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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