Why is aliasing inherently non-linear?
I think you mean "images", not "aliases".  They become aliases if there is foldover from resampling. 

It's because you are not adding two signals, $x(t)$ and $\operatorname{III}(t)$, you are multiplying them that these images appear.

$$\begin{align}
x_\text{s}(t) &amp; \triangleq x(t) \cdot \operatorname{III}(t/T)  \\
&amp;= x(t) \cdot \sum\limits_{n=-\infty}^{+\infty} T \delta(t-nT) \\
&amp;= T \sum\limits_{n=-\infty}^{+\infty} x(t) \cdot \delta(t-nT) \\
&amp;= T \sum\limits_{n=-\infty}^{+\infty} x(nT) \cdot \delta(t-nT) \\
&amp;= T \sum\limits_{n=-\infty}^{+\infty} x[n] \cdot \delta(t-nT) \\
\end{align}$$

where $x[n] \triangleq x(nT)$ and

$$ \operatorname{III}(u) \triangleq \sum\limits_{n=-\infty}^{+\infty} \delta(u-n) $$

$$\begin{align}
\operatorname{III}(t/T) &amp;= \sum\limits_{n=-\infty}^{+\infty} \delta\left(\tfrac{t}{T}-n\right) \\
&amp;= \sum\limits_{n=-\infty}^{+\infty} \delta\left(\tfrac{t-nT}{T}\right) \\
&amp;= \sum\limits_{n=-\infty}^{+\infty} T \delta(t-nT) \\
&amp;= \sum\limits_{k=-\infty}^{+\infty} e^{j k \frac{2 \pi}{T} t} \\
\end{align}$$

The last line is from doing fourier series.  Now, if you use the shifting property of the Fourier Transform, then the Fourier Transform of $x_\text{s}(t)$ is

$$\begin{align}
X_\text{s}(f) &amp; \triangleq \mathscr{F}\{x_\text{s}(t)\} \\
&amp;= \mathscr{F}\{x(t) \operatorname{III}(t/T) \} \\
&amp;= \mathscr{F}\left\{x(t) \sum\limits_{k=-\infty}^{+\infty} e^{j k \frac{2 \pi}{T} t} \right\} \\
&amp;= \mathscr{F}\left\{\sum\limits_{k=-\infty}^{+\infty} x(t) \, e^{j k \frac{2 \pi}{T} t} \right\} \\
&amp;= \sum\limits_{k=-\infty}^{+\infty} \mathscr{F}\left\{ x(t) \, e^{j k \frac{2 \pi}{T} t} \right\} \\
&amp;= \sum\limits_{k=-\infty}^{+\infty}  X\left(f-\tfrac{k}{T}\right) \\
\end{align}$$

where $ X(f) \triangleq \mathscr{F}\{ x(t) \} $ .

That non-linear process of multiplication generates frequency components than did not previously exist in $x(t)$.  Those new components are simply shifted versions of $X(f)$ and are called "images".