The Language of Mathematics

We learned from the last chapter that mathematics is indispensable. People need it to survive and to catch up with the fast-paced world. Hence, we need to understand mathematics. We need to speak mathematically.

Learning a new language takes time. A student who wants to speak a new tongue, say Spanish or Korean, needs to dedicate part of his or her time in learning its vocabulary and grammar. Most likely, this new language has its own way of communicating expressions and sentences. For R.E. Jamison [3], the mathematical language has three main features:

The following are definitions of a "subset". The first one is taken from an online English dictionary (Oxford Learner's Dictionary); the other one is from the book Discrete Mathematics and Its Applications by K.H. Rosen.

1. subset - "a smaller group of people or things formed from the members of a larger group" [1];

2. A set A is a subset of a set B if and only if for any element x of A, x is also an element of B [2].

The first definition is given in common English. This is how that important terms in other fields would be defined; but not in mathematics. In mathematics, definitions need to be precise and straightforward. The first definition is full of ambiguities and may only result to inconsistencies later in the subject. For example, with the first definition, the words "smaller" and "larger" would signify that the set B has more elements than A. This would be inconsistent in a later idea that every set is a subset of itself. Further, the first definition will not be able to logically imply that the empty set is a subset of any set.

From this point forward, we will refer to the usual human languages such as English, Filipino or any dialect as "common language" or "ordinary speech".

Sometimes, due to these unique features, young students find it hard to adjust into speaking mathematically at first because most of them are used to the common language which is time-dependent, full of emotions, and ambiguous. With a little time and effort, everyone could surely understand and speak the mathematical language.

Furthermore, C.F. Burns says that mathematical language has three characteristics [4]:

• precise (exact and clearly identifies differences);

• concise (short but comprehensible); and,

• powerful (able to express complex thoughts).

In addition to being a unique feature, precision is also a characteristic of the mathematical language. Precision refers to the exactness of mathematical sentences and expressions without the presence of ambiguities. Ambiguity is common in ordinary speech. For example the statement "foreigners are hunting dogs" is ambiguous because it is not clear whether dogs are being hunted by foreigners or that 'hunting dogs' describes the foreigners. This example was taken from LiteraryDevices.net and you can see more examples from here.

In mathematics, the most important thing to check to guarantee precision in our statements is differentiae specificae which literally means "specific difference". In other words, when we define terms, or establish assumptions or state theorems, we have to specifically point out the difference that distinguishes one from another.

A concise statement is a statement that is short but comprehensive. A mathematical idea has to be stated completely but using the minimum number of words or symbols needed. In ordinary speech, we sometimes add words or phrases to make our statements sound more sophisticated [6] but sometimes the addition just makes the statement confusing.

For example, "At this point in time, let us welcome the man that I am about to call, Mr. Kenneth Lim." This could be greatly improved with, "Now, let us welcome, Mr. Kenneth Lim."

Some pointers to be concise:

• If long phrases could be simplified into fewer words, do so.

• Try to remove phrases or words in the sentence and see if it does not change the meaning. If it doesn't then you could remove them.

"If an odd number is added to another odd number, then their sum is an even number."

Not concise: The way the first phrase is stated is a long way of saying "the sum of two odd numbers". Also, since it is clear that we are all talking about numbers here, there is no need to say the word the second time.

A concise version would be"

"The sum of two odd numbers is even."

The idea is, for two statements with exactly the same mathematical meaning, choose the simpler one.

Another way that mathematics becomes concise is the use of symbols and notations. For example, the Pythagorean theorem, if stated in just words is:

"The sum of the squares of the two shorter legs of a right triangle is equal to the square of the hypotenuse."

But with the aid of symbols it could be simplified:

In the common language we are used to hearing "no words can explain" from someone who thinks that what he thinks is too complicated to express it verbally. This does not happen in mathematics (provided the math is already understood). A powerful language is a language that could express complex ideas with relative ease.