Tue Jan 28 10:38:19 CET 2014

Integers don't exist

Time for some wacko intuitive math speculation..

I've always found the tension between integers and smooth functions
quite interesting.  Integers are "meta", in that they appear in the
theory _about_ smooth functions, but not in their domain.

The real world is a smooth domain, while the _study_ of the real world
is a discrete domain.  Integers have no reality!  <cough>

An interesting rule of thumb: whenever you run into integers while
doing signal processing, you're making a jump to "interpretation",
i.e. talking about discrete patterns in the behaviour of smooth maps.
In DSP or AI, domains that abstract measurments to interpretation,
integers and discrete structures are sometimes called "features".

Floating point math is best interpreted as an (apporoximate)
representation of smooth functions over smooth domain.  The smoothness
maps well to the idea of rounding errors in floating point: errors
will not amplify as much, and equality of numbers has no precise
meaning, while equality or equivalence of smooth maps is

When thinking about singnal processing, integers should always be
understood in relation to topology.  I wonder if this idea of link
between floats and smooth domains can be made precise.