Forbidden Truths from Within
Implications of Gödel’s Incompleteness Theorem on our Personal, Professional, and Cosmological Worlds
This essay is for the truth-seekers out there, those looking to squeeze as much knowledge out of this world as possible, even when that knowledge feels out of reach, like a satiating juice wrapped in stone. Sometimes, this feeling never goes away, and your knowledge feels incomplete no matter how hard you try. Other times, a consistent understanding eludes us when our knowledge inches forward only to slide back two more inches due to the unearthing of further complexities. Here, we will explore Gödel’s Incompleteness Theorem and what it means for those in search of truth.
Gödel’s Incompleteness Theorem
Gödel’s Incompleteness Theorem states that no sufficiently complex formal system can be both complete and consistent. This leads to one of two outcomes within each system:
Consistent but not complete: some truths will never be able to be proven within the system.
Complete but not consistent: all truths can be proven within the system but the truth is muddied with contradictions. In other words, paradoxes are spotted left and right.
Don’t believe it since it’s just a theorem? Let’s reflect on Gödel’s famous self-referential statement:
“This statement is not provable.”
If it can be proven, then the system proves a falsehood, making it inconsistent. If the system can’t be proven, then the statement is true but unprovable, making the system incomplete. Either way, the system fails to be both complete and consistent. In much more rigorous mathematical terms, Gödel illuminated this exactly: logic has its limits.
The overarching implication here is that we must step outside of a system if we strive for truth that is consistent and provable with a system defined as a connection of parts that coordinate to fulfill a higher purpose beyond those of the individual parts. In the context of Gödel, a system is characterized by a distinct boundary of what lives within the system vs what lives outside the system. This insight compels us to recognize that every system has boundaries - and those boundaries limit what can be known from within. This applies to mathematics, medicine, personal habits, and even the universe itself. Let’s begin with the implications in a small system and progressively zoom out.
Gödel on Biological Systems
As biological systems, we are all composed of many organ systems with lines drawn between them for their different purposes: blood circulation, digestion, and transmission of information between the brain and body through nerves. Within these systems exist subsystems, like the fight-or-flight sympathetic nervous system vs the rest-and-digest parasympathetic nervous system. If a failure or risk within your circulatory system is detected, you visit a specialist with expertise in that organ system. Per Gödel, the only way a cardiologist can gather a complete or consistent understanding of the failure mode, its root cause, and an optimal resolution is through sufficient understanding of the overarching system (i.e. the human body). This is why specialized expertise is layered on top of an anatomical foundation built in medical school.
In theory, this works great. Physicians have the adequate knowledge of an organ system and its overarching biological systems to gain a complete and consistent understanding of a patient’s issues. The issue can be understood and effectively resolved, leaving everyone happy, including the patient, physician, and Gödel.
But in practice, this is not how things shake out. The individuals involved forget about Gödel’s Incompleteness Theorem and we find ourselves in a world where our medicine fixes the issues in one organ system while wreaking havoc in another, creating a game of pharmacological whack-a-mole, where each mole gets whacked with an evermore intricate cocktail of FDA-approved drugs.
Let’s look at an example: an overweight 45 year old visits his cardiologist for a routine appointment. After collecting a few measurements, the cardiologist concludes the patient has elevated cholesterol levels, putting him at risk of heart disease. Since the patient is younger, the cardiologist suggests that this can be turned around through lifestyle improvements like a well-balanced diet and more movement.
Fast forward a year: the patient returns to his cardiologist who again takes some measurements. He observes similar results, so he asks his patient how his diet and exercise program went. The patient shared that he was not able to sustain it, so the cardiologist, caring about his patient, decides that a statin is the best course of action to prevent any catastrophic cardiac episodes, particularly at a young age. He knows that statins adversely impact the digestive system, but he reasons that a minor yet chronic risk to the digestive system is worth accepting over the major risk facing his patient’s circulatory system.
Such stories happen time and time again, leading to a healthcare system of symptom management rather than elimination of root cause. The same outcome can be observed with slightly modified plots: perhaps the cardiologist operates in a hospital where he is incentivized to prescribe a certain drug. Or he isn’t all that concerned about his patients, but tends to overprescribe drugs for fear of litigation. Or maybe the physician himself forgot about Gödel’s Incompleteness Theorem either through hubris or sheer ignorance, leading to a reductionist view that every issue within the cardiovascular system can be resolved within the cardiovascular system.
