"One follows this method..."
So begins the explanation of the Babylonian solution to a pair of linear equations in two unknowns (Example 4.8 in our handout). What strikes me about the examples we considered in this handout are that they are steps - almost a sort of numerical recipe - for a given scenario. These steps are described not using symbols, but in words. And from what I can tell, it seems the aim in these problems is often to obtain a "standard form" from which familiar steps can be taken. That is, these examples seem to show ancient algorithms.
General mathematical principles are thus stated using concrete, narrative word problems that are then worked out using a sequence of precise steps. But rather than use any formulaic symbols, the Babylonian scribe worked with words and sometimes abbreviations. For example, to describe what we today call variables, the Babylonians used everyday physical dimensions. So, in the example 4.8 mentioned above, the unknows are represented by the Babylonian words for "length" (ush) and "width" (sag), while the product of these two unknowns is called "area" (asha). Thus, the language employed in the Babylonian problems, while corresponding conceptually in some ways to our use of these terms, is also different. One can, for example, subtract "length" from "area," and so on.
One thing that strikes me is that even when a problem has nothing to do with land measurements, the problem is still framed as finding the "length" and "width" of a rectangle (or a field, perhaps). I have not thought a lot about how students start to learn algebra in grade 6-7 or so, but I would not be surprised to learn there are parallels to this approach. It would seem natural to have an intermediate step before arriving at full abstraction. Certainly before any abstract symbols are introduced, young students encounter word problems that describe a situation and ask for a numerical answer, but they are not manipulating abstract variables!
Another things that seems to be happening in the Babylonian approach to solving these problems is a kind of "cut-and-paste" geometry. Rather than performing abstract operations on variables, the Babylonian approach seems to be the rearrangement of areas. For example, in Example 4.7, it seems they visualized a rectangle with an unknown length and width, then cut off the protruding "excess" length, slice it in half, and paste one half onto the bottom of the rectangle to form a perfect, larger square missing a corner. By calculating the area of that missing corner, they could figure out the dimensions of the entire shape. This was done by adding the "corner" to the existing area of the rectangle to get the total area, then working backwards to find the dimensions of the large square. Such a "physical" geometric approach would also be helpful for students learning algebra; simply the insight that there can be physical explanations for why certain mathematical processes "work" is encouraging to students who might struggle to "trust" the abstract representations.
The Babylonians have shown us that mathematics indeed is not all about abstraction and generalization. The examples we considered show us that math began as a practical, concrete tool that was taught through specific, concrete examples. The focus here is on the procedure to get a correct answer, rather than the universal law behind why the procedure worked -- but it is still mathematics, the mathematics of counting, measuring, and navigating the physical world. It is from such foundations that later abstract and structural mathematics could be developed.
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