Jefferson wrote: > Andrew Janssen wrote: > > >>I had some other concerns, like how the Scirus gleam handles the fact >>that number theory is omega-incomplete (a fancy way of saying that you >>can make true statements of number theory which cannot be proven using >>number theory), or how it deals with Euclidean v. Non-Euclidean >>geometry, but that's probably too esoteric. > > > Another yep. It's not that I don't understand what's involved as that I > don't understand the _implications_, and so have no idea how Feroze would > handle that sort of authority. If it becomes important I'll make a > decision then. Well, to summarize in reverse order, the Euclidean v. Non-Euclidean geometry has to do with the parallel postulate and the shape of space. Take a line and a point not on the line, A. In Euclidean or plane geometry, there is exactly *one* line which passes through point A and does not intersect with the first line. In Non-Euclidean spherical geometry, there are *zero* lines which pass through point A and do not intersect the first line. In Non-Euclidean hyperbolic geometry there are *at least two* lines which pass through point A and do not intersect the first line. Another way to describe it is that in plane geometry, the sum of the angles of a triangle is *exactly* 180 degrees; in spherical geometry, the sum of the angles of a triangle is *greater than* 180 degrees; in hyperbolic geometry, the sum of the angles of a triangle is *less than* 180 degrees. For most purposes, Euclidean geometry is fine, but when you're working with the surveying of meridians on a planetary surface, you're getting into spherical geometry. Hyperbolic geometry is just plain weird, and is used mostly by mathematicians and theoretical physicists. As for the omega-incompleteness of number theory, it boils down to the fact that no matter how sophisticated your theory there will always exist at least one true statement which cannot be proven using the theory--it's the mathematical equivalent of saying, "This statement is false." Paradox is inescapable. However, I doubt most espiri will ever run across this barrier--at least, not in the near future. Andrew > Jefferson > http://www.picotech.net/~jeff_wilson63/rpg/Exq_Main.html > > > > ---------------------------------------------------------------- > To unsubscribe, send mail to celandra-off@phoenyx.net. > ---------------------------------------------------------------- To unsubscribe, send mail to celandra-off@phoenyx.net.
