The Gödelian Foundations of Language: How Incompleteness Reflects the Limits of Formal Systems and Human Understanding.

Kurt Gödel’s incompleteness theorem, published in 1931, is a landmark discovery in mathematical logic with profound implications for mathematics, the philosophy of mind, and artificial intelligence (Sabinasz, 2017). Gödel demonstrated that any formal system capable of basic arithmetic contains true statements - known as Gödel sentences - that cannot be proven within the system itself (Goldstein, 2006). This means no formal system can be both complete (able to prove all truths) and consistent (free of contradictions) (Goldstein, 2006; Sabinasz, 2017). Gödel achieved this by constructing a self-referential statement that essentially asserts, “This statement is not provable within this system.” If the system could prove the statement, it would create a contradiction, rendering the system inconsistent (Penrose, 1994). Conversely, if the system cannot prove the statement, the statement is true but unprovable within the system (Penrose, 1994; Sabinasz, 2017). Thus, Gödel’s theorem reveals that sufficiently complex formal systems always contain true but unprovable statements and cannot prove their own consistency (Goldstein, 2006). This discovery highlights the inherent limitations of formal systems, challenging the notion that they can fully capture all truths or provide absolute reliability in fields like mathematics and computation (Goldstein, 2006; Penrose, 1994; Sabinasz, 2017).

Gödel’s Incompleteness Theorem consists of two interconnected results. First, any consistent formal system F capable of performing basic arithmetic is incomplete, meaning there are true statements in its language (for example, mathematics) that cannot be proven or disproven within the system (Penrose, 1994; Sabinasz, 2017). These findings demonstrate that absolute consistency proofs are impossible for axiomatic systems capable of arithmetic. Such systems will always contain statements that cannot be resolved through their formal rules but are nevertheless true (Goldstein, 2006). Thus, Gödel’s Theorem reveals that any sufficiently strong formal system has inherent limitations, with truths that lie beyond its scope of provability.

Gödel’s theorem can be challenging to grasp, but it becomes clearer by examining its implications. Consider the analogous statement, “This formula is unprovable in the system.” If it were false, it would mean the statement is provable, leading to a contradiction: a provable statement that claims to be unprovable (Lucas, 1996). In a consistent system, anything provable must be true, so the statement must be true but unprovable to avoid inconsistency (Lucas, 1996). This self-referential logic demonstrates that in any consistent system capable of basic arithmetic, there will always be true statements that cannot be proven within the system itself (Penrose, 1994; Lucas, 1996; Sabinasz, 2017). Gödel’s rigorous proof establishes this without flaw, showing that these unprovable truths exist, and from outside the system, we can recognise them as true (Lucas, 1996; Penrose, 1994).

Humans, unlike machines, can recognise why Gödelian sentences are true, despite the inability for formal systems to prove the truth of these Gödelian sentences. This ability to "step out" of the formal system and analyse its structure is uniquely human, allowing us to identify why the contradiction occurs and understand the system’s limitations (Lucas, 1996; Penrose, 1994). Machines, constrained by their formal rules, cannot reflect on their own limitations in the same way (Lucas, 1996). This indicates Gödel’s incompleteness theorem, where a Gödel sentence - a true statement a formal system cannot prove - reveals similar limitations of machines governed by formal rules (Sabinasz, 2017; Goldstein, 2006). While machines are bound by their programming and unable to recognize such truths, humans can reason about the system as a whole and understand why the Gödel sentence must be true without leading to contradiction (Lucas, 1996; Sabinasz, 2017). Thus, Gödel’s Theorem illustrates that machines are fundamentally limited by the rules of their formal systems. They cannot solve certain problems or recognise specific truths (Lucas, 1996; Penrose, 1994). Humans, however, engage in meta-reasoning, using intuition and conceptual understanding to transcend these limitations (Penrose, 1994). This capacity to “step out” of formal systems supports the argument that human reasoning is non-computational and fundamentally distinct from machines (Lucas, 1996).

I’ve been reflecting on a Gödelian perspective on language and how it challenges logical positivism and Wittgenstein. The logical positivist model of language ultimately treats certain foundational words - like “meaning” - as pointless or empty because they cannot be empirically validated. This view rests on the assumption that the mind functions purely as a computational system.

In this framework, defining words is seen as a process of expressing specific algorithmic instructions that collectively form the definition of a word. Just as a computer follows binary instructions made up of 0s and 1s, language is treated as a system that can be broken down into smaller components, with the overall meaning of a word emerging from the sum of these parts.

