I will try to compare natural human languages and mathematical language in an attempt to answer the question: "Is Maths really a language?"
This is a very big question that would require a great deal of writing. My purpose, however, is not to make you feel bored, but to spark your curiosity about the topic so that you will continue exploring it on your own. That is why I will make this comparison only briefly.
Mathematics is very often called a language, and it is true that it is used as a tool in many other areas of human knowledge. Think of engineering, science, geography and almost every other discipline—they all use mathematics in one way or another. It seems that the language of mathematics exists to serve other fields by offering them a wide range of tools: arithmetic, statistics, graphical methods of representing knowledge, probability for risk assessment, geometry for construction and engineering, and algebra, which is sometimes so annoying with its constant effort to find x in our lives.
But this is exactly what we do very often in our own lives—we become detectives trying to solve our own little mysteries. So we can agree that mathematics is present in every aspect of our personal and professional lives.
What is a natural human language, then? By natural, I mean languages that are spoken by people and have developed over many centuries. They may be official national languages or dialects, which are spoken locally and do not have official status. There are more than 6,000 languages in the world, and many of them are disappearing, mainly because of migration. Some languages are spoken by hundreds of millions of people, such as English, Arabic, Russian and Spanish. Others are much smaller, and some are even micro-languages.
I spent part of my university career researching one very small Slavic micro-language—the Upper Sorbian language—and I even wrote two books about it. It is spoken in eastern Germany and is considered an endangered language, with only about 50,000 speakers. Most of its speakers are bilingual and also speak German. Their cultural centre is the town of Bautzen, about 60 kilometres from Dresden. The Sorbian Institute is located there, and I visited it many times while carrying out research for my books.
What is the definition of a natural human language? Since we want to compare mathematical language with human language, we need a starting point. We first need to make sure that the things we are comparing are, in fact, comparable.
Let us use the following as a working definition:
"Natural human languages have a structure consisting of many systems of symbols and signs. There is a hierarchy among these systems. The main function of human language is communication; people use language to communicate with one another."
This is not a perfect definition, but it gives us the main characteristics of language, and we will try to discover whether mathematical language possesses the same characteristics. In other words, we are looking for similarities and differences.
Human languages consist of systems of symbols organised in a hierarchy. Their structure is like a pyramid. If we fail to learn one level, we cannot successfully build the next. We begin with the alphabet, which gives us letters, but behind every letter there is a sound. We combine sounds and letters to form words. We connect words according to the rules of grammar, semantics and syntax to create sentences. We then connect sentences to create a text.
The text is the highest level of language. One possible definition of a text is: "A text exists when at least two sentences are connected both semantically and syntactically."
To summarise, the hierarchy of language includes letters, sounds, words, sentences and texts. These levels are connected through the rules of grammar, semantics and syntax. This is, of course, a simplified explanation, but we do not need anything more complicated here.
To say that you know a language, you need to be able to listen and understand, to read and understand, to speak and to write. A teacher who teaches a language should understand all of these levels. Otherwise, it would not be fair to the students.
One more important point in our definition is the communicative function. This is the primary function of natural human languages—they developed because people needed to communicate through systems of symbols. Languages also have other functions, such as the creative function, but we are not concerned with those at the moment. Human languages are highly complex systems that develop according to their own internal rules. Most of the time, linguists simply observe and record these changes; they cannot control or direct them.
What about mathematical language? Does it resemble natural human languages? There are different opinions, but we can certainly identify both similarities and differences.
Let us begin with the similarities.
Mathematics also uses symbols and signs, and it has a hierarchical structure. Just as in natural languages, if we fail to learn one level, we cannot successfully build the next. We begin with digits and numbers. These combine with operation symbols—such as addition, subtraction, multiplication and division—to form expressions, equations, formulae, identities and inequalities. Behind all of these lies meaning. They communicate ideas, relationships and intentions.
In both natural human languages and mathematical language, we can even identify something similar to parts of speech. Think about the parts of speech in English. In fact, they are broadly similar in most languages, although there are some differences: nouns, verbs, adjectives, pronouns, adverbs, prepositions, conjunctions, particles and interjections.
We can speak about the "parts of mathematical speech" in a similar way because they perform comparable functions.
Digits and numbers are similar to nouns because they represent objects or quantities.
Operation symbols represent actions and therefore resemble verbs.
Relation symbols, such as equals, greater than and less than, express relationships and comparisons. In natural language, they are closest in function to adjectives and adverbs.
Grouping symbols, such as brackets, organise mathematical ideas in much the same way that punctuation organises written language.
Variables, or placeholders, represent unknown quantities and therefore resemble pronouns.
Mathematical language is precise and exact, yet at the same time highly abstract. When we write the number 4, we are not referring to any particular object. It may represent four tiny particles or four galaxies. The symbol itself is independent of the object it describes.
Problem-solving is similar to a complete text in a natural language because it demonstrates how mathematical knowledge and skills are applied together to communicate a complete idea.
What, then, are the functions of mathematical language? Can it be used for communication? Certainly. Mathematical language allows us to communicate knowledge, intentions, logical arguments, complex ideas and abstract concepts. That is precisely why it is used across so many different fields of human knowledge.
Now let us consider some of the differences.
By convention, mathematical language is not normally used for everyday communication. If people agreed on a syntax capable of expressing every meaning, perhaps we could even write poetry in mathematical language. That would be a fascinating challenge because poetry relies so heavily on metaphor, imagery and comparison. It is certainly not the fault of mathematics that it developed along a different path from natural human languages. We could say that mathematical language exists primarily in written form.
So, together, natural human languages and mathematical language allow people to express the full range of human thought—from the most rigorous logical reasoning through mathematics to the richest poetic imagination through natural language.
I hope this brief comparison has made you think about questions you may never have considered before. If it encourages even a few of you to look at mathematics from a different perspective, I will be happy to know that I have helped open your eyes to some of the more unusual aspects of mathematical knowledge. Thinking, together with love, is one of the greatest joys we can experience.
I have one activity for you.
Use your own native language and compare it with mathematical language. Choose several features to compare, make a list of them, and present your ideas in a Venn diagram. For example, if you speak French, your Venn diagram should compare the French language with mathematical language. It will be interesting to see your diagrams. The intersection of the two circles should contain the features that both languages have in common.
In our lives, we are like walking human Venn diagrams. Relationships can sometimes be complicated, especially when there is no area of intersection between two people. It is a sad situation when two people simply do not fit well together because they are like two circles in a Venn diagram with no overlap. Yet we can change this by making a conscious effort to build that shared area in our relationships—the things that connect us, the things that unite us. Peace in the world depends on the size of the intersection between us.
Let us hope that you have learned something new about mathematics and that you now love both mathematics and languages even more.
(E. S. Lyubenova, LoveMaths Story for My Students)
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