It's also possible that the question, although seemingly made up of
ordinary English language words used in a logical way, is actually
incoherent.

If I say, proposition P is both true and false, that is a sentence made
up of English words, but it does not really make sense.  I could then
demand to know whether P is true or false, and whatever answer you give,
I say that it is the opposite.  If you say P is true, I point out that
we just agreed that P was false, and vice versa.

This is a trivial example because the paradox is so shallow, but the
same thing is true for deeper paradoxes.  The problem is not a failure
of our reasoning tools, but rather that the question has no meaning.

So you can't always take a sequence of words and expect to get an
unambiguous and valid answer.  You must always consider the possibility
that your question is meaningless.  The fact that people can't necessarily
answer it does not imply that mathematics is unknowable or that there
is no such thing as mathematical knowledge.  There may be other reasons
to think so, but it does not follow merely because a given sequence of
words has no consistent analysis.

Hal Finney

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