We all learned about homonyms in elementary school. They're words that have the same spelling (and often the same pronounciation) but mean different things. For example, the word "bear" can be a noun, meaning a big furry animal, or a verb, with the approximate meaning "to carry". (Actually, you can also define homonyms to mean "any two words that sound the same; a homophone", but I'm dealing with the more restrictive definition in this post.) For people learning English as a second language, homonyms are among the more frustrating facets of our language. The fact that a single word can have several distinct meanings might make it difficult for people just learning the language to make sense of our sentences. For this reason, it's sometimes said that the existence of many homonyms is a major problem with the English language.

Furthermore, in English, such factors as the homonym problem can cause ambiguity because sometimes we're not sure what a person means when he or she uses a certain word. In everyday life, we normally get around this well enough; we are generally able to communicate and be understood. But when it comes to really important matters, such as the design of a spaceship or a nuclear power plant, where small errors can result in catastrophe, the imprecision of English is unacceptable. The Doctor Rocket Surgeons of our society must use an unambiguous language -- the language of mathematics -- to ensure, well, "mathematical precision".

As I understand it, a primary reason that a mathematical language is used, especially in physics, is because it eliminates ambiguity. Furthermore, it can be argued that mathematics is a language; certainly, it is a language to the extent that it is a self-consistent means by which phenomena can be communicated and understood. I'm not going to debate the extent to which mathematics /is/ a language; the fact that it is used to self-consistently communicate and understand concepts is good enough for my purposes.

But -- /if/ mathematics can be considered a language, then the homonym problem in mathematics is much worse than that in English.

Consider what may be the simplest of English words: "I", referring to the author or speaker of a sentence. What does this simple word mean in the language of mathematics and physics?

- The square root of negative one.

- The set of numbers created by multiplying all real numbers by the square root of negative one.

- Any interval of numbers [a, b] on a number line.

- The location of an image created by reflection or refraction.

- The component of a vector expressed in vector notation along the x-axis.

- An atom containing 53 protons; Iodine.

- The moment of inertia for an object.

Plus I probably forgot some.

A similar exercise could be carried out with pretty much any letter in the Roman alphabet. Conclusions? I think that they should teach mathematicians and physicists kanji.

-SB