[Fis] New Discussion on Complex Self-Other Interactions and Gödelization of Biology

Fri Nov 11 00:35:49 CET 2022

Dear Andrei and dear Colleagues

Again, many thanks for the lively discussions that have followed from my post on Self-other Interactions and Gödelizaton of Biology.  I apologize for the delay in responding. But be assured that while attending to routine tasks like setting exam papers (took most of last week), I’m mulling things over prompted by comments being posted and responding to them as soon as I can.

I agree with Andrei that Barbieri’s notion of codepoiesis may help refocus efforts toward code based explanations rather than solely bio-physical explanations, often thermodynamic ones such as minimizing entropy, for genomic information processing.

(i)So why Gödel ?

Andrei and I seem to think that Gödel Incompleteness Results may have some bearing on how code based systems can endogenously produce novelty in the form of syntactic objects (code based again) that lie outside listable sets.  Why this and not random shocks qualify to be the model for novelty in biology ?

As recently pointed out by Mikhail Prokopenko et. al (2019) if one wants to utilize such an explanation for novelty,  then we need to demonstrate that the system in question has embedded the necessary conditions that Gödel used to constructively produce such a syntactic object.   Gödel numbering is the first feature as a means of giving unique (integer) identifiers to finite strings based on a fixed and finite set of alphabets.  Self-reference and negation operators are the two others that Prokopenko et.al. identify.

We need to rewind to the humble antecedents for the classic steps involved in Gödel Incompleteness Results that may throw light on both the above noted ingredients (self-reference and negation operator) and more importantly the idea that there is always something novel in that it will exceed enumeration.  In the so called Cantor Diagonal Lemma there is a sure fire way of showing that no listing is complete and there is something that has been left out.

The (informal ) proof of finding a set that is left out of any enumeration of sets W1   , W2 ……  Wy , ….. , given along the rows of a matrix indexed by integers y = 1,2,3, … ,  which are also listed along the columns. The membership of each  set Wx  is given as 0 or 1s indicating if an integer belongs to the set in row x , y Wx or otherwise.  In order to identify a set that cannot be in the list  W1   , W2 ……   Wx , ….. , consider  the  set along the diagonal.  The self-referential predicate • (x,x) = Diag(x) follows as it refers to x Wx  or whether Wx  contains its own index x.   On applying a negation operator along the diagonal array, changes 0’s to 1’s and 1’s to 0’s, we now have a set (the anti-diagonal set denoted as D¬) that is different from all listed sets Wx  by at least one element along the diagonal. The steps given here for the set D¬  that cannot belong to any enumeration of sets are informal.

The genius of Gödel (1931) is to ‘mechanize’ these ingredients to produce a syntactic object which typically takes the form of the undecidable  proposition, aka Gödel Sentence,  along the lines that Louis Kauffman has expertly shown.

The typical rendering of the first Gödel (1931) result is as follows

                            PA ͰA ↔ ¬Prov(A). Here, A effectively says of itself that it is not provable (¬Prov(A)), with this undecidable
proposition being a theorem (Ͱ) in the formal system of Peano Arithmetic (PA).

As things stand -  it is very hard to show how this relates to Biology.  I call it a funky, esoteric construction in the foundations of maths, to date of little relevance to anything in the real world…

 (ii) We need far more machinery than Gödel (1931) to show how Diag, Negation & Theoremhood relate to Biology

In my opinion as Gödel (1931) predates the full developments of what is called Recursive Function Theory, the notion of Church Turing thesis of an algorithm, hence of codes/programs of self-assembly machines.
We need (Roger’s) Second Recursion Theorem to identify the fixed point of a negation software that will alter the codes of the self-assembly machines.

Most of all we need the Emil Post (1944) set theoretic proof of Gödel Incompleteness  result which
Smullyan (1961, p.58) has noted plays  a fundamental role in modern approaches to incompleteness and undecidability.   I  quote Fefferman of saying the same in my 2017 (AIMS paper).  But  I do not know of anybody who has used the Emil Post (1944) proof with creative and productive sets in any application.  The sets of theorems and non-theorems, which is implicit in Gödel(1931)  becomes explicit in Post rendering of the problem.

It is only when things are laid out in Hartley Roger format of Gödel Meta Representation Theorem in his proof for his Fixed Point Theorem does one see direct evidence such as the mirror structures in the immune system and in the brain.
I will elaborate more on this in my response to Louis Kauffman.

