The Boundaries of Communication: What Defines a Language? Can Mathematics, Logic, and Music be Considered Languages?

Josh Farr

IB2 Theory of Knowledge

Dr. D. Churchward

Final Word Count: 1,435

Mathematics, logic, and music are three very different disciplines with one common thread between them: their role in transferring information. In terms of the acquisition of knowledge, all three of these fields can aid an individual’s intellectual enlightenment. However apt mathematics, logic, and music are at conveying information, they are not means of direct communication per se. This is what separates these media from verbal language. In order to test any of these three subjects, one must ask the questions, “Can I have a fluid and continuously variable dialogue using this form of expression?” and, “Can I express and argue my opinion through this medium?” In doing this, it is possible to specify the definition of “language” and therefore assess any of the three subjects’ validity as a language. When one applies this process to the question at hand, one finds that although all three methods of communication have their limitations, logic and music can almost definitely be categorized as a form of language, while mathematics is a more challenging entity to classify.

Mathematics is a difficult expression of information to define since it fulfills some of language’s requirements, but fails in others. Although it has been dubbed the “universal language,” it cannot accomplish the same things that verbal language can. When the aforementioned questions are applied to mathematics, the first question’s answer is “no.” The second question yields a positive response with one caveat: mathematics cannot express emotion. This partial satisfaction of a language’s requirements muddles mathematics’ status. While it is possible to express almost anything mathematically (with the exception of emotion), it is not possible to have an instantaneous and adaptive exchange of ideas through this medium. It should be noted, though, that it is at least somewhat possible to accomplish this in a more basic way when dealing with theoretical mathematics. For example, if one mathematician devises a formula ‘proving’ the existence of a theoretical entity such as anti-matter, he can ‘prove’ his assertions mathematically, thus validating them. That is to say, he can express his concept mathematically, and use established mathematical truths to make a compelling mathematical statement regarding his theorem. However, it is entirely possible for another mathematician to ‘disprove’ the first’s theorem by introducing a new variable or scenario that the first’s formula cannot account for. In this regard, it is possible for mathematics to harbour the essence of dialogue, but not engage in it as instantaneously as with verbal language. Mathematics cannot spontaneously digress from its objective like natural language can. Moreover, in the above example of anti-matter, one cannot ‘prove’ or ‘disprove’ anything beyond the shadow of a doubt, since the subject matter is intangible to begin with and cannot be studied without mathematics. Returning to the governing rules or “truths” of mathematics, these limit mathematics’ ability to provide what is necessary for adaptive conversation. While it is easy for any skeptic to argue that one plus one does not equal two, by using the nebulous argument that truth is an individual concept and cannot account for each person’s own perception, the real-world application of such essential truths debunks such dubious sentiments. In an elementary example, if one was in possession of a banana and purchased another banana from a fruit vendor, one would have two bananas to eat. One banana plus another banana equals two bananas. Since the world’s counting system has been standardized over its thousands of years of existence, one plus one equals two is undeniably a universally accepted truth. While such truths are essential to mathematical and real-world calculations, they nonetheless impose limits on this field’s flexibility as a conversational and interpretive language. Even though mathematics can be employed as a worthwhile tool to hypothesize logically about the indefinable and indescribable, verbal language offers looser parameters in the gathering of information due to its argumentative qualities, and is therefore most often better suited to the acquisition of knowledge. In this regard, mathematics is ultimately not a language.

This is a preview of the whole essay

The difficulty in determining logic’s classification as a language is that it is most often used as an extension of natural language. Oftentimes during an argument, people will use logic in conjunction with verbal language since logic is a natural way for people to express and contend their opinions. Despite it being mostly embedded in verbal language, logic is in fact a language unto itself. Logic, when used in conjunction with verbal language, correlates various elements of an argument in order to reach a reasoned conclusion. Therefore, logic can be used to express one’s opinion cogently and effectively. If we apply the aforementioned “test” questions to logic, “can I express myself through this medium?” and “can I use logic in a fluid conversational context?” we find both answers to be, “yes.” Logic is usable by opposing sides of an argument, and can be applied almost instantaneously. A simple form of a logical statement is the “since, then” statement. In a scientific context, one may say:

Since the pot of heated water reached a maximum temperature of 100 degrees Celsius before emitting steam, and we know that the boiling point of a liquid is the maximum a liquid can be heated before it changes state to a gas, then the boiling point of liquid water is 100 degrees Celsius.

