Incompleteness in Biology and its Implications for Bioengineering
This year (2021) marks the 90th anniversary of Kurt Gödel’s seminal and math-shattering paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I" (1931) that irrefutably proves the incompleteness of formal logic (initially for first-order logic, but extended to all higher forms). After encoding the formal limits by defining mathematical completeness, he went on to show the absolute limits of any mathematical system in a most ingenious way, creating and using what we often hear described as Gödel Numbering. His paper deeply impacted the vision for mathematics as defined by-then current leader David Hilbert, demonstrating that not all Truths that can be stated in a logic system are Decidable or Provable (there is a subtle distinction between these, but not relevant for us). This forever altered the direction mathematics would take including the newly emerging field of computer science (see Turing's Halting Problem).
Now that we have arrived to the 21st-century exploring the mechanisms behind life, Gödel's work may be about to influence the directions biology and biomedicine are heading in, including molecular genomics, viral adaptation, molecular basis of diseases, and genetic/molecular engineering. All these areas and more are affected by biological constraints and opportunities; they are also affected by the intrinsic incompleteness of biological systems and their built-in (logic) capacity to handle all situations, from aging, to inflammatory disorders, to infections, and eventually to cancers.
The argument to firmly establish this would take more room than is available here, but the essence can be described briefly as the inherent limitations of any organism along with its genomics, to be prepared and able to survive any situation and even handle genomic alterations. A particular aspect that arises, as it relates to Gödel’s original proof, is the challenge of any living system to distinguish Self (innate as well as what arises by random recombination) from Non-Self. Here undecidability is replaced by Self-verification. This is quite universal for all living things, from protection against bacteriophages, to corals competing, to pollen rejection implants, to immune responses. Vertebrates put a lot of emphasis on this with their histocompatibility genes and antibodies, as well as the energy required by this subsystem. More recently, research into immuno-oncology attempts to leverage what normally could/should take place: a subsystem to safeguard against attacks by microorganisms or genomically-altered tissue. A gödelian view would suggest that auto-immune disorders and cancer's evasion of immune responses are two sides of the same Self vs Non-Self problem.
The formal concept of completeness is described as the ability to derive all valid statements (e.g., theorems from proofs) or downstream derivations from a finite set of axioms and rules; the concept of soundness is the converse and requires all statements to be logically consistent without any contradictions. Gödel proved if a formal system is sound, its statements (reifications) can be true, but they cannot all be derived from initial assertions. This interpretation is clear for mathematics, but what does it mean in biology? What for example is incompleteness or an inconsistency in a biological system?
Recall, Gödel’s logic proof doesn't only apply to some mathematical or logical systems, but all formally defined logic system (FDLS). One may assert biology is not an FDLS, but that is most likely not true: life's logic is formal (consistent and reproducible) since each example of life has rules it has engendered, encoded, and follows via the genetic coding and the regularized causality of molecular interactions all life relies on. It certainly isn't trivial logic, and must support a logic (meta)language that can be demonstrated in various ways, such as insertion of new genetic codes by random natural processes or human design. We have cracked some of that language, but still just a tip of the iceberg.
As a metamathematical aside, living systems can even perform some of what we associate with basic math: they can count and process sets of elements (e.g. Y chromosome presence, copies of DNA, operons, number of cells in fly's ommatidium, nerve tracks), perform negation (using gene or protein regulation), do multiplication by cell division or by regulation via molecular interactions (product of concentrations) of 2 or more reactants, and importantly store symbolic states (cell states in differentiation, phosphorylated proteins, recombined DNA). A complete equivalence isn't easy to show or even necessary, but a correspondence may be provable.
As we enter this new age of bioengineering and genomic therapies, we will need to understand the ability and consequences of biosystems to determine what is (newly engineered) Self and what should be Non-Self. The Incompleteness theorem implies that for logical systems this is not always possible, and some genomic versions of a biosystem (e.g., mutations) may have downstream effects that are not be foreseeable. Clearly some diseases have been anticipated by our immune response, but it cannot be guaranteed to work (from within) 100% of the time. However, the future role of a computational physician will be able to augment and overcome these limitations much of the time. And this corresponds to Gödel's extension of logical systems by inserting new axioms and rules, introduced by agents outside of the biologic system.
Admittedly, this is a different approach to thinking about biomedicine where one typically applies all the latest biomolecular technology, elucidates reductionist biological cause-and-effects, and trains machine-learning algorithms to find therapies for diseases our society deems relevant (either by public, researchers, or investors) for a focused assault.
There's however an alternative approach, which I feel is more appropriate for this century that relies less on biohacking and chance. We will need to take a new take on defining fundamental bioprocesses and how these are encoded, not modeled as contemporary computer programs but as biological algebras based on the fundamental molecular and cellular principles that can be composed together. How things go wrong might also be abstracted from this, such that understanding diseases may not just be about collecting sets of genotypes/phenotypes but spaces around normal functions that can get perturbed in multiple directions!
In actuality, diseases may not even be separate challenges, but tied together by the underlying cross-domain logic. We already see this in efforts to understand/control cancer, immune, and infectious diseases. I specifically suspect that many/most diseases will be related to the different forms of completeness/consistency errors Gödel had already categorized, e.g., undecidable yet True (attack Self ~ false positive) and inconsistent but False (ignore Not Self ~ false negative). I've even come to wonder if the oncogene model in cancer is not the best way of looking at cancer.
So stepping back for a moment, what activities could a gödelian perspective influence?
Bioinformatics? Yes, but certainly in positive ways by expanding how we model and represent biological systems. Perhaps some of the deeper laws reside undetected in yet to be discovered genomic patterns and structures.
Computational Modeling? Very likely, since it would offer us new formalisms (e.g. algebras) that can be defined and even proved logically, and then evaluated (disproven, not proven) experimentally. Perhaps these would enable us to model different immune responses and non-responses.
Biomedicine? Very much so, I think it will offer more reusable concepts and mechanisms for defining diseases and potential therapeutic paths that go beyond ontological definitions (which mainly address data and less about models).
Bioengineering? Indeed, and this is where I think things will get really interesting. The world of CRISPR and other bioengineering technologies bear a subtle echo with the goals set out by David Hilbert a century ago. As a quick example, recall that for every k (specific) interactions a protein p or its gene g (total N) has with other cellular factors, a consistent biosystem (organism) must AVOID interactions with ~N*(N-k) other components, as well as disrupt their interactions. This is a tough requirement for newly designed biomolecules (~half a billion per each bioentity).
There are also some insights from biology into metamathematics that may come to surface. Although there are limits to how a (bio)system can compute decidability or general solutions of repairing itself or treating a disease, it also points to how non-deterministic processes like evolution that can circumvent gödelian limitations via selection. Evolution is how living things get around previous constraints, and the growing genome is basically an accumulation of more necessary axioms; it could explain why organisms with small genomics still survive quite well, while others have continued growing the genomic size to overcome constraints. Indeed, even the power of adaptive immune responses by the evolution of lymphocytes itself creates the opportunity for a novel class of tumors: lymphomas. Each new genetic axiom often requires another axiom to keep it in check! As for Gödel's numbering system and its role, evolution came up with its own: DNA/RNA.
A gödelian formalism for biology will have many implications: Technologies and knowledge will continue to advance and have great importance to mankind, but we will better realize how biology is directed by bio-logic to take advantage of faster and hopefully safer biomedical solutions that rely less on lengthy investments and reduce errors. Time will tell if that will have an impact on the research and industry.
