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 Feb 24 comment Are proofs by contradiction really logical? Let's try a specific statement; say, the Continuum Hypothesis (CH). Are you asserting that CH is definitely either true or false in the "real world"? I am not sure I even know what that means, much less that it is "intuitively clear". And isn't calling something "intuitively clear" basically the same as making it an axiom? In which case, your argument reduces to "if you believe the law of the excluded middle then you should believe the law of the excluded middle"... Feb 23 comment Are proofs by contradiction really logical? It sounds like your definition of "real world" assumes what you are trying to demonstrate. Feb 19 comment Logic puzzle: Which octopus is telling the truth? @SebastianRedl: More accurate to say: "With four different answers, obviously at most one can be telling the truth." Which is a rather different statement... Jun 9 comment Control Points of Bézier Curve? @Peter: Thanks, that is more clear. Although I still do not see how it answers the question... I thought the question was to prove that the control points (apart from the first and last) do not lie on the curve. Which is not the case in general, so the interesting question is, when is it true? Your expression just says for any t not 0 or 1, there exist control points such that P(t) equals none of the points. Jun 8 comment Control Points of Bézier Curve? @Peter: I still do not get it. For any particular $t$, obviously the value cannot equal all of the $P_{i}$... Jun 8 comment Control Points of Bézier Curve? I mean all of the points being collinear. That is, I believe the statement is true provided (a) each control point lies on the convex hull of all control points and (b) not all control points lie on a single line. Jun 8 comment Control Points of Bézier Curve? @Peter: I am sorry, but then I do not see what your answer has to do with the question? Obviously the curve is not "identically" anything for all $t$... Jun 8 comment Control Points of Bézier Curve? @Peter: Well, obviously, since I pointed out the collinear counterexample in my reply to you :-). I have updated my answer. Any other counterexamples? Jun 8 comment Control Points of Bézier Curve? @Peter: By "identically", do you mean for all $t$? If not, what do you mean, exactly? Jun 8 comment Control Points of Bézier Curve? @Peter: Where does your proof use the assumption that (e.g.) the control points are not collinear? And that they form a convex set? (Because the statement is not necessarily true otherwise...)