Reading about Babylonian mathematics made me realize how much I take algebraic notation for granted. I was honestly surprised to learn that the Babylonians even had a concept of algebra at all. I had assumed algebra came much later, with the Greeks. But what the Babylonians show us is that algebraic thinking, reasoning about unknowns and relationships, existed long before algebraic symbols were invented. Instead of using letters like x or y, they described the unknowns through words, images, or geometric interpretations. That makes me wonder if mathematics is really about generalization and abstraction at its core, rather than the symbols we use to express it.

One thing that stood out to me was their way of solving systems of equations by introducing new variables:

• $x=\frac{1}{2}s+w$

• $y=\frac{1}{2}s-w$

At first, this felt counterintuitive. Why replace two variables with two new variables? But the more I thought about it, the more fascinating it became. In some ways, w functions almost like an “error term” or a way of capturing deviation from the mean. By reframing the system in terms of a sum (s) and a difference (w), the Babylonians simplified the relationships without needing our modern notation. It’s a powerful reminder that general mathematical principles can be expressed through clever structure, even without symbols. That said, part of me still resists this approach. It feels strange to assume that x and y must be equidistant from the mean. Still, I can see how powerful it would have been as a systematic way to solve equations in the absence of algebraic symbols.

Thinking about this makes me appreciate that mathematics has always been about patterns, relationships, and structure, not just abstraction through notation. Algebraic symbols make it compact and general, but the Babylonians show us that the ideas themselves have been around much longer than the tools we now use to express them.