A Topological Rant

Jeffrey Achter

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 cneturies to define itself, to discern its limitations, and to know what it knows. To a large extent, this project has been successful.

So for me, mathematics represents a bastion of relative sanity. My hackles rise when I see writers from other endeavors (mis)using the terminology of mathematics. Often they do this without regard to the tehcnical meanings of the words they throw around. All that is left is the appearance of mathematics, without any of the substance; the aura, but not the spirit. This truly disturbs me. I would wish for the claims of mathematics to be distinguishable from the claims of "mathematics." But I have seen this distinction abused by political scientists, sociologists, and the Maharishi Mahesh Yogi. And now, apparently, Barthes and Derrida have joined their ranks.

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 {Ui} for i in some index set I, whose union is X and which is closed under finite interseciton and arbitrary union. A continuous map f: X -> Y of topological spaces is a function such that, if V is an open subset of Y, then f-1 (V) in X is open, too. 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.

So what are we to make of Barthes' use of the word topological? A typical example comes to us from page 8 of S/Z:

Topologically, connotation makes possible a (limited) dissemination of meanings, spread like gold dust on the apparent surface of the text (meaning is golden).

What does Barthes mean by topological? I'm not sure, but I suspect it hasn't anything tto do with the mathematical denotations and connotations of the word. Which is okay, I suppose; where I take offense is where authors try to imbue their texts with the rigor of mathematics by quoting it. Page 246 of Dissemination, by Derrida, gives us:

We must here quote a long and beautifully written page of the Preace. Inquiring into "the very notion of a theme, on which [our] whole enterprise is based," Richard has just noted the "strategic value" or "topological quality" of the theme. "Any thematics will thus derive both from cybernetics and from systematics. Within this active systemk, the themes will tend to organize themselves as in any living structure: they will combine into flexible groupings governed by the law of isomorphism and by the search for the best possible equilibrium."

It's difficult to discern what, exactly, Barthes and Derrida believe to be a topological property.

Some responses from contributors to Technoculture, a once-active online discussion group: