Wittgenstein's Hinges and Gödel's Unprovable Statements
By: @Sam26
Abstract

In Ludwig Wittgenstein's final notes, published posthumously as On Certainty (1969), Wittgenstein introduces the concept of hinge propositions as foundational certainties that lie beyond justification and doubt (OC 341-343). These certainties support our language-games and epistemic practices, offering a distinctive perspective on knowledge that challenges traditional epistemology's demand for universal justification. I argue for a structural parallel between Wittgenstein's hinges and Gödel's 1931 incompleteness theorems, demonstrating that consistent mathematical systems contain true statements that cannot be proven within those systems. Both thinkers uncover fundamental limits to internal justification: Wittgenstein shows that epistemic systems rest on unjustified certainties embedded in our form of life, while Gödel proves that mathematical systems require axioms that cannot be demonstrated within the system itself. Rather than representing failures of reasoning, these ungrounded foundations serve as necessary conditions that make systematic inquiry possible. This parallel suggests that foundational certainties enable rather than undermine knowledge, pointing to a universal structural feature of how such systems must be grounded. This analysis has implications for reconsidering the nature of certainty across epistemology and the philosophy of mathematics.

Introduction

We often perform actions without hesitation, such as sitting on a chair or picking up a pencil, without questioning the existence of either. This unthinking action illustrates Wittgenstein's concept of a hinge proposition, a fundamental certainty that supports our use of language and epistemological language-games. Wittgenstein compares hinge propositions to the hinges that enable a door to function; these certainties provide the underlying support for the structures of language and knowledge, remaining unaffected by the need for justification.

Wittgenstein's hinges bear a remarkable resemblance to Gödel's incompleteness theorems, revealing unprovable mathematical statements. This resemblance points to deeper questions about how both domains handle foundational issues. Both Wittgenstein and Gödel uncover limits to internal justification, a connection I will examine.

Traditional epistemology often misinterprets hinges by forcing them into a true/false propositional role, neglecting their foundational status embedded in our epistemic form of life. These bedrock assumptions precede argument or evidence, forming the foundational elements of our epistemic practices. Similarly, Gödel's incompleteness theorems showed that any consistent arithmetic system contains true statements unprovable within the system and cannot demonstrate its own consistency.

This connection is significant because it highlights the boundary between what counts as bedrock for epistemic and mathematical systems. Both rest on certainties that lie beyond justification, certainties that are not flaws in reasoning but necessary foundations that make knowledge claims possible. This paper argues that ungrounded certainties enable knowledge, rather than undermining it, and that hinges and Gödel's unprovable statements serve a similar purpose. By examining the parallels between Wittgenstein and Gödel, particularly the role of unprovable foundations and the need for external grounding, this paper sheds light on the nature of certainty in our understanding of both epistemology and mathematics.

Section 1: Hinges and Their Foundational Role

Wittgenstein's concept of hinge propositions is crucial to his thinking, particularly in the context of epistemology. In On Certainty, Wittgenstein introduces the idea of hinges as certainties that ground our epistemic practices. While Wittgenstein never explicitly distinguishes types of hinges, his examples suggest a distinction between nonlinguistic and linguistic varieties, revealing different levels of fundamental certainties.

Nonlinguistic hinges represent the most basic level of certainty, bedrock assumptions that ground our actions and interactions with the world. These are not expressed as propositions subject to justification or doubt but embodied in unreflective action. For instance, the certainty that the ground will support us when we walk is a nonlinguistic hinge that enables movement without hesitation. Similarly, our unthinking confidence that objects will behave predictably, that chairs will hold our weight, that pencils will mark paper, represents this bedrock level of certainty. These hinges operate beneath the level of articulation, forming the silent background against which all conscious thought and language become possible.

Building upon this bedrock foundation, linguistic hinges operate at a more articulated but less fundamental level. These are certainties embedded within our language-games and cultural practices, often taking the form of basic statements like "I have two hands" or "The Earth exists." Unlike nonlinguistic hinges, these can be spoken and seem propositional, yet they resist the usual patterns of justification and doubt. Other examples include statements such as "I am a human being" or "The world has existed for a long time," assertions that appear to convey information but function more as structural supports for discourse than as ordinary claims requiring evidence.

These two types of hinges show how certainty operates at different levels in grounding knowledge. Nonlinguistic hinges form the deepest stratum, revealing the unquestioned backdrop that makes any form of questioning possible. Linguistic hinges, while still foundational, represent a layer above bedrock that anchors shared discourse within specific contexts. Both types resist justification, but their resistance stems from different sources: nonlinguistic hinges from their pre-rational embodiment in action, linguistic hinges from their structural role within our language-games.

