  Classical mathematics are founded within set theory, but sets don't have
\emph{symmetries}. We conjecture that if we allow sets with symmetries, then
many problems such as \emph{Mirror symmetry} or \emph{Homological mirror
symmetry} can be explained. One way to do this is to embed sets to higher
categories and especially into higher groupoids as already envisioned by
Grothendieck, and more recently by Voevodsky. We simply outline this idea in
these notes.
