# [FOM] Buckner's Question on the number line

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Tue Nov 15 22:59:04 EST 2005

```Buckner asks
>  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?

>
>I have a quotation that says that Goedel also found this idea
>problematic.  Is this true?
>
--- I don't know about Gödel, but lots of other intelligent people
have wondered about it.  Roberto Casati and Achille Varzi in their
"Holes  and other Superficialities" (MIT Press 1994; the paperback,
ISBN 0-262-53133-X is sometimes referred to as "the holey book,"
since it came with illustrative holes punched in the cover) mention a
number,  including Leonardo da Vinci and C.S. Peirce.
>From da Vinci's "Notebooks" they quote
"What is it...that divides the atmosphere from the water?
It is necessary that there  should be a common boundary
which is neither air nor water but is without substance,
because a body interposed between two bodies prevents
their contact, and this does not happen in water with air
...Therefore a surface is the common boundary of two bodies
which are not continuous, and does not form  part of either
one or the other, for if the surface frmed part of it, it
would have divisible bulk, whereas, however, it is not
divisible and nothingness divides these bodies the one from
the other."
(C&V's gloss on this is that Leonardo "put forward the view that
surfaces are some sort of abstraction.")
>From Peirce they cite "The logic of quantity" (1893) (in v. IV of
"Collected Papers") on the puzzling question of whther the line of
demarcation between a black spot and a white background is black or
white:
"It is certainly true, First, that every point of the area
is either black or white, Second, that no point is both
black and white, Third, that the points of the boundary are
no more white than black, and no more black than white....
This leaves us to reflect that it is only as they are connected
together into a continuous surface that the points are colored;
taken singly, they have no color, and are neither black nor
white, none of them."

((Personally, I find Leonardo's view attractive, though his argument
seems to have several  suppressed premisses.  A continuous line can
usefully be thought of as an ordered set of points for mathematical
purposes, but the  points are "abstractions," and should not be
thought of as corresponding to real "parts" of  any actual
continuous "substance." ...  With all respect to America's greatest
philosopher (and one of the founders of  modern logic), I find