So a stepwise derivation for MLP (multilayer perceptron) starts with
From line to surface to hypersurface $a: \mathbb{R}^{n_0} \rightarrow \mathbb{R}$
$$a(x)=w_0+w_1x_1+...+w_{n-1} x_{n-1} +w_n x_n=w^T \tilde{x}$$
Where $x_i$ are obviously the observations and $w$ are (I suppose) coefficients to be solved for.
What does "from line to surface to hypersurface" mean in MLP?
I believe that this is more of a turn of phrase than something to get too bogged down in. However, the author is meaning to convey that the problem of finding $\omega^T$ in the case of $n_0=1$ is the case of finding a line to separate two observations, the case of $n > 1$ is to solve for a service, and in order to best solve for the best separating surface, one must optimize the hyper-surface defined by all $\omega^T$.