Response to "Babylonian Algebra"

When I went through this reading, I felt like there was a relatability here even across the millennia. While I was struggling in my university math classes and using roundabout, borderline brute force yet creative strategies, the Babylonians were working through their word problems with a similar level of grit. As such, I continue to have respect for these ancient mathematicians, going into hard mathematical battles with nothing but a long spear and broad rocks (see what I did there?)

Thinking about it, there were still mathematical principles at play in these pre-algebraic times. Linear, quadratic, and higher order equations were still there to be solved, after all. The Babylonian mathematicians in this reading gave some examples of how one would do this, which I could sum up as plenty of formula and table of values usage. For example, the solution to (what is basically) ax = c -- they had a formula ready to use, and a table of reciprocals. I imagine that the wording would be arcane for these things (I had to reread the word problem descriptions multiple times to even understand), but with the use of geometric terms like "length", "width", "volume", "area", and "volume", even though it wasn't declaratively algebra, they had something to work with along the lines of an algebra.

Which gets to the next point: I believe that at the core, mathematics is mostly about generalization and abstraction. Throughout history, we have tried to simplify mathematical processes and solutions by proving and deriving things like rules and defining terms to describe mathematical entities. In the reading, we see many instances of this with formulas, such as the solution to ax = c or the quadratic solutions, or unknowns being defined with the list I quote-marked above. Even these stem from lower level abstractions; multiplication, for example, can be seen as repeated addition. Exponentiation comes with many laws to address things like integers and fractional powers and bases. With number theory, I remember learning about the Chinese Remainder Theorem. We had formulas to solve systems of congruences for many different cases, which stemmed from relational experimentation in order to give us abstracted instruments. All of this talk reminds me of the computer science concept of abstraction, where complex details are "abstracted away" in order to give users and programmers easy-to-use instruments.

Of course, there are some parts of mathematics that cannot simply be generalized or abstracted. Even with the Chinese Remainder Theorem, it had to be split up into various cases which reduced its generalization. There are also atomic principles such as 1 + 1 = 2, which doesn't lend itself much to abstraction. Regardless, it cannot be overstated how important generalization and abstraction are to the field, as they are at the core of foundation building that allows mathematics to grow so much while abiding by a universal sense of logic.

As such, I find it a bit harrowing to state general/abstract relationships without algebra. So many of our definitions and formulas rely on the symbols of algebra, so to write down something like the quadratic equation's solutions in the following manner would be cumbersome and excessively verbose: "First, you get the area of a square with the breadth given. Take away four of the area formed by the given length and height. Then, get the square root (and so on. This was certainly difficult, as I caught myself saying "let's call this square root n"!)". If I think about it, it would still be possible to describe these relationships with language, but it would just be unnecessarily verbose and hard to decipher. We would still be able to use visuals, which I surmise the Babylonians used considering their affinity for geometrical terms. Visuals are already a key part of understanding things like geometry (showing shapes and their dimensions), calculus (graphs of functions) and especially graph theory (showing vertices and edges, and reasoning with them visually), though I suspect that with visuals we'll still say "this" and "that" a lot when describing the relationships. Regardless, I can definitely say after struggling through the article and writing this that algebra is a powerful tool that I cannot be more thankful for. Long live algebraic notation, and may it continue to transcend cultural and language barriers as a universal script.