Wittgenstein breaks with traditional epistemology here. Rather than viewing these certainties as beliefs requiring justification, he recognizes them as the ungrounded ground that makes justification itself possible. He notes, "There is no why. I simply do not. This is how I act" (OC 148). Doubting these hinges would collapse the very framework within which doubt makes sense, like attempting to saw off the branch on which one sits.

A crucial distinction emerges between subjective and objective dimensions of these certainties. While our relationship to hinges involves unquestioning acceptance, this certainty is not merely psychological. These assumptions are shaped by our interactions with a world that both constrains and enables our practices. The certainty reflected in our actions has an objective component, as it emerges from our shared engagement with reality and proves itself through the successful functioning of our practices.

This interpretation of hinges as operating at different foundational levels finds support in recent Wittgenstein scholarship, though it diverges from some prominent readings. Danièle Moyal-Sharrock argues that hinges are fundamentally non-propositional, existing as lived certainties rather than beliefs or knowledge claims (Moyal-Sharrock 2004). While my distinction between nonlinguistic and linguistic hinges aligns with her emphasis on the embodied, pre-propositional character of our most basic certainties, I suggest that some hinges do function at a more articulated level within language-games, even if they resist standard justification patterns.

Duncan Pritchard's interpretation emphasizes hinges as commitment-constituting rather than knowledge-constituting, arguing they represent a distinct epistemic category that enables rather than constitutes knowledge (Pritchard 2016). This view supports the parallel with Gödel's axioms: both hinges and mathematical axioms function as enabling commitments that make systematic inquiry possible without themselves being objects of that inquiry. The mathematical case strengthens Pritchard's insight by showing how even formal domains require such commitment-constituting foundations.

This analysis extends beyond epistemology to reveal a striking parallel with Gödel's incompleteness theorems, which demonstrate analogous limits within formal mathematical systems. Just as Gödel showed that mathematical systems rely on axioms that cannot be proven within those systems, Wittgenstein's hinges reveal that epistemic systems rest on certainties that cannot be justified internally. This comparison suggests a fundamental structural limitation in rational grounding, whether in mathematics or human knowledge, and invites reconsideration of what it means for knowledge to be properly grounded.

Section 2: Gödel’s Unprovable Statements as Mathematical Hinges

Gödel's incompleteness theorems, published in 1931, establish fundamental limits within formal systems, revolutionizing our understanding of mathematical foundations. Gödel demonstrated that within any consistent system of arithmetic, there will always be statements that are true under the standard interpretation but cannot be proven within the system itself. For instance, the statement asserting the system's own consistency, a meaningful mathematical claim about the system's properties, cannot be demonstrated within that system, even if the system is indeed consistent. Moreover, no such system can demonstrate its own consistency. Such statements are meaningful propositions with definite truth values that reveal structural limitations inherent to formal systems. This limitation persists even when systems are extended. Adding new axioms to prove previously unprovable truths creates strengthened systems that, if consistent and sufficiently powerful, generate their own sets of true but unprovable statements. The cycle of incompleteness is thus perpetual, revealing not a flaw in particular systems but a structural feature of formal mathematics itself.

This limitation mirrors Wittgenstein's hinges in important ways. Just as hinges are certainties that cannot be justified within the epistemic systems they support, Gödel's results show that mathematical systems require axiomatic starting points that cannot be proven within those systems. The Peano axioms, which establish the foundation for arithmetic, exemplify this necessity. These axioms are not accepted because they are provable; they cannot be proven within the systems they generate. Rather, they are adopted as systematic starting points that enable mathematical development, chosen because they make possible coherent, productive systems.

The parallel extends to the necessity of external acceptance. Gödel's systems require axioms accepted from outside the formal system itself, while Wittgenstein's hinges are certainties not arrived at through investigation but accepted as part of our form of life (OC 138). In both cases, what enables the system lies beyond the system's internal capacity for justification. Mathematical axioms and epistemic hinges both function as ungrounded grounds, foundational elements that make systematic inquiry possible precisely because they are not themselves subject to the forms of scrutiny they enable.

Yet there is an important difference here: mathematical axioms are typically chosen for their elegance, consistency, and power to generate interesting mathematics, while hinges appear more embedded in contingent cultural and biological practices. Yet this difference strengthens rather than weakens the parallel. If even mathematics, often considered the paradigm of rigorous proof, requires unjustified foundational elements, how much more must everyday understanding rely on unexamined certainties? The universality of this structural requirement across domains as different as formal mathematics and lived experience suggests a fundamental feature of how systems of thought must be organized.

