On 5/31/2013 10:35 AM, Bruno Marchal wrote:

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.

But those are not beliefs in the mathematicians sense, they are beliefs in the common sense. They don't just believe the axioms and that the theorems follow from them. Scientists usually call them hypotheses or models to emphasize that they are ideas that are held provisionally and are to be tested empirically.

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.

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.

What question is that?

But the theaetetus' idea works in arithlmetic, thank to incompleteness, and that's is deemed to be called, imo, a (verifiable) fact.

But does it work outside arithmetic?




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