(iii) Evidence
My quibble with Andrei is that unless we produce evidence such as even for Gödel numbering in Biology, things are still up in the air.  My first inkling for this is from the work of Gunter Blobel discovery of peptide ‘zip codes’ in proteins which even tell them where to go !! I believe  transcription factors in gene expression, which require a number of genomic components to be turned on together, are identified by having a common identifier (class mark) etc.

My idea of what is entailed in Godel numbering is quite prosaic.  Say we have a string of alphabets – mapped to an integer m. Now apply another software operation f on it (some strings from elsewhere via copy paste) – this this composite f (m)  string has g.n q etc.   This is called S-M-N theorem in RFT, no clue as to how biology does it…

Best Regards

Sheri

From: Andrei Igamberdiev <a_igamberdiev at hotmail.com>
Sent: 31 October 2022 17:07
To: Markose, Sheri <scher at essex.ac.uk>; fis at listas.unizar.es
Subject: Re: New Discussion on Complex Self-Other Interactions and Gödelization of Biology

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Dear Sheri and Colleagues,

we are discussing the central and most fundamental issue of generation of novelty and complexification in biological world. The main point here is the generation of Gödel numbers, i.e. the assignment of previously undefined statements in the natural computing system of the cell by meanings. To understand progressive biological evolution, we need to make a move from the description of biological system as self-sustaining (autopoiesis) to its understanding as generating evolutionary novelty (codepoiesis). While the term autopoiesis arises to Maturana and Varela (although first the autopoietic structure was suggested by Tibor Gánti in 1971 or even earlier - he conceived the basic idea in 1952), the concept of codepoiesis was introduced by Marcello Barbieri (in detail in his book of 2015 and earlier in his papers). For many years generation of novelty was considered as a random process, while it can be described as a continuous realization of the proof of Gödel's theorems. Why this happens - it is an inevitable consequence of natural computation, in other words, it is the consequence of the incompleteness of a sufficiently rich formal system, i.e., of the presence of undecidable statements within the system. The simple prokaryotic system can be insufficiently rich for advanced complexification (although at certain stage the process of complexification generated the eukaryotic cell), while in eukaryotes the capacity for complexification significantly increases.

The process of Gödel numbering introduces a new code to the system which proves that certain statements are undecidable within the system. This code can be externally applied but it can be efficiently internalized in the system itself. The resulting hierarchical structure, consisting of the formal system and its reflection in the course of the proof of undecidability, corresponds to a new organization where Gödel numbers represent a new information system that emerged from the original system by assigning codes to it. The undecidable statements become the basis of the evolutionary unfolding of the system by assigning certain values to them. The details can be found in my paper published in BioSystems 207 (2021) 104454, https://urldefense.com/v3/__https://doi.org/10.1016/j.biosystems.2021.104454__;!!D9dNQwwGXtA!UyCtjIKq6bLofcL6dBQ6z-6R4kA6nftw0YH6vVFZUKODBcUEs1j7q0VcLQrp00gchhkHyXZ3YnrePcYYCLM$ <https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=https*3a*2f*2fdoi.org*2f10.1016*2fj.biosystems.2021.104454&c=E,1,EahD_AZmPoOx690SC8rsCzcPdE7d1D96ZEy32L2jizRYPlUhBsh9jq_8YaaoEkrBSmJwdzckQLb6klSv94sqUziYLX1k_E6IzS-vGxB5_d5H&typo=1__;JSUlJSU!!D9dNQwwGXtA!UyCtjIKq6bLofcL6dBQ6z-6R4kA6nftw0YH6vVFZUKODBcUEs1j7q0VcLQrp00gchhkHyXZ3YnrejIttHi0$ >, although the original idea arises to my early paper published in 1986. More detailed metamathematical analysis of this process of assigning codes is presented in Shelah, S., Strüngmann, L., 2021. Infinite combinatorics in mathematical biology. Biosystems 204, 104392. https://urldefense.com/v3/__https://doi.org/10.1016/j.biosystems.2021.104392__;!!D9dNQwwGXtA!UyCtjIKq6bLofcL6dBQ6z-6R4kA6nftw0YH6vVFZUKODBcUEs1j7q0VcLQrp00gchhkHyXZ3YnreMB8Ct3Q$ <https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=https*3a*2f*2fdoi.org*2f10.1016*2fj.biosystems.2021.104392&c=E,1,r32d1oKsRGye-9Ya_4O662cEGxP0muGCKCPPj1DTtUKAc2hVYM07MXG1f4DI-MGZR7R7lT5Or1dBWj2fEJsBO2V-x9oZyFwyoIo-bwJBia0sJehAzqDqd_CYt0Y,&typo=1__;JSUlJSU!!D9dNQwwGXtA!UyCtjIKq6bLofcL6dBQ6z-6R4kA6nftw0YH6vVFZUKODBcUEs1j7q0VcLQrp00gchhkHyXZ3Ynrete8XQ2w$ >.

