|Original post by Fenrisulvur|
I'm not exactly sure which terms and notation you're taking as "common", here, or in what sense. Are you rejecting structure like metric spaces and definitions like that of the circle I gave in favour of some unstated common-sense interpretation, or are you familiar with such structural abstractions and objecting to having it all rehashed?
More the latter. For example, this says the same thing in three ways: "the set of ordered pairs of real numbers, in other words
". What seems strange to me is that I believe the first would reach the largest audience since the Cartesian product is likely to be introduced later in any math program. It's maybe justified by the use of
in the mapping notation to make the connection more immediate, but functional notation with implicit (co)domain and range would've worked as well.
|Original post by Way Walker|
(you even got fed up with it and just said you would be using "vector space axioms", which usually goes without saying)
Oh, no it doesn't. That line would've been torn apart in a more formal setting, I hadn't defined any structure other than a metric at that point. Do you have any idea how many vector spaces can be constructed on R*R?
How many are isomorphic? Or, more to the point, how many are isomorphic to the Euclidean space?
I maintain that an answer to the troll's paradox is going to have to be structural, it's going to have to illuminate criteria which govern whether a sequence of bounding shapes do or do not coverge to give the circumference of the circle, and we're probably going to need a formal definition of what a circumference actually is.
I agree, but the simplest definition would be to simply integrate the length along the curve.
|Original post by JoeCooper|
I don't think we could possibly have a procedure that can validate any and all possible shapes you could throw at it, for all manner of procedures, with my philosophy-student level math.
Maybe true, maybe not. I was hoping something would come out of the discussion. [smile]
As I understood the specifications originally, all I felt I needed to do was to show that this particular gizmo isn't approximating a circle.
Actually, the problem is that it is approximating a circle (defined as the set of points) but not all of its properties are good approximations of the properties of a circle. In particular, the perimeter, but also, for example, the first derivative. The n-gon method also doesn't provide good approximations of all the properties of a circle, like the second derivative.
I don't mean this to sound hostile in any way, but you seem confident that you have a superior handle on the maths to everyone else - why not pitch a fitness test?
Doesn't seem hostile and I almost certainly don't have a superior handle on the math. For one thing, I know very little about fractals.
I've been trying to think of a good fitness test, but I'm having trouble finding a necessary condition. One problem being that the limit of the derivative is not just incorrect, but undefined.
|Original post by Eelco|
'the derivatives'? Only the zero'th and first, actually. That still leaves the nagging question as to why it is only those two that matter.
On a more generic level to nilkn's answer, the different derivatives have different meanings. Having a continuous first derivative makes a function smooth, so, since the lack of smoothness is part of the problem, it wouldn't be a surprise if the first derivative is important. Likewise, if curvature isn't an issue, then the second derivative probably isn't important.
Emphasis mine. The second derivative and upwards are all zero for a polygon, so they do not converge at all.
Technically, they do converge, just not to the same value as the derivatives of the figure the polygons converge to. I wonder if they have to converge to the same value, or if convergence is enough (e.g. it just needs to be smooth). Of course, I'm not sure that you can have the zeroth derivative converge to the right value and the first derivative converge without the first derivative converging to the right value.