This statement utilizes logic in combination with universally accepted scientific truths (the definition of “boiling point”) through verbal or written language – a.k.a. natural language – to reach a conclusion and prove an opinion about boiling water. Conversely, it is possible (but difficult in this example) for the interlocutor to counter-argue:

Since the pot of heated water reached a maximum temperature of 100 degrees Celsius, and we know the definition of the term “boiling point,” and we know that boiling point is a universal scientific standard measured at a universal constant: sea level, and we know that this experiment was conducted in Toronto, a city approximately 600 feet above sea level, then the boiling point of water is not necessarily 100 degrees Celsius.

Although this is a weak argument since an elevation of 600 feet does not substantially affect the boiling point, it is nonetheless valid. Both of these statements are logical, but opposed in conclusion. This virtue of logic gives it the necessary attributes to define it as a language.

Of these three forms of expression, music is by far the most difficult to define as a language. One reason for this is that music takes on many forms. It can involve only one instrument or human voice, many instruments and voices, verbal language or nonsensical noises, and even a simple or complex rhythm alone can be considered music. An accepted boundary of music is that it must contain at least one of the three following elements: rhythm, melody, and harmony. Modern “jungle” music, for example, is nothing more than a complex rhythm with no instruments other than digitally manipulated drums; yet it is still regarded as music. Some rap and hip-hop songs are as simple as rhythms accompanied by a singer reciting – not singing – words atop the beat. There is a plethora of genres that continue to challenge what defines music, and because of this it is most important to examine how music as a language expresses itself internally as well as to the audience. When one asks if music can facilitate a free-flowing and adaptive dialogue, it is possible to say that it does, so long as this dialogue is contained within the composition itself. If one considers musical instruments to be extensions of their players’ own voices, it is possible in jazz improvisation for example, to have a ‘conversation’ between the instruments, and therefore their human operators. This dialogue is not articulated into words, of course, but an assemblage of human players can express themselves through their instrument via musical methods. Different combinations of notes – harmonies, dissonants, minor and major scales – evoke different emotions from listeners, thus giving music a sense of ‘mood’. When this is coupled to the spontaneity of improvisational music, it is possible to represent a dialogue through this medium. It is important to note the significance of the word “represent,” since the ‘mood’ of a song is merely a representation of a feeling through sound, nothing more. Barber’s “Adagio for Strings,” for example, uses combinations of notes that usually prompt a sorrowful and melancholy response from the listener. However, the piece is nothing more than a particular arrangement of notes – it itself is neither sorrowful nor melancholy, but is merely a cluster of sounds. Although music cannot specifically phrase emotions or opinions since it is without a detailed and specialized lexicon, it is both expressive and conversational; albeit in a more basic form than natural language. Because of this, it is possible to classify music as a language.

All three of these means of expression – mathematics, logic, and music – are forms of communication. While could be possible to classify all three as languages in one interpretation of the word, the concept of “language” can be nebulous and therefore requires a stricter definition in order to arrive at any sort of conclusion. For this reason, asking whether these forms of communication are expressive, and whether they have the capacity for spontaneous and adaptive dialogue, is necessary. In applying this argumentative filter and examining examples of each, one finds that only two of the three subjects, logic and music, provide clear “yes” answers to both the above questions. Upon further scrutiny, one discovers that all three of these forms of communication are limited in one sense or other. Mathematics cannot describe emotion, logic cannot be expressed without written or verbal language, and music lacks the descriptive articulation necessary to directly communicate with its audience. Although each of these three items have the capacity to express themselves in one form or another, they are ultimately limited and leave natural language as the superior form of communication, and only true language.