It is definitely true that the immune system has a great capacity for generation of Gödel numbers. The significant increase of capacity for complexification occurs at certain level of organization, corresponding to eukaryotic cell. The prokaryotic progressive evolution is limited by the size of formal coding system, and most likely most living structures in the Universe do not overcome this limit, so the chance of finding the eukaryotic life outside the Earth is extremely low.

The simplest way of generating new codes (especially at early stages of evolution) is the activity similar to the enzyme polynucleotide phosphorylase (EC 2.7.7.8), which can catalyze the casual rearrangement of nucleotides and operates without energy cost. It realizes the RNA-based multiplication of generativity and thus can serve as an elementary tool for codepoiesis. A new rearranged sequences generated via such polynucleotide phosphorylase equilibrium can potentially acquire biological functions and correspond to the evolution of autopoietic systems.   At higher evolutionary levels, combinatorial rearrangements represent a powerful tool for evolutionary changes; they can be realized at the level of meiotic crossover and lead to the expansion of the genome.

Sheri has suggested certain concrete examples of Gödelization of code-based information such as bio-peptide unique identifiers, e.g., signal peptide sequences, antibody production in response to non-self antigens, etc. We can discuss the particular cases and outline possible ways of generation of such new information. Comparing bacteria-phage arms race with antibody production will clarify the complexification of the eukaryotic system in comparison with the prokaryotic.

In conclusion, the understanding of Gödelization of code-based information will represent an overall fundamental principle of evolving natural-computing systems operating towards the appearance of the structures with maximum power efficiency, and the well-known principles (and their limitations) such as natural selection, variability, adaptability, etc. will represent a consequence of the basic phenomenon of the codepoietic nature of the evolutionary biological phenomena. It is worth to consider and discuss the particular examples of codepoiesis as well as the "codification" of the codepoietic events in the individual development of multicellular organisms during cell differentiation. This could be a very valuable discussion.

All the best,
Andrei Igamberdiev

________________________________
From: Fis <fis-bounces at listas.unizar.es<mailto:fis-bounces at listas.unizar.es>> on behalf of Markose, Sheri <scher at essex.ac.uk<mailto:scher at essex.ac.uk>>
Sent: October 25, 2022 6:42 PM
To: fis at listas.unizar.es<mailto:fis at listas.unizar.es> <fis at listas.unizar.es<mailto:fis at listas.unizar.es>>
Subject: [Fis] New Discussion on Complex Self-Other Interactions and Gödelization of Biology

Dear Colleagues -  I look forward to your valuable comments.

Gödelian Self-Referential Genomic Information Processing: Complex Self-Other Interactions and Novelty Production

Overview for Foundations of Information Science Discussion –

Sheri Markose         25-10-2022

  1.  Background

Complexity, evolvability, novelty production and ‘thinking outside the box’ in biology and humans have for most part relied on models of randomness or on statistical white noise error terms.

                  The digitization of inheritable information in the genome encoded in a near universal alphabet  (A,T,C,G/U) has been called the ‘algorithmic takeover of biology’ by Walker and Davis (2013).  The Faustian pact at the genesis of life colourfully portrayed by Freeman Dyson  as ‘ the takeover of a replicative digital virus of an analogue metabolism’ accords with the perspective of Forterre, Zimmer, Villareal, Koonin and others.  This underpins the remarkable fact that in nature only life and biology as we know it and the artifacts of genomic intelligence (GI) thereof are explicitly code based digital systems.