However, I reject this view. I believe that language—and the mind—are not purely computational. Certain foundational concepts cannot be reduced to algorithmic instructions, suggesting that the mind operates on a level beyond mere computation.

My claim is that natural language, like mathematical language, has a Gödelian nature. This means that certain words cannot be broken down into smaller algorithmic instructions but are instead understood as true simply through our innate understanding of them. This parallels Gödel’s Theorem, which asserts that mathematics is based on foundational axiomatic truths that cannot be proven solely through the formal rules of the system. The same applies to language: there are certain words that we recognize as true, even though we cannot explicitly define them in an algorithmic or fully structured way.

For example, words like "God" and "truth" aren’t meaningless constructs to be discarded - they form the foundation of truth itself. Language can be understood as inherently hierarchical, with more fundamental words sitting at the very foundation of this hierarchy. A concept like “truth” or "God" for instance, can be seen as one of the most foundational elements of language - a bedrock upon which the definitions of all other words ultimately depend. The deeper a word sits in this hierarchy, the more fundamental it is to the structure of language itself.

Take the word “car” as an example. Its meaning is predicated on the meanings of other, more basic terms - such as “vehicle,” “transportation,” or “movement” - which in turn depend on even more foundational concepts. Eventually, this chain of dependency leads down to the most fundamental concepts, such as “truth,” “meaning,” “purpose,” “good,” and “evil.” These foundational terms are not only conceptually basic but also existentially necessary for the coherence of the linguistic and conceptual system as a whole. Without these foundational words, the entire hierarchical system of language would collapse. For example, consider the simple proposition, “the table is there.” The meaning of this statement depends on the semantic mapping of each individual word to other words, which themselves are mapped to yet more words in a complex web of interconnected definitions. This interdependence means that even a basic proposition like “the table is there” ultimately relies on foundational Gödelian language axioms — fundamental, irreducible linguistic truths that underpin the entire conceptual structure of language.

If these Gödelian axioms were absent from the base of the linguistic hierarchy, it would be impossible to understand even the simplest statements. When someone says, “the table is there,” the ability to comprehend this statement depends on the existence of certain foundational linguistic truths — such as the concept of “truth” itself — which cannot be broken down further into simpler components. Without these foundational axiomatic statements, the entire process of understanding language would break down. In other words, comprehension is not just about recognizing individual words, but about grasping the deeper axiomatic truths that provide the conceptual foundation for language. If these fundamental truths were absent, it would be impossible to make sense of any proposition, because the underlying semantic framework that gives language coherence would be missing.

Thus, this hierarchical nature of language parallels Kurt Gödel’s incompleteness theorems, which demonstrate that within any formal mathematical system, there are certain true statements that cannot be proven using the formal rules of the system itself. Gödel showed that at the foundation of mathematics, there are truths that cannot be reduced to or justified by formal rules alone - yet we as humans can still perceive them as true.

This entire discussion closely parallels my current work on the Intrinsic Nature Problem in physics. The Intrinsic Nature Problem refers to the limitation of physics in revealing the intrinsic nature of matter. While physics can describe the properties and characteristics of an electron — such as its charge and structure, including the orbit of positive and negative charges around a nucleus — it cannot explain what an electron fundamentally is, nor the true nature of any form of matter.

Consider an electron’s properties - such as mass and charge – which are defined in terms of their behaviors: mass involves gravitational attraction and resistance to acceleration, while charge describes interactions with other charges (Goff, 2019).  These descriptions focus exclusively on relationships and interactions, leaving the intrinsic nature of the electron unexplained (Eddington, 2019). Another example can be given to simplistic explain this point. Consider a chess piece. Knowing how a bishop moves tells us what it does on the board, but not what it is - whether it is made of wood or plastic (Goff, Seager, and Hermanson, 2001). Thus, there must be more to an electron than its external behaviors, an intrinsic reality that physics does not address (Eddington, 1920; Goff, 2019). Therefore, this limitation indicates that while physics is a powerful predictive tool, it cannot explain the true nature of matter (Goff, 2019; Eddington, 2019; Jarocki, 2024; Jaki, 2018).

