Date: Fri, 8 May 1992 11:15:29 -0400

From: David Durand

Subject: situated knowledges

In-Reply-To: Jeff Achter's message of Thu,
7 May 1992 19:49:31 -0400 <9205072353.AA04008@BU.EDU>

Let me freely admit some of my biases from the beginning. I am continually frustrated when other disciplines attempt to borrow the ethos of mathematics. Implicit in this -- or even explicit, I suppose -- is a special position for mathematics in the network of disciplines. Mathematics has worked hard over the centuries to define itself, to discern its limitations, and to know what it knows. To a large extent, this project has been successful.

Take Barthes' use of the word topological. Please. Now, as a mathematician, I might define topology as follows:

Let X be a set. A topology on X is given by a collection of open subsets U(i) of X whose union is X, and which is closed under finite intersection and arbitrary union. A continuous map of topological spaces is a function such that the inverse image of an open set is open. A homeomorphism is an invertible continuous map. A property which is invariant under homeomorphism is called a topological property.

Clearly, this definition is of limited utility in the analysis of literature. One might make analogies with the decomposition of a set into open and closed sets; but such an analogy wouldn't instantly transfer mathematical knowledge about topological objects onto a text. An intuitive definition of topology might read:

Topology studies the properties of an object which don't change when you smoothly squish that object; it studies the structures of connections, not sizes and shapes. That's why a topologist can't tell a donut from a coffee cup.

This might actually have some applications for literary theory and hypertext. One can imagine trying to analyze the complexity of interconnections in a text just as one might analyze the genus of a graph. Bear in mind that this is, at most, a metaphor; one is not actually doing anything in the way of mathematics.

Well, it seems that by not responding I've let the literature peopel in for a bashing -- and I'm a ringer. A computer scientist, and thus at least supposedly sympathetic to mathematics, even if CS people are also noted abusers of the terminology and methods of mathematics. I was thinking of topology in hypertext as relating to the shape of possible readins as experessed via the links. The open sets of documents would be the sets of documents linked by 1 level of linking, by 2 levels of linking, on to the set of equivalence classes of nodes under the transitive closure of the "linked-to" relation. Thre will of course be more than one, only if some protions of the text are disconnected from other portions of the text. I think that taking the open sets as signleton nodes, and then successive sets of linked nodes, actualyl does create a topology in the sense you mean, so that the analogy is reasonably precise. However, I was thinking of the second sense of the term, in which we are considering the connections of documents in a hypertext without the3 notion of _any_ coordinate system -- thus I used the term topologizing to denote such an organization -- since the only relations between nodes are defined by the links.

The notion of using coordinates to indicate relations among documents need not imply a particular attitude, but from a literary and historical perspectivee Euclidean geometry and Cartesian spaces are linked to the Renaissance, Newton, and other "hard-nosed rationalist" conceptions.

Metric non-Euclidean spaces (the sort most people probably think of in this kind of context as axiomatic approaches seem to have little to offer in the way of analogies) have other notions of distance imposed on them -- so that given the same uniform Catesian coordinates, the metric is not the normal one that you get from the pythagorean theorem. So, perhaps we can imagine a hypertext system, where items are sorted by some kind of coordinate space, but the visual representations reflext a non-Euclidean metric. Some texts, Like Shakespeare's works or the Bible, say would be defined to be like black holes -- they would stretch space around them, so that the densely packed parasitic texts are farther apart. In a sense, massive or "heavy" texts would stretch hyperspace around them to accomodate additional sattelite texts.

Cool, perhaps, but would this be useful? (BTW does anyone know of work done on rendering of views from inside various spaces using the appropriate non-Euclidean optics? It's too weird not to have been worked on, but I've never seen anything on it...

-- David Durand