                     Genomic intelligence is a concept introduced in Markose (2021)<https://urldefense.com/v3/__https:/doi.org/10.3390/e23040405__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqV3O-kDY$> specifically to characterize the Gödelization of code-based information processing in genomic systems with the distinctive self-referential conditions of Gödel Incompleteness results that appear to have been acquired for complexification over the course of evolution of multicellular eukaryote life, Markose (2022)<https://urldefense.com/v3/__https:/doi.org/10.1016/j.biosystems.2022.104718__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqPn2Rp5k$>.  To explain the latter in contrast to the relative stasis of prokaryotes, which has remain unchanged since inception, Barbieri (2012<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib10__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqdTKw3lg$>; 2018)<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib13__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSq0lIhVMs$>,for instance, has resorted to Code Biology and Codeopoiesis, viz. biotic code based explanations to complement  the more popular thermodynamic explanations of self-organization and autopoiesis<https://urldefense.com/v3/__https:/www.sciencedirect.com/topics/mathematics/autopoiesis__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqDFaksjk$>. Igamberdiev (2021)<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib59__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqIQhnjCw$> gives general principles, but no concrete evidence, for why “living systems during evolution continuously realize the proof of Gödel's theorems (1931).”

                   Significantly, while debunking the idea that the primary source of evolutionary changes arise from random transcription/replication errors, the epochal discovery by Nobel Laureate Barbara McClintock (1984)<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib88__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqajIC2C4$> of viral transposable elements that conduct cut-paste (transposons) and copy-paste (retrotransposon) gives a code based explanation  for genomic evolvability, brain plasticity and novel phenotype primarily in eukaryotes.  This underscores the truism that primarily only software can change software and also that viral hacking by such internal and external biotic malware is the Achilles heel of genomic digital systems.

  1.  Unpacking the Evidence for Gödelization of Biology

                        While operations on encoded information fall under the purview of Computation Theory and Recursion Function Theory (RFT), till recently there was no evidence for how this and Gödel's theorems apply to biology.  Markose (2022<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib61__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSq6qlj9z0$>) unpacks the recent breakthroughs here.

                     Gödelization of information processing starts, firstly, with unique identifiers or Gödel numbers for digital entities well known in the digital economy and taking the form of bio-peptide unique identifiers including ‘zip codes’ in organisms as discovered in the Nobel prize winning work of Blobel (2009). It appears all signalling in bio-ICT relies on peptide identifiers from transcription factors to neuron-neuron links.

                     Two other distinctive Gödelian features found in genomic intelligence, using epithets from Hofstadter (1999) are self-reference (Self-Ref) or the online machine execution involving the Diagonal operator and offline virtual self-representation (Self-Rep) of the former.  The breakthrough on the significance of these staples of RFT found in textbooks such as Rogers (1967) and Cutland (1980), starts with the insight from Gershenfeld (2012<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib48__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqGSrl56A$>, 2017<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib49__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqpGCiPPQ$> Chapter 3 p. 109) as to what the self-referential/Diagonal operator means for biology, where a program g builds machine • to run g (typically denoted as •g(g)). Gershenfeld (2012<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib48__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqGSrl56A$>) says what 21 century digital fabrication is trying to achieve is something biology solved 3.7 billion years ago with the self-assembly programs associated with the ribosome and other transcriptase machinery involved in gene expression for the morphology, somatic identity and regulatory control of the organism.

                      Despite the central role assigned to self-reference for the sentient self in advanced organisms (Gardenfors  2003<https://urldefense.com/v3/__https:/scholar.google.se/citations?view_op=view_citation&hl=sv&user=ZW8RWdAAAAAJ&citation_for_view=ZW8RWdAAAAAJ:Y0pCki6q_DkC__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqclXIy_8$>, Northoff et. al, 2006<https://urldefense.com/v3/__https:/doi.org/10.1016/j.neuroimage.2005.12.002__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqKaUbmro$> , Newen (2018)<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib94__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqkgZx_Ng$> , Miller et.al., 2018<https://urldefense.com/v3/__https:/doi.org/10.1016/j.pbiomolbio.2018.10.002__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqM9CMK80$> , etc) only Tsuda (2014)<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib114__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqPf630Ss$> and Markose (2017, 2021,2022) have noted how the evolutionary development of Self-Rep mirror structures as in the Gödel Meta-Representation Theorem (Rogers,1967) is necessary for biotic elements to make statements about themselves, first having self-assembled themselves.  This offline embodied Self-Rep, which contrasts with no such structures in prokaryotes, was a paradigm shift in the Adaptive Immune System (AIS) some 500 million years ago in the lineage of jawed fish, called the Big Bang of Immunology, Janeway et. al (2005).  This latterly appears as Mirror Neuron System (MNS) mostly in primate brains first discovered by the Parma Group of neuroscientists.

