In class the other day, a student wondered, “Why don’t they just use symbols? Didn’t Euclid or Eudoxus notice how much easier it would be to use symbols and pictures? All of this language is so difficult.”
I believe no one would deny that mathematics is its own language. In fact, I think most people would categorize modern mathematics as a very foreign language. Symbols, operations, variables, etc. are mashed together in such a way that only a mathematician can understand what it is expressed or being questioned. This is absolutely necessary on some level. Mathematicians, and other users of mathematics, need an efficient and effective way to represent operations and the language of computation. On the other hand, it is quite sad to me that a “universal” language is so foreign. Indeed, if mathematics is a universal language, why is it understood by so few, and even despised by so many?
Mathematics didn’t start out being a foreign language. As described in Boyer and Merzbach, the earliest algebra descriptions did not use symbols. It was referred to as rhetorical algebra and an equation like x+2-z=3 would be written similar to: a thing plus two and minus a thing is equal to three. Sure, this description seems a little weird and clunky, but for most people, I argue it is easier to figure out and less “scary” or off-putting than the typical symbolic equation given above.
In Book I, proposition 41 from Euclid’s Elements, it describes something in language that all people could attempt to understand.
Thus, if a parallelogram has the same base as a triangle, and is between the same parallels, then the parallelogram is double (the area) of the triangle.
The above sentence may be difficult to understand, but it is approachable by anyone. All of the words can be defined or looked up. Representing mathematics with words allows everyone to be mathematically literate. I believe the loss of language in mathematics is a hindrance to it accessibility, popularity, and literateness.
Why was Euclid’s Elements so popular? Everyone seems to have read it, from President’s to peasants, for at least 1000 years. Part of the reason for this may be that it was actually readable. Anyone who could read, could read Euclid’s Elements. It is hard to imagine my calculus text being written in such a way, but it is important to ask, is there value in writing it in that way? Would it be a worthwhile undertaking to attempt to translate mathematics into words? Would that make it more accessible, more enjoyable, and more relevant to everyone? Would non-mathematicians be able to pick up a book and attempt to understand it? I know very few people who could pick up this and understand it:
However, most people might understand or could at least attempt to understand: write out one-half, eight times. Between each one-half write a plus symbol. Now, starting at zero and continuing until seven, raise each iteration of one-half to a power. The first one-half is raised to zero and the last one-half is raised to 7. Compute each iteration, and then add it all together. This is called summing a sequence of numbers from zero to seven. If I read this description when I first saw the above notation, I would not have been so afraid of it. I also might have tried to do it. Instead, seeing this foreign language scared me off. It just seemed too complicated to understand and figure out.
Most of my favorite math books do just that (Zero: The Biography of a Dangerous Idea; 100 Essential Things you didn’t know you didn’t know). They are books about math that contain numerous verbal descriptions of operations, ideas, and calculations. Sure, they also contain real formulas, but that is a supplement to the written description of the idea. Finding ways to use language to write about and describe math is important. Math should be accessible to all even if it is not computable by all. Wouldn’t approaching math in this way allow people to be at least as literate in mathematics as they are in English?
The Thirteen Books of Euclid’s Elements; translated from the text of Heiberg with introduction and commentary by Sir Thomas L. Heath, K.C.B., K.C.V.O., F.R.S., SC.D. Camb., Hon. D.SC.
Oxford, Volume III, Books X-XIII and Appendix: https://archive.org/details/JL_Heiberg___EUCLIDS_ELEMENTS_OF_GEOMETRY
A History of Mathematics, Jan 11, 2011 by Carl B. Boyer and Uta C. Merzbach
equation taken from http://www.math.utah.edu/grad/exam/DiffEquatF2014.pdf