These two are fundamentally different problems.
The first is impossible by definition. The definition of triangle is,
"a three-sided polygon". Asking for a "four-sided triangle" is akin to
asking for "a value of three which is equal to four".
That's right. But see below.
The second is only "impossible" because it contradicts our understanding
(based on observation) of how the physical universe works. Our
understanding could simply be wrong.
And arithmetic could be inconsistent, in which case it might be possible
to prove that 3 equals 4. We don't know for sure that arithmetic is
consistent, and according to Godel, there is no way of proving that it is
consistent. There's no evidence that it isn't, but then, unless the
inconsistency was obvious, how would we know?
http://www.mathpages.com/home/kmath347/kmath347.htm
We've certainly been wrong before,
and we will undoubtedly be proven wrong again in the future. When it
comes to things like electromagnetic theory, it doesn't take too many
steps to get us to the fuzzy edge of quantum physics where we know there
are huge questions yet to be answered.
No. I worded my question very carefully. The discovery of magnetic
monopoles, as predicted by the fuzzy end of quantum physics, would not
invalidate my claim. Magnets don't generate magnetic fields by the use of
monopoles, and the discovery of such wouldn't make it possible to cut an
ordinary magnet in two to get an individual north and south pole. That
would like taking a rope with two ends (an ordinary rope, in other
words), cut it in half, and finding that each piece has only a single end.
Now, you could counter with a clever solution involving splicing the rope
to itself in such a way that it had one end and a loop at the other, er,
end. And such a solution might be very valuable, if we needed a way to
get a rope with a loop at one end. But it isn't solving the problem of
cutting a rope in two and getting only two ends instead of four. It's
solving a different problem.