Both domains thus reveal that functioning without such foundational elements is implausible. Mathematical systems risk incoherence without axiomatic starting points, just as epistemic practices risk collapse without the bedrock certainties that Wittgenstein identifies. The parallel illuminates a shared structural necessity: systematic thought requires ungrounded foundations that enable rather than undermine the possibility of reasoning within those systems.

Section 3: Beyond Internal Justification: A Cross-Domain Analysis

Both Wittgenstein and Gödel reveal that justification operates within boundaries, where certain elements serve as foundations that cannot be further justified within their respective systems. Both thinkers expose a basic structural feature of systematic thought: the impossibility of a complete system of justification in either domain.

Traditional approaches to knowledge often assume that proper justification requires tracing claims back to secure foundations that are themselves justified. This assumption generates the classical problem of infinite regress: any attempt to justify foundational elements through further reasoning creates an endless chain of justification that never reaches secure ground. Both Wittgenstein's hinges and Gödel's axioms reveal why this demand for complete internal justification is not merely difficult but impossible in principle.

As Wittgenstein observes, "There is no why. I simply do not. This is how I act" (OC 148). This insight captures something crucial about the nature of foundational certainties: they are pre-rational in the sense that they precede and enable rational discourse rather than emerging from it. Hinges are not conclusions we reach through reasoning but lived realities that make reasoning possible. Similarly, mathematical axioms are not theorems we prove but starting points we adopt to make proof possible.

There is an important difference between these domains. Hinges emerge from contingent practices embedded in particular forms of life, while mathematical axioms are selected through systematic considerations within formal contexts. Hinges reflect the biological and cultural circumstances of human existence, whereas axioms reflect choices made for their mathematical power and elegance. If anything, this difference makes the parallel more compelling by demonstrating its scope: if even the most rigorous formal disciplines require unjustified starting points, the necessity of such foundations in everyday knowledge becomes even more apparent.

This cross-domain similarity reveals what appears to be a universal structural requirement. Systems of thought, whether formal mathematical theories or practical epistemic frameworks, cannot achieve complete self-justification. They require external elements that are not justified within the system but make systematic inquiry within that framework possible. Rather than representing failures or limitations, these unjustified foundations function as enabling conditions that make coherent thought and practice possible.

Recognizing this structural necessity transforms how we understand the relationship between certainty and knowledge. Instead of viewing unjustified elements as epistemological problems to be solved, we can understand them as necessary features that allow knowledge systems to function. Both mathematical proof and everyday understanding depend on foundations that lie beyond their internal capacity for justification, yet this dependence enables rather than undermines their respective forms of systematic inquiry.

Conclusion

I have argued for a fundamental parallel between Wittgenstein's hinges and Gödel's incompleteness results: both demonstrate that systematic thought requires ungrounded foundations. By examining how epistemic and mathematical systems share this structural feature, we gain insight into the nature of foundational certainties across domains of human understanding.

The parallel between these seemingly distinct philosophical insights suggests that the limits of internal justification are not accidental features of particular systems but necessary conditions for systematic thought. Recognizing this gives us a more realistic picture of how knowledge actually functions, not through endless chains of justification reaching some ultimate ground, but through practices and formal systems that rest on foundations lying beyond their internal scope.

Rather than viewing these limits as philosophical problems requiring solutions, this analysis suggests embracing them as structural necessities that make knowledge possible. Wittgenstein's hinges ground our epistemic practices in the lived realities of human existence, while Gödel's axioms ground mathematical systems in choices that prove their worth through the coherent theories they generate. Both reveal that the search for completely self-grounding systems is not merely difficult but misconceived.

I believe this perspective has broader implications for understanding certainty and knowledge. It suggests that the interplay between grounded and ungrounded elements is not a flaw in human reasoning but a fundamental feature of how systematic understanding must be structured. By recognizing this necessity, we can develop more nuanced approaches to foundational questions in epistemology, philosophy of mathematics, and potentially other domains where the relationship between systematic inquiry and its enabling conditions remains philosophically significant.




References

Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38, 173-198.

Moyal-Sharrock, D. (2004). Understanding Wittgenstein's On Certainty. Palgrave Macmillan.

Pritchard, D. (2016). Epistemic Angst: Radical Skepticism and the Groundlessness of Our Believing. Princeton University Press.

Wittgenstein, L. (1969). On Certainty (G. E. M. Anscombe & G. H. von Wright, Eds.; D. Paul & G. E. M. Anscombe, Trans.). Basil Blackwell.