Tuesday, June 28, 2016

Labyrinths and Problems

In Semiotics and the Philosophy of Language, Umberto Eco identifies three kinds of labyrinth.

(1) Classical labyrinth: The classical labyrinth is unicursal, with a single path and no branches. (The famous mythological labyrinth of Daedelus seems to have been branching, and is occasionally represented as such in very, very early representations, but at some point the representation became invariably of a single, intricately folded path to a center.) Such a labyrinth involves no choice; if a path is possible, it is the path to take. However intricate its folds, it can be perfectly represented by a single line. Eco notes that Ariadne's thread is useless in such a labyrinth, or rather, "the labyrinth itself is an Ariadne thread" (p. 80). Any challenge, like the Minotaur, is distinct from the labyrinth as such.

(2) Maze: Except for rare cases, real mazes are almost never represented in art before the Renaissance. Structurally, it would be represented not by a line but by a tree. Unlike a classical labyrinth, a maze is multicursal. It offers choices: paths are not necessary paths, a path can be a mistaken path, and thus Ariadne's thread, the clue that shows where to go next, can be important. The challenge is built into the very structure:

A maze does not need a Minotaur: it is its own Minotaur: in other words, the Minotaur is the visitor's trial-and-error process. (p. 81)

(3) Meander: If we took the tree-structure and started connecting nodes, we would et a net. "A net is unlimited territory" (p. 81): there may still be wrong answers, but there may be more than one right answer, and anywhere could, at least potentially, take you anywhere. Meanders also allow for loops.

At one point Eco makes the interesting observation that the network of a meander can be seen as a system of hypothetical trees. If I want to go from Austin to Milan, there are any number of ways I can go; different ways I could go would lead me to be faced with different choices, and eventually, by the time I got to Milan, I would have narrowed it down to one particular series of choices, which would be one tree out of many. many of the trees making up the meander of possible air routes won't get to Milan at all; there will, on the other hand be many that do. Of course, if one thinks about it, the tree of a maze is a system of hypothetical lines, routes through the maze. One could perhaps imagine (although Eco does not) a labyrinth that was a system of hypothetical networks. (Eco suggests that the rhizomes of Deleuze and Guattari are meanders, but since rhizomes are able to change through time, it seems to me that they would be a better candidate for precisely such a system of possible meanders.)

We can adapt Eco's classification of labyrinths to a form a classification of problems we might have to solve.

A plain labyrinth-problem is one in which the rule for solution (Ariadne's thread) is the only option for working through the problem (the labyrinth itself). It may have obstacles to create a challenge (Minotaurs), but the way to solve the problem, however long and intricate, is set by the problem itself. Examples of problems like these would be purely mathematical or logical problems in which one simply has to work through the options, or apply the axioms, to get the right answer -- that is, you just have to solve in a straight line.

A maze-problem, then, would be a problem in which there are alternative ways one might go about trying to solve the problem, not all of which get you to the end. We have a branching tree of choices, or (what is the same thing) a set of possible paths. The basic challenge of the problem (Minotaur) is built into it, although, of course, additional challenges could arise from other things. In order to solve the problem directly, at least with any consistency, we need information from outside the problem itself (Ariadne's thread). Otherwise, we have to try options until we find the ones that work -- any problem that is solved by trial and error is a maze-problem.

A meander-problem is one in which we have more than one possible series of alternatives. Like a maze-problem, the Minotaur is built in; like a maze-problem, any Ariadne's thread requires additional information from outside the problem itself; unlike the maze-problem, there may be multiple Ariadne's threads heading in different directions, some of them better than others. A good way to think of the difference here is by thinking of what happens if the labyrinth is infinite. An infinite labyrinth-problem is insoluble -- you never finish the task required to solve it. This is true of an infinite maze-problem, as well. But it is not necessarily true of an infinite meander-problem -- it depends on where one starts and ends and on the topology of the network and on the actual process of choosing routes. Or, to put it in other terms, since the structure of a meander is that of a system of hypothetical trees, even if some, or most, of those trees are infinite (and thus lead to no solution), some of them could still be finite. An infinite network has finite as well as infinite trees for its parts. One often finds meander-problems with searches (although not all searches are meander-problems); and they come up a lot when we are beginning to explore ideas or possibilities about which we know very little (and thus do not really know what the best way to try to solve the problem might be).

Umberto Eco, Semiotics and the Philosophy of Language, Indiana University Press (Bloomberg: 1984).

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