"I am not told. I am the verb, sir, not the object,"
—Alan Bennett, The Madness of King George
One of the regular frustrations of studying for this blog comes from the number of papers I read by people who argue as though, because language and mathematics both manipulate symbols, they can both be described by the same generalizations. They cannot. Take for example the differences that arise from the presence of verbs in language and their absence in mathematics.
If we think of a sentence as working like a solar system, then the sentence's sun is its verb. The sentence gets its dynamism from the verb's gravity and all parts of the sentence are related to one another through the verb. Mathematics lacks that kind of unifying authority..
By "the verb's gravity," I mean the verb's ability to enliven a group of static phrases. To see this dynamism at work, take three noun phrases (in alphabetical order)—Dave, John, the ball—and add a verb: John feeds Dave the ball. It is the verb that determines which phrase goes where and clarifies the relation between the noun phrases.
Now find a mathematical formulation that does the same thing. Equations are so static that the task might seem hopeless, but if we replace the equals sign with an arrow to indicate the changed result, we can get somewhere. Let J stand for John, D stand for Dave and B for the ball: J -B-> D. We can easily imagine some notation conventions that uses the arrow to show that B begins at J and ends at D. The arrow might be interpreted as indicating a number of motions: feeds, throws, gives, bounces, tosses, fires, dribbles, sends, shoots, chucks, pegs, flings, flicks, hurls, lobs, and so on.
All those choices reminds us that verbs have an amazing specificity that goes beyond naming an action. The choice of verb can place an action within a moral, social, psychological, editorial, or intimate context. If the verb is feeds, we understand that the recipient is expected to do something with the ball: John feeds Dave the ball, who dunks it in the basket. Whereas John lobs Dave the ball puts the event in a more leisurely context: John lobs Dave the ball, and Dave waits impatiently for its arrival. The action of both of these verbs can be depicted with the same –B-> formulation, but the choice of verb enriches the action in a way denied the mathematical formulation. That enrichment is part of what I include when speaking of the verb's gravity.
Another difference between the sentence and the formulation is that we can change the subject and object, as in Dave obtains the ball from John and dunks it in the basket. This sentence describes the same event depicted in the previous paragraph, but obtains is not a synonym of feeds. The difference is in the way the verb assigns roles. Both sentences are outside the scene, looking on, but the first sentence makes John the actor while Dave is the actor in the second. Could we make the ball the actor? Sure. The ball traveled from John's hand to Dave's mitt. So assigning relations is another result of the verb's gravity that is not duplicated by mathematical symbolism.
If we change the formulation J -B-> K to K -B-> J we change its truth value. But what about the sentences John feeds Dave the ball; Dave obtains the ball from John; and The ball traveled from John to Dave (or …to Dave from John)? They might all be true or all be false, but clearly we have not finished considering a sentence when we have decided on its truth or falsity.
Mathematics is often praised for its objectivity and absoluteness, qualities that have proven very useful and which appeal to many tastes, but they come not from adding features to mathematical notation but by removing verbs and their gravitational effects. Without the selection of actors, the imposition of dynamic roles, and the enrichment of the action's context, any system would be left with a series of objective, static relationships.
Language has been around much longer than mathematics has been known. When we think about language, especially about old language and its origins, we should remember that language is not merely less abstract than mathematics. It is contextually richer and its propositions are far more dynamic. To focus on its use of symbols, its veracity, or its displacement is to examine language's least-special traits. Let us try to understand language, in its utility, its structure, and its origins, by keeping our eyes on how it is different from other ways humans communicate.
What brought on this rant? I have read one-too-many papers which take as proven the assumption that until non-verbal symbols appear in the archaeological record there can have been no language as we know it today. Granted, that paper was but a straw and a feeble one at that. But those straws have been piling on my head for years and at last my neck has snapped. Aough!