[FOM] Buckner's Question on the number line

[A.P. Hazen]

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?

and adds

>
>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 
Peirce's remarks less helpful!))

Allen Hazen
Philosophy Department
University of Melbourne.