Well, yes and no. I had the same initial reaction to the strip, but on
further reflection decided the business guy's attitude was probably
appropriate.
Note that mathematical impossibility/undecidability results all depend on a
close specification of the tools available for the solution.
(I won't address the piano tuning example. I don't know the physics, I
have not looked at the math, and I am basically tone deaf - three strikes
is more than enough.)
In non-Euclidean geometry it is certainly possible to construct a triangle
whose angles total more or less than 180 degrees. According to relativity,
non-Euclidean geometry provides a better description of physical space. So
it is possible to construct a _real_ triangle with angles different from
180 degress.
My favorite example of the interplay between the tools available for
solution and impossibility results is Goedel's Incompleteness Theorem: In
any formal system strong enough to express arithmetic, there are sentences
which can neither be proved nor be refuted (refutation =3D proof of the
negation).
Goedel proved this theorem by (giving the method for) constructing such a
sentence. The sentence says, in effect, that it has no proof. So,
assuming first-order logic is consistent, the sentence is true. Goedel's
proof proves the very sentence that it proves can't be proved.
It's not really so paradoxical as it sounds. Part of the proof is a close
specification of formal proof. (A formal proof is a sequence of sentences;
each sentence in the sequence either has the form specified by one of the
axiom schemata or follows, by one of the rules of inference, from a set of
sentences which appear earlier in the proof.) The Goedel sentence cannot
be proved by the form of proof so specified, but Goedel's proof does not
have that form.
So, what does this have to do with Dilbert?
Problems are often solved (or perhaps resolved) by changing the tools
available for their solution. For many centuries it was considered
impossible for a human being to fly. Human beings just can't flap a wing
fast enough. So, we don't flap the wing; we blow air over it. And, of
course, we use mechanical/chemical resources beyond muscle power.
It is impossible for anything (even information) to travel faster than
light. If a star is twenty light years away, we cannot travel there -
cross the intervening space - in less than twenty years. If we can find an
appropriate wormhole, however, we can skip the intervening space and get
there faster.
It strikes me that a business, or any enterprise (including teaching)
involving the real as opposed to the theoretical, should find
impossiblility results basically irrelevant - except as warnings that new
tools are needed.
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