Word Problems in the History of Mathematics

For me, practicality in math has three requirements: one, the math involved should be relatively simple, even if the problem is not; two, the problem being considered should be non-mathematical in nature; three, the problem should be quotidian. I don’t believe that my interpretation relies on my familiarity with contemporary algebra, beyond the fact that being able to represent the situation in algebra rather than words makes it more tractable. Babylonian mathematics appears to have been developed, at least initially, to address real-world problems in farming, accounting, and surveying, and in that sense, Babylonian math was often practical. As Susan Gerofsky notes, however, sometimes the “impractical nature of [Babylonian word problems] […] casts doubt upon the serious practicality of even those most plausible problems” (Gerofsky 2004). 

My notion of abstraction, on the other hand, is heavily reliant on contemporary algebra. For me, math is abstract when it is one degree removed from a practical problem. The further one gets from a concrete scenario, the more one has to rely on symbols; I don’t know, for instance, how one could describe modern group theory without the language of contemporary algebra. However, some abstract math can be represented without algebra, as we see in the case of Babylonian mathematics. Hoyrup, quoted in Gerofsky, points out that Babylonian mathematics was “pure in substance, [but] […] applied in form” (Hoyrup, as cited in Gerofsky). Similarly, I would content that Babylonian mathematics is always practical in form, even if it is abstract in substance, as some of the exercises for scribes appear to be.

For me, math is pure when it is being used to solve a mathematical problem, and applied when it is used to solve a physical or real-world problem. I recognize, however, that this more a rule-of-thumb than an analytic definition. Calculus, in the form developed by Newton to solve concrete physical problems, is undeniably not applied mathematics despite being used to solve a physical problem. Tying back to the reading, I think that it can be difficult to identify Babylonian math as ‘pure’, even when it is pure by my definition. I think this problem occurs precisely because Babylonian math is always applied in form, even when the actual substance of the math is about more abstract methods.   

When I did the reading this week, the notion that Babylonian word problems often “look[ed] like real-world problems at first” (Hoyrup, as cited in Gerofsky)  only to be revealed to be artificial upon closer inspection, resonated with me. I can only assume that this is because that has been my experience with word problems in the past: superficially applicable problems that are really just dressed-up vehicles meant to elicit a specific technique or method. The distinction here, though, is that in the current day we are also able to ask the problems without any window-dressing, in purely algebraic form; the Babylonians could not do that, and so they had no choice but to couch even abstract content in the form of an applicable word-problem. 

Gerofsky, S. (2004). A man left Albuquerque heading east: Word problems as genre in mathematics education. Peter Lang Publishing Incorporated.