Thus, we have briefly established that science is limited to describing the quantitative aspects of reality. It can measure properties like size and position, but it cannot reveal the intrinsic nature of matter or explain what matter fundamentally is. Science describes how matter behaves, but not what it is.  However, this gap in the scientific worldview presents an opportunity for a parsimonious solution - one that addresses this explanatory gap in a coherent and elegant way. The solution to this gap is presented as followed. As human beings, we do have a small window into the intrinsic nature of matter: the consciousness we directly experience in our own brains (Goff, 2019; Eddington, 2019). While physics provides "pointer readings" (measurements of external properties), the subjective reality of consciousness reveals something about the intrinsic nature of the matter within us (Eddington, 2019). While physics cannot reveal the intrinsic nature of matter, human beings have a unique insight into it through our own brains, which generate the subjective experience of consciousness (Eddington, 2019).

However, pertaining to the Intrinsic Nature Problem in physics, there is a counterargument, known as causal structuralism. It is contended that the concept of "intrinsic nature of matter" is misleading and inherently problematic (Hawthorne, 2001). Instead, it is proposed that there is no intrinsic nature of matter; rather, reality is defined solely by the relationships between things (Sepetyi, 2023; Hawthorne, 2001; Schneider, 2017). Consequently, if intrinsic matter does not exist, the idea that human experience provides a unique insight into matter - and that this perspective can be extended to all matter - is fundamentally flawed.

Thus, causal structuralism asserts that there is nothing more to the nature of a physical entity, such as an electron, than how it is disposed to behave (Eagle, 2009; Sepetyi, 2023; Hawthorne, 2001). Therefore, if you understand what an electron does, you then know everything that there is to know about its nature (Sepetyi, 2023; Hawthorne, 2001). On this view of causal structuralism, things are not so much beings as doings (Alter and Pereboom, 2023; Saatsi, 2017). If one assumes causal structuralism, it becomes possible that the models of physics can completely characterise the nature of physical entities; a mathematical model can capture what an electron does, and in doing so will tell us what the electron is (Hawthorne, 2001; Alter and Pereboom , 2023). This is because causal structuralism asserts that there is nothing more to the nature of a physical entity than how it is disposed to behave, rather than any intrinsic nature (Hawthorne, 2001; Saatsi, 2017; Eagle, 2009).

One of the central arguments against causal structuralism is the issue that causal structuralists attempt to characterise the nature of matter as merely beings, rather than things, lead either to a vicious regress or a vicious circle (Goff, 2019). According to causal structuralists, we understand the nature of a disposition only when we know the behaviour to which it gives rise when it is manifested (Hawthorne, 2001; Eagle, 2009). For example, the manifestation of flammability is burning; we only know what flammability is when we know that it’s manifested through burning (Goff, 2019). However, assuming causal structuralism, the manifestation of any disposition – how it behaves under certain conditions - will be another disposition, and the manifestation of that disposition will be another disposition, and so on ad infinitum (Goff, 2019; Eagle, 2009; Azzano, 2023). The buck is continually passed, and hence an understanding of the nature of any property is impossible (Goff, 2019). In other words, a causal structuralist world is unintelligible (Goff, 2019; Azzano, 2023). Therefore – if the causal structuralist are correct – then the world is unintelligible, insofar as every property is defined by another property, creating an infinite regress with no foundational explanation (Goff, 2017).

This point can be expressed best through an example. According to general relativity, mass and spacetime stand in a relationship of mutual causal interaction: mass curves spacetime, and the curvature of spacetime in turn affects the behaviour of objects with mass (as matter tends, all things being equal, to follow geodesics though spacetime) (Hobson, Efstathiou and Lasenby, 2006; Wüthrich and Huggett, 2020). What is mass? For a causal structuralist, we know what mass is when we know what it does, i.e. when we know the way in which it curves spacetime (Ryder, 2009; Goff, 2017). But to really understand what this amounts to metaphysically, as opposed to being able merely to make accurate predictions, we need to know what spacetime curvature is (Goff, 2017; Berenstain, 2016). What is spacetime curvature? For a causal structuralist, we understand what spacetime curvature is only when we know what it does, which involves understanding how it affects objects with mass (Goff, 2019; Ryder, 2009; Saatsi, 2017; Dorato, 2000). But we understand this only when we know what mass is. And so, we find ourselves in a classic “Catch 22”: we can understand the nature of mass only when we know what spacetime curvature is, but we can understand the nature of spacetime curvature only when we know what mass is (Goff, 2017; Azzano, 2023; Berenstain, 2016).

This is all getting very abstract, so let’s take a ludicrously simple example. Suppose I have three matchboxes, and I tell you the first contains a “SPLURGE,” the second a “BLURGE,” and the third a “KURGE.” You innocently ask me, “Oh really, what’s a SPLURGE?” I answer, “A SPLURGE is something that makes BLURGES.” Now, you can’t really understand my answer until you know what a BLURGE is, so naturally your next question is, “Fine, so what’s a BLURGE??” I respond, “Oh, that’s easy, a BLURGE is a thing that makes KURGES.” But, in a similar way, you can’t understand this answer until you know what a KURGE is, and so - starting to get a bit irritated - you now demand to know: “What on earth is a KURGE???!!” My response: “It’s something that makes SPLURGES.”

Another way to frame the circularity objection for the Intrinsic Nature Problem — which parallels my argument about language — is through the idea that language must have Gödelian sentences at its core. Without these foundational terms, the entire semantic structure of language would collapse, as all words, definitions, and concepts ultimately depend on these fundamental Gödelian terms that underpin the structure of language. The argument is as follows: if every word was defined in terms of other words, then all definitions would ultimately be circular, and language could never reach beyond itself. In order to get meaning going, we need to have some primitive concepts that are not defined in terms of other concepts. The concepts of physical science are not primitive but inter-defined: mass is characterized in terms of distance and force, distance and force are characterized in terms of other phenomena, and so on until we get back to mass.

Thus, the central argument derived from my analysis of causal structuralism is that matter must possess a fundamental intrinsic nature. Without such a foundation, the universe would ultimately be unintelligible. The same principle applies to language. We require certain foundational terms that form the absolute basis of our conceptual framework; without them, the entire semantic structure of language would collapse. Therefore, without these foundational Gödelian terms — such as "God" or "truth" — meaningful discourse would become impossible, as all language ultimately depends on these core concepts that underpin our conceptual structure.

Furthermore, this suggests that the mind must be more than computation, since algorithmic deductions alone cannot grasp the truth of these Gödelian sentences - as Gödel proved. Several intriguing observations emerge from this assertion. First, what does it truly mean for the mind to be more than mere computation? How does that arise, and how does it function? Second, assuming my thesis is correct, why do we possess these primitive concepts that seem almost innate to our understanding of reality? One possible answer to the latter question aligns with Jung’s insights about the mind. Jung argued that we are guided by a hierarchy of values — that is, our lives are organized according to a structured set of values, with whatever sits at the top of this hierarchy functioning as our "deity." For example, for a narcissist, the pinnacle of this hierarchy is the self, whereas for a religious person, it would be God. I believe the same principle applies to language. The existence of these core Gödelian terms at the foundation of our semantic structure appears to be fundamental to human nature. Why these terms arise and how they come to structure our thinking remains a profound mystery — assuming my analysis holds — but it is certainly perplexing and seems to hint at something divine. The fact that we are oriented toward truth itself is remarkable.

Returning to the claim that Gödelian terms sit at the foundation of our semantic language hierarchy, the implications for language are profound. If there are foundational truths in mathematics that transcend formal proof, the same may apply to language. Foundational linguistic concepts—such as “truth” or “good” - may function as Gödelian elements within the structure of language itself. They cannot be defined with complete precision within the system of language itself, yet they remain necessary for the system to function coherently.

This also reveals a crucial insight about the nature of the mind. If human understanding of foundational truths extends beyond what can be formally defined or algorithmically processed, it suggests that the mind is not merely a computational machine. The logical positivists, including Wittgenstein in his early work, attempted to define language with mathematical precision. They ultimately encountered deep difficulties when trying to define terms like “meaning,” “purpose,” “good,” and “evil” with such precision.

Their conclusion was that these terms were empty or meaningless - linguistic artefacts without real utility.

However, this conclusion overlooks the Gödelian nature of foundational concepts. Just as the truths at the foundation of mathematics are irreducible yet essential, the foundational words of language - "truth", "meaning", "good", "evil" - are not meaningless constructs but rather the axiomatic ground upon which the entire hierarchy of language rests. Their resistance to precise definition is not evidence of their emptiness but rather a reflection of their fundamental nature.

This suggests two key conclusions:

1. The human mind is not reducible to mere computation. If it were, the inability to define foundational concepts precisely would collapse the linguistic hierarchy - but it does not. This points to the existence of a non-computational, irreducible element in human cognition.

2. Foundational words such as “”god”, truth,” “good,” and “evil” are not merely true; they are more than true - they are the necessary foundations upon which all other definitions and concepts are built. Their indefinability is not a sign of conceptual failure but a marker of their essential, irreducible nature within the structure of meaning itself.

Thus, language, like mathematics, rests on axiomatic foundations that transcend the capacity of formal definition - indicating that both human cognition and linguistic truth extend beyond the bounds of computation and formal systems.