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> [ Words to be emphasized (usually by printing them in italics) here
> will be put between asterisks * ]
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> On Sun, 4 Jul 1999, Gregory Moore asked:
>
> > Does anyone on the list have any information about when and how lines
> > (also curves, planes, etc.) started to be treated as sets of points?
>
> The formulation "treated as sets" is open to many interpretations.
>
> The most narrow one is that in which a line (curve) is said "to be"
> a set of points (with particular properties) in its definition. That
> mode of procedure I find 1960 in Borsuk/Szmiliew "Foundations of
> Geometry" [the Polish original is slightly earlier], where the set
> notation is used throughout (e.g. defining the meeting point of two
> lines as the intersection of the two sets). This way of speaking is,
> of course, in the spirit of the "Fundamenta Mathematica", and you might
> wish to look at the earlier volumes of this journal for articles
> employing this point of view. Presumably Kuratowski speaks of curves
> as of sets of points in those parts of his "Topologie" devoted to
> the topology of the plane.
>
Dear Walter Felscher,
It is your first meaning that I have in mind. There are clear references
much earlier than Kuratowski. The best that I know is Peano's book of
1889: I principii di geometria logicamente expositi. He begins his
axiomatization of plane Euclidean geometry by taking point as a primitive
term and also segment with endpoints a and b. One of his axioms states
that the segment ab is a class of points. He proceeds to develop a
treatment of geometry in which rays, lines, etc. are classes of points.
I also have a reference to Bolzano in which he treats lines as sets of
points, but here it is a bit more ambiguous.
Many thanks for your thoughts on the matter.
Cordially,
Greg Moore