Subject: Re: Reply to Do Points Have Area

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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