                   The AIS demonstrates virtual offline mirrored self-representation (Self-Rep) in the MHC1 T cell receptors of ~85 % of expressed genes viz. halted machine executions of genomic self-assembly codes that determine the somatic and phenotype identity for the organism.  As will be seen, these Self-Repped gene codes in the Thymus, called the Thymic Self  (Ramon and Faure, 2021<https://urldefense.com/v3/__https:/www.frontiersin.org/articles/10.3389/fpsyt.2020.540676/full__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqRtkoQzk$>) or ‘the science of self’ (Greenen, 2021<https://urldefense.com/v3/__https:/link.springer.com/article/10.1007/s00281-020-00831-y__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSq71KRW4U$>) are primarily used to identify the hostile other, viz. negation function operators of non-self antigens.  In turn, the Mirror Neuron System (MNS) reuses codes of self-actions from the sensory-motor cortex for social cognition and inference regarding conspecifics via virtual simulations in the MNS (Fadiga et al. (1995)<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib35__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqek3WNP4$>; Gallese et al. (1996)<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib46__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqHLWVVsM$>; Rizzolatti et al. (1996)<https://urldefense.com/v3/__https:/www.sciencedirect.com/science/article/pii/S0303264722001022*bib102__;Iw!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqmxeFzqA$>).

                 It is conjectured that an identical RFT machinery is involved in the self-other nexus in both the AIS and MNS.  The graphics for Self-Rep Mirror Mapping in the AIS and MNS is given in Figure 1 in Markose (2021)<https://urldefense.com/v3/__https:/www.mdpi.com/1099-4300/23/4/405__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSq8Rs91D0$>.

  1.  Detection of Non-Self Antigen in AIS: New Diversity-Selector Model and Gödel Sentence in Genomic Blockchain Distributed Ledger (BCDL)

                     The most significant of all breakthroughs here is the one made by the game theorist Binmore (1987) who raised the ‘spectre of Gödel (1931)’ in the form of Gödel’s Liar who will negate or falsify what can be computed/predicted. Binmore effectively mooted the adversarial digital game which is co-extensive with life itself (Markose, 2017, 2021). This constitutes the fourth condition of Gödel systems and involves an adversarial agent in the form of a virus or a hacker whose actions cannot be constrained in anyway.

                The Gödel Incompleteness result that generates the Gödel Sentence permits a code qua biotic element to self-report  “I’m under attack”, when it has been hacked/negated by a novel malware. This marks an endogenous exit from listable sets, a necessary condition for novelty and the production of novel antibodies, to avoid the irrational state of logical inconsistency of formal systems (Smullyan,1961<https://urldefense.com/v3/__https:/www.jstor.org/stable/j.ctt1b7x7ww__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqtrLoY_E$>).  The genomic Gödel Sentence in terms of 21 century nomenclature is a hash that helps demonstrate endogenously that the outputs of expressed genes have been maliciously altered. Note, that halting self-assembly gene codes, g • G that create the organism are theorems of the genomic system theorems and is disjoint from known non-theorems or what the immunologist Burnett (1958) famously called ‘forbidden codes’ if allowed to run will ‘negate’ the theorems and cause harm.

                Indeed, how the AIS identifies novel software attacks on own gene codes, g , by malware/parasite negating functions fp¬ ! , which belongs to an uncountable infinite set that cannot be mechanically listed, is stupendous case of uber bio-cybersecurity.

              The AIS implements ‘out of the box’ astronomic anticipative search for novel non-self antigens necessary for novel anti-body production and cognition in humans manifests unbounded proteanism for novel extended phenotypes, Dawkins (1987), in the form of artifacts outside of ourselves.  This facility found only in the AIS relying on the Recombination Activator genes (RAG 1 and 2) and in the human brain for neural receptor diversity runs into orders of magnitude of 1020 – 1030 that exceed the pre-scripted germline genome size many times over.

