On 18 Feb 2011, at 17:13, benjayk wrote:
Bruno Marchal wrote:
Hi,
What do you mean by Platonia?
The kind of Platonia in Tegmark or in Peter's (1Z) post does not make
sense for mathematicians. Even if you are using a theory like Quine's
NF, which allows mathematical universes, you still have no
mathematical description of the whole mathematical reality.
Do you have to have a description of the whole mathematical reality to
assert it exists?
You need it to make sense of it. Mathematical attempts lead to either
inconsistent theories, or to a definition of a putative mathematician
(like with the theory of topos), which is very interesting but not
quite "platonic".
As a figure of speech Platonia can make sense, but it is doubtful in a
theoretical context, like when we search for a TOE.
Isn't it enough to say everything that we *could* describe
in mathematics exists "in platonia"?
The problem is that we can describe much more things than the one we
are able to show consistent, so if you allow what we could describe
you take too much. If you define Platonia by all consistent things,
you get something inconsistent due to paradox similar to Russell
paradox or St-Thomas paradox with omniscience and omnipotence.
And then when you try to convey something which is counter-intuitive
and against the current main paradigm, like your poor servitor, you
have to base things on what the most agree (but this is not an
argument, just a methodological remark).
Bruno Marchal wrote:
Like in Plotinus, the ultimate being (arithmetical platonia) is
not a
being
itself (nor is matter!).
Could you explain what you mean with that?
Platonia, the platonia of Plato, is the Noûs, also called the
Intelligible Realm, or the World of Ideas, with the idea that Ideas
are more true/real than any of their terrestrial approximation/
incarnation. For example the perfect circle is in Platonia, together
with PI, but any natural circle is a gross and "less real" imitation
of the "eternal ideas". What we see is conceived as being only the
shadow of that "intelligible reality".
But in the Parmenides, Plato understood that the Intelligible Realm
has to come from something completely unified, and Plotinus attributes
his notion of ONE to the Parmenides of Plato. In neoplatonism the ONE,
which is really without name, nor description of any kind, truly
ineffable, is the principle from which both the Intelligible Realm
will "emanate" 'followed' by the "Universal Soul". The Universal Soul
is a sort of product of both the ONE (the soul keeps its umbilical
cord uncut with "GOD" (the ONE), and the Intelligible Realm, also
called the Divine Intellect. That are the three primary hypostases of
Plotinus: the One, the Divine Intellect, and the Universal Soul. They
correspond more or less to the origin, the reason, and the experience,
but are presented as three Gods, in the usual greek manner. The One
has many things in common with the "God" of the monotheist religion,
and the Universal Soul has many things in common with the Inner God of
the mystic and many schools of Eastern religions.
There is a inevitable tension between the Divine Intellect and the
Soul, and eventually the Soul will fall, and that is how Matter, a
quasi synonymous of Evil, rises. The notion of existence or being is
defined by the Divine Intellect. What exist is what the Divine
Intellect can talk about, and it cannot talk about the One, because of
its absolute ineffability and inaccessibility, and it cannot talk
about Matter, which cannot belong to the Intelligible Realm, because
it is so much unintelligible that even God (the one) has no control on
it. This makes the One, and Matter outside 'existence' or being. They
are the antipode of the intelligible existing things. Intelligible
by ... the divine intellect, note, which has to be distinguished from
"Man", i.e. the terrestrial intellect, or discursive reasoner, which
is the one who dies and pays taxes, and try to understand.
Now, it has been shown that if you give to a universal machine some
provability and inductive inference abilities (easy to do), and ask
such a machine to introspect itself, the machine is able to
distinguish truth, belief (proof) and knowledge (proof of truth). She
can know that a truth encompassing herself is not nameable or
describable. She can distinguish the terrestrial believer from the
divine believer, and even guess a part of the "divine discourse", with
"divine" meaning "true" on that level where truth is not definable.
She can understand and feel (accepting some definition already in
Plato and Plotinus) the inevitable tension between the "Divine
Intellect" and the "Universal Soul", she can understand (believe,
proof) that the Universal Soul (which actually is also unnameable) has
already "a foot in matter', and that the Soul will fall (by connecting
inappropriately the terrestrial intellect with the divine intellect),
and the soul glues itself in that part of the internal border of
reality where God loose control. That generates a logic which should
correspond to the logic of the observable.
Technically, albeit roughly, with p arithmetical proposition, you have
the following arithmetical interpretation of Plotinus' three primary
hypostases, which I put in a lozenge:
The
ONE = arithmetical truth (p)
The terrestrial believer = Gödel's provability (Bp, logic G) ----------
the divine Intellect (the truth on the Gödel's provability) (Bp, but a
different logic (G*)
The
Universal Soul (Bp & p); logic = S4Grz
That represent the happy harmonic state "before the fall". The fall
comes from the fact that although G* proves that Bp is equivalent with
Bp & ~B~p, G cannot prove that, but the one who want to bet on a
reality has to follow the Bp & Dp logic (D = ~B~) and this gives the
two Matter of Plotinus, which both inherit from the G/G* splitting
(due to the disticntion between proof and truth, that is due to
incompleteness).
the terrestrial intelligible matter (Bp & Dp, controlled by G)
-----------the divine intelligible matter (Bp & Dp,controlled by G*)
the terrestrial sensible matter (Bp & Dp & p, controlled by
G)----------the divine intelligible matter (Bp & Dp & p, controlled by
G*)
You can see a lozenge above a square. It is good memo for the 8
variants of provability (arithmetical hypostases). Again:
p
Bp------- Bp
Bp & p
Bp & Dp----------- Bp & Dp
Bp & Dp & p------Bp & Dp & p
At the left you have the terrestrial (effective) realm. At the right
you have the divine (true) realm. The lozenge gives the three primary
hypostases on the right. And the poor terrestrial man on the left.
