Sunday, July 24, 2011

Critique of abstractions

As I read through Philosophy in the Flesh, I am constantly reevaluating some old thoughts I've been mulling over for a long time. Lakoff and Johnson have helped to confirm my suspicion that perception and imagination are inextricably linked, that abstract thought is inherent to human perception, and that we often mistakenly assume that our abstract thought really is quite concrete. For instance, when we view a ball flying through the air, our minds immediately conjure up this sense of trajectory. The concrete physical details are unimportant to us; what matters is the continuous trajectory our minds imagine. Similarly, when I see another person's face, my mind immediately recalls who it is, without me having to consciously acknowledge the concrete details of that person's face. It is really the abstraction that I'm interacting with; if I had to notice all the concrete details and consciously reconstruct the person's identity, personal interaction would be next to impossible.

This insight leads me to a question I've been mulling over, due to my experience teaching mathematics: why is "abstract thought" often considered difficult and even terrifying? Mathematics is certainly not the only instance where this fear arises. Philosophy is perhaps even more vilified for its abstractness than mathematics, and really every academic pursuit comes under fire from this sort of complaint. Many people just do not see what any of this theory has to do with real life. And yet these same people, who profess that they prefer the concrete to the abstract, are also affected by the most abstract ideas during a political campaign speech or an advertisement on television. I will not attempt a list of the mindlessly repeated abstract nouns which so easily embed themselves in the public consciousness. The point should be clear: the world simply does not run on concrete understanding alone (although I will have to come back later to qualify this point). Moreover, even the things most people consider common, everyday activities actually depend on the use of abstractions, not concrete details. Thus, most people actually use abstract concepts quite casually, without really noticing it. How, then, can they be afraid of abstract thought?

It occurs to me that people aren't really afraid of abstract concepts. What they are afraid of is instead the critique of abstract concepts. In other words, rarely do people like to question that which comes naturally.

There are two ways (that I've thought of so far) to critique the abstract concepts we are accustomed to using. The first is through the use of logic; this is what we would normally call "abstract thought," and it is what many people hate. Logic is the foundation for both philosophy and mathematics, as well as the sciences. Logic critiques abstract concepts by showing what they're really made of: it pursues the end results of those concepts when they are forced to behave as consistently as possible. Socrates is the poster boy for this kind of critique: he would seek to precisely define an abstract idea, such as justice, and continue to press the question until it became clear that we really do not know what justice is at all. But at least we could know what the logical results of our assumptions were.

In mathematics, this method works very well, because it turns out that there are many concepts--specifically quantitative ones--which have endlessly fruitful logical consequences. If we come to a point where we realize our rigorously defined concepts don't actually describe what we want them to describe, we strive toward defining new concepts. For instance, when one comes across a puzzle such as the Banach-Tarski Paradox, the thing to realize is that not all sets in Euclidean space can really represent volumes in the way that we want them to. Thus mathematics provides a critique of our intuitive concept of objects extended in space.

As an aside, I confess I once thought of mathematics as a rather self-contained body of intellectual progress, blessed with a degree of certainty of which other forms of inquiry were deprived. I now realize that on the whole mathematics would be meaningless without its ability to critique our intuition. If we looked at the Banach-Tarski Paradox and did not see a paradox, would we see any need to study measure theory? That is to say, most of our problems in mathematics seem to really be putting concepts on trial: if the concepts we have can be shown logically to have absurd or otherwise undesirable properties, then we have to define new concepts. Any mathematician should be able to think of numerous examples of this: the absurdity of the delta function implies the need for distribution theory, Bertrand Russell's paradox is circumvented by category theory, the paradoxes caused by the axiom of choice (including the Banach-Tarski paradox) are resolved by measure theory, and so on.

There is a second way to critique our abstract concepts, which goes in the opposite direction of logic. I'll call this second way "art." This way is followed when a person insists on examining things closer to see them in their concrete detail. Indeed, a visual artist is a fine example of this. I mentioned before that most of us do not need to consciously examine all the concrete features of a person's face to recognize the person. Similarly, we do not need all the concrete details in order to enjoy a beautiful piece of scenery, or to admire a beautiful object. But the artist critiques our abstract recognition by consciously examining the details and then reproducing them, often in ways that are jarring or perplexing. One can recognize people in a Picasso painting; and yet the arrangement of their features are so out of the ordinary that it forces us to contemplate these features in their concreteness. We cannot simply pass over these faces as mere examples of a general phenomenon.

This form of critique is not limited to the visual arts. It is abundantly manifest in music, where sounds are carefully placed together in order to convey a message. We are used to hearing all kinds of sounds and perceiving them as abstract concepts: talking, chirping, honking, buzzing. Indeed, I love to go work in public places where I am surrounded by "white noise;" it's so easy to process the noise as one abstract entity. But music (good music) critiques our relationship to sound by organizing it consciously, forcing us to notice sound in all of its individual life.

Likewise poetry, the art of words, critiques our view of language as a mere conveyor of meaning, forcing us to notice words in their concrete existence. Prose and storytelling have their own way of doing this, as well, although the manner may not be so obvious. Often this is why storytellers have to defy the conventions of their genre, in order to get people to notice a story in its concrete detail. But in any case, even if the audience does not perceive the critique going on, the artist himself is delving much deeper into the language than a normal speaker or reader would in order to create something new, and that alone is an implicit critique of the abstract way in which we normally encounter language. Similar points go for music and visual arts, as well.

It appears to me, though this is just a thought, that all progress we make really is in one of these two directions. Both of them are hard. I don't know that people more often prefer the second direction (art) to the first (logic); it would appear that any great endeavor requires some combination of the two. Think of an entrepreneur. It is one thing to have an idea. But the entrepreneur must first understand whether the idea is sound--is it possible? is it useful? is it original?--and he must then understand all the concrete details which will lead to the idea's fruition. Is he then more of a logician or an artist? It really isn't clear. What is clear is that the path to success is always difficult and requires a great deal of self-critique.

As a closing thought, I should return to what I mentioned earlier, that the world doesn't run on concrete thought alone. I find it helpful to be aware that most of what we experience must be filtered through a fairly limited set of abstract concepts. A crowd of people, a length of road, an amount of time--these are all abstract concepts which our minds unconsciously generate (whether through biology or social conditioning) in order to make sense of the world. If we really had to be consciously aware of all the concrete details of our world in order to make sense of it, well then we never could make sense of it.

But progress does require that we critique these abstract concepts which come naturally to us. The critique is where expertise comes from, and it is certainly not limited to the first approach, the logical approach. Experts are the people who care enough about a particular thing in its concrete existence that they will study it until they have mastered it. Without expertise, we would have no progress. It is in the concrete implementation of good ideas that we make material advances. This requires people immersing themselves in the details of particular tasks, which in turn requires a division of labor--and here I find myself once again getting back to Adam Smith. It is important to realize that this division of labor is necessary precisely because we normally don't think concretely about very many things, nor can we. Otherwise, we'd be experts in everything.

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