[FOM] Question on the number line
William Tait
wwtx at earthlink.net
Wed Nov 16 17:19:37 EST 2005
On Nov 15, 2005, at 2:34 PM, Dean Buckner wrote:
> 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?
This was Aristotle's argument in *Physics* that the line is not
composed of points.
About the notion of a continuum, he writes (227^a9 )
\bq
A thing that is in succession and touches is `contiguous'.
The `continuous' is a subdivision oof the contiguous: things
are called continuous when the touching limits of each become
one and the same and are, as the word implies, contained in
each other.
\eq
The ``in succession'' would seem to indicate that Aristotle
had in mind here only linear continua. Suppose that we cut
such a continuum, say a line segment, into two
successive segments A and B. The `limit' or boundary p
of A and the boundary q of B are points; and the
continuity of the original line segment A+B requires that
these points coincide---otherwise A+B would have a gap
between p and q. (Much as I hate to say good things about Aristotle,
this is not a bad analysis of the notion of a continuum).
At the beginning of Book VI, he goes on, on the basis of this
analysis, to argue that a
continuum cannot be composed of indivisibles. In the case of
line segments he writes
\bq
For the extremities of two points can neither be *one*
(since of an indivisible there can be no extremity as
distinct from some other part) nor *together* (since
that which has no parts can have no extremity, the extremity
and the thing of which it is the extremity being distinct).
\eq
I think the argument is this: when we cut the line A+B
into the two segments A and B, regarding the line as a
set of points, the corresponding extremities p and q
must coincide. But, thinks Aristotle, p, the extremity of
A must be in the set of points A and q, the
extremity of B must be in the set of points B. But then
A and B have the point p = q in common, contrary to the
supposition that they are separate parts of A+B.
Dedekind noticed that there are always two cuts corresponding to a
point: in one the lower cut contains the point and, in the other, the
upper cut.
Regards,
Bill Tait
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