[TPF Essay] Wittgenstein's Hinges and Gödel's Unprovable Statements
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: Gödels 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.
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: Gödels 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.
Comments (73)
Thank you for this well-presented OP. While I agree that Godels incompleteness theorems can lend themselves to the assumption of groundless grounds akin to Wittgenstein hinges, I dont believe Godel would have been comfortable with such a relativistic, pragmatist conclusion. He considered himself a mathematical platonist. As Roger Penrose says about Godel:
To clarify, this is not an OP. It is part of the Philosophy Writing Challenge - June 2025.
https://thephilosophyforum.com/categories/55/phil-writing-challenge-june-2025
It is an essay whose author will be identified later. To keep anonymity, Moliere has posted all 13 entries in his name.
Everything is about objectivity and subjectivity, actually. It's not merely a psychological issue, but simply logical. We can easily understand subjectivity as someone's (or some things) point of view and objectivity as "a view without a viewpoint". To put this into a logical and mathematical context makes it a bit different. Here both Gödel and Wittgenstein are extremely useful.
In logic and math a true statement that is objective can be computed and ought to be provable. Yet when it's subjective, this isn't so: something subjective refers to itself.
Do note the self-referential aspect Gödel's incompleteness theorems, even if Gödel smartly avoids direct circular reference of Russell's Paradox. Yet I would argue that Wittgenstein observes this even in the Tractatus Logicus Philosophicus as he thinks about Russell's paradox:
Here I think it's very important to understand just what is objective and what is subjective in this context. An objective model can is true when it models reality correctly and can be written as a function like y = F(x). But what then would be a subjective model, that couldn't be put into the above objective mold?
Let's take one example. Let's assume that the market pricing mechanism is dependent on the aggregate actions of all market participants. This obviously is true: trade at some price happens only when there is at least one participant willing to sell at the price and at least one willing to buy with the similar price. At first this looks quite objective and we can write as a mathematical function like y = F(X). But then, if we want to use this model, let's say to forecast what prices are going to be in the future and then participate in the market, this isn't anymore an objective function. Now actually the function is defining itself, which as Wittgenstein observed, cannot contain itself. Us using the function is self-referential, because the model is the aggregate of all market participants actions, including us. How are we deciding our actions? Because of the function itself.
Quoting Moliere
If I understand correctly what you mean by grounded / ungrounded foundations, I would say it differently: Not all systematic thought can be brought back to grounded foundations. Usually we can use axiomatic systems and get an objective model, but not allways.
Just as there is also Gödel's completeness theorem, that theorem doesn't collide with the two incompleteness theorems.
Splendid composition.
Isn't this rather a long-winded way of saying that there are indeed necessary truths? That necessary truths can't be, and don't need to be, justified in other terms - that's what makes them necessary. As Thomas Nagel remarks on an essay on the sovereignty of reason, 'the epistemic buck must stop somewhere'; there are thoughts we can't 'get outside of', or judge according to some other criterion, without thereby undermining their necessity ('contingent cultural and biological practices').
I think what's interesting about this whole line of thought is why it's interesting. Why is it we presume that foundational ('hinge') propositions can be or need to be justified by further analysis, and what are the implications of their not being so justified?
This would make hinges quite a bit different from many axioms. But must hinges involve unquestioning acceptance? Isn't that the whole history of skepticism, questioning such foundations? And for most of the ancient skeptics at least, this questioning wasn't an epistemic exercise, so much as a practical one. One questioned one's bedrock beliefs so as to attain equipollence, a sort of detached equilibrium between beliefs such that one was not concerned about anything and could attain apatheia.
Anyhow, I figured this might have relevance for reason as such (as opposed to any [I]one[/I] system), since, as Hegel says, to have ever recognized a limit is to have already stepped over it. The fish doesn't know where the water ends; it's only the frog, who has actually broken the surface, who sees it as a limit. The sorts of hinges we accept unquestioningly would seem