[Fis] CODE DISCUSSION. Recursion & ANNOUNCEMENT

Sun Sep 26 06:21:47 CEST 2021

Dear Pedro,
(This copy is sent to fis list.)
A comment on your comment.
I do not, in making examples, claim that one must restrict one’s formalism to the form of those examples.
The examples are intended to highlight conceptual aspects of the discussion.
The objection that a given formalism is not complete can always be made, particularly if the formalism is believed to be consistent and is rich enough to do arithmetic (Goedel).
Therefore, I do not attempt to produce complete formal systems. 
I intend to produce the simplest formal system that illustrate the ideas I am discussing.

The issue that I am addressing in my previous two letters is the issue of the cybernetics of 

Meaning giving rise to Syntax 
and 
Syntax giving rise to Meaning.

We understand this issue from the point of view of distinctions made by observers who themselves are part of the systems being observed.
When one takes the point of view of human cognition, meaning has a large role.
When one looks at molecular biology we see a fantastic array of syntactical behaviour and it is very difficult to see the locus of meaning except in ourselves as scientific observers of the biology.
And yet we are ourselves such biological systems.
This leads to a host of questions and many discussions.
To look at the simplest references such as
2x —> xx
and so 22 —> 22 giving self-reference,
is helpful in gaining an understanding of these issues.
Next simpler formalisms such as the von Neumann building machine
B,x —> X,x
whence
B,b —> B,b
are structures that can be enfolded in the more complex systems that you describe, and indeed the variety of ways that this can happen is part and parcel of the discussion.
For example, it is very significant that molecular biology, in order to achieve self-replication of DNA in the context of cellular systems must overcome a host of complex topological and geometrical problems that a logician would never imagine from the abstractions of self-reference above.
Best,
Lou

