, as we can see that even many proofs in these topics are already done only in words, drawings, and ideas, rather than symbols and algebra. For example, many geometry proofs and principles can be discussed without algebra, in that we can construct the ideas instead with basic methods of compass and straight edge.