chalidze
Cantoresque
by Valery Chalidze, Defender Of Faith In the Obvious
As often happens, after years one can see that his thoughts can be
expressed more simply.
In Euclidean geometry one can place a point between any two points on
a line. That is proven. So there are no two points next to each other
without a hole between them.
This is enough to declare that it is impossible to cover a line's
interval by points continuously without holes. Yet the road to
the paradise of contemporary mathematics, using Gilbert's colorful
expression, is paved by such "continuous" points.
Suppose there are some fantastically non-Euclidian points, but still
with no dimensions. Then let's locate any pair of neigboring
points with no hole between them. If they both have no size, then
the pair of points is indistinguishable from one point; those two are
simply one point. One might say that the one point contains two points.The same for three, four and so on to infinity -- countable infinity, that is. But then, each point of this infinity also contains an infinity of points. The reader is invited to continue this line (or point) of reasoning as a mental exercise. In this manner, one might try to knock at the gates of uncountability. One might search for the alef of points within a point.
Should I continue? Perhaps not. That old enemy of all thinkers -- the ugly
question, "so what?" -- is waiting for me behind the next sentence.
This is an appendix to my "Infinity or Not? An Arithmetical Satire"
[Now in public domain.
http://www.worksofvalerychalidze.com/My e-mail VRAZHEK43 at Google com.]
by Valery Chalidze, Defender Of Faith In the Obvious
As often happens, after years one can see that his thoughts can be
expressed more simply.
In Euclidean geometry one can place a point between any two points on
a line. That is proven. So there are no two points next to each other
without a hole between them.
This is enough to declare that it is impossible to cover a line's
interval by points continuously without holes. Yet the road to
the paradise of contemporary mathematics, using Gilbert's colorful
expression, is paved by such "continuous" points.
Suppose there are some fantastically non-Euclidian points, but still
with no dimensions. Then let's locate any pair of neigboring
points with no hole between them. If they both have no size, then
the pair of points is indistinguishable from one point; those two are
simply one point. One might say that the one point contains two points.The same for three, four and so on to infinity -- countable infinity, that is. But then, each point of this infinity also contains an infinity of points. The reader is invited to continue this line (or point) of reasoning as a mental exercise. In this manner, one might try to knock at the gates of uncountability. One might search for the alef of points within a point.
Should I continue? Perhaps not. That old enemy of all thinkers -- the ugly
question, "so what?" -- is waiting for me behind the next sentence.
This is an appendix to my "Infinity or Not? An Arithmetical Satire"
[Now in public domain.
http://www.worksofvalerychalidze.com/My e-mail VRAZHEK43 at Google com.]