Extended abstract
1. Background
For some 95 odd years since his epochal 1931 paper, Gödel’s Incompleteness Theorems (GITs) have been the focus of more heat than light. In this time, there has been very little in the way of evidence that has shown that GITs have any bearing on real-world phenomena.
I aim to unpack recent assertions by Igamberdiev (2021) for why “living systems during evolution continuously realize the proof of Gödel's theorems”, or Miller/Torday (2018)'s statement : “As self-referential cognition is demonstrated by all living organisms, life can be equated with the sustenance of cellular homeostasis in the continuous defence of 'self'.”
It should be noted that Miller/Torday give centrality to self-referential information processing in genomic systems, specifically to detect and mitigate adversarial changes to self-codes, but they make no reference to the Gödel logic or foundational mathematics of Computation Theory, also known as Recursion Function Theory (RFT). I will draw on my recent work, Markose ( 2021, 2022), that the Gödel logic and, in particular, the Gödel Sentence, far from being esoteric constructions in the foundations of mathematics, are ubiquitous to genomic information processing in eukaryotes.
Genomic intelligence is a concept introduced in Markose (2021) 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 the evolution of multicellular eukaryote life (Markose (2022)). This is a very tall order of computational intelligence that corresponds with so-called Type IV Dynamics in the Wolfram–Chomsky Schema, which are unique in producing novelty. To date, complexity, evolvability, novelty production and ‘thinking outside the box’ in biology and humans have, for the most, part relied on models of randomness or on statistical white noise error terms. Advances in gene science, neuroscience and security of software systems (cryptography) are beginning to provide evidence that only code-based systems with self-referential recursive machinery in place can produce open-ended novelty production whilst maintaining self-codes hack-free from internal and external malware.
I will first briefly address the two canards that are associated with Gödel Incompleteness Theorems that have posed a stumbling block to the necessary breakthroughs on genomic information processing. The canards are that GITs prove that human cognition is not computational and self-reference leads to paradox. That the latter is not so in the Gödelian setting as unlike the Cretan Liar Paradox This is False, Gödel’s painstaking application of the Cantor Diagonal Lemma as a means of mechanizing self-reference leads to a theorem. Systems that can mechanize the exit route from listable sets pave the way for syntactic objects (encoded objects) that self-referentially say of themselves a true statement that they cannot be computed if the system is consistent. Influential commentators like Roger Penrose have used GITs to conclude that the human cognition can outstrip what Turing Machines can do. As Rescorla (2020) says :”It may turn out that certain human mental capacities outstrip Turing-computability, but Gödel’s incompleteness theorems provide no reason to anticipate that outcome.”
- Unpacking the Evidence for Gödelization of Biology
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 thereof are explicitly code-based software systems.
Significantly, while debunking the idea that the primary source of evolutionary changes arises from random transcription/replication errors, the epochal discovery by Nobel Laureate Barbara McClintock (1984) 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.
While operations on encoded information fall under the purview of Computation Theory and Recursion Function Theory (RFT), until recently there was no evidence for how this and Gödel's theorems apply to biology. I will unpack the recent breakthroughs here.
The 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 that 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 the 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, 2017, Chapter 3, p. 109) as to what the self-referential/Diagonal operator means for biology, where a program g builds machine ɸ to run g (typically denoted as ɸg(g)). Gershenfeld (2012) says that what 21st century digital fabrication is trying to achieve is something biology solved 3.7 billions 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. The domain of these self-assembly diagonal machines are the theorems for the eukaryote organism, viz. the self-codes that have to be maintained free from adversarial hacking.
Despite the central role assigned to self-reference for the sentient self in advanced organisms (Gardenfors 2003, Northoff et. al, 2006 , Newen (2018) , Miller et.al., 2018 , etc.), only Tsuda (2014) 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. Tsuda (2014) makes an explicit observation that unless the mirror Self-Rep recursive structures are in place, it is unlikely that statements about self can be made, let alone make out the other. One can also include here the adversarial other in the form of negation to self-codes. 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 a 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) or ‘the science of self’ (Greenen, 2021), 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); Gallese et al. (1996); Rizzolatti et al. (1996)).
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 are given in Figure 1 in Markose (2021).
3. 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 any way.
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. In other words, using the Second Recursion Theorem, the index for the fix point for the novel negation function for the self-code/Theorem for the genomic system is generated. This marks an endogenous exit from listable sets, a necessary condition for the production of novel antibodies, to avoid the irrational state of logical inconsistency of formal systems (Smullyan,1961). The genomic Gödel Sentence in terms of 21st century nomenclature is a hash that helps demonstrate endogenously that the outputs of expressed genes have been maliciously altered.
Indeed, how the AIS identifies novel software attacks on own gene codes, g , with malware/parasite negating functions fp¬ ! , which belongs to an uncountable infinite set that cannot be mechanically listed, is a stupendous case of uber bio-cybersecurity.
The AIS implements an ‘out-of-the-box’ astronomic anticipative search for novel non-self antigens necessary for novel anti-body production and cognition in humans which 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 of the Second Recursion Theorem for yet-to-happen fp¬ ! attacks by the non-self antigens are generated in the AIS in the 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 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 life’s building blocks and no novel blocks can be added that are not consistent with extant blocks.
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 from adversarial viral agents; geared toward unbounded proteanism from the get-go, starting with the 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.
Selected References
#Blog Essex University Website: How we became smart – a journey of discovery through the world of game theory and genomic intelligence https://www.essex.ac.uk/blog/posts/2021/10/26/how-we-became-smart
#Markose, S.M, 2022, Complexification of eukaryote phenotype: Adaptive immuno-cognitive systems as unique Gödelian blockchain distributed ledger, Biosystems, https://doi.org/10.1016/j.biosystems.2022.104718
#Markose, S.M, 2021, “Genomic Intelligence as Über Bio-Cybersecurity: The Gödel Sentence in Immuno-Cognitive Systems”, Entropy. 23(4), 405; ttps://doi.org/10.3390/e23040405
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, doi: 10.3934/jdg.2017015
