On 2/15/2011 1:01 PM, 1Z wrote:
The difference is I can choose what are/who are/the behavior of...
> > Sherlock holmes/pink unicorn/whatever... not the numbers once an
> > axiomatic system is chosen.
>
> No, it's only a difference of degree. You can't choose Sherlock Holmes
> to be an American or a bus driver. He "exists" in a looser axiomatic
> system than integers, but he is still defined by being consistent with
> the character in the stories by Conan Doyle. Similarly, you can't
> imagine a pink unicorn that is blue and has two horns.
>
> Brent
The ontology of fiction can be true of mathematics even if the
methodology isn't.
It seems that fictional characters exist in a different domain than
Platonia. One of the attributes of fictional characters that
distinguishes them from real people is that there questions about them
that would have factual answers if they were real but which don't
because they are fictional. For example, did Sherlock Holmes have a
mole on his left arm? If I asked that of say, Conan Doyle, we wouldn't
know the answer but we would suppose there is a definite fact of the
matter.
Because numbers are wholly defined by a set of axioms, it seems that
they are more real than fictional characters. Whatever question you can
ask about a number has a factual answer, although you may not know it or
how to find it. But when you consider arithmetic as a whole this no
longer holds. There may be questions that aren't decidable and whose
answer could be added as an axiom; the way a writer could add a mole to
Sherlock Holmes' arm.
Brent
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