Re: [Fis] _ Re: _ Gödel discussion

[Bruno Marchal]

Dear Alex, dear friends,


On 04 May 2016, at 02:49, Alex Hankey wrote:


Dear Friends,

I was so struck by the group's focus on Gödel's theorems that I went  
back to John R. Lucas who originated the idea that Gödel's insights  
imply that the human miind is not a machine - and therefore capable  
of genuine phenomenal experience. You may find the ideas in the  
following informative and useful

https://en.wikipedia.org/wiki/John_Lucas_(philosopher)
https://en.wikipedia.org/wiki/Minds,_Machines_and_Gödel

I noted particularly that I have used complexity (which Lucas  
mentions towards the end of MM) to establish that organisms are  
not machines, and out of that I identify the form of information  
which may explain various aspects of experience.



I work on this since a long time. It is my domain of investigation  
actually. Indeed, it is Gödel's proof which made me decide along time  
ago to become a mathematician instead of a biologist, when I saw that  
Gödel's technic gave a conceptually clear explanation of how something  
can self-duplicate, self-refers, self-transforms, etc.


The first to get the idea that incompleteness can be used to "prove"  
that we are not machine was Emil Post in 1922, wen he anticipated  
incompleteness. Then Emil Post was also the first to see the main  
error in that argument, and he saw what can still be derived from it  
(mainly that iF we are a machine THEN we cannot know which machine we  
are: a key that I have exploited in the derivation of physics from  
arithmetic and mechanism).


Note that Gödel's first incompleteness theorem can be rigorously  
derived in very few line (indeed just one double diagonalization) from  
the Church-Turing thesis. This has been seen by Kleene, and is  
exploited by Judson Webb in his 1980  book to illustrate that not only  
Gödel's theorem does not refute mechanism, but Gödel's theorem is an  
incredible chance for mechanism. I have written myself  a lot on this,  
and my work extends this at the extreme, as it shows that mechanism  
makes incompleteness the roots of both the appearance of qualia and  
quanta, and this in a precise and unique way, making mechanism  
empirically refutable.


I intended to give here the proof in a few line of Gödel's  
incompleteness from Church-thesis, but that might wait.


Someone (Lou?) said "Proving” that we are not machines is somewhat  
quixotic from my point of view, in that it should be obvious that we  
are not machines!"


The statement "we are machine" is ambiguous. Does "we" refers to our  
souls or to our bodies? Does it refers to our third person describable  
relative bodies or to our private unnameable and non describable first  
person view.


Here, what the Gödel-Löbian machine can already prove about themselves  
is that IF they are self-referentially correct machine, and if they  
survive a digital substitution at some level of description, they  
their soul is not a machine, once we admit to identify the soul with  
the knower, and translate the Theaetetus' definition of knowing (true  
opinion, []p & p) in arithmetical terms. If my body is a machine, then  
my soul is not (says Peano Arithmetic!).


I explain the main things in my two JPMB contributions. The key idea,  
related to Gödel, is that incompleteness separates clearly what is  
true about the machine and what is justifiable by the machine on one  
part, and on the other part, it separates clearly the justifiability  
([]p), knowability ([]p & p), "observability" ([]p & <>t), sensibility  
([]p & <>t & p) with []p for Gödel's probability predicate, and <>t =  
~[]~t = ~[]f = consistency (t = constant true, f = constant false). I  
predicted that []p & p, []p & <>p and []p & <>t & p should give a  
quantum logic (when p is "computably accessible, or Sigma_1), which I  
manage to prove after 30 years of research. The whole things leads to  
a many dream interpretations of arithmetic, from which a derivation of  
quantum mechanics should be possible (and is partially done).


All this has been made possible by Gödel's theorem and its many  
generalizations in many directions: Löb's theorem, Kleene's theorems  
and mainly Solovay's completeness theorem for some modal logics, which  
capture everything in the undecidability field in two modal logics (G  
and G*). I will just refer everyone interested to my papers.


