[Fis] The Limits

Bruno Marchal marchal at ulb.ac.be
Tue Feb 26 18:29:42 CET 2019

Dear Pedro, dear colleagues,

> On 25 Feb 2019, at 19:42, Pedro C. Marijuan <pcmarijuan.iacs en aragon.es> wrote:
> Dear Karl and FIS Colleagues,
> Your message has made me think a couple of subjects. First, I have acknowledged several times, both publicly and privately, that your approach to estimating the multidimensional partitions on limited sets (the limit of distinctions when multiple qualities are piled upon elements of finite sets) is highly original and may find application in different fields. I think particularly in hippocampus' space/time organization of our spike sequences into binding percepts; probably in fields of physics too. But on the other hand, I have always disagreed on your (over)extension to DNA triplets, which has received a strong emphasis from your part ... Well, it is my personal opinion, and it may be quite wrong, of course.
> Anyhow, the above has taken me to the next reflection, somehow outlandish, that concerns "limits". I have some vague memories of a reflection in C.Booker (2004; or was it in Bonnet 2006?) on why we are not conscious of our own limitations and incur in quite many idiosyncratic biases, which are so well captured in narratives. I will try to put it in a more conceptual way: our thinking limitations do not let us establish the limits of our thought. It has individual consequences in our terrible inclination to overextend paradigms, but also a more "abstract", collective lack of final anchors. There is a false closure attempted that fails, and inevitably reappears later on in strange but fundamental principles: Godel, Heisenberg, Church-Turing... They basically consist in limits of thought put to the foundations of universalistic disciplines. In other more restricted fields, particularistic ones, those principles do not appear, or better, they are not needed. In the case of information science, which in my view is also universalistic, that kind of principled limit is needed too. Once properly established, or at least intuited, we could better discuss on the kinds of general theories that may be comprehended within a really multifarious enterprise such as info science.
> I will appreciate hearing opinions on these baseless comments.

If we assume the Mechanist Hypothesis in the cognitive science, or if we simply studied what narratives the machines develop about what it can, and cannot, prove, guess, observe, believe and/or know about itself, in the third and first person sense, we get immediately many results. 
What many people (like Lucas and Penrose, to name a few) ignores, but was already well seen by Gödel in his 1931 paper is that a classical sound machine, having enough beliefs in arithmetic, will be able to prove its own “Gödel incompleteness theorem”. The machine is able to prove that if she is consistent, then she is unable to prove its consistency, and that it is consistent that she is inconsistent. So the machine is aware of its own limitation following its intuition that she might be consistent, and more generally such a machine will understand that the many intensional variants of provability obeys different logics, despite being extensionally not distinguishable. 

The main variants corresponds nicely with the hypostases of the Neopythagorean Moderatus of Gades (first century) or with the hypostases of Plotinus, including in both case the theories of Matter (Intelligible Matter and Sensible Matter). Those variants are, with “[]” representing Gödel’s provability predicate, and  <>p denotes ~[]~p, t denotes “1 = 1” and f denotes "~(1 = 1”). So <>p = ~[]~p = consistent(p). 

1) p (the truth, the one, the Outer-God)
2) []p (the provability, the rationally justifiable beliefs; the intellect)
3) []p & p (the knowability, the soul, the Inner God) (true belief, cf Theaetetus’ theory of knowledge.

4) []p & <>t (the “bettable” or “guessable” the intelligible matter)
5) []p & <>t & p (sensible matter)

That provides 8 types of narrative, which, by a theorem of Solovay, can be axiomatised (at the modal propositional level) by 8 different modal logics, describing 8 weakening of classical logic. Why 8 and not 5? Because three of those logics get distinguished on truth and provability.

The “2)” above give G and G*
The "3)”above leads to intuitionist logic, and does not get separated in two (like “1)"
The “4)” above get split into Z and Z*
The “5) above get separated into X and X*.

So the machine knows its limitations, and among them there is an impossibility to justify rationally a lot of truth about herself, that she is still able to know true, but only in a surrational way: those truth belongs either to G* minus G, or to Z* minus Z, or to X* minus X. The limitations of the provability get inherited by the Z and X variants, leading to non-observable reality and non-sensible realities as well. To be sure, here we needs the computationalist restriction, and p is for any sigma_1 sentences (they obey “p -> []p” when the machine is “enough rich” or Gödel-Löbian.

G and G* are awkward modal logic, highly counter-intuive. But “3)” lead to Intuitionist logic, and “4)” and “5)” lead to quantum logics, using G and G* + p -> []p, and indeed a “theory of matter” has to be derived from them, as stable information accessible by observation (first person measure one). 

