I cannot summarize here the thousands of pages written about the concept of abstraction. Instead, I will focus on abstraction in mathematics and discuss some generalizations that naturally arise in the dialogue. Before you set aside this blog post thinking it will get technical, rest assured I will be exploring concepts, not mathematics per se.
The primary question in this blog post is why mathematics is so tightly connected to concrete phenomena. When I was a young teenager I simply assumed that mathematics was like a language that describes reality. Of course, any description of reality would have a counterpart in the physical world.
But in high school, I became fascinated with Einstein's general theory of relativity. I learned that he used mathematics to predict phenomena that had never been observed. In 1915 he predicted that light would not travel in a straight line but would be bent and twisted by gravity even though it had no mass. Later, in 1919 the phenomenon was actually observed to be true.
This shocked me at the time. If mathematics were just a language that describes reality, how could it describe things that had never been observed in reality? How could it predict future observations?
In my first year of college, I took my first course in philosophy and encountered the concept of abstraction. The answer to my dilemma was there all the time, I had focused too much on physics and too little on philosophy (probably because I spent too many teenage years reading science fiction).
Mathematics is indeed a language that can describe the laws of nature. But, it is not a description of reality, It is a description of reality that strips away irrelevant specifics. For example, the mass of an object does not affect the rate at which it falls in a vacuum (you can drop two bricks and they will fall at the same rate when you glue the two bricks together and drop twice the mass as a single object). Thus, mathematics is an abstraction of related events.
Mathematics "predicts" the same will be true of falling rocks, falling rain, etc. We don't need to observe these events to describe them, mathematics conveniently extrapolates to unobserved events.
Abstraction is, for present purposes, simplification. It works in situations that go beyond mathematical predictions. It allows the categorization of things in general and in specifics. We can parse the universe into that which lies outside the earth's atmosphere and that that lies inside. The inside stuff can be divided into living and non-living things and further distinguish cats from dogs. Things get more specific when I identify our cat, Tikvah among living things; more so yet when I say Tikvah sleeping on my lap as I write this, and so forth.
By forming layers of abstraction, we come to understand reality. We know we shouldn't: try to walk through brick walls, tease mama bears, or leap onto a street from a third-story balcony. We need not treat one brick wall differently from another and we don't need the address of the balcony to determine if we can fly from it. In the blog post, "A Conversation" we can thank abstraction for knowing a brick is a brick.
Evolution has tuned our senses, and those of other animals, to maximize our respective chances of survival and reproduction. Useful abstractions form around these senses. Humans see details in daylight that Tikvah does not see and Tikvah sees details at night we cannot see; she has better peripheral vision and is nearsighted so a mouse has little chance of survival close to her. Below is a comparison between human and cat vision in daylight and at night.
Tikvah's senses categorize her nighttime world into mouse/no mouse and movement/no-movement. Humans see more detail in daylight to spot edible berries on a bush or a hungry predator in the woods. All animals pay special attention to abstract categories that matter most to their survival and the survival of their species. Evolution has embedded in our respective physical bodies the ability to sense what we need to survive and filters out unnecessary information. If we depended on chasing mice at night, we would have better night vision, and our abstract art would be, well, less interesting in my human eyes. It makes sense to me that abstraction is defined by the eyes and mind of the beholder.
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| The top panels simulate what a human sees, the bottom panels simulate what a cat sees. |