> Of course, I have no real evidence for my views...
> On the other hand, the people who talk about the great difficulty of factoring have equally little evidence...
This is a classic antinomy (paradox): one can argue indefinitely in either direction, because the question lies along the bounds of human reason (or so says Immanuel Kant).
The two sentences above, in themselves, provide a bit of evidence of the impossibility of solving the problem, and at the same time provide evidence for the possibility of handling this problem as a significant phenomenon of pure mathematics.
:)
EDIT: I mean only that the insolubility of the problem may itself be of mathematical use: it may (insofar as it is unsolvable, and insofar as it appears to be soluble) amount to a kind of 'anchor' for mathematics, a marker that indicates the boundary of the mathematical sciences, and that such a boundary would be of tremendous import to mathematicians and philosophers. Why is _this_ problem, _this_ problem specifically, unsolvable? (Rather than some other problem that has been solved?)
tl;dr The question of "why have we have trying to solve this problem for millennia?" is perhaps more significant for mathematics than the solution to the problem.
> On the other hand, the people who talk about the great difficulty of factoring have equally little evidence...
This is a classic antinomy (paradox): one can argue indefinitely in either direction, because the question lies along the bounds of human reason (or so says Immanuel Kant).
The two sentences above, in themselves, provide a bit of evidence of the impossibility of solving the problem, and at the same time provide evidence for the possibility of handling this problem as a significant phenomenon of pure mathematics.
:)
EDIT: I mean only that the insolubility of the problem may itself be of mathematical use: it may (insofar as it is unsolvable, and insofar as it appears to be soluble) amount to a kind of 'anchor' for mathematics, a marker that indicates the boundary of the mathematical sciences, and that such a boundary would be of tremendous import to mathematicians and philosophers. Why is _this_ problem, _this_ problem specifically, unsolvable? (Rather than some other problem that has been solved?)
tl;dr The question of "why have we have trying to solve this problem for millennia?" is perhaps more significant for mathematics than the solution to the problem.