I write more for myself than for an audience. The act of writing flushes out ideas and provides a record of what I was thinking so that I do not spend time reinventing the wheel (by wheel, I mean my personal wheel, not new ideas to the world, which I am sure are few). Sadly, I often get hot about an idea, start writing about it, and then get too busy with other things to finish. As a result, my ideas are lost.
I spend a few minutes every few months erasing incomplete posts. Today, while I was clearing several such posts, I pondered about the wasted effort of the process and the added entropy to the universe each time I hit "delete." So, I decided to share this one with myself and anyone else who cares to read it.
To place the state of my mind in perspective, I was writing this post in the first half of August, 2011, just before the start of the semester, when I was slated to teach Classical Mechanics. I recall being excited about my insights on the topic, but sadly, I no longer recall the punchline. Perhaps one of you can help me out.
Here it is:
Language is often inadequate to describe what we are feeling. A far greater problem is that language permits imprecision and inconstancy. As a result, we are falsely lulled into a sense of meaning when there is none.
The classic example of self contradiction is the sentence: "This sentence is false." We can reject this construction for obvious reasons. However, consider the statement, "His action was immoral." The first three words are well defined, but the last is not.
At issue is the fact that many concepts in language are based on subjective feelings, that when assigned a word, may be imprecise or nonsensical yet carry an absolute sense of its existence. When analyzed dispassionately, we can surmise that the sense of morality is an inbred feeling that was shaped by evolution, and helped the survival of our species. Thus, when someone cheats, our sense of distaste stems from our collective disapproval of behaviors that weaken the group.
However, our gut assigns to the concept of morality a sense of absoluteness of "right" and "wrong" of actions - two additional words of the same ilk. Morality is thus elevated to an absolute standard that cannot be questioned. It is wrong for women to vote. Why? Because it is an absolute, and absolutes cannot be questioned. Thus, the sense of morality can lead to concepts such as women being mere property to serve at the pleasure of men, homosexuality as an evil, drawing a cartoon of certain individuals an objectionable action deserving of death, etc.
One may argue that without an absolute morality, humans would be lost and unable to decide what is right. Humans have been making the "right" decisions for ages without religion; but, this is not the point of this post. Instead, I want to speak about a different kind of language that does not suffer through the same pitfalls; but ironically, is responsible for the development of imprecise language; and that is, mathematics.
One of the earliest incarnations of mathematics was counting. Shepherds wanted to make sure that all their flock was accounted for by the end of the day. The simple act of counting may seem trivial and lack meaning. As it turns out, it is the basis for everything.
Mathematics became more sophisticated with the introduction of multiplication, which is the act of counting groups of groups of things. Three families of four make twelve people. Division is then the inverse of multiplication as is subtraction to addition. Aside from keeping track of cattle and assigning value to property, mathematics in this guise appears devoid of any deep meaning.
But, mathematics progressed. Variables were introduced to assign unknown quantities, functions to describe relationships between variables, etc. The growth of mathematical structure grew hand in hand with applications. Exponential functions could be used to describe the growth of livestock, the degree of hotness was associated with temperature, etc.
Then operations on functions were introduced. Derivatives gave slopes of curves and integration, the reverse of a derivative, yields the area under the curve. It was a stroke of intense insight when someone recognized that numbers associated with various physical quantities behaved in correspondence with what the operations predicted. This is a subtle point that deserves more - later.
Then came abstract mathematics that deals with groups theory, linear algebra, differential geometry. Even these seemingly non-practical concepts describe things such as the curvature of space-time, that governs the the motion of spacecraft and makes the GPS system possible, and predicts the groupings of elementary particles.
All of this leads to the obvious conclusion that there is no dichotomy between quantity and quality. Quality is in fact described in terms of quantity.
For example, we may say that gold has the qualities of being soft, yellowish, and shiny. Silver, on the other hand, is harder, greyish or some would say colorless, is shiny and half as dense as gold. As it turns out, the difference between the two atoms is in the numbers of protons, neutrons and electrons. Silver has 47 protons in a tiny nucleus and 47 orbiting electrons Gold, on the other hand, has 79 protons and 79 electrons (we ignore the neutrons since they don't affect an atom's chemical properties). It is the number of electrons and protons that determines the quality of the material. Thus quantity determines quality.
The numbers of various atoms in a molecule determine its properties. As we go up the later and make complex molecules, cells, organs, people, communities, and the universe, the properties of each object is determined by numbers that quantify the underlying things.
I have failed to mention forces, which determine how matter "sticks" together. The forces, which behave according to simple laws, determine the structures of molecules, galaxies, and nuclei. The simple laws that describe forces are formulated in terms of equations that represent numbers. So there are numbers everywhere that determine the quality of things.
However, the macroscopic universe is so complex, that it is difficult on human scales to express its properties in terms of the numbers that quantify the smallest units. This is where it is easier to think in terms of quality. You would prefer to think of your winter coat as warm and cozy rather than describe it in terms of the thermal conductivity of its parts, the chemical reactions in your body that create heat, etc.
-- This is where my post ends, with typos and incomplete thoughts. Perhaps someone can figure out what I had in mind. If so, please send me a note.