According to my lecture, Fuzzy c-Means tries to minimize the following objective function:
$$J(X,B,U)=\sum_{i=1}^c\sum_{j=1}^n u_{ij}^w \, d^2(\vec{\beta_i},\vec{x_j})$$
where $X$ are the data points, $B$ are the cluster-'prototypes', and $U$ is the matrix containing the fuzzy membership degrees. $d$ is a distance measure.
A constraint is that the membership degrees for a single datapoint w.r.t. all clusters sum to $1$: $\sum_{j=1}^n\, u_{ij}=1$.
Now in the first equation, what is the role of the $w$? I read that one could use any convex function instead of $(\cdot)^w$. But why use anything at all. Why don't we just use the membership degrees? My lecture says using the fuzzifier is necessary but doesn't explain why.