I am not sure Karl will concur with this, but it seems to me that the frontier between equilibrium and non-equilibrium is akin to the computer scientist frontier between the computable and the non computable, or between the provable and the non-provable-albeit-true. In between the rationally justifiable and the irrational/false, there is a corona of the surrational, given by G* minus G, making some machine searching for numbers which do not exist. Such machines typically don’t halt, assuming that they are immortal (which eventually they are from their first person perspective.

In that setting, Socrates argument against Theaetetus’ theory of knowledge is no more valid. But his argument of the immortality of the soul remains valid. In fact, all sound machine knows (in the sense above) that they have a soul, and they know that her soul is not a machine, not even describable in *any* 3rd person ways. That is useful to obtain a theory of consciousness. Indeed, if we characterise consciousness by true, non provable, immediately knowable, indubitable, and non definable, then, we can show that all universal machine are conscious, and that the sufficiently rich machine, having those “8” selves notions, are self-conscious, and know well their (conditional) limitations.

How to make a link with information theory?  I think that the nuances given in the non provable part suggests a way to tackle the differences,ce between Shannon quantitative information theory and a theory of qualitative personal information, which is not communicable in-between machine, and proceeds from consciousness. 

Note that I use the Mechanist hypothesis only for simplicity. Most of the “theology of machine” remains valid for strong weakening on the notion of machines. It is valid for most notion of oracle-machines, super-Turing machines, and even classes of Gods/oracle (i.e. highly non computable set of numbers, seen as Gödel numbers of propositions. Now, by using algorithmic information theory, maybe all this could be made more precise, but here, we do see that the “information” theory abstract completely from the meaning. So, we are left with the usual denotational, or connotational theories of meaning. 

I hope those few remark can help, perhaps to suggest a way to make some interesting argument in this list in more appreciable (by mathematicians, say). 
Digital Machines/numbers have a very rich (mathematical) theology, decidable at the propositional level, but highly undecidable when we add the quantifier (first order modal logics). If qX denotes the quantified extension, qG can be shown PI_2 complete, and qG* is PI_1 complete in the oracle of the whole arithmetical truth (the “God”!, here), which means that the Intellect, Plato’s Noùs,  completely overwhelms the “God of the Machine”. 

It is a Skolem-like situation: the arithmetical reality is far bigger when seen from inside than from outside, a bit like “a brain”, in some sense.

With the Mechanist hypothesis, that theology is testable, by comparing the physics derive from self-reference with the actual observation. The fact that we get quantum logic, and also the intuitive semantics of Quantum Mechanics without reduction of the wave, is a good encouragement for Digital Mechanism. Note also that the physical reality becomes not entirely computable, and not entirely “discretisable": there should be at least one continuous operator playing some role in physics. So Digital indexical Mechanism is not compatible with digital physics or digital physicalism, despite the discrete "fundamental reality" which here can be taken as the elementary arithmetical sigma_1 truth (a formula is sigma_1 if it has the shape ExP(x) with P decidable (they obey “p -> []p” for the universal machine, and are provable by the “rich” Gödel-Löbian machines and entities.

To sum up: not only a (universal, Gödel-Löbian) machine is aware of its limitations, but it can study the mathematical structure of its ignorance, and that one plays a crucial role for both consciousness and the appearance of a material reality.



> El 19/02/2019 a las 12:08, Karl Javorszky escribió:
>> Dear Pedro,
>> please allow me to raise a dissenting voice to the content of following citation:
>> “…On the other hand, no general theory for large non-equilibrium systems exists.  The legendary Hungarian mathematician John Von Neuman once referred to the theory of non-equilibrium systems as the “theory of non-elephants” meaning there could be no unique theory of such a vast area of science.” (Per Bak, How Nature Works)
>> In fact, the theory has been brought to you since some 24 years, as a sequence of suggestions, proposals, models, initiatives, encouragements, requests and so forth,  that observing the interaction between sequences and mixtures  is opening up a new door to a completely fresh view of the interrelations among the parts of the world. The principles deducted from models that employ such elements which are distinguishable and concurrently both contemporaneously and sequentially labeled (as opposed to all models known hereto, which each use elements that are indistinguishable and either sequential or contemporary), these principles are valid and actually at work in Nature, on all echelles, from the subatomar to the galactic .
>> I include the abstract I submitted to IS4SI, as part of the FIS track, and hope that the colleagues will participate in bringing recgnition to the collaborative work that has gon on in this FIS chatroom since 1997. The abstract describes, in the form of a general theory, large non-equilibrium systems. By including that part of the world, which is not the case, the theory encompasses elephants and non-elephants concurrently.
>> Karl
> -- 
> -------------------------------------------------
> Pedro C. Marijuán
> Grupo de Bioinformación / Bioinformation Group
> pcmarijuan.iacs en aragon.es <mailto:pcmarijuan.iacs en aragon.es>
> http://sites.google.com/site/pedrocmarijuan/ <http://sites.google.com/site/pedrocmarijuan/>
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