The Language of Mathematics

Lecture 04

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:

It is non-temporal.

In other words, it is time-independent. It has no past, present, or future tense. When stating definitions, axioms, or theorems, we always use "is" --- and this "is" doesn't necessarily stand as a present form of the verb.

For example, when reading the equation x = 1, we always say "x IS equal to 1" or "x equals 1". We do not have to say "x WAS equal to 1" or "x WILL equal 1" but if we do, these statements have absolutely no distinction from the first.

It is devoid of emotions.

Mathematical literature is different from artistic literature. In the latter, we write in a way so that our readers would feel the way we want them to. For example, the Arnold Schwarzenegger quote "if I can see it and believe it, then I can achieve it!" is full of hope and determination and maybe courage. But in mathematics, statements are emotionless: "if is n even, then n + 1 is odd."

This may sound dull and boring for some but this lack of emotions in mathematical statements helps it to be more plain and straightforward. However, this may only apply to written mathematics such as books and research articles. In oral mathematics, such as lectures or presentations from math instructors or speakers, statements may be added color to make them lively. For example, a math teacher might say, "multiplying x by y will ZERO IT OUT" to mean the equation xy = 0 (the phrase "zero it out" is not an actual math term).

It is precise.

"Ordinary speech is full of ambiguities, innuendoes, hidden agendas, and unspoken cultural assumptions." Mathematical expressions and sentences are always clear. It is not like the common language wherein there is no clear distinction between synonymous words such as "heavy", "massive" and "hefty", or "big", "large", and "great". In mathematics, we can always identify the difference between closely related terms such as "homomorphism" and "isomorphism", or "maximum" and "supremum".

In this lecture we will see how mathematical language differs from common languages like English or Filipino. Then, we will start expressing ideas in the mathematical way.

Characteristics of the Mathematical Language

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.

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

Mathematics: a Precise Language

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 square is a polygon with four sides."

Not precise: Trapezoids and kites are polygons with four sides too but they are not squares.

"A square is a quadrilateral whose sides are all congruent."

Not precise: A rhombus is a quadrilateral with four equal sides too but it's different from a square.

The precise version of this statement would be:

"A square is a rectangle with all four sides congruent."

The above definition clearly distinguishes the square from all other polygons. Classifying it as a rectangle identifies the square a quadrilateral with all interior angles being right angles. Then, the second part of the statement "...all four sides congruent" tells how a square distinguishes from all other rectangles.

Six quadrilaterals of different types | (c) jim.belk - Wikipedia

Mathematics: a Concise Language

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:

"If a and b are the legs of a right triangle and c is its hypotenuse, then

Mathematics: a Powerful Language

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.

For example, before 1687, the idea of gravity was still so complicated. People did not understand why objects fall and the moon does not. Galileo already had some findings and other physicists already have their theories but despite these, the concept was still so difficult to explain. See here to read more about how people tried to explain "gravity" before Newton. But, when Newton decided to express the concept of gravity mathematically, it was so much easier:

Notice that Newton was able to pack thousands of years of trying to explain gravity into a single equation not more than one inch long.

References

[1] Oxford Learner's Dictionary. Subset. from https://www.oxfordlearnersdictionaries.com/us/definition/english/subset?q=subset

[2] Rosen, K.H. (2012). Discrete mathematics and its applications 7ed. McGraw Hill.

[3] R.E. Jamison. Learning the Language of Mathematics. Clemson University

[4] C.F. Burns. The Language of Mathematics. One Mathematical Cat! Please.

[5] Literary Devices. Ambiguity. from https://literarydevices.net/ambiguity/

[6] University of Illinois Springfield. Writing Clear and Concise Sentences. from https://www.uis.edu/cas/thelearninghub/writing/handouts/grammar-mechanics-and-style/writing-clear-concise-sentences/

Review Questions

What are the three features of the mathematical language according to Jamison?
What does it mean for a language to be precise? Give an example of mathematics being a precise language.
What does it mean for a language to be concise? Give an example of mathematics being a concise language.
What does it mean for a language to be powerful? Give an example of mathematics being a powerful language.

Activity

Activity 04 - Math vs Common Language

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