Ask a working mathematician what it is they are \emph{REALLY} talking about, and the 1st most common answer is to point at Sets. The 0th most common answer is "Oh f#&! you, why are you even at this analysis seminar?!"
There are only so many Kinds of things that a mathematician is interested in, and they need a langauge to tell each other, and to remind themselves, which Sorts of objects are which. But what makes Sets the Type of Stuff that a mathematician might be using?
We are going to take an intuitive overview tour of the mathematical multiverse and make a few landmark stops. Worlds where all functions R->R are continuous, worlds where every set is constantly changing, and worlds where everything is pointless. We'll construct universes to make a noncommutative thing become commutative, and universes with a (multi)universal group or ring. And even if we don't find ourselves drawn to a mathematical home, we'll at least leave with a better appreciation for the one we have.