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:
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
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
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 ->
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
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
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.
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
To post to this group, send email to email@example.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.