## The Topology of Syntax

### Iain A. Plicable

Lecturer in Mathematical Linguistics

University of Ledworth

A key theorem of Universal Grammar is that lines do not cross in tree diagrams. However, critics of Universal Grammar challenge even such a basic result as this on the basis of sentences such as the following, taken from Virgil’s *Eclogues:*

*ultima*

Adj.F.S.Nom

last

*Cumaei*

Adj.M.S.Gen

Cumean

*venit*

V.3.S.Pres

come

*iam*

Adv

now

*carminis*

N.M.S.Gen

song

*aetas*

N.F.S.Nom

age

“Now the last age of Cumaean song comes.”

*Figure 1: An invalid tree diagram*
Leaving aside the fact that such forays into historical linguistics tell us nothing about the necessarily synchronic nature of Universal Grammar, we must address the fact that a naive attempt to parse this sentence leads to profoundly unsatisfactory results, as seen in Figure 1.

Sceptics claim that this shows that Latin phrases are bound together by dependency relationships rather than constituency relationships. This hypothesis, however, assumes that morphology is governing syntax, in a clear violation of the natural hierarchy of information structures. A more rigorous, mathematical approach provides a better solution.

While the surface structure of language is linear, the existence of tree diagrams shows that this structure is embedded in a 2-dimensional surface. Were it not so, we should be incapable of any grammar other than clause chaining. For convenience, this surface is usually depicted as a simple plane. However, just as the linear structure is embedded in a 2-dimensional surface, the surface may itself be embedded in 3-dimensional space. This allows us to envisage it as a curved surface, such as a torus, familiar as the shape of a doughnut, or alternatively a coffee cup.

*Figure 2: A torus and its representation as a topological diagram*

The topological diagram in Figure 2 represents how a torus may be constructed from a flat sheet by joining opposite edges. That corresponding arrows point in the same direction indicates that the torus is an orientable surface, non-orientable grammar being an issue for future research. Using the curvature of the torus, we can now produce the sentence diagram in Figure 3.

*Figure 3: Using a toroidal surface produces a valid tree diagram*
Nothing could be simpler.