Bijan Parsia writes:
At 1:45 PM -0500 2/2/98, Patrick Logan wrote:
(3) Point is a two-dimensional value. It cannot be made linear.
(I'm assuming that by 'linear' one means 'having a one-to-one correspondence with the natural numbers'.)
(Smalltalk) points are ordered pairs of integers (not of reals), correct? Thus, they have a mapping to the rationals.
1@1 --> 1/1 1@2 --> 1/2 1@3 --> 1/3 etc.
I think the point is that Magnitude assumes that: (a > b) | (a < b) | (a = b)
But when you consider, say, the points 2@4 and 1@2, they are neither less than or more than each other -- but they are not equal either. (the same problem would occur in a mapping x@y --> x+y, and any other reasonable mappings I can think of from points onto numbers).
The ordering brought up before: a > b ==> (a x > b x) | ((a x = b x) & (a y > b y)) satisfies the requirement (even though it isn't a map onto numbers). However, it's rather arbitrary -- useful for, say, sorting, but not much else. It's consistent but not meaningful. The current implementation is (sort of) meaningful but not consistent.
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