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gwern comments on Rationality Quotes: October 2009 - Less Wrong
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There are ways to trisect the angle; did your method break the rules and use one of them, or was it just wrong?
It was just wrong.
I think he meant that he tried to trisect an angle in general, by construction; this has been proven impossible.
Which is perhaps what you meant by "break the rules" (of construction), by using a marked ruler, for instance.
Right. There are some constructions like Archimedes's use of a marked ruler (which is covered, actually, in the 'Means to trisect angles by going outside the Greek framework' section) which work correctly & are not immediately obviously breaking the rules. So I had to ask before I could know whether he had broken the rules or broken his proof (if you follow me).
Couldn't you trisect a right angle by making an equilateral triangle with one of the right angle's lines for a side, then bisecting that angle of the triangle? It wouldn't generalize to other angles, but you wouldn't need a ruler.
Of course you can trisect some angles, just not all of them. For example, you can't trisect the angle of an equilateral triangle (60 degrees).
Just like you can solve the Halting Problem - for particular Turing Machines. The interesting impossibility results are always general.
The analogy isn't perfect because the halting problem can in principle be solved for each particular machine, but trisection can't be solved for each particular angle.
It can only be solved in that a correct answer can be given; there are some Turing machines that do not halt for which there is no proof that they do not halt.
Yes, this is correct, but the halting problem can still be solved with a simple algorithm for each such machine, and even for all of them at once :-) But okay, we understand each other.
Prove it.