Topology and Verb Classes
Feature Column Archive
2. Thom's Axioms
Here is a list of statements that can serve as axioms for the catastrophe-theoretic study of syntax. Some of these are in fact theorems. Compare with Thom's Résumé des thèses on p. 321 of Stabilité ....
- Processes of interest in the world about us are governed by internal dynamics of the following type:
- The process is driven along the gradient lines of an energy-type function, so as to decrease the ``energy.''
- Besides internal variables, the function may also depend on one or several external parameters.
- At almost every internal configuration this gradient is non-zero, and the process is driven to a configuration giving a local minimum of the function. (The set of points driven to a given local minimum is its basin of attraction.)
- The internal dynamics are rapid in the observer's time coordinate, so when observed, the process is almost always in the steady state, in a local-minimum-energy configuration.
- The only local-minima which can be perceived are those which are stable: a small perturbation in the internal variables or in the parameters leads to only a small perturbation in the state.
- As the function parameters change, what was once a local minimum may find itself in the basin of attraction of another one. The process will then suddenly switch from the old local minumum to the new. This is a catastrophe.
- The set of points in parameter space at which catastrophes occur is the catastrophe locus. To a large extent, catastrophe loci (edges, changes) are the data furnished by our senses. For us to perceive a catastrophe in our 4-dimensional space-time, there must be a 4-dimensional sheet running through parameter space which intersects the catastrophe locus transversely, i.e. crosswise. Otherwise an infinitesimal displacement would make the catastrophe disappear, and we would have no chance of seeing it.
- Up to topological equivalence, there are only 7 elementary catastrophes which admit 4-dimensional transversal sheets in their parameter spaces. These catastrophes have algebraic representatives given by polynomials. With internal variables x, y and the number of parameters a, b, c, ... required for stability, they are (with the addition of the simple minimum which is stable, and therefore does not generate a catastrophe set, but which is of interest in the context of language):
| the simple minimum|| x2|
| the fold|| x3 + ax |
| the cusp|| x4 + ax2 + bx |
| the swallowtail ||x5 + ax3 + bx2 + cx |
| the butterfly|| x6 + ax4 + bx3 + cx2 + dx|
| the parabolic umbilic ||x2y + y4 + ax2 + by2 - cx + dy|
| the elliptic umbilic ||x3y - 3xy2 + a(x2 + y2) - bx - cy|
| the hyperbolic umbilic ||x3 + y3 + axy - bx - cy|
- If a process in space-time can be characterized by one of these catastrophes, then the mental process which apprehends it will mimic that catastrophe, and the syntax of a verb phrase describing it will correspond to the topology of a one-dimensional section through its parameter space. (Thom lists sixteen such topologies in Topologie ...). In particular the number of arguments of the verb (subject, object, instrument, destination) corresponds to the number of minima that can simultaneously coexist. In all of Thom's sixteen examples this number is less than or equal to four; this corresponds to the linguistic observation that in general a verb can have at most four arguments.
In the rest of this column we will examine the first four critical points on the list, the one-dimensional sections through their parameter spaces, and the corresponding verb classes.