# [FOM] Question on the number line

Hartley Slater slaterbh at cyllene.uwa.edu.au
Wed Nov 16 21:36:49 EST 2005

At 12:00 PM -0500 16/11/05, Steven Ericsson Zenith wrote:
>therein lies the problem, the number line is
>not a geometric line in the world.  The number line has epistemology,
>but no ontology - it can be known, but it does not exist.

That makes it sound mysterious.  The point about what might be called
'non-existence' is most clear in connection with Zeno's paradoxes.

Halving the 0-1/2 number line, to progressively reach 1/4, 3/8, 7/16,
15/32, etc. only gets one through the open interval [0, 1/2).  It
therefore leaves one at no place on the line from which to 'move the
rest of the way' to cover the closed interval [0, 1/2].   But that is
not because it lands one 'right next to' 1/2, for instance.  It is
because there is an infinity of half-way points, and so 'where you
end up' is no place in space, nor are you 'there' at any point in
time.  For there is no end point within that infinity: the infinity
of half-way points is endless.

Hence Achilles never reaches the Tortoise?  Not at all; he does so in
a perfectly straightforward, finite operation.  To cover the closed
interval [0, 1/2] with closed sub-intervals, one, for instance,
simply goes through some finite number of the previous half-way
points, and then jumps to the end point.  Q: how does one get from
one of the half-way points to 1/2?  A: the same way one gets from 3/8
to 7/16, from 7/16 to 15/32, etc.
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater