Truths unproven: Gödel's incompleteness theorem
and the potential impact on our understanding of truth and knowledge
Can life be understood? I don’t know.
Here are some things that have been fascinating…and they mostly revolve around uncertainty and the state of indifference.
It’s also research I’m doing for some other content, so some of this will be left out of that more “scripted” content.
Gödel's Incompleteness Theorem (And maybe why it matters.)
For starters, I’m not a mathematician…
However, I enjoy hyper-fixing on this type of stuff, so here’s my explanation.
Mathematical logic in the early 20th century aimed to formalize all of mathematics into a set of axioms, from which every mathematical truth could be derived. This was part of a broader project called "Hilbert's Program," named after the mathematician David Hilbert.
Basically, they were trying to create a closed circle of mathematics. But maybe the reality is that the circle will always have gaps…a circle built with dots instead of bold lining.
The hope was to prove the consistency (i.e. no contradictions can be derived) and completeness (i.e. every true statement can be derived) of these axiomatic systems.
Gödel's First Incompleteness Theorem: In 1931, Kurt Gödel shook the foundations of mathematics with his first incompleteness theorem. It states that for any consistent, formal mathematical system that's capable of expressing basic arithmetic (like the natural numbers and their operations), there are true mathematical statements that cannot be proven within that system.
In simpler terms, there are true mathematical statements that are "undecidable" using the rules and axioms of the system. No matter how you expand the system, as long as it remains consistent, there will always be such undecidable statements.
How Gödel Proved It: Gödel ingeniously constructed a statement that essentially says, "This statement cannot be proven to be true." If the statement is true, then it's unprovable (as it claims), but if it's false, then it leads to a contradiction. This self-referential technique is often likened to the "liar's paradox" ("This statement is false").
He achieved this using a technique called "Gödel numbering," where he assigned unique numbers to each statement, proof, and rule in the system. This allowed him to "talk" about statements and proofs within the system itself.
Gödel's Second Incompleteness Theorem: Building on the first theorem, Gödel's second incompleteness theorem states that if a formal system is consistent, then its consistency cannot be proven within that system.
This was a blow to Hilbert's Program (or this is what I’m claiming), which aimed to prove the consistency of mathematics using mathematical means.
But what are the implications?
Gödel's theorems don't imply that mathematics is incomplete in a practical sense.
However, the theorems do show the inherent limitations of formal systems and challenge our understanding of truth and what is provable within mathematics.
In essence, Gödel's incompleteness theorems highlight the profound and somewhat paradoxical nature of formal mathematical systems: while they aim to encapsulate all of mathematical truth, there are truths that inherently lie beyond their grasp.
Something I find interesting about this is how our philosophical investigations into logic, rationality, and reasoning are deeply tied to questions about the nature and limits of knowledge. We face this constant interplay of mathematics jostling with our basic assumptions and observations within epistemology.
We are forced to be mindful and confront our epistemological limitations (and their implications).
For an even deeper dive check out Gödel’s incompleteness theorem in the Stanford Encyclopedia: here
But what could this mean for our understanding of knowledge and truth?
Gödel's theorems don't suggest that mathematics is inherently flawed. Most of mathematics operates seamlessly, and the undecidable statements Gödel pointed out seem somewhat of a more abstract area of thought in areas of epistemology (this is my understanding at least), far removed from daily mathematical endeavors. However, his work does highlight the limitations of our formal systems and challenges our perceptions of truth and provability.
Which is why it’s so interesting. It seems as though everywhere we turn, we are reminded of the endless uncertainties and unseen potentialities that exist all around us.
But before someone reading starts thinking Gödel is some form of subjectivist or doesn’t believe in some objective truth…well:
Gödel, a Platonist, believed that mathematical truths exist objectively, independent of human thought. His theorems don't dispute the existence or knowability of these truths. Instead, they emphasize the constraints of our tools, the axiomatic systems, in uncovering them. It's a crucial distinction that underscores the difference between the nature of truth and the methodologies we employ.
Learning and reading about this made me wonder if at our core, we begin to recognize, or have a “feeling” of this theorem already. We know that our systems and structures of knowledge are incomplete and will always be incomplete. We are the illusion doing the interpreting of this formula and structure. We become the inherent contradiction playing itself out into the universe.
So, to me, the more profound implications of Gödel’s work extend beyond mathematics. It nudges us to confront the epistemological foundations of our beliefs. If even in mathematics, a discipline celebrated for its precision and clarity, there are truths beyond our grasp, what does it say about other domains of knowledge?
I think it forces us to be mindful of our potential limitations, while simultaneously, being mindful that those very limitations are only potentially true.
We must be willing to constantly shift and redefine the boundaries of our perceived knowledge.
Anyway, the reason I found all of this so interesting is it suggests that our understanding of the world is not a static set of truths but a dynamic, interconnected web of knowledge.
Down with the hierarchy (or something)!
Now, here is one of my favorite film clips that I was reminded of while reading about this…
A video that I thought explained this in a fun way….
And for some current events…
Although this video is a couple of years old, I think it explains the situation in Israel and Gaza very well:
Stay curious.
You sure hit a lot of my buttons hi to interested in these arcade ideas even though I am no mathematician. I love thinking about this stuff and it makes me realize that we need more patience with our vocal adversaries who take different political stances. Nobody is all right are all wrong sometimes that’s harder to except that others. Thanks for a great article.
I really love the themes of this article and how you treated them. My obsession with this topic and the wider questions of epistemology led me to pursue a degree in cognitive science where I was able to plumb deep into this rabbit hole at the foundations of mathematics and I think you captured it nicely. As you say, Gödel’s theorems are not a problem for mathematics per se, but they were the death knell for Leibniz’ and later Vienna Circle reductionist programs of logical positivism which sought to ultimately ground all epistemology on the foundations of mathematics and then mechanically churn out ALL possible scientific truths from the organ grinder works of an axiomatic system. And as you say Gödel’s findings are incredibly exciting in how they point at the paradox at the very heart of all semiotic reference, language, cognition, and epistemology in general. Reference is all about letting one thing represent another, like letting these scratches on paper “Apple” stand in for a crisp red juicy fruit of the rose family that is wonderful baked in a pastry with a little cinnamon and a side of vanilla ice cream. It involves setting up a pseudo identity relationship like between a map and the territory which ultimately boils down to saying that something is what it is not. And that’s absurd from a logical excluded middle standpoint. But human cognition seems to work precisely because it is able to see both the binary excluded middle perspective and the holistic included middle perspective. Often our most surprising discoveries are when these two perspectives are found to both be correct. It’s akin to finding that parallel lines can be simultaneously perpendicular. Which is impossible by definition if you live on a flat earth. But if you live on a sphere it’s not so crazy when you follow lines of latitude from the equator to the poles. I’m excited by your article (I will wipe the foam off my chin now) and I look forward to your explorations in this are in the longer piece you mentioned!