Points & points
Subject: Re: Reply to Do Points Have Area?
Author: John Conway <conway@math.Princeton.EDU>
Date: Thu, 22 Jan 1998 16:44:27 -0500 (EST)
 Jesse, I have tried, and tried very hard, to understand what you're 
saying, but have reached the point at which I'm about to give up. Before 
I do so, I'm making one last try (I will make more "last tries" if I
get something out of this one!). 
 Let me say that I am entirely happy with your basic idea of getting rid 
of Euclid's fiction of "points with no magnitude". It's just that I have 
not managed to get any kind of understanding of what you think you are 
putting in its place. Several times I have asked you direct questions, 
but the answers have been no help in telling me what you're thinking about.
 The closest I got was when you told me that the Points in your plane 
looked like:
 o o o o o o o o o o o o o 
o o o o o o o o o o o o o 
o o o o o o o o o o o o o 
o o o o o o o o o o o o o 
(but magified so that they touch). But then you went on to define a Circle
to be a continuous string of these, from which I deduced that in fact 
there couldn't be any non-trivial Circles. Then it turned out that it
wasn't the set of all Points in your plane that looked like the above 
figure, but only those in the coordinate-system (or something). Forgive 
me if I'm getting this wrong, but but I really am confused. 
 So I got the idea that there were more Points besides those in the 
coordinate system. It seemed to me that (taking a suitable unit), the 
points in the coordinate system were discs of unit diameter centered at 
(Euclid's) points with integer coordinates, while perhaps there were
also other Points (which were also discs of unit diameter) centered at 
other points. This would then allow there to be Circles in the sense in 
which you defined that term, i.e., continuous closed loops of Points all 
at the same distance from a given Point, namely the discs of unit 
diameter centered at the vertices of one of Euclid's regular polygons of 
edgelength 1. So I asked you explicitly whether there was any 
difference between this model and your geometry, and you said something 
like "well, let's try that". Well, I don't want to just try something.
I'm perfectly capable of studying all kinds of geometry and working out 
their properties; but what I want to know is precisely what you are 
thinking about, and you don't seem to be capable of telling me. I really 
don't know just what it is. 
 I ask again. Are all the Points of your plane arranged in an array 
like the above, or are there others? Can two Points overlap withouyt 
being equal? Is there any difference between your kind of plane geometry 
and the set of all unit discs in Euclid's geometry, and if so, just what 
is this difference? Or have you not yet understood your own ideas in 
enough detail to be able to give answers to these questions?
 John Conway




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