On 19 Jun 2015, at 18:36, Terren Suydam wrote:
On Mon, Jun 15, 2015 at 3:23 PM, Bruno Marchal <[email protected]>
wrote:
On 15 Jun 2015, at 15:32, Terren Suydam wrote:
On Sun, Jun 14, 2015 at 10:27 AM, Bruno Marchal <[email protected]>
wrote:
We can, as nobody could pretend to have the right intepretation of
Plotinus. In fact that very question has been addressed to
Plotinus's interpretation of Plato.
Now, it would be necessary to quote large passage of Plotinus to
explain why indeed, even without comp, the "two matters" (the
intelligible et the sensible one) are arguably sort of hypostases,
even in the mind of Plotionus, but as a platonist, he is forced to
consider them degenerate and belonging to the realm where God loses
control, making matter a quasi synonym of evil (!).
The primary hypostase are the three one on the top right of this
diagram (T, for truth, G* and S4Grz)
T
G G*
S4Grz
Z Z*
X X*
Making Z, Z*, X, X* into hypostases homogenizes nicely Plotinus
presentation, and put a lot of pieces of the platonist puzzle into
place. It makes other passage of Plotinus completely natural.
Note that for getting the material aspect of the (degenerate,
secondary) hypostases, we still need to make comp explicit, by
restricting the arithmetical intepretation of the modal logics on
the sigma-& (UD-accessible) propositions (leading to the logic
(below G1 and G1*) S4Grz1, Z1*, X1*, where the quantum quantization
appears.
The plain language rational is that both in Plotinus, (according to
some passage----this is accepted by many scholars too) and in the
universal machine mind, UDA show that psychology, theology, even
biology, are obtained by intensional (modal) variant of the
intellect and the ONE.
By incompleteness, provability is of the type "belief". We lost
"knowledge" here, we don't have []p -> p in G.
This makes knowledge emulable, and meta-definable, in the language
of the machine, by the Theaetetus method: [1]p = []p & p.
UDA justifies for matter: []p & <>t (cf the coffee modification of
the step 3: a physical certainty remains true in all consistent
continuations ([]p), and such continuation exist (<>t). It is the
Timaeus "bastard calculus", referred to by Plotinus in his two-
matters chapter (ennead II-6).
Sensible matter is just a reapplication of the theaetetus, on
intelligible matter.
I hope this helps, ask anything.
Bruno
I'm not conversant in modal logic, so a lot of that went over my
head.
Maybe the problem is here. Modal logic, or even just modal notation
are supposed to make things more easy.
For example, I am used to explain the difference between agnosticism
and beliefs, by using the modality []p, that you can in this context
read as "I believe p". If "~" represents the negation, the old
definition of atheism was []~g (the belief that God does not exist),
and agnosticism is ~[]g (and perhaps ~[]~g too).
The language of modal logic, is the usual language of logic (p & q,
p v q, p -> q, ~p, etc.) + the symbol [], usually read as "it is
necessary" (in the alethic context), or "it is obligatory" (in the
deontic context), or "forever" (in some temporal context), or "It is
known that" (in some epistemic context), or it is asserted by a
machine (in the computer science context), etc...
<>p abbreviates ~[] ~ (possible p = Non necessary that non p).
All good here.
Thus my request for "plain language" justifications. In spite of
that language barrier I'd like to understand what I can about this
model because it is the basis for your formal argument AUDA and
much of what you've created seems to depend on it.
In AUDA, the theory is elementary arithmetic (Robinson Arithmetic).
I define in that theory the statement PA asserts F, with F an
arithmetical formula. Then RA is used only as the universal system
emulating the conversation that I have with PA.
Everything is derived from the axioms of elementary arithmetic (but
I could have used the combinators, the game of life, etc.). So I
don't create anything. I interview a machine which proves
proposition about itself, and by construction, I limit myself to
consistent, arithmetically sound (lost of the time) machine. This
determined all the hypostases.
It is many years years of work and the hard work has been done by
Gödel, Löb, Grzegorczyck, Boolos, Goldblatt, Solovay.
I think it's debatable that you didn't create anything. I think
reasonable people could disagree on whether the 8 hypostases you've
put forward as the basis for your AUDA argument are created vs
discovered.
Not only they are discovered, but I show that *all* self-referentially
correct machine discover them when looking inward.
I'm coming from an open-minded position here - but trying to assert
that you're not creating anything strikes me as a move to grant
unearned legitimacy to it.
