On 31 May 2013, at 01:19, meekerdb wrote:
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.
They believe that they publish papers, and usually share the
consensual believes, like in rain, taxes, and death (of others).
All humans have many beliefs. A genuine scientist just know that those
are beliefs, and not knowledge (even if they hope their belief to be
true). So they will provides axioms/theories and derive from that, and
compare with facts, in case the theory is applied in some concrete
domain.
Of course they believe things in the common sense that they are
willing to act/bet on something (at some odds).
Yes. For example most believe that there is no biggest prime numbers.
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.
Very often, alas. But the israelites and the taoists encourage the
comments and the discussion of texts. So there are degrees of dogmatic
thinking.
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.
http://www.ditext.com/gettier/gettier.html
It is equivalent with the dream argument made by someone who believes
he knows that he is awake.
Gettier is right, but he begs the question.
But the theaetetus' idea works in arithlmetic, thank to
incompleteness, and that's is deemed to be called, imo, a (verifiable)
fact.
Bruno
http://iridia.ulb.ac.be/~marchal/
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