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.
But apart from the Fundamenta-authors, even in the 1940ies you will
find people writing that they "identify" a line with its set of
points, expressing that there are different things which then are "identi-
fied". J.A.Todd, in his most beautiful "Analytic and Projective Geometry"
from 1947 for instance, writes about a line and the "collection of its
points" [where "collection" is used in the sense of "set"]. So *is* a
line the set of its points, or does the latter *correspond* to it ?
[In this connection, keep in mind that for Dedekind the real numbers
*correspond* only to his cuts, while for Weber they *are* the cuts.]
In the case of geometry, this really brings us back to Descartes and
the introduction of coordinates. A point - *is* it the tuple of its
coordinates [or an equivalence class of those], or does it only
*correspond* to that ? It seems to me that it required the development
of Dedekinds set language [as achieved e.g. in Weber-Wellstein] in order
to define geometric objects as set.
In the case of the real line, it brings us back to Simon Stevin's
principle of correspondence who wrote 1685 in his L'Arithm/etique
la communaut/e & similitude de grandeur et nombre, est si
universelle qu'il resemble quasi identit/e [p.3 , l ] .
But then recall Hankel and du Bois-Reymond, who as recently as 1882
in "Die allgemeine Funktionenlehre" vigorously objected to the
identification of the continuum with the collection of real numbers.
[For more historical details on the classical continuum problem cf.
the second volume of my Naive Mengen und Abstrakte Zahlen, Mannheim 1968 ]
W.F.