The main point relevant here is that incompleteness saves machine from  
all reductionism. It shows that a machine (or any effective theory)  
like Peano Arithmetic (say) is already quite clever. PA can already  
refute all reductionist theories about its soul, and indeed can  
already derive physics from Mechanism + Arithmetic.


Unfortunately, this uses mathematical logic, which is not well know by  
non-logicians. Mathematical logic contains important sub-branches,  
like computation theory, computability, theory, proof theory, and  
model theory. Model theory is the study of meanings and semantics of  

[Fis] _ Re: _ Gödel discussion

[Alex Hankey]

Dear Friends,

I was so struck by the group's focus on Gödel's theorems that I went back
to John R. Lucas who originated the idea that Gödel's insights imply that
the human miind is not a machine - and therefore capable of genuine
phenomenal experience. You may find the ideas in the following informative
and useful
https://en.wikipedia.org/wiki/John_Lucas_(philosopher)
https://en.wikipedia.org/wiki/Minds,_Machines_and_Gödel

I noted particularly that I have used complexity (which Lucas mentions
towards the end of MM) to establish that organisms are not machines, and
out of that I identify the form of information which may explain various
aspects of experience.

I found Minds, Machines and Godel very useful to read since it seemed to
confirm aspects of my offering to you all.

On 3 May 2016 at 07:49, Francesco Rizzo <13francesco.ri...@gmail.com> wrote:

> Cari Terrence, Louis, Maxine e Tutti,
> I state that I am a "exponential poor" who do not claim to lead any claim
> whatsoever. But I think I have understood from the triangulation of three
> colleagues, I do not think it is concluded, that:
> - Rosario Strano, a mathematician at the University of Catania, lecturing
> on "Goodel, Tarski and liar" in the end he said: "In closing, we conclude
> with a remark 'philosophical' suggested during the conference by fellow F.
> Rizzo : a result that we can draw from the theorems of exposed is that the
> search for truth, both in mathematics and in other sciences, can not be
> caged by mechanical rules, nor reduced to a formal calculation, but it
> requires inspiration, intuition and genius, all features own human
> intellect ( "Bulletin Mathesis" section of Catania, Year V, n. 2, April 28,
> 2000);
> - Also it happened to me, to defend the economic science from encroachment
> or domain of the infinitesimal calculus, to declare that the mathematical
> models resemble simulacra, partly true (according to the logic) and partly
> false (according to reality) : see. lately, F. Rizzo, ... "Bursars (c)
> to" Arachne publishing, Rome, April 2016;
> -in theory and in economic practice the capitalization rate "r" of the
> capitalization formula V = Rn. 1 / r can be determined or resorting to
> "qualitative quantity" of Hegel ( "The Science of Logic") or to complex or
> imaginary numbers that, among other things, allowed the Polish
> mathematician Minkowski, master of A. Einstein, to adjust the general
> theory of relativity, so much so that I wrote: "the imaginary and / or
> complex numbers used to conceive the 'Minkowiski of' universe that
> transforms time into space, making it clearer and more explicit the
> isomorphic influence that space-time exercises the capitalization formula
> and equation of special relativity, maybe they can illuminate with new
> lights the function of the concept of co-efficient of capitalization "(F.
> Rizzo," from the Keynesian revolution to the new economy ", Franco Angeli,
> Milan, 2002, p . 35).
> How you see the world looks great but basically it is up to you and even
> to poor people like me to make it, or reduce it to the appropriate size to
> understand (and be understood) by all.
> Thank you for the opportunity you have given me, I greet you with
> intellectual and human friendship.
> Francesco
>
> 2016-05-03 5:28 GMT+02:00 Louis H Kauffman :
>
>> Dear Folks
>> I realize in replying to this I surely reach the end of possible comments
>> that I can make for a week. But nevertheless …
>> I want to comment on Terrence Deacon’s remarks below and also on
>> Professor Johnstone’s remark from another email:
>>
>> "This may look like a silly peculiarity of spoken language, one best
>> ignored in formal logic, but it is ultimately what is wrong with the Gödel
>> sentence that plays a key role in Gödel’s Incompleteness Theorem. That
>> sentence is a string of symbols deemed well-formed according to the
>> formation rules of the system used by Gödel, but which, on the intended
>> interpretation of the system, is ambiguous: the sentence has two different
>> interpretations, a self-referential truth-evaluation that is neither true
>> nor false or a true statement about that self-referential statement. In
>> such a system, Gödel’s conclusion holds. However, it is a mistake to
>> conclude that no possible formalization of Arithmetic can be complete. In a
>> formal system that distinguishes between the two possible readings of the
>> Gödel sentence (an operation that would considerably complicate the
>> system), such would no longer be the case.
>> ”
>> I will begin with the paragraph above.
>> Many mathematicians felt on first seeing Goedel’s argument that it was a
>> trick, a sentence like the Liar Sentence that had no real mathematical
>> relevance.
>> This however is not true, but would require a lot more work than I would
>> take in this email to be convincing. Actually the crux of the Goedel
>> Theorem is in the fact that a formal system that
>> can handle basic number 

