[FOM] Question on the number line
Dean Buckner
d3uckner at btinternet.com
Tue Nov 15 15:34:49 EST 2005
Forgive me if this is a naïve question. It occurs in the chapter in
Suarez (the 16C philosopher & mathematician) that I mentioned in a
previous posting. Suppose one takes a line segment of length 1 and
break it in half (assuming we can break it "in half"). Intuitively, you
would imagine you get two identical segments of length 0.5. But if you
"identify" that line segment with the reals from 0.0 to 1.0, you have to
ask what happens to the point at 0.5 -- which "half" does it belong to?
If it stays with only one side, then you don't have two identical line
segments. Instead, you have one segment that is closed at both ends and
another segment that is open at one end.
Suarez discusses a wall, originally all one colour, half of which has
now been painted another colour. Which colour is the boundary between
the one colour and the other? It can't be both colours. To resolve the
problem he distinguishes between a "terminus intrinsecus" – a boundary
which includes the end-point, and which seems to be the (late) medieval
equivalent of a closed segment, and a "terminus extrinsecus" which does
not include the end-point, and which seems to be the late-medieval
equivalent of an open segment. (He says, in one of those charming
passages where Theology meets mathematics, that an angel, and much more
God, sees clearly in what way bodies really touch).
But Suarez says there is still a problem, however, because there seems
no reason why the boundary should lie on the one half rather than the
other.
I have a quotation that says that Goedel also found this idea
problematic. Is this true?
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