If we learn from our friend in cardiology and embrace the implications of Gödel’s work, we arrive at startling breakthroughs within the human body:
Without Gödel, we futilely obsess over supplements and diet to fix the digestive system, spend endless hours ruminating in our minds with therapists without considering our screen time and diets, or we maximize calcium intake for osteoporosis when we just need to use our muscles. We miss the bigger picture because we hyperfocus on what’s inside the boundaries of a system.
Gödel on Personal Systems
Zooming out one step from our biological systems, many of us view health itself as a personal system to maintain, complete with regimented diets and exercise programs. To Gödel’s point, we must look outside this system to know whether a given health protocol fulfills its goal of overall health. Otherwise, we fall into silly debates like whether a push-pull-legs split is a superior strength training program to an upper-lower split. Or whether playing sports is a better form of HIIT than organized classes. Or if cycling is a better way to maintain zone 2 cardio compared to running. A routine for optimal health depends on numerous other things outside of your health system. Is your job stressful? What type of gym is most accessible to you? Do you prefer your movement to be more social? Or do you simply enjoy some forms of exercise more?
Similarly, many of us view our work life itself as a personal system that starts and ends at a certain time each day - for most, this is 9am to 5pm. As dramatized by the TV series Severance, what happens within that system can be totally isolated from what occurs outside that system.
To understand the objective of this work system, we must again nod to Gödel and look outside of it; it could be a means to support yourself or your family with as little work as possible, it could be a source of meaning, or maybe it’s a ladder to climb. If we don’t take the time to understand the objective of that system, it’s nearly impossible to understand basic questions like: does my current work system serve me well?
For an accounting of which goals are fulfilled by the various activities and habits we employ ourselves with, we may find it helpful to compartmentalize everything that serves a personal goal into different systems: health, earning money, relaxation, family time, and socialization. This probably feels very natural to many “Type A” readers - to have a sort of task list to ensure we tick all of those boxes in a given day, week, or month.
This reductionist approach that once helped us organize our lives can become suffocating when all of our systems intended to fulfill different goals compete for the same 24 hours in a day. Now we’ve reached Gödel’s incomplete or inconsistent state, where we serve our systems rather than vice versa or our systems sit in competition with each other rather than in coordination.
This concept may look familiar to those who have read my most popular essay The Three Body Problem of the Mind, where we explored balancing the competing fundamental desires of hedonism, individuation, and duty. I like to think Gödel would have arrived at a similar conclusion as that in the essay: we must step outside of our individual desires and their supporting systems to understand them. As an example, to know whether my fitness system serves me well, I need to understand how it interfaces with my work system. Is work stressful? Maybe low intensity is better for my particular situation. Do I travel frequently? Maybe I need to meal-prep food that I can easily travel with.
I recently read an essay that articulated Gödel’s implications on our personal systems slightly differently through a concept called de-atomization. The author,
, advocates for a better life through viewing your life as one cohesive system rather than several siloed systems. The ultimate goal of that system is for you to decide. And orienting all aspects of your life towards that purpose can create a cohesive overarching system functioning in one direction. Work, play, relaxation, physical health, and connection work as sub-systems in one well-oiled machine.
Gödel on Interpersonal Systems
Now zoom out: what’s the boundary of you as a system? We can draw a line between what lives within your system versus outside your system. Since our memories and experiences shape how we interpret what we see, hear, feel, smell, and taste; the bounds of this system can be fairly clearly drawn between different sets of sensory organs, memories, and experiences.1 Or to put it in plain English, we can consider each individual a unique system.
As Gödel would predict, we run into a limit when multiple truth-seekers interact with each other. This is because our sensory organs are the measurement devices we use to discern what is true, and they can deceive us. Here is a simple image illustrating that what is true to one is not true to another.