[Fis] re Gödel discussion

[Maxine Sheets-Johnstone]

Many thanks for your comments, Lou and Bruno. I read and pondered,
and finally concluded that the paths taken by each of you exceed
my competencies. I subsequently sent your comments to Professor
Johnstone—-I trust this is acceptable—asking him if he would care to
respond with a brief sketch of the reasoning undergirding his critique,
which remains anchored in Gödel’s theorem, not in the writings of others
about Gödel’s theorem. Herewith his reply:


Since no one commented on the reasoning supporting the conclusions 
reached
in the two cited articles, let me attempt to sketch the crux of the case 
presented.


The Liar Paradox contains an important lesson about meaning. A statement 
that says of itself that it is false, gives rise to a paradox: if true, 
it must be false, and if false, it must be true. Something has to be 
amiss here. In fact, what is wrong is the statement in question is not a 
statement at all; it is a pseudo-statement, something that looks like a 
statement but is incomplete or vacuous. Like the pseudo-statement that 
merely says of itself that it is true, it says nothing. Since such 
self-referential truth-evaluations say nothing, they are neither true 
nor false. Indeed, the predicates ‘true’ and ‘false’ can only be 
meaningfully applied to what is already a meaningful whole, one that 
already says something.


The so-called Strengthened Liar Paradox features a pseudo-statement that 
says of itself that it is neither true nor false. It is paradoxical in 
that it apparently says something that is true while saying that what it 
says it is not true. However, the paradox dissolves when one realizes 
that it says something that is apparently true only because it is 
neither true nor false. However, if it is neither true nor false, it is 
consequently not a statement, and hence it says nothing. Since it says 
nothing, it cannot say something that is true. The reason why it appears 
to say something true is that one and the same string of words may be 
used to make either of two declarations, one a pseudo-statement, the 
other a true statement, depending on how the words refer.


Consider the following example. Suppose we give the name ‘Joe’ to what I 
am saying, and what I am saying is that Joe is neither true nor false. 
When I say it, it is a pseudo-statement that is neither true nor false; 
when you say it, it is a statement that is true. The sentence leads a 
double life, as it were, in that it may be used to make two different 
statements depending on who says it. A similar situation can also arise 
with a Liar sentence: if the liar says that what he says is false, then 
he is saying nothing; if I say that what he says is false, then I am 
making a false statement about his pseudo-statement.


This may look like a silly peculiarity of spoken language, one best 
ignored in formal logic, but it is ultimately what is wrong with the 
Gödel sentence that plays a key role in Gödel’s Incompleteness Theorem. 
That sentence is a string of symbols deemed well-formed according to the 
formation rules of the system used by Gödel, but which, on the intended 
interpretation of the system, is ambiguous: the sentence has two 
different interpretations, a self-referential truth-evaluation that is 
neither true nor false or a true statement about that self-referential 
statement. In such a system, Gödel’s conclusion holds. However, it is a 
mistake to conclude that no possible formalization of Arithmetic can be 
complete. In a formal system that distinguishes between the two possible 
readings of the Gödel sentence (an operation that would considerably 
complicate the system), such would no longer be the case.



Cheers,
Maxine
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