[FOM] Question on the number line

[Alasdair Urquhart]

The puzzle raised by Suarez is one that shows
that it is problematic to make a naive identification of
physical space with abstract Euclidean space.  The distinction
between an open and closed ball in Euclidean space, though
of crucial importance in mathematics, has no clear physical
sense.  

There is a rather long report of Gödel's thinking on related
topics in Section 3.1 of Hao Wang's book "From Mathematics
to Philosophy."  In particular, on p. 86, Wang writes:

"In set theory we think of points as parts of the continuum
in the sense that the line is the set of the points on it
(call this the `set-theoretical concept').  In space intuition
we think of space as a part of fine matter so that each point
has zero weight and is not part of matter (but only a limit
between parts).  Note that it is not possible to cut a material
line segment or a rod in two ways at the same point or surface P,
once with P on the left part, once on the right, because there is
nothing between the two completely symmetrical parts.
According to this intuitive concept, summing all the points,
we still do not get the line, rather points form some kind of
scaffold on the line."  

The whole section, entitled "Gödel on mechanical procedures
and the perception of concepts" is very interesting and repays
close study.