Many point to this image and rightfully recognize the limits of our knowledge. Where they fail is by forgetting about the implications of Gödel - truth can be accessed through scale. Instead, they fall into naive belief systems like postmodern factual relativism where they think all perspectives are equally valid, as if each person operates in their own system disconnected from others. Or worse, they fall into solipsism and become infatuated with only their experiences.
Based on what we’ve contemplated thus far, the reader knows better: truth exists at a higher order above competing perspectives. Either that 6/9 was next to some other sequential numbers and we can arrive at the truth, or the artist intended it to be a specific number. Of course, someone could find more meaning in the opposite interpretation since we all interpret art differently, but that doesn’t make such an interpretation any less false just because it was meaningful in a one-person system. Maybe the artist left it intentionally ambiguous as a nod to Gödel and the truth exists in the recursive debate itself rather than a particular conclusion.
Gödel on Large Scale Human Systems:
Once enough people come together to work towards a common goal, we see new systems formed, characterized by something more than just the interpersonal relationships of those involved. The most common of these types of systems is a business. In this sense, everyone contributing to the business is a cog in a machine intended to generate revenue while anyone not employed by the business sits outside the system. Applying Gödel’s theorem, this means that anyone sitting within the business cannot truly understand it - not without creating contradictions bound to confuse others within the system or missing the bigger picture for office politics.
It shouldn’t surprise us that we’d have blind spots about a system that we are so intimately tied to, one that puts food on our table from the stable income it provides. Not to mention there are emotions like pride or fear - hidden or not - preventing you from objectively looking at the system that puts food on your table via your stable income.
This is part of the reason why consulting is popular, particularly with highly complex businesses employing large numbers of people. An outsider can peer into the system and see the influence of office politics rather than let their judgment be affected by those very political machinations. If you think of a business or highly complicated system like a maze, those occupying the maze may have a much harder time finding the exit than someone with a bird’s eye view.
As previously alluded to, the culture and politics of a business are hidden yet critical aspects of the system, so an objective view of the system presented by insiders - who may be perceived competitors or subordinates - may not be received well by leaders battling all the human emotions of fear, pride, and greed. Consultants exist outside this system so they are less likely to threaten any senses of identity from those within the system. If you’ve ever been disappointed that your company paid exorbitant amounts of money for a consultant, just to say what some employees have been saying all along, now you know why.
The caveat to this of course is that the consultant must be capable - they must be able to see the machinations of the system: including inputs and outputs amongst people, processes, tools, and culture. In a world where companies hand out large sums of money to tell them what needs fixing, charlatans lick their chops. They say all the right things, but a modicum of questioning reveals that behind the charlatan consultant’s intellectual verbiage sits nothing but vapor - purely regurgitation of past lessons from an MBA that may or may not apply to a given situation. They inhabit the corporate archetype of those who orient themselves around sounding valuable rather than being valuable to others.
I digress, let’s take a look one layer deeper into our business. Within this system sits several smaller subsystems, or departments. R&D sees the business from a different perspective than Commercial, which is different from Operations, which is also different from Quality Assurance.
Gödel’s Incompleteness Theorem explains why generalist roles exist, ranging from the high level general manager to low level project coordinators. They exist outside any one functional area, making it easier to see the connections between the various expertise supporting the business. So long as they gather sufficient inputs from each function, they can see departmental interdependencies that cannot be seen from the vantage of the production specialist in Operations or the engineer in R&D. Between the incentives driving each function, the narratives of those leaders, departmental politics, and siloed expertise; attempting to piece together an understanding of how the business works from the vantage of each department would resemble blind scientists studying an elephant.
The main takeaway for those in search of truth within complex organizations: we need to synthesize each other’s knowledge and expertise; lest we mistake an elephant for a tree, snake, rope, or spear.
We can continue to scale this logic to even larger human-powered systems: multiple businesses build an industry. Without knowing industry trends, it becomes difficult to completely understand a given business’s strategy. Multiple industries create an economy, so it becomes nearly impossible to predict the direction of an economy without understanding the culture of the people within the system and how that may change. And multiple economic and political systems join together to form a geopolitical climate. As a rule of thumb, we should always extrapolate one level higher to understand the context surrounding the system we study.
