On 30 Dec 2013, at 15:25, Alberto G. Corona wrote:




2013/12/30 Bruno Marchal <marc...@ulb.ac.be>

On 30 Dec 2013, at 12:39, Edgar L. Owen wrote:

All,

In response to the discussion of the possibility of a "Final Theory" I'm starting a new topic on the Nature of Truth since this is an important and separate issue from previous discussions.


1, it is impossible to directly know the external fundamental reality, we know external reality only filtered through the structures of our own minds. What we really know is only our own mental model of external reality which is provably very very different than actual external reality.

2, However we can easily prove that we do know external fundamental reality to an extent sufficient for us to function reasonably effectively within it. If we didn't have some actual true knowledge of external reality we could not even function within it and thus could not exist. So our very existence in actual reality demonstrates we do have some true knowledge of it. (This true knowledge consists of snippets of logical structure rather than the physical world we believe it to be.)

That are belief, not knowledge.

Then, what is knowledge? the one derived from mathematical deductions based on the belief on + and succ ?


That one is still on the type belief (a consequence of Gödel's incompleteness).

To know that 1+ 1 = 2, you need to

1) believe that 1 + 1 = 2, but you need also that

2) it is the case that 1 + 1 = 2   (in your "reality")

If you put arithmetical realism on the table, anyone believing that 1 + 1 = 2, knows that 1 + 1 = 2. This needs some "reality" satisfying the fact that 1+1=2, and we do suspect its existence indeed, as the structure (N, 0, s, +, *) taught in high school.

Usually "rational belief" in a large sense is axiomatized by the modal axiom K

B(x -> y) ->(Bx -> By),

with or without the necessitation rule (inferring Bx from x), but (almost) always with the modus ponens (inferring B from A -> B and A).

Then a form of self-awareness is captured by the possible axioms Bx -> BBx.

Gödel provability obeys that. That are the K4 reasoners. 4 is the name (sic) of the formula Bx -> BBx, as it was the main axiom of the fourth system by Lewis (S4).

S4 is the knowledge theory. It is K4 together with the axiom Bx -> x. By definition of knowledge, if you know x, x is true. If p were not true, i.e; if it was not the case that p, you would just be believing wrongly.

Gödel's provability obeys K4 (indeed K4 + B(Bx->x)->Bx), but does not obeys Bx -> x, at least from the machine 3p points' of view on itself.

But the conjunction of Bx & x does obeys S4 (indeed S4 + B(B(x->Bx)- >x)->x, the Grzegorczyk formula).

Set theoretically, knowledge is the intersection of your beliefs and truth.

It can be explained that some machine, like PA and ZF, already understand (prove, or prove from some Dt conditional, or more) that their *personal* knowledge escape all possible 3p definitions. They can't believe they are any machine. They still can bet on it, like "nature" apparently already did.

Bruno



http://iridia.ulb.ac.be/~marchal/



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