I wasn't going to spend much time on this but the problem may be more subtle
than you seem to appreciate. The solution for one or two points on a curve
contains less information than the equation of the curve. And that made me
realize that a curve or even a straight line has an infinite number of points.
A fundamental idea underlying Chaitin's, Kolmogorov's and Solomonoff's theories
is that there are some algorithms that can be used to generate information;
that is, they can generate more information than is input into them, and it is
not simply repetition, although repetition is used in some multiplying effect.
There are costs involved, but can we find ways to cut through the ordinary
costs in order to get special (and narrow) results that would be interesting to
us. Thank you for talking about this to me. Although I don't have solutions I
(at the least) feel that I am learning a little about looking at this question.
So, is there another short cut that could find solutions (individual points) of
higher order equations without resorting to trial and error solutions or the
existing equations which involve finding roots of coefficients and so on? This
solution suggests that there may be - and probably are. That would be
spectacular because there are no formula solutions for equations over the order
of 7 (i think it is 7). The solutions, if feasible, would become longer as they
the order of the equation increased. But impossible is not a fact, because
there is no underlying principle or mathematical abstraction that establishes
an impossibility theorem. A possible linear solution to a cubic may force
imaginary solutions into the problem but that is not a big deal. I think a
"solution" to the cubic equation is a search for 3 points (but I can't
remember for sure). I don't see any abstract reason to say that finding 2, 3,
4, ... points on equations of stepped order is impossible because there is no
free lunch (to put it in words that I can easily muster-or mustard to stay with
the metaphor). There is a slight similarity to n=np?
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Artificial General Intelligence List: AGI
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