I think that abstraction and pure/applied math rely on familiarity with contemporary algebra, yet practicality and generality naturally emerged in the whole development of math.
Moreover, I believe that practicality and generality are two main cause-factors of the development of math. The origin of mathematical language is to generalize and solve real-life problems. For instance, Babylonians invented the sexagesimal numeral system based on their observations and generalizations of daily lives.
In contemporary school, math is separated into different subjects based on abstraction. Also depending on the focus of the research area, there are two categories of math: pure and applied. From my understanding, the goal of pure math scholars is to explore more theoretical concepts, while applied math researchers devote themselves to promote mathematical procedures into the industry to solve or optimize possible problems. In addition, I feel in the context of contemporary math, practicality is the goal of applied math and generality is the light to pure math.
Also, my definitions of pure and applied math could be different from others. These differences may lead to arguments that are similar to whether Babylonian math is practical. How would people from thousands of years after, know what is "practical" to Babylonians? Last class, when thinking about reasons for Babylonians to have age-guessing like word problems, my first thought was there was a chance they lost the clay tablets (breaking, burying into mud etc). They wanted to train students in this way in case of a situation of losing certain parts of information happened.
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