Mathematicians need to be clear and concise when they communicate. As a result, mathematicians are extremely attentive to the foundations of language and logic. Over the years, they developed their own conventions for communications. These conventions have evolved to a point that mathematics can be viewed as a distinct language.
The conventions developed by mathematicians are extremely powerful. The language of mathematics is better at communicating quantitative information than day to day language. As a result, scientists have adopted the conventions of mathematics, and you will often hear professors describe mathematics as the language of the sciences.
In this article, I will explore this concept of mathematics as a language. I will begin by briefly skimming over a few of the elements of communication. Then drop a few lines about more complex subjects such as symbolic logic, equations, and computer programming.
For the most part, I see the evolution of mathematics into a language as a positive development. However, there are a few excesses. Far too many people see mathematics as a language. They view it as a foreign language. In many ways the language of mathematics is an elitist language, and as an elitist language it can do more to hamper communications than to enable communications.
The goal of descriptive mathematics is to find ways to bring this extremely powerful tool of mathematics into our day to day lives. To accomplish this goal, we need to understand how mathematics works as a language, and how we can best make use of this knowledge.
The structure of a language is limited by the medium of the language. Prehistoric man learned to communicate with their natural abilities and senses. The first languages used the innate senses of hearing, sight, feeling and smell. As a result, prehistoric man learned to communicate with sound, gestures, sensations, and smells.
Of course, some forms of communication proved more successful than others. Smell is an extremely limited medium. It is difficult to control, and has a tendency to linger. Smell is okay for communicating very simple ideas such as I want sex, I am weak and sickly, or please don't eat me--I taste bad. Smell is not a good way to express complex ideas, such a proof of an integral equation.
Feeling is fun, but you have to be in close proximity to communicate through touch alone, and again it is difficult to communicate complex ideas with a massage. People have learned to read in Braille. Unfortunately, my sense of feel is weak, and I find it difficult distinguish between the different letters. Since reading with feel is more difficult than reading with sight, it is less common to communicate with feel alone.
Sight is, without a question, the most powerful of the senses. We can take in more information with our eyes than we can through any other medium. With sight, we can communicate in many different ways: we communicate ideas through gestures, drawings, and writing. Sight is the most powerful sense, but the power is somewhat overwhelming, and it is difficult to develop and express ideas through sight alone. Sight also stops working at night, when it is dark.
Hearing is not as strong of a sense as sight. However it is easy to control the sounds we make, and our primary languages evolved around the sense of hearing. People learned to make different sounds with their vocal cords and lips. From these distinct sounds, we learned to string together words, sentences and paragraphs, stories, poetry and songs.
Spoken language is linear in nature. As a result it is a great way of communicating information about linear events, like stories and history. Because there is a defined start, middle and end to a speech, we can create communicate logical inferences between the different parts of a story.
Our primary language is the spoken word. However, people have always found creative ways to communicate with with sight. Prehistoric people carved images into walls, and wove symbols into their tapestries, and scraped crude maps in the sand.
Straight graphical representation is pretty much like the art of perspective drawing. When you create a map, or pictographs on the wall, you are pretty much trying to create a full image of the subject, or at least a fair representation of some attribute of the images. Early pictographs would show a creature with four stick legs, and curly horns to represent a mountain goat, or a spiral next to a stick figure to indicate a person bit by a snake.
Full graphical representation tries to create a picture to represent an idea. As an amateur artist, I admit, this act of trying to communicate with images is extremely time consuming and difficult.
When I visit the desert canyons of Southern Utah, I imagine that artists used pictographs as the focal point for telling stories. Many of the most interesting pictographs are on the walls that can serve best as a stage for acting out dramas and plays. I imagine the artist as a singer using a combination of visual representation and sound in tribes ceremonies and rituals.
As people struggled to communicate with graphical images, they began to develop symbolic languages. In a symbolic written language, the symbols have very tight, well understood meanings. Often symbols would still carry a significant attribute of the thing they represented.
In many symbolic language, there are symbols, or combinations of symbols for all the words in the language. Many of these symbolic languages survive to this day. For example Mandarin and Japanese use symbolic languages.
Please note, all languages are symbolic in nature. A spoken languages uses words and sounds as the symbols. There are many books which explore the history and nature of symbols.
One of the greatest strengths of a symbolic language is that you can often convey meaning through the mechanical manipulation of the symbols. For example, when you do long division on a chalkboard, you are basically using the mechanical manipulation of the symbols to simplify the work.
