IB Theory of Knowledge (TOK) Paper
Are Mathematics, Logic and Music Languages?
Mathematics, logic and music all play rather important roles in the lives of most people and are used quite frequently to communicate certain things. However, does the mere use of these forms of admittedly limited communication qualify them as languages? I shall attempt to show that two, music and mathematics, act as languages by themselves, as well as become incorporated in the larger scheme of spo ken language, for which they serve integral roles. Logic is not a language by itself, but rather aids in the language process and provides a type of playing field for all types of language.
Mathematics is perhaps the more straightforward of the two to understand as a language by itself. It has a set of symbols which are used to convey certain messages and has rules which it follows in which one can convey these messages. These messages, whether they are as simple as “1+1=2,” or more complex like “º 3×2 + 2x + 3 dx = x3 + x2 + 3x + C,” can be understood by anyone in the world who has a basic working knowledge of mathematics. Needless to say, a more advanced knowledge of math is required for the more complex aspects, but the point remains the same: math fulfills the requirements of being a language. It works with a set of symbols to convey a message which can be understood by anyone who knows how to interpret the symbols.
Most may not think of math as a language, since it doesn’t work in the same manner that spoken languages work. While it is true that math cannot convey things such as emotion, it is quite useful in explaining most everything else. Superficially viewed, one might think of numbers as only being useful for describing quantity. However, when reminded of the role that physics plays in the natural world (which math in an integral part of), it goes much deeper. Math can describe motion, color and sound, among other things, in a purely mathematical form, which again, can be understood by anyone who is fluent in the language of math. Math reflects the aspects of human existence which are pure reason.
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Music is perhaps a little less concrete in its existence as a language. To best discuss music, I will divide it up into two parts: musical notation and actual sound produced. Musical notation is a lang uage in much the same way that math is. On a sheet of music, one finds symbols which represent actions: how fast one moves one’s fingers, how loud a tone one produces, what frequencies of tones are produced, how long those frequencies will last and, in forms of music such as jazz, what frequencies one can choose to play from. Interestingly, one can see aspects of mathematics creeping into music through musical notation. Again, like math, these symbols are recognizable to anyone who can understand the “language” of musical notation.
The actual musical tones are on a different plane altogether. The mere notes on a sheet of paper do not comprise music—any musician can tell you that. Music is the personal interpretation of those notes. Where mathematics reflects the pure reason which can be found the human experience, music reflects the pure emotion which is present in the human experience, and it does so in a manner indescribable in spoken or written words. In so ngs such as Percy Grainger’s arrangement of “An Irish Tune from County Derry (Londonderry Air/Oh, Danny Boy),” and Claude Debussy’s “Claire de Lune” emotions are stimulated by the juxtaposition of different tones, the dynamic contrast of said tones and the personal inflection placed upon them by the musicians, which evoke feelings which can best be associated with “sadness” or, in the case of Grainger, the more vague “so beautifully stated that one could cry” feeling. In contrast, Sousa’s “Liberty Bell” and Tchaikovsky’s “1812 Overture” produce feelings related to being spirited and proud or “patriotic.” However, using words to describe these emotions is pointless, as they cannot adequately describe the exact emotion felt by the listener. For this reason, music becomes a language almost by default—the feelings expressed in music are best understood merely by listening to the music and these fee lings cannot be replicated by spoken or written language. Much in the same way that music cannot equal the physically descriptive power of spoken language or math (i.e. musical notes cannot accurately describe a red pencil box, whereas one can verbally describe it or express it in numbers which represent the dimensions of the box, the frequency of light which it absorbs and the number of pencils in the box), spoken language and math cannot equal the emotionally descriptive power of music. To attempt to write about music is about as useless as dancing about architecture.
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Music and mathematics are not confined to their fields, however. Both frequently play a part in verbal language. When one speaks, certain changes in pitch are noticed. When one states something, the voice slightly drops in pitch. When one asks a question, the pitch rises slightly. Here we see music entering into our spoken language; change in pitch occurs throughout everything we say and gives our speech color as w ell as meaning. Were a person to speak in strict monotone, not only would he be excruciatingly boring, but his ability to communicate would be lost. In the same manner, mathematics enters into spoken language through rhythm. While change in pitch gives speech color, rhythm gives it order. The necessity of rhythm in language is reflected in how we perceive speech with no meter. When one speaks with no pauses between words, clauses, phrases and sentences, language is soon turned incomprehensible. A perfect example would be a Gilbert and Sullivan patter song. While the song “I am the Very Model of a Modern Major General,” from The Pirates of Penzance, has pitch present, the words are strung together in sixteenth note patterns and sung rather quickly, leaving the audience confused about what exactly the Major General is trying to say.
Both pitch and rhythm are aspects of music and mathematics in spoken language and they lend their traits to what would otherwise be in comprehensible, boring sounds and lines of symbols on paper. The presence of music in spoken language does more than make what we say interesting; it gives what we say emotion. The presence of mathematics makes language comprehensible and gives it meaning in a strictly scientific way. Here enters logic.
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As I have briefly mentioned before, logic provides a playing field for music and mathematics. It relies too much on actual spoken and written language to be classified as a language on its own, however, it works as a filter through which everyone experiences aspects of music, mathematics and natural language. Earlier, I mentioned that math works within a system of symbols and rules. Logic is what governs those rules. Were it not for logic, “2+2” could equal 42. Logic can also be found in music. In most traditional music, a song ends on the tonic, or first tone of the scale. Anything else sounds “off.” Chords, in music, have certain structures and go with o ther chords, again showing a set of rules inherent in the language of music as a result of logic. Even in modern music when a song doesn’t end on the tonic, or a chord resolves into a chord less mellifluously than expected, the composer works with some kind of logic in order to produce desired emotional results from the listener. Sometimes, however, the logic of the listener does not exactly match the logic of the composer and one is left with hearing noise, as many people experience listening to music by John Cage or “Revolution 9,” by the Beatles.
Having now examined mathematics, music and logic we can clearly see that the first two are languages of sorts by themselves; mathematics describing the scientific aspects of the human experience and music describing the emotional aspects. Both operate within sets of rules which help them function and facilitate understanding of both. The two play into spoken language through pitch and rhythm, which add color and coherenc e to the words humans use. Again, we see logic’s important role. While speech can include rhythm and pitch, these things need to happen at certain times in speech, or else the words spoken would be as incomprehensible as those without pitch or rhythm. Thus, one can see that while music and mathematics are important to language, logic dominates, turning random numbers into the scientific laws which govern the human experience, arranging random pitches in some sequence to express the emotions felt in the human experience and transforming nonsensical babble into words to describe the human experience itself.
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