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Re: Cantor's absurdity, once again, why not?
Posted:
Mar 14, 2013 9:49 PM
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david petry <david_lawrence_petry@yahoo.com> writes:
> On Thursday, March 14, 2013 2:03:35 PM UTC-7, Jesse F. Hughes wrote:
>
>> Is the negation of Fermat's theorem a falsifiable statement? If so,
>> how might one show that it is false?
>
>
> Didn't you just answer your own question?
No.
>> Is it the case that if a given mathematical statement is
>> "falsifiable", then so is its negation?
>
>
> I think you've already answered your own question.
We don't seem to have the same ideas of falsifiability[1].
Here's my intuitions. The statement
All ravens are black.
is falsifiable, since the discovery of a white raven would prove it
false.
The statement
Not all ravens are black.
is not falsifiable, except in exceptional circumstances[2]. If I see
many ravens and all of them are black, then I still can't conclude
that this statement is false, since the next raven I see might be
white. So, although the statement is verifiable, it is not
falsifiable.
Obviously,
(An>2) NOT (E a,b,c)( a^n+b^n=c^n )
is like the first statement and hence plausibly falsifiable, while its
negation is like the second statement and hence not falsifiable.
But maybe I'm wrong. How would you set out to prove that
Not all ravens are black.
is false?
Footnotes:
[1] Here, I assume we're using the usual, fairly naive notions of
falsifiability which actually have not been fashionable in philosophy
of science for quite some time, due to a number of issues which need
not detain us.
[2] If there is a finite number of ravens and you know when you have
found them all, then it is falsifiable.
--
Jesse F. Hughes
"It is not as satisfying to disagree with a book."
-- Russell Easterly, on why he argues against set theory without
reading a book on set theory.
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