Topology and Verb Classes
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 ....
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.
- 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
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.