                The Rogers (1967) fixed point indexes for yet to happen fp¬ ! attacks by the non-self antigens are generated in the AIS  in most ingenious fashion: a large number of codes/indexes purported to be of different fp¬ ! on each self-repped  g are generated in the T-Cell Receptors. This is the most spectacular case of predictive coding. Suppose that the g.n for the specific tuple { fp¬ , gn } denoted by g¬. When the attack by fp¬ !  takes place in real time in the periphery involving the said pair { fp¬ , gn }, the experientially driven peripheral MHC1 receptor must record this and if this ‘syncs’ with the one that was speculatively generated in the thymic MHC1 receptors, two parts of the fixed point come together to construct a genomic Gödel Sentence.

               This molecular genomic diversity-selector model follows a unique self-referential blockchain distributed ledger that is different, in terms of the self-referential design, from man-made BCDLs first invented circa 2009.  The genomic BCDL manifests secure digital and decentralized record keeping where no internal or external bio-malware can compromise the immutability of the life’s building blocks and no novel blocks can be added that is not consistent with extant blocks.

                    There are no doubt macroscopic environmental and population level pressures of the Neo-Darwinian style natural selection arising from conspecific or multi-species competition for survival in terms of those which reproduce more and those which die out. There is growing evidence that at level of bio-digital smart controls of the eukaryote genome and homeostasis, selective pressures are based on the principles of a unique self-referential BCDL

                    In conclusion, genomic intelligence in vertebrates that has reached its apogee in humans is highly empathic as the conspecific/other is the projection of self; greatly Machiavellian having co-evolved with adversarial viral agents; geared toward unbounded proteanism from the get-go starting with transposon based diversity of RAG genes in the immune system and stringently self-regulated by a BCDL driven by the principle of autonomy of the life of the organism and an agenda to be hack-free.

Points for discussion and areas where I would like collaborators and those I can learn from as in my day job I’m an economist

#1 I fully concur with the latter part of Miller/Torday (2018)<https://urldefense.com/v3/__https:/doi.org/10.1016/j.pbiomolbio.2018.10.002__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqM9CMK80$> statement :

“As self-referential cognition is demonstrated by all living organisms, life can be equated with the sustenance of cellular homeostasis<https://urldefense.com/v3/__https:/www.sciencedirect.com/topics/biochemistry-genetics-and-molecular-biology/homeostasis__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSq8F9h1Jc$> in the continuous defence of 'self'.” However, Miller/Torday do not use any of the code based RFT Gödel style Self-Ref mirror structures for genomic information processing.

                No doubt prokaryote CRISPR based identification of non-self antigens is highly sophisticated and so are the innate immune system strategies which primarily entail analog defences of setting up barriers, toxicity, raising temperature by inflammation and ingestion by phagocytes. Prokaryotes distinctively do not have the mirror Self-Rep offline structures of self as in the Rogers (1967) RFT fixed point theorems which permit bio-molecules to self-report changes to their own codes.  My take on homeostasis in eukaryotes post jawed fish is that bio software data security and hence the autonomy of self was taken to the next level: the design of how codes can self-report they have been changed by bio malware by utilizing hashes like the Gödel Sentence.

                It will be interesting to compare the bacteria-phage arms race in prokaryotes with the one in the AIS antibody production outlined above. I have reached out to John Mattick and Silvia Sanchez Ramon on these issues.

#2.  In Markose (2021) I indicate that the generation of genomic Gödel Sentences for detection of novel the non-self antigen is necessary for novel antibody production is a testable hypothesis. Those who died of Covid 19 were found not to be capable of antibody production and instead were subjected by a cytokine storm brought about by the analog defences of toxicity and inflammation of the innate immune system.

# 3 A common Self-Rep mirror RFT machinery for self-other recognition in the AIS and social cognition following from the Mirror Neuron Systems in primates has been conjectured. Similar conjectures have been made at least since Irun Cohen, but without any RFT to guide the discussions.  Figure 1 in Markose (2021)<https://urldefense.com/v3/__https:/www.mdpi.com/1099-4300/23/4/405__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSq8Rs91D0$> shows how a knock-out of the circuitry for self’s updates on non-self activity vis-à-vis self (the RHS entry in the offline Self-Rep •(g,g) operator ) known to be controlled by interferon gamma as in Jonathan Kipnis Group 2016<https://urldefense.com/v3/__https:/www.nature.com/articles/nature18626__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqF1KRYvk$> experiment on rats, could lead to loss of immune capabilities as well as their sociability.  Note, the Markose RFT machinery for the self-other nexus is a different explanation to the evo-devo one given by Kipnis et.al<https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=https*3a*2f*2fet.al&c=E,1,gM9IH0l2_WyCH4oxGyy4qH5YHFHLfyPWaXVy6cuQ8D9BarKsxnYxmIv85rzwVEYx3t674fcYC9xwPHX9REjolv82s-JTTmjlFPERtC4ZphcjW1Y,&typo=1&ancr_add=1__;JSUl!!D9dNQwwGXtA!UyCtjIKq6bLofcL6dBQ6z-6R4kA6nftw0YH6vVFZUKODBcUEs1j7q0VcLQrp00gchhkHyXZ3YnreRaiVXtM$ >