Well, I am not sure what I would have created. The existence of the
hypostases is a consequences of incompleteness. Some of them have been
discovered independently by me, Boolos, and Goldblatt.
I still am not clear on why you invent three "new" hypostases,
granting the five from Plotinus (by creating G/G*, X/X*, and Z/Z*
instead of just G, X, and Z),
This is not a choice. G does really split in two: the provable part
by the machine, and the true part on the machine (that the machine
can prove or not). same for Z*, and X*.
But that is the chance, beacuse the notion of Plotinus are
theological, and side with the truth.
In Plato the distinction between Earth and Heaven becomes a
distinction between effective/constructive and True. The Noùs is
guven by G*. G is the "man", or the discursive reasoner, that I take
too as an hypostases, although the man is a bit despise by Plotinus
(which is normal when talking about God, by the Platonists).
OK, I'm with you here.
except that you say "[it] homogenizes nicely Plotinus presentation,
let us say that the math shows that there are 8 hypostases (roughly
speaking, as it is more like 4 + 4 * Infinity).
Three describes well the three primary hyposates of Plotinus (divine
one!), and two describes what Plotinus called sensible matter, and
intelligible matter, and where the comp quantum logic appears, and
should apper, by the UDA.
But they did split, which is not something we can avoid or hide.
OK.
and put a lot of pieces of the platonist puzzle into place."
Symmetry isn't an explanation. What I'm looking for would be
something along the lines of "It makes sense to split the intellect
hypostases into G & G* because ...."), likewise for X, and for Z.
The splitting of G and G* comes from Gödel's theorem. It is the
splitting between what a machine can prove (notably on itself) and
what is true about the machine.
Typical example: "I am consistent", it is the same as "I don't say
the false". For the correct machine, this belongs to G* minus G. It
is true, but the machine cannot prove it.
So it is not "It makes sense to split the intellect hypostases into
G & G* because ....". It is: mathematician believed that a machine
as a provability predicate equivalent with a truth predicate until
they discover that provability and truth are different. Now it is a
mathematical theorem that provability obeys to the modal logic G
(and indeed that G is complete for it), and that the truth on
provability obeys the modal logic G*.
The same for the modal nuances, which existence are themselves
consequences of incompleteness.
OK. My only confusion lies with the distinction between X and Z.
And then relative to that distinction, what is the difference
between X and X*. And the difference between Z and Z*.
Z is what the machine can say about the []p & <>t points of view (like
the bet that you will have coffee in the modified step 3 protocol).
[]coffee means you get coffee in all consistent extensions (which in
this protocol are W and M), and <>t is the explicit conditioning that
there is at least one consistent extension, which does not follow from
[]p due to incompleteness. You can see that []p & <>t is a weakening
of the []p & p move. Incompleteness forces the machine to provides
different logics for those nuances.
X is the usual theaetetus idea re-applied to []p & <>t. That is []p &
<>t & p. Again, incompleteness makes those nuances existing.
Then, X* and Z*, provides the logic of true propositions (about those
points of view), as opposed merely to what the machine can justified.
Again, it is incompleteness which justifies those distinction.
All the math part, AUDA, is based on:
- Church's thesis and the discovery of the universal machine
- Gödel technics of translating meta-arithmetic (with notion as
"provable") in arithmetic,
It ought to be possible to justify the hypostases in a non-
technical way.
Read Plotinus, or any mystic remaining rational.
I was referring to the 8 you've identified.
They are in Plotinus. G corresponds to the discursive reasoner. G* to
the Nous. S4Grz to the Soul.
X-X* and Z-Z* correspond to the sensible and intelligible matter
respectively, but Plotinus does not see the "splitting", (well, in
some passge he sees it, but in other he made confusion, and admit the
difficulties). He has he no tool for describing them, (we have the
universal machine and its dicourses!). He is aware of the
difficulties, which happens to be clarified when we take the necessary
computer science distinctions into account.
G is the just the logic of the correct 3p self-reference by numbers or
machine. It is decidable at the modal propositional level, and all
other modal logics can be emulated, and indeed are defined, *in* the
logic G.
For example G emulates G*, as G* proves A iff G proves that the
conjunction of []# -> #, with # being a subformula beginning by a box,
entails #.
G* proves <>t because <>t <-> []f -> f, and the conjuction of the []#
-> # is []f -> f and G proves, indeed even K proves, or even CP itself
that ([]f -> f) -> ([]f -> f).
So G emulates all hypostases, at the terrestrial effective level, and
at the proper theological level with the x* minus x logics for the
hypostases which inherits the G/G* splitting.