> On Sep 24, 2021, at 5:17 AM, Pedro C. Marijuan <pcmarijuan.iacs at aragon.es <mailto:pcmarijuan.iacs at aragon.es>> wrote:
> 
> Dear All,
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> -----------------------------------------------------------------------------------------
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> 
> Then, answering to Lou (and acknowledging other philosophical arguments), there is an additional point that could be relevant:  heterogeneity.
> Maybe I am wrong, but when I read the mssg below, my impression is that we are only considering a kind of homogeneous alphabets, that can grow via recursion, but that cannot go beyond the formal confines of that alphabet-world.
> 
> Machine codes in computers (and the cascade of codes built upon them) are ultimately due to the physical heterogeneity of their processing architectures. I mean, CPU, ALU, Central memory, peripherics, etc. are structurally different and the strings of 0s and 1s circulating within the system do need codes (*addresses*) so that they can circulate via the central bus towards the specific component-architecture. This is the final meaning of computer codes: matching between heterogeneous processing architectures--and ascending upon multiple levels so that languages, etc. can be designed.
> 
> In the biological world something similar occurs: we have sequential architectures (DNA & RNA) based on complementarity, structural architectures (membranes, cytoskeletons) based on identity, and a general "diluted" processing architecture (enzymes and proteins) based on supplementarity. Whatever function we may observe implies multiple ups and downs, back and forth, among these architectures and they need the recognition of quite specific motives arranged in well organized sets. Once the biological genetic code was in place, some complexity addicts (eukaryotes) progressively developed more and more codes & complex functions----and finally here we are. The final meaning of biological codes, like in computers, would be matching between heterogeneous architectures. About what agents helped in the code multiplication of eukaryotes, the response nowadays looks clear and clear: viruses and their complex retinue in the now internalized RNA world. See Luis Villarroel: ex virus omnia.
> 
> So, there are formal, logical, philosophical, semiotic arguments to make about codes--OK, but if they do not consider this real world aspect of heterogeneity of processing architectures am afraid they will be of limited usefulness to approach the reliance of computers and living beings on codes.
> 
> Best regards,
> --Pedro
> 
> El 22/09/2021 a las 19:29, Louis Kauffman escribió:
>> Dear Joseph,
>> 
>> (I am sending this again without graphics so it can end up on fis.) 
>> 
>> The RD construction is very general, but we have articulated it with particular models. 
>> 
>> One starts with a domain where distinctions can be made and there is a notion of locality.
>> In that domain one makes distinctions about given entities 
>> and replaces the entities by the corresponding signs of distinction. 
>> These signs amalgamate to become new entities to be distinguished at the next round. 
>> 
>> Lest this seem abstract, look again at the model.
>> <RD.png>
>> The sign = is an amalgamation of empty word on left and empty word on rght.
>> The sign ] is amalgamation of empty word on left and  vertical bar | on rght.
>> The sign [] is amalgamation of vertical bar | on left and  empty word on rght.
>> The sign O is meant to be a box and so is an amalgamation of vertical bar on left and right.
>> 
>> The point is that the alphabet arises by the amalgamations of previous distinctions and so new symbolic entities arise from the process of distinguishing by these rules.
>> After the body of this email, I will show you how the two dimensional RD alphabet arises.
>> 
>> This level of model includes notions of language and description in a very elementary formal framework. 
>> I have not included a logical language or a language that even begins to
>> have self-reference and ordinary reference. Thus meta-levels and thinking about thinking are not fully reflected in this small model. 
>> But the action of the model gives one the opportunity to reflect on these
>> larger issues. I am in the business of finding significant minimal models. 
>> These models are not going to be complete articulations of the whole situation of thinking about thinking or
>> reflecting on reflecting. They are intended as ways to help thinking about that. 
>> 
>> Note also that the RD above engages in “mitosis” or “self-replication”. It does so without a reflective level. This is of interest
>> for thinking about coding in biology and in thinking about what coding would mean in physical situations “below” biology.
>> 
>> Let me give here another example: the audio-active sequences studied by John Horton Conway.
>> 1
>> 11 - read “one one” and it describes the line above.
>> 21- read “two ones” and it describes the line above.
>> 1211- read “one two, one one” and it describes the line above.
>> 111221- read “three ones, two two’s, one one” and it describes the line above.
>> 312211- read “one three, two two’s, one one” and it describes the line above.
>> …
>> This recursion is an example of "describing describing" and the coding issues are somewhat different from the RD.
>> We retain locality of interaction and have a fixed alphabet that is less iconic. Counting is needed at the descriptive level.
>> This recursion requires more structure to run, but it is very very interesting.
>> Note that 22 is the only self-referential sequence.
>> 
>> Note also that we could write nx —> xx…x (n x’s) meaning that nx describes xx…x. 
>> In the audio active line we would have
>> xxxx…x
>> nx
>> 
>> as in
>> 777
>> 37.
>> 
>> So we have nx —> xx…x
>> and 2x —> xx
>> and so
>> 22 —> 22.
>> 
>> This pattern fits into the whole of 20th century logic since Russell.
>> Let me tender persuasions.
>> Russell:  Rx = ~xx
>> This is the definition of the Russell set if you take AB to mean “B is a member of A” and ~ is “not”.
>> Then: RR = ~ RR is the Russell paradox.
>> In this formalism, the Russell paradox becomes the production of a fixed point for negation.
>> 
>> Church and Curry generalized this to an abstract formalism (lambda calculus) where they could write
>> gx = F(xx) for an arbitrary F.
>> Then
>> gg = F(gg) and we produce a fixed point for F.
>> This is exactly how we got to the self reference of 22 by
>> 2x —> xx
>> 22 —> 22 
>> where here equality is replaced  by reference.
>> 
>> The notion of replacement of reference is integral to Goedelian self-refefrence.
>> g ——> P(x) now interpreted as “g is the Goedel number of the proposition P(x) with free variable x.
>> Then let #g be the Goedel number of P(g) so that 
>> #g —> P(g).
>> Let the language be rich enough so that # is an operation in the language.
>> h —> P(#x)
>> Then
>> #h—> P(#h)
>> and P(#h) talks about its own Goedel number.
>> Then with Goedel let ~B(x) mean that “there is no proof of the statement with Goedel number x.
>> Then with 
>> h —> ~B(#x)
>> #h —> ~B(#h)
>> and we have produced the proposition ~B(#h) that asserts its own unprovability!
>> This is the core of Goedel’s incompleteness Theorem.
>> 
>> I hope you see that we have arrived Goedel’s Theorem on a track leading directly from the self-reference of 22, by way of the Russell Paradox.
>> All this comes from the capacity of language to speak about language and thinking to think about thinking.
>> Goedel in fact shows us that all formalisms that we build that are rich enough and are consistent will be incomplete.
>> So there is no intent here to create complete formalisms.
>> I want to look at how the simplest non trivial formalisms behave, and how even very similar ones such as the RD and the audio activity are related to each other and to 
>> the larger issues of reference, self reference  and the generation of such dialogues at all levels.
>> 
>> Note also how small formalisms can summarize wide ideas.
>> The von Neumann Universal Building Machine B acts as follows.
>> B,x —> X,x.
>> Give B a blueprint x and B will produce X (the entity described by the blueprint).
>> Hence if b is the blueprint for B, then B can build itself!
>> B,b —> B,b.
>> This is another variant of 2x —> xx so 22—> 22.
>> 
>> Another thing that happened in the 20th century is that everyone got frightened about Rx =~xx giving RR = ~ RR,
>> and the story I am telling here of the central role of reference and self-reference is still not fully appreciated by mathematical practitioners such as
>> economists and even physicists.
>> Very best,
>> Lou
>> 
>> 
>> 
>> 
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> 
> -- 
> -------------------------------------------------
> Pedro C. Marijuán
> Grupo de Bioinformación / Bioinformation Group
> 
> pcmarijuan.iacs at aragon.es <mailto:pcmarijuan.iacs at aragon.es>
> http://sites.google.com/site/pedrocmarijuan/ <http://sites.google.com/site/pedrocmarijuan/>
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