Much of the interest in developing symbolic logic was based on the affects you could achieve through the manipulation of the symbols. When you study symbolic logic, you will find ways to reduce arguments to small set of symbols. By studying these symbols, you can determine if the argument is logically consistent.
The fact that you can mechanically manipulate symbols is not that surprising. The physical symbols we scribe on a piece of paper often share the same logical principles as the subject we want to study.
Mechanical manipulation of symbols can often uncover mistakes that we made when we try to reason through a complex problem. However, we must realize that mechanical manipulation is not in itself superior thinking. It is a tool that we can use to broaden our understanding of a subject.
Think of it this way. We can understand machines. Machines can't understand us. When you work with mechanical devices you are in charge. You define a meta language that control the overall action of the machine.
Symbolic languages are great way to learn and communicate. In my opinion, though, one of the greatest advances in communications came with the development of phonetic written languages. With a phonetic alphabet, you use a small set of characters to denote the different sounds that people make while they speak.
Writing in a phonetic language can bee seen as the act of transcribing the spoken word on to a piece of paper. A phonetic written language is really just a way of transcribing a spoken language on to paper. As I write, I move my lips and even sound out the words. I could take this work, and read it in front of a class as a lecture. Simply by sounding the words, others could read the article out loud as well.
One of the big advantages of a phonetic written language is that it allows us to study our spoken language in greater detail. The phonetic written language and our spoken language support enhance each other.
It is interesting to note, that after the development of the printing press, and formalization of the phonetic language, the actual evolution of the language slowed dramatically. Fifteenth century French is much as it is today. English, on the other hand, was in a state of flux when people first formalized the rules for the phonetic written language. As a result, English as a very arcane set of pronunciation rules.
Phonetic written languages help support and enhance the spoken language, and have been highly successful in communicating information. Since the alphabet has fewer characters, it is easier to print and inscribe into computers.
There are advantages and disadvantages to both symbolic and phonetic languages. When you get down to it. Both types of writing are capable of expressing extremely complex ideas. I would even go as far as to say that there are no ideas that can be expressed in a phonetic written language that could not be express in a symbolic language, and visa versa.
In the western hemisphere. People standardized on phonetic language for most communications.
In mathematics, however, people learned to use a combination of symbolic means for representing numbers, and graphical representation for expressing geometrical ideas.
When you use a limited set of symbols, you are able to convey more information in a smaller space than with a standard written phonetic language.
Again, I should repeat. Mathematicians strive to find the most effective ways to communicate their ideas. By optimizing their conventions, they can drill to the heart of complex matters more quickly than they could if they tried to write all of their work out long hand.
During a lecture
Communication is Key
Mathematics makes use of both phonetic and symbolic languages. A good mathematician will find the best means of communicating their ideas to their audience. A good communicator will use all tools at their disposal. They will write equations on the chalkboard. They will scribble notes on paper. They will gesticulate wildly, and use the vulgar language of day to day life in their lectures.
In some cases, symbolic logic is more efficient than vulgar spoken languages, and you can often explore ideas simply by the mechanical manipulation of the symbols.
I attended several lectures where the professor introduced symbolic logic as a superior language to the common tongue. It was not subject to the vagaries of ill defined words. It was accepted as an international convention...
However, I could not help notice that, despite all the praising of symbolic logic as a language, the professors still used the common language for describing their work. Yes, the professor wrote all of the proofs on the chalkboard in symbols, but the professor described the proof with a spoken language.
The conventions of symbolic logic are extremely powerful, but the spoken language encapsulates the symbolic language.
To ask which is superior, a symbolic, or phonetic language. We are fortunate to be able to use both. The questions I face is, how best do I communicate my work. The answer is to use a combination of written phonetic words, graphical representation of information, and certain symbolic conventions of mathematics.
Which is the best combination. Well, I think it is a matter of trial and error to find what combination best conveys meaning to my audience.
Mathematicians pay careful attention to the language they use to communicate their ideas. Since they have the need to communication a large amount of extremely complex information in a short amount of time, Mathematicians work to find the most powerful ways to communicate. They take advantage of spoken language, phonetic languages, graphical images and symbolic language.
Mastering mathematics is as much about learning a new language as it is about exploring ideas. However, mathematicians may have gone too far in the development of mathematics as a separate, foreign language. The challenge of the mathematician is not simply thinking up harder and harder proofs, but the challenge of finding ways to communicate information.
The next essay is the sad, sad fable of the Emperor's New Language.
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