On 02 Jun 2017, at 00:29, David Nyman wrote:
On 1 June 2017 at 18:00, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 01 Jun 2017, at 16:42, David Nyman wrote:
On 1 Jun 2017 15:20, "Bruno Marchal" <marc...@ulb.ac.be> wrote:
On 01 Jun 2017, at 15:59, David Nyman wrote:
On 1 Jun 2017 14:01, "Bruno Marchal" <marc...@ulb.ac.be> wrote:
On 01 Jun 2017, at 12:44, David Nyman wrote:
<snip>
Yes, but doesn't the comp theory itself assume the relation as if
true, in order for mechanism to make sense (which was my caveat)?
The point here is very subtle. We cannot assume Mechanism is true,
we can only assume Mechanism. We can assume it as a sort of meta-
level hypothesis. It is indeed the reason why I insist on an act of
faith, and why it is a theology.
Now, you might ask me what is the difference between assuming X, and
assuming X is true. The difference is that when we assume X, but
don't invoke the full semantic of X, and we can preserve our
consistency. By saying "mechanism is true", even in an hypothesis,
you refer to the God of mechanism at a place that is impossible.
We have something similar, but slightly simpler, for the notion of
self-consistency. Take PA. It has 6 axioms + the infinity of
induction axioms
1) Ax (0 ≠ s(x))
2) AxAy (s(x) = s(y) -> x = y)
3) Ax (x+0 = x)
4) AxAy (x+s(y) = s(x+y))
5) Ax (x*0=0)
6) AxAy (x*s(y)=(x*y)+x)
7) the infinity) (F(0) & Ax(F(x) -> F(s(x)))) -> AxF(x)
PA is consistent (I hope you are OK with this, as all mathematicians
are, except nelson)
Imagine I add to PA the axiom 8) PA is consistent. That is
8) consistent(1+2+3+4+5+6+7)
In that case there is no problem that new theory, PA+con(PA) is
consistent, and even much more powerful than PA. It proves new
theorems and it shorten many proofs.
But imagine we add the following to PA
8') consistent(1+2+3+4+5+6+7+8')
That is self-referential, but using the diagonal lemma (the D'X' =
'X'X'' trick), we can build that formula in arithmetic and add it to
PA as axiom.
In that case, we get a theory which can prove, indeed in one line
proof, its own consistency, and so, by Gödel II, that theory is
inconsistent.
The case above is more complex to describe, because it refers to
"arithmetical truth", which cannot even be defined in arithmetic. It
is almost the nuance between (Bp & p), and (Bp & V(p)). For each
partuclar p, you can write Bp & p, but you cannot define knowledge
by a general (Bx & x). It is the nuance between assuming that x + 0
= x, and using only that, and assuming True('x + 0 = x'), which
cannot be express in the arithmetical language.
Mechanism is awkward on this. Yet, this sustains your intuition
that the ultimate 3p, the outer God view, is a 0p pov.
"Out of sight, out of mind" as the proverb has it.
"Loin des yeux, loin du coeur", we say in french (which has not
exactly the same intent).
It works well with Plotinus idea that God does not exist, as it is
the ultimate reason why everything exists without mentioning ever
itself in the creation (making the idea that God can talk with us in
a direct public way nonsensical). Only the first person has access
to this, but can only stay mute.
Another example is that even if someone survives the classical
teleportation experience, he/she cannot claim that he/she knows that
mechanism is true.
OK
Still another example, even closer to what is alluded above is the
case of the notion of sigma_1 truth, or simply sigma-truth. The
notion of sigma-truth *is* definable in arithmetic, and the self-
referential sentences saying that she is sigma-true appears to be a
sigma proposition, yet it can be shown to be ... false! This is very
unlike the sigma proposition saying "I am provable", which is always
true! All this despite G* proves (sigma-true <-> sigma provable),
but again, only G* says this, the machine has to be mute.
The machine would appear to take Wittgenstein's dictum to heart.
But its guardian angel is able nonetheless to intuit a truth in what
it cannot say.
I would not use "intuit" for the guardian angel. he is not a subject.
just a machinery who knows everything about the machine(s). At the
propositional level, he knows/proves the whole truth about the self-
referential sentence. We know it is true, only because we limit
ourselves to true machine, by decision.
I cannot say "if mechanism is true then the truth of mechanism go
without saying", but I can say at the meta-level "If mechanism then
the truth of mechanism go without saying". I think.
Perhaps "if mechanism is true" is close to what Brent calls a
reification.
Hmm... Brent does not use "reification" correctly. He accused me of
reifying arithmetic, when I said that he reified primary matter. but
arithmetic is assumes, not reified in anyway. Primary matter is
reifed, as it is an appearance only, and its assumption is uded to
avoid solving a question. It is like "God made it".
To say "mechanism is true" is more a logical mistake, although a very
subtle one, based on Tarski theorem of the undefinablity of truth.
"If mechanism" on the other hand is a conjectural point of departure
from which we are then at liberty to reason. That reasoning may then
draw us towards an intuition of truth.
OK with this. technically, it is really the difference between saying
that 2+2=4, and saying that the sentence "2+2=4" is true. Or between
saying "Darwin is perhaps right" and saying God made "Darwin perhaps
right".
I am not sure I could have grasp those nuances without the help of
mathematical logic. They are quite counter-intuitive.
Bruno
(I fall myself in that trap often, but *sometimes* it is just to
avoid nuances which at some stage is a bit like a 1004 mistake with
respect to beginners ...). In our context, I told you that we are
always close to inconsistency. You seem to like the panorama we can
get from climbing a very high cliff.
Yes. Oddly enough, I'm fond of what I call "C. D. Friedrich
moments". It's dizzying sometimes.
David
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