# 019 Catastrophe Theory

The Shape of M

Suppose *p* is a parameter point on a conflict line. We translate the sociological notion of „conflicting opinions at *p*“ into the mathematical notion of bimodality of F_{p}; in other words, we assume F_{p} has two local maxima, and so *M* is two-sheeted over a neighborhood of *p*. Now globalize this notion by assuming that *F* is bimodal over neighborhoods *N*_{1}, *N*_{2} of the two conflict lines *L*_{1}, *L*_{2} , and is unimodal otherwise. We call *N*_{1}, *N*_{2} the conflict regions; they are shown shaded in Figure 20.3. As we leave either side of a conflict region one of the two modes is preserved and the other has to disappear; we assume, further, that one of the modes is preserved on one side and the other on the other side. Therefore *S* has an S-shaped fold over *N*_{1}. Meanwhile *L*_{2} terminates in the point K, and by Hypothesis 3 the only way an S-shaped fold can terminate is in a cusp (Markus 1977; Zeeman n.d.); therefore, *S* has a cusp singularity at *K*. Summarizing our assumptions and deductions into a single hypothesis we obtain Hypothesis 4.

The projection x: *S* → *P* *has fold curves and a cusp singularity as shown in Figure* 20.3. The reader will observe that we have drawn *S* in Figure 20.3 as if it were sitting in three dimensions rather than in the (2 + *n*)-dimensional space *P* x *X*. We justify this in a later section. Meanwhile notice that our discussion of folds and cusps is independent of the ambient space in which *S* happens to be sitting. The following deductions from Hypothesis 4 are also independent of the ambient space. *M* has two components *M*_{1} and *M*_{2}, whose projections overlap on *N*_{1}. The projection of *M*_{2 }overlaps itself on *N*_{2}. The complementary subsurface *S-M* has two components *S*_{1}, *S*_{2 }shown shaded in Figure 20.3, that project onto *N*_{1}, *N*_{2}; however, the components *S*_{1}, *S*_{2 }are sociologically meaningless because they represent saddle points of *F*, around which no opinion points cluster.