Consider a glass of water in your hand and a table to place it on ...
How many locations are there to choose from ? Is two-dimensional
space quantized ? Are there less than an infinite number of "places"
on the tabletop ? Any spot you pick, I (for example) relocate the glass
to a new spot 1/sqrt(3) from the left side if the original distance
was "1" . Starts to look like an infinite number of locations on the
table top .. even using only one dimension. And then there's the
"front-back" dimension ... and the rotation of the logo on the glass ...
each of these "variables" seem to have an infinite number of states.
As for the amount of water in the glass ... if composed of something
like quarks, sure, there's a finite number of ways to partition this
quantity, but the spatial relationships between these quarks exhibit
something like "uncountably infinite" number of states.
The language of mathematics shapes the concepts easily framed.
The language of an aborigine may not contain a word for "snow", but
this does not mean that snow does not exist.
... the home for non-verbal memes ...