Sentences have parse trees, maybe similar limitations hold. In English words should not be re-used in the same sentence, a non-self-intersection rule. This could be explained by a sentence being represented in the brain by the activation of a connected set of neurons. Such a setup could not handle conceptual self-intersections [1].
So one can potentially ask "what is the dimension of the mind that produced those sentences?", using Wolfram's definition of network dimension.
Similarly one could observe the eye-saccade patterns of a person to try to infer the dimension of the mental model they are building of a scene -- does it use only short range interactions (like a Markov Random Field), or does it use long range correspondences too (like a wavelet decomposition or feature hierarchy -- abstraction)? This model could be idealized as a Bayesian network (though, if possible, using a non-causal representation).
[1] Except by using pronouns to explicitly note them.