Halliday & Matthiessen (1999: 70-1):
What is the relationship between these two perspectives on meaning? So far in work on semantics only the typological perspective has been able to be realised in a formal system; the power of the topological perspective still derives largely from the metaphor of space within theories of meaning. For present purposes, we can say that the topological organisation construes a semantic space, creating the correspondences shown in Table 2(4).
Both perspectives are valuable. The typological perspective allows us to gain insight into the organisation of meaning through the network, both as theoretical metaphor and as a system of formal representation. The topological perspective gives us complementary benefits — in the first instance, the general notion of a multidimensional elastic space. We have indicated in the rightmost column some implications that derive from adopting the topological perspective. The general motif here is that of indeterminacy. For example, we can show how regions of meaning overlap (e.g. doing & happening overlapping with sensing in an area of 'sensing as activity'). We will use informal topological diagrams at various points in our discussion to bring out this central feature of the ideational system.The last row in the table above also represents indeterminacy, but this is indeterminacy of a particular kind, relating to the probabilistic nature of the semantic system. The types in the semantic system are instantiated according to probability values; these are manifested as relative frequencies in text. The equivalent in the spatial interpretation of meaning would be curvature or chreodisation. Chreodisation embodies time and represents the change of systemic probabilities over time (see e.g. Waddington, 1977, Sheldrake, 1988: Ch. 6, for discussion).
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Waddington modelled biological development as unfolding in an abstract space in which there were canals (curvatures) that represented the necessary path of development ('chreode'), which, crucially, the process of development could return to after being perturbed. Here Halliday & Matthiessen are only concerned with using space curvature as a means of representing probability differences topologically.