Attempts were once made to show that Euclid's fifth postulate could be deduced from the others by assuming it was false, and seeing where that led. A perfectly valid technique, as you know. Several assumed that it succeeded, because the conclusions were so obviously ludicrous, and from that it was assumed that Euclid's geometry was the only one.
Of course, now we know that the conclusions weren't contradictions, and are "simply" non-Euclidean geometries.
"Counter-Intuitive" is not the same as "Contradiction."
Just because the Axiom of Choice leads to such counter-intuitive results it doesn't mean they're not true. Just because infinities lead to bigger infinities (via Cantor), flasks of finite volume but infinite surface area,
Attempts were once made to show that Euclid's fifth postulate could be deduced from the others by assuming it was false, and seeing where that led. A perfectly valid technique, as you know. Several assumed that it succeeded, because the conclusions were so obviously ludicrous, and from that it was assumed that Euclid's geometry was the only one.
Of course, now we know that the conclusions weren't contradictions, and are "simply" non-Euclidean geometries.
"Counter-Intuitive" is not the same as "Contradiction."
Just because the Axiom of Choice leads to such counter-intuitive results it doesn't mean they're not true. Just because infinities lead to bigger infinities (via Cantor), flasks of finite volume but infinite surface area,
http://community.tes.co.uk/forums/t/277351.aspx
and hotels that never run out of room,
http://mathforum.org/kb/message.jspa?messageID=5584228&t...
it doesn't mean that they aren't useful, or don't, in some sense, exist.