on abstraction

I’ve come across the word “abstraction” in exactly three contexts. The first is in computer science, where “abstraction” is the hiding away of details within a black box whose interfaces are well defined. The second is in mathematics, where “abstraction” is the induction from special cases. The third is in art, where “abstraction” seems to be independence from concrete meaning. Abstract is “drag away” in Latin, so it is likely to be defined by its opposition — that which it is “dragged away” from. No wonder it has so many different meanings. Interpreted thusly, the first context for “abstraction” may better be termed “opacity” in opposition to “transparency”; the second, “genericity” in opposition to “specificity”; and the third, “notionality” in opposition to “concreteness.”

What is the point of abstraction? Is there something terrible about transparency, specificity, or concreteness; are they not qualities that we praise, for the clarity that they provide?

There must be some limitation to these generally positive qualities. Maybe, once something is transparent, specific, or concrete, we “understand” it. Then it is not much of a challenge and we get bored. The seemingly obfuscating nature of “abstraction” instead challenges us to tackle more difficult questions by removing the easiest ones already answered. Why are abstracted questions more difficult? Because they must give “larger” (I hesitate to use the phrase “more powerful”) answers, since they begin with fewer informational constraints. So in the first case, opacity is used to build larger systems; in the second case, genericity is used to demonstrate more applicable truths; and in the third case, notionality is used to elicit a wider range of reactions.

Still, what does “larger” mean? And what is the relation between “abstraction” and “precision”: are the two complementary, or a tradeoff? They seem to be at first, yet there is nothing imprecise about mathematical abstraction, for example, but there is something imprecise about artistic abstraction. Maybe what I mean is best represented by this:

\(\underbrace{…1001101001001}_\textrm{precise}\overbrace{01010101…}^\mathrm{imprecise}\)
\(\underbrace{…10011010}_\mathrm{unabstracted}\overbrace{0100101010101…}^\textrm{abstracted}\)

Here I have some bits. To the left are MSBs and to the right are LSBs. “Precision” is commonly understood to be how many bits into LSBs I can measure. Similarly, “abstraction” as defined here is how many bits I hide. So immediately, the two are describing complementary perspectives, one from an receiver’s POV, and the other from a transmitter’s POV. It is easy to see where complementarity could arise.

But in fact, “precise” and “abstracted” bits can overlap; and when they do as here, then the transmitter has done something significant by hiding away bits the receiver would otherwise be capable of measuring. Moreover, what the receiver gets, the “unabstracted” part, is entirely precise. This is the case with mathematical abstraction. In fact, mathematics has no “imprecision” at all, so the unabstracted part is as precise as the abstracted part. In other situations where imprecision exists, the part of abstraction that does not overlap with precision are bits the receiver will always be uncertain about. (Those are stuck-up, elitist, phony abstraction bits as far as the receiver is concerned, haha…)

\(\underbrace{…1001101001001}_\textrm{precise}\overbrace{01010101…}^\mathrm{imprecise}\)
\(\overbrace{…10011010}^\mathrm{abstracted}\underbrace{0100101010101…}_\textrm{unabstracted}\)

However, abstraction does not need to occupy the LSB end. It could start on the other end, too. When there is some overlap between “imprecise” and “unabstracted” bits, the result is some degree of imprecision in the conveyed object. This is the case with artistic abstraction, I believe. Here, as abstraction rises from the transmitter, we get less precision in the receiver.

So now we can answer the question of what becomes “larger” in abstraction: the number of possibilities in the abstracted and precise part — the part that the receiver is capable of measuring but which the transmitter deliberately does not send — rises as abstraction gets higher. Yet, what happens to the precision of the unabstracted part is not guaranteed. Precision and abstraction are not complementary.

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