If not, then it strikes me as a weak spot of the argument, even if
the argument is technical.
Let me make a try. UDA is the formulation of the mind-body problem,
in "human term".
AUDA is the translation of UDA in the language of the machine.
Now, I ask the machine "will you believe this or that". What the box
[] represent is the thrid person self of the machine: it is the
explanation of the functioning of the machine, in the language of
the machine, so that I can ask her question about itself. This is a
third person self. []p means that the machine asserts p. To get the
first person I use the Theaetetus method, or variant, and this gives
rise to the horizontal layers in the diagram. The logics are
different thanks (technically) to incompleteness. Then three among
the hypostases split in two, (x and x*) again due to incompleteness.
Tell me if this helps.
I was asking for a plain-language justification of the 8 hypostases,
not the AUDA argument itself.
I would say they come from the machine/subject ability to
distinguish the following notions/views
- truth
- personal belief
- personal knowledge
- personal observable
- personal sensitive/sensible
Then for machine, the math shows that for the universal machines
arithmetical truth is enough, (and sigma_1 arithmetical truth is
enough for machine believing in computationalism) and you get
- (sigma_1) arithmetical truth
- Gödel's Beweisbar
- Theaetetus applied on Beweisbar
- Weak-Theaetetus applied on Beweisbar
- Theaetetus applied on the Weak-Theaetetus
That is what we can translate in arithmetic, with p any (sigma_1)
arithmetical proposition:
p
[]p
[]p & p
[]p & <>t
[]p & <>t & p
But the math shows that the logic/theory of three of them splits on
the incompleteness gap, so those five "hypostases" are really eight.
I think I'm mostly there.
I can explain more. A good text is the Theaetetus of Plato.
It does make me want to get into the actual logics, but
unfortunately I don't have the time to dedicate to that right now.
The logic G is really an incredible shortcut. But of course this means
understanding the basic of self-reference logic and its relation with
machine. Smullyan's Forever Undecided is a gentle introduction, even
if "The heart of the matter" is quick. Smullyan's betrays his
aristotelian prejudice also there (exercise: where?).
In my mind, UDA *is* the layman explanation/argument of AUDA,
somehow. You might just miss a detail. []p means only the machine
asserts that p. I use the Dennett intentional stance. A machine
believes p when the machine asserts p. This is enough, as we limit
ourself to ideally correct machine.
I find it amazing that the UDA and AUDA end up leading to the same
conclusions, as the arguments are so different on the surface. I
understand the UDA argument but not AUDA, so my amazement there
leads to skepticism, but again, my mind is open to the possibilities.
(I guess you were not on the list when I explained a bit of the
modal logic, and its relation with incompleteness). I can explain
more (but not in June as I am busy).
I was, and wished I could have followed along in detail.
We will have opportunity to dig deeper. I got UDA in flash when I was
a kid, but AUDA took 30 years of work, and that has been possible
because the hard work has been done by Gödel, Löb and others.
If you want to play the role of the candid, I can explain all the
details, but by its very nature AUDA is technical: it is exactly UDA,
yet explained to a machine which believes no more than in successor,
addition and multiplication of non negative integers.
The first thing to understand is that we can explain the entire
functioning of the theory:
0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
in that very theory. That's basically what Gödel did, and the main
result is that arithmetical (and sigma_1 complete) beweisbar
predicate. It is the 3p name (self) of the machine in its own
language. To be sure, we add the induction axioms to get the
interviewable machine.
And once you get G, you get the seven other one, and of course still
many other interesting variants.
In fact G can be used as a polymodal logic, as you can mix the
different intensionnal variants. You can, and should for futire uses,
add boxes invariant on transfinite autonomous progression, like some
Russian (Beklemishev) have already axiomatized.
I might be naive, and this might be a toy theology, but it already
predicts, and thus justfies, what we thought was weird about nature.
Obviously, it *is* a bit shocking, and it is important to remind that
comp can still lead to a contradiction, but once assumed, the problem
of the origin and stability of the sharing persistent experiences is a
problem of arithmetic (which does not mean simple).
The bomb is not the incompleteness of PA. The bomb is that PA can
prove that if PA is consistent then PA cannot prove its consistency.
PA and Löbian machines are already quite clever, and indeed, share a
non trivial theology, at least as long as they remain mute on the x -
x* things, which is a blaspheme, unless communicated them through
hypotheses, like computationalism, and the work on ideally correct
machine "by definition".
Bruno
Terren
Bruno
Terren
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