There's a thing in mathematics called ``duality''. You can
probably get a feel for what it is by just the name. One of the
best examples of it in mathematics is that in Euclidean (normal,
run-of-the-mill, you-learned-it-in-high-school) geometry the
concept of *point* is dual to the concept of *line*.
Every theorem in geometry has a dual theorem which you can
obtain by swapping the notions *point* and *line*.

Two points determine a line. Two lines determine a point.

Another place duality crops up readily is in category theory where a theorem has its dual by simply reversing all of the arrows (inverting all of the morphisms).

Duality appears all over the place in mathematics. Another
ubiquitous concept in mathematics is *Symmetry*. A symmetry
of a system is a transformation which takes the system back to itself.
For example, rotate a square by ninety-degrees or flip it over one
of its diagonals. Or, rotate a circle by any amount.

It occurs to me that *Duality* is just *Symmetry*
in the semantic space/ conceptual space/ what-have-you. In the case
of the *point*/*line* duality of Euclidean geometry,
this is something like flipping the square over one of its diagonals.
Technically, it's more like the the symmetries of a line segment.
There's a complete change of 180-degrees, but everything still works
(including another complete change of 180-degrees).

**( A little more musingCollapse )**