Gödel on Cosmological Systems:
Finally, we zoom out enough to reach the end of our line: the universe as one massive system that houses everything we know. Within this system, we enter the world of physics governed by the behaviors of matter and its theoretical counterpart: anti-matter. To better understand our universe, physicists paradoxically zoom in as far as possible to study matter’s most basic building blocks.
A few years before our friend Gödel identified the subject of this essay, Werner Heisenberg, a Nobel prize winning physicist for his work in quantum mechanics, discovered his uncertainty principle. It essentially states that the devices we use to measure the most basic elements of the universe alter their behavior. Much like how Hawthorne observed that people behave differently when they are observed, our mere observation of the building blocks of the universe alters their behavior, blocking a complete or consistent understanding of their behavior. His work specifically revolved around understanding both the position and momentum of a quantum particle.
Heisenberg foreshadowed Gödel by discovering the limits of what we can observe in the world of physics. Heisenberg’s discovery can explain previously discovered contradictions about the universe itself like light behaving like both a wave and a particle. Knowing what we know about Gödel, this isn’t a surprise, but the implications lead us to an unsettling conclusion: we will never reach a complete or fully cohesive understanding of our universe unless we can find a miraculous way to transcend the universe itself. Perhaps this transcendence is a rite of passage we all get to experience after death. But until then, we’re left speculating, fumbling about in an incomplete or inconsistent understanding of the universe.
What Now?
This begs the question: How do we cope in a world like this - in a world where our understanding is never complete or teeming with contradictions?
Armed with the applications of Gödel’s Incompleteness Theorem, you’ll find many people susceptible to what we’ll call Gödel’s Trap: where they believe a self-contained system holds all the answers. In other words, they mistake a complete map for a complete territory. Ralph Waldo Emerson echoed Gödel’s Trap when he stated, “A foolish consistency is the hobgoblin of little minds.” If many fall for Gödel’s Trap by clinging to a foolish consistency, it makes the case for the title of Emerson’s essay: Self-Reliance.
Self-reliance in this context can be described as a reliance on systems thinking from first principles rather than accepting the assumptions of a system defined by others. This mindset enables an escape from Gödel’s trap since it forces you to define the boundaries of the system yourself so you can navigate in and out of it - between the bird’s eye view and the mouse’s view of the maze - to determine what is true. Without self-reliance, you run the risk of listening to those who fall for the seduction of seemingly complete or consistent systems, completely unaware of their blind spots.
This self-reliance must paradoxically be balanced with humility since you too have blind spots. Such balance can be achieved by seeking out the perspectives of others, but not necessarily accepting them. Many self-reliant types fall into Gödel’s Trap by viewing their own perspective as law, but a humble attitude towards new perspectives can combat this. Self-reliance with humility looks like a quote widely attributed to Aristotle:
“It is the mark of an educated mind to be able to entertain a thought without accepting it.”
The bearer of the educated mind has the humility to seek and entertain new thoughts, but he remains steadfast in his self-reliance by retaining his veto power over the validity of the thought. He neither automatically accepts nor automatically rejects the thought, both slippery slopes into Gödel’s Trap.
In our humble search for differing perspectives, we must also recognize that others operate from different semantic systems. With different semantic systems come different personal definitions despite exchanging the same words, creating semantic noise in which understanding becomes difficult. This makes the case for regularly employing patience, empathy, and precision in our communication. To truly understand someone’s perspective, we must be willing to take a step back to the basics and ask, “What specifically do they mean when they say X? Oh, they associate X with Y? What experiences and mental models led them to associate X with Y?” Otherwise, we may fall into the trap of thinking a person with a differing perspective is a fool. Likewise, we too must be precise in our communication lest our counterpart finds us to be a fool.
And in the end, I hope the readers who read this far will heed this advice to consider differing perspectives living in differing systems since their pursuit of truth may stem from a desire to not be a fool. It certainly appears on my bingo card for potential explanations for why I consider myself a truth-seeker.
Until next time,
- VC
We’re going to conveniently exclude nuances like false memories and non-empirical knowledge (i.e. intuition) which can be argued as proof that we cannot draw an arbitrary line between us and the rest of the world. These nuances fall in line with Gödel's Incompleteness Theorem since no system can be fully complete and consistent.