Magically, the Universal Soul belongs to both the terrestrial and
divine realm, but splits in two (terrestrial/divine) in the fall. This
is not easy to prove. The first person pov (the "soul") confuses
naturally provability/knowability and truth, like an intuitionist (and
this can be made very precise). The lozenge is the harmonic state of
the universal machine, and the square will glue the machine in the
realm of the consistent (Dp) extensions (histories). Note that 5
variants of provability lead to 8 "hypostases" dues to the G/G*
splitting of three of them.
Then, you can model computationalism, or interview the machine on
comp, by restricting the arithmetical interpretation of p to the
Sigma_1 sentences, which are the arithmetical equivalent of the
"border of the universal dovetailer", and this gives you 8 more
refined logics. By the UDA argument, The universal Soul, the divine
intelligible matter, and the the divine intelligible matter (with
some other variants of them) can provide the logic given by the
measure one for the observation and sensation. There, we get the truth
of DDf (the possibility of the possibility of the false), which is
similar to the early Lewis modal logics. The white rabbits seems to
disappear but remain very close, perhaps.
Gödel's theorem, which originates that "Bp" logic (G), is often used
as an argument that we are not machine (Lucas, Penrose), but what
people rarely take into account is that machines can prove their own
incompleteness, making the left part of the diagram provable by the
machine, and the right part inductively inferable, 'bettable',
'hopable', 'fearable', etc. (doesn't look like english, but you see
the point).
This gives a TOE, (necessitated by Comp + the classical theory of
knowledge), which is ontologically just (sigma_1) arithmetical truth
(which is really weaker than most formalism), and which admits as
internal epistemology provability and its variants. Each variants is a
different view of the same unique tiny arithmetical reality. But those
views from inside imposes rich topologies and measures, but also
complex mathematical problems.
Basically, for the ontology, you need only classical logic + the
axioms of addition and multiplication
x+0 = x
x+s(y) = s(x+y)
x*0 = 0
x*s(y) = (x*y) + x
In that system you can define the Löbian Bp and variants (a very long
and tedious fact shown by Gödel & Co), which "believe" in more things
already (the axioms of addition and multiplication + the axioms of
induction, which makes them Löbian and obeying, like all their sound
extensions to the 8 hypostases).
There is a subtle tour de force here, if you indulge me to say, which
consists to use Tarski notion of truth (p) in the place of the
impossible (by a theorem of Tarski) task to define truth in the
language of the machine, by some predicate V('p'). The same for
knowledge and sensibility. Quanta and qualia should appear (and does
appear a little bit already) at the extreme bottom right of the
diagram. All this is a sum up of "AUDA" the arithmetical UDA. It is
not needed for understanding UDA (physics is arithmetic seen from
inside). But UDA is useful to motivate AUDA and to relate it to the
mind-body problem.
Any universal programming system can be used instead of numbers. With
the combinators, the ontology is given by an even shorter theory: the
laws of elimination and duplication:
Kxy = x
Sxyz = xz(yz)
I mean, the choice of the initial universal system is free. You can
take a quantum computer, but this is cheating with the goal of solving
the mind body problem, which by UDA needs a derivation of the local
universal observable structure. This one has to be justified properly
to get both the quanta and the qualia, avoiding the elimination of the
person and consciousness.
The god of Plotinus is not omniscient, nor omnipotent. "He" is
overwhelmed by its first emanation, the Divine Intellect, and then
both the ONE and the Divine Intellect are overwhelmed by the Universal
Soul, which truly put the mess (matter, notably) in the (arithmetical)
Platonia.
Like in Pratt's Chu transform (cf Stephen), or like in Galois
connections, Plotinus dynamics go in two opposite directions:
emanation and conversion.
- God (the One) by a sort of excess of generosity let the Divine
Intellect emanates from him, and then let the Universal Soul emanate,
which eventually generates nature up to matter (where God lose control
and man needs a "bastard calculus" on the non determination)). That's
the emanation part, from the One to the soul and matter.
- The terrestrial soul then tries hard to leave matter, and eventually
succeeds in reaching the Divine Intellect (by math, music, astronomy,
notably), and eventually recovers The inner God and the ONE phase.
That's the conversion path, from Matter and Soul to the One.
The two processes correspond to the same truth, again seen from
different views (the One and the Soul)
Bruno
http://iridia.ulb.ac.be/~marchal/
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