Babylonian word problems

 Word problems can be both practical and imaginary (pure math). For example, the first problem in our reading is “I have added seven times the side of my square to eleven times its area, and it is 6;15”. If we think from plane geometry point of view with assumption that the numbers ‘seven’ and ‘eleven’ do not have dimension (unit-less), then we would say there is no practical value.

7(side of square) + 11(Area of the square) = 6;15

It is because the equation representing the statement does not satisfy dimensional equality to be practical. The left hand side is the sum of two objects with the first object in dimension of [length] and the second object in the dimension of [area] or square of [length]. Some people would treat it as a puzzle. In fact, the term ‘square’ is not necessary. ‘Square’ and ‘side’ can be replaced by ‘circle’ and ‘radius’ or, more generally, ‘f(x)’ and ‘x’. Thus, if we use symbolic algebra, it can be written as:

7x + 11 f(x) = 6;15

In this statement the function f can be an abstraction such as a measure of an n-dimensional structure built from a component object of value x. We can further generalize it by replacing the constants with other objects (words, parameters) such as:

ax + b f(x) = c

On the other hand, we can assign the length dimension (7 meter, 7 feet, etc.) to the number seven; assign no dimension (unit-less) to the number 11. Then the problem becomes “the area of the rectangle made of seven meter side and the side of my square added to eleven times my square is 6;15”. 

The problem now has a practical value. Therefore, when we talk about a word problem or a set of word problems such as writings on clay tablets found in Babylon archaeological extraction from the so-called lens of ‘practicality’, ‘generality’, or ‘abstraction’ while sitting at our time, we would see that they are connected or we can build the connections. In other words, in the language of contemporary (symbolic) algebra in which we have a store of not only natural language but also mathematical symbols, we see the connection. We can see them separate, and we see them together because we can specialize, generalize, and replace some objects with abstract objects. Any applied problem can be generalized, then add abstraction to it. When we studied ‘vector space’ (abstract idea) in linear algebra, we could also transform in opposite direction.

 However, if we can transport ourselves to the time of Babylonia, what would be the case? We would not have symbolic algebra. We could start from ‘generality’. Generalization is possible using words. The above same example can be reworded into: “I have added seven times the size of a side of my object to the eleven times the size of my object. It is 6;15”. Here the word ‘object’ can be square, circle, cube, etc.

For abstraction without symbols, it is possible but it would be limited by the skills in using words and sentences, and experiences either real or imaginary. In above rewritten word problem, we could build the ‘object’ as an abstract object. In “Babylonian algebra (Crest of the Peacock)” we learned that Babylonians did power of four and eight even though they could only have power of two (area) and three (volume) in applications. Therefore, it is safe to assume that limited abstraction is possible without using symbolic algebra.

By this discussion, we find that by learning the history of mathematics (Babylonian and Egyptian), we are studying applied and pure mathematics of those days, and we could learn how generalization and abstraction can be possible and to what extent without using symbolic algebra.