Representation

The representation of a task can radically affect our reasoning abilities and performance. For example, the game of tic-tac-toe (opposing players mark X's and O's in a 3x3 grid) can be equivalently represented as a game of drawing numbered cards with the goal of selecting three that sum to 15. From a computational perspective, these two problems are isomorphic. However, the tic-tac-toe representation is significantly easier to work with because the representational form of the problem makes visible the most relevant constraints implicit in the problem. As Simon writes, in mathematics, "solving a problem simply means representing a problem so as to make the solution transparent"

We need to get some freedom from context so that we're not stuck repeating a single performance, without ending up so free that we have no idea what the music is supposed to sound like. This is the space where notation comes in.

There are two different vocabularies I like for thinking about this tension. One is Brian Cantwell Smith's idea of "the middle distance" in On the Origin of Objects, which I wrote about here. Useful notation is somewhere in the middle distance between the spectrogram and the cat, not rigidly causally linked and not completely unrelated either.

The other is super obscure. I got it from this paper by Gordon Bearn that I fished up from Google Scholar. Bearn talks about how representations must broach the possibility of meaning, by providing some link at all to the thing they represent (this is what the cat fails at). At the same time they must also breach their context by getting some distance from it (this is what the spectrogram fails at).

People count by tens and machines count by twos—that pretty much sums up the way we do arithmetic on this planet. But there are countless other ways to count. Here I want to offer three cheers for base 3, the ternary system. The numerals in this sequence—beginning 0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101—are not as widely known or widely used as their decimal and binary cousins, but they have charms all their own. They are the Goldilocks choice among numbering systems: When base 2 is too small and base 10 is too big, base 3 is just right.

...The cultural preference for base 10 and the engineering advantages of base 2 have nothing to do with any intrinsic properties of the decimal and binary numbering systems. Base 3, on the other hand, does have a genuine mathematical distinction in its favor. By one plausible measure, it is the most efficient of all integer bases; it offers the most economical way of representing numbers.

At the end of the 18th century, Spain's official geographer, Tomás Lopez, was asked by the King to create an accurate map of the kingdom. In an attempt to delegate the herculean labour required, Tomás drew a series of circles, picked the town in the center of each circle, and asked the local priest to answer a questionnaire and draw up a map of their province. The goal was to amalgamate the responses into a single map. But none of these priests were trained in cartography, and many of them would have had limited access to maps at all. Nonetheless, 500 of them tried. In one map, the entire region is represented simply by a series of letters (“A” for church, “B” for hermitage, “C” for house, “D” for tree, and so on). Another represents the surrounding villages as if they are orbiting planets. In some, the handwriting forms the topographies. In others, descriptive columns of text take center stage, as if the language itself is a landmark. Each priest implicitly reveals how they see the world around them, and the relative importance of its constituent parts: nature or people, religion or trade, architecture or landscape, precision or vibes.

[Brian Cantwell] Smith’s first example [from On the Origin of Objects] is fanciful but intended to quickly give the flavour of the idea:

…imagine that a species of “super-sunflower” develops in California to grow in the presence of large redwoods. Suppose that ordinary sunflowers move heliotropically, as the myth would have it, but that they stop or even droop when the sun goes behind a tree. Once the sun re-emerges, they can once again be effectively driven by the direction of the incident rays, lifting up their faces, and reorienting to the new position. But this takes time. Super-sunflowers perform the following trick: even when the sun disappears, they continue to rotate at approximately the requisite ¼° per minute, so that the super-sunflowers are more nearly oriented to the light when the sun appears.

A normal sunflower is directly coupled to the movement of the sun. This is analogous to simple feedback systems like, for example, the bimetallic strip in a thermostat, which curls when the strip is heated and one side expands more than the other. In some weak sense, the curve of the bimetallic strip ‘represents’ the change in temperature. But the coupling is so direct that calling it ‘representation’ is dragging in more intentional language that we need. It’s just a load of physics.

The super-sunflower brings in a new ingredient: it carries on attempting to track the sun even when they’re out of direct causal contact. Smith argues that this disconnected tracking is the (sunflower) seed that genuine intentionality grows from. We are now on the way to something that can really be said to ‘represent’ the movement of the sun:

This behaviour, which I will call “non-effective tracking”, is no less than the forerunner of semantics: a very simple form of effect-transcending coordination in some way essential to the overall existence or well-being of the constituted system.