
Re: NonEuclidean Arithmetic
Posted:
Sep 3, 2012 12:32 PM


>When a teacher says "added x number of times" they write the multiplicand x number of times, not the addition symbol. When I ask you what is the sum of 12, 34, 16 and 7 (4 addends) I am asking what is SUM(12,34,16,7) and the algorithm to do that is to start with 12, then add 34, then add 16, then add 7.
Sometimes I've seen the wording like so: multiplication is the *taking* of one quantity as many times as there are units in the other.
The "addition", *if* any, (if that's what to be done) is left implicit, according to the problem at hand. I can multiply the contents of my cookie jar without necessarily ever adding it all up.
Further, it would be nice to know as literal translation as possible  you can see the Greek here:
http://farside.ph.utexas.edu/euclid/Elements.pdf (Book 7, def. 15)
The English translation alongside it is full of parenthesized words and phrases, perhaps put it by the translator? One of the phrases is "to itself". Can anyone here give us a very literal account of the original Greek?
Finally Euclid didn't see the correspondences between geometry and arithmetic the way we do, as a mapping that can be used or ignored at will. Read the next few definitions down. When one multiplied two quantities, the result was of a different type: plane. All the discussion that went on a while back about unit square being magnitude 1  comes into play here. 2D areas are a different "type" (i.e., in Hansen's language, they are not dimensionless!) a least by this view of things.
These papers try to explain a bit the differences in thinking between then and now:
http://www.lamath.org/journal/vol5no2/Ratio_Proportion.pdf
https://gismodb.fi.ncsu.edu/gismodb/files/articles/574997accfc1c3b56e72f57d39a9af55.pdf
Bottom line, though, is almost every school child and the occasional adult knows what is meant in today's understanding. Devlin proposes radical change to circumvent some supposedly ponderous problem that doesn't really exist, and Mr. Crabtree proposes a trivial change to effect some imagined radical improvement.
Joe N