#4 Finally, it is a matter of incredulity that over 90 years have passed since Gödel (1931) for evidence  to be found that the staples of Self-Ref and Self-Rep and the Gödel Sentence are ubiquitous in Biology.  A number of factors can be adduced. Even those who espouse that code-based operations are relevant in cognition such as in the Computational Theory of Mind never mention Self-Ref and Self-Rep, let alone that the Gödel Sentence is like a hash which permits a software document to endogenously self-report it has been hacked.  There is a strong anti-machine view which overlooks the encoded basis of the genome and biology and claims that biology is a non-digital natural process that is creative in some vitalistic way.

References

#Blog on University of Essex Website: How we became smart – a journey of discovery through the world of game theory and genomic intelligence https://urldefense.com/v3/__https://www.essex.ac.uk/blog/posts/2021/10/26/how-we-became-smart__;!!D9dNQwwGXtA!UyCtjIKq6bLofcL6dBQ6z-6R4kA6nftw0YH6vVFZUKODBcUEs1j7q0VcLQrp00gchhkHyXZ3YnreUl841iA$ <https://urldefense.com/v3/__https:/www.essex.ac.uk/blog/posts/2021/10/26/how-we-became-smart__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqVgSsoLo$>

#Markose, S.M, 2022, Complexification of eukaryote phenotype: Adaptive immuno-cognitive systems as unique Gödelian blockchain distributed ledger, Biosystems, ISSN: 0303-2647, Vol: 220, https://urldefense.com/v3/__https://doi.org/10.1016/j.biosystems.2022.104718__;!!D9dNQwwGXtA!UyCtjIKq6bLofcL6dBQ6z-6R4kA6nftw0YH6vVFZUKODBcUEs1j7q0VcLQrp00gchhkHyXZ3YnreG8D8AFs$ <https://urldefense.com/v3/__https:/doi.org/10.1016/j.biosystems.2022.104718__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqPn2Rp5k$>

#Markose, S.M, 2021, “Genomic Intelligence as Über Bio-Cybersecurity: The Gödel Sentence in Immuno-Cognitive Systems”<https://urldefense.com/v3/__https:/doi.org/10.3390/e23040405__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqV3O-kDY$>, Entropy. 23(4), 405; ttps:////doi.org/10.3390/e23040405  <https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=https*3a*2f*2f*2f*2fdoi.org*2f10.3390*2fe23040405*26nbsp*3b*26nbsp*3b&c=E,1,5ljJnDyIM9_72M8MI5wXv03g8R_xcMpRIJlprpoJlE3VxeysqCeuaM5BD03I_cDZuFkptB_foj9qlcTVhWq8_mCu8nrDmJk5UpevoDwqAG-mS9ZcGA,,&typo=1&ancr_add=1__;JSUlJSUlJSUlJSU!!D9dNQwwGXtA!UyCtjIKq6bLofcL6dBQ6z-6R4kA6nftw0YH6vVFZUKODBcUEs1j7q0VcLQrp00gchhkHyXZ3YnreTR3KzjA$ >

Markose, S.M, 2017, Complex type 4 structure changing dynamics of digital agents: Nash equilibria or a game with arms race in innovations. Journal of Dynamics & Games, 2017, 4 (3) : 255-284. doi: 10.3934/jdg.2017015<https://urldefense.com/v3/__http:/dx.doi.org/10.3934/jdg.2017015__;!!D9dNQwwGXtA!RRmmHQCs_Cav9UXiPz27dODNI-Kke3ky-PhpVLOvgOETwB-s3hTK7egkhJHf5AFKr0UuxLxzixSqC_UOvRc$>

-------------------

Professor Sheri Markose

Economics Department

University of Essex

UK
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