Symmetries

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.