Sept 30 - Babylonian Algebra

 How could one state a general mathematical principle in a time before the development of algebra and algebraic notation?

    As examined in the reading, the Babylonians used words instead of algebraic variables in order to describe and solve problems. This methodology seems perfectly analogous to how we verbally explain a concept like area or volume to students, before setting them to the equation. I think the primary disadvantage would be actually setting out to solve it. Algebraic notation for us is a way of quickly quantifying unknowns and performing operations on them, with minimal writing required. In Example 4.7, the Babylonian solution given is effectively performing the BEDMAS steps, without having them written out line by line. A consequence of this is that it is likely much harder to solve more complicated problems, which we see with the way the Babylonians tackled higher degree polynomials.

Is mathematics all about generalization and abstraction?

    I would not say mathematics is “all about” any one thing in particular. Generalization and abstraction are certainly a facet of it, but there is also very concrete application, such as what the Babylonians used it for, that do not require abstraction to function. 

Thinking about various areas of mathematical knowledge -- number theory, geometries, calculus, graph theory, etc. etc. -- how could you imagine stating general or abstract relationships without algebra? 

    I think describing geometry, number theory and graph theory without algebra would be challenging but doable, due in large part to either ease of visualization or illustration, or ease of using language to replace the need for algebra. I struggle to imagine describing the rate of change of one entity with respect to another which it is also dependent on without using algebraic notation.