Reflection on Babylonian Algebra

 General mathematical principles can be described with words rather than using symbols. I’ve read bits of Euclid’s Elements of Geometry, and I regularly had to adjust my reading to deal with Euclid’s style of explanation and argument, as it was written it what may be considered a more “rhetorical” approach. There was also less notation, and he didn’t use degree measurements, instead describing angles using fractions and multiples of right angles.

A big part of math is generalization and abstraction; however, it is not all of it. Being able to apply a general-situation tool to a specific situation is also important. For example, Babylonian mathematicians appear to have general strategies for solving types of polynomials but then require tables of values to substitute in for particular situations.

I mentioned how Euclid discussed geometric principles and proofs without much symbology. He started with a written description of building blocks (these “axioms” include the definition of a line, point, and plane) and then used these foundations to expand his ambition. 

I’m still shaky on the differences between rhetorical, syncopated, and pure algebra, particularly on the distinction between the first two. My current understanding is that syncopated is something of a transition stage between rhetorical and pure, where terms, written out in words, occupy the position of the unknown. When teaching story problems in an algebraic or pre-algebraic context, I would often describe an intermediate step between the story and the fully abstracted equation. I didn’t have a great name for it, sometimes calling it a “word equation” in contrast to the “story problem” and the “symbolic equation”. It would be a blend of symbols and words (for example, 2 oranges + 3 apples = $5.50), and some students appreciated this extra step. Perhaps this is like syncopated algebra and served a similar purpose: bridging the gap between words, symbols, and algebraic thinking.