On 5/30/2013 3:43 PM, Russell Standish wrote:
On Thu, May 30, 2013 at 12:04:13PM -0700, meekerdb wrote:
You mean unprovable? I get confused because it seems that you
sometimes use Bp to mean "proves p" and sometimes "believes p"
To a mathematician, belief and proof are the same thing.
Not really. You only believe the theorem you've proved if you believed the axioms and
rules of inference. What mathematicians generally believe is that a proof is valid, i.e.
that the conclusion follows from the premise. But they choose different premises, and
even different rules of inference, just to see what comes out.
I believe in
this theorem because I can prove it. If I can't prove it, then I don't
believe it - it is merely a conjecture.
In modal logic, the operator B captures both proof and supposedly
belief. Obviously it captures a mathematician's notion of belief -
whether that extends to a scientists notion of belief, or a
Christian's notion is another matter entirely.
I don't think scientists, doing science, *believe* anything. Of course they believe
things in the common sense that they are willing to act/bet on something (at some odds).
The Abrahamic religious notion of 'faith' is similar to that; the religious person must
always act as if the religious dogma is true (at any odds). This precludes doubting or
questioning the dogma.
When it comes to Bp & p capturing the notion of knowledge, I can see
it captures the notion of mathematical knowledge, ie true theorems, as
opposed to true conjectures, say, which aren't knowledge.
Gettier (whom I know slightly) objected that one may believe a proposition that is true
and is based on evidence but, because the evidence is not causally connected to the
proposition should not count as knowledge.
But I am vaguely sceptical it captures the notion of scientific
knowledge, which has more to do with falsifiability, than with proof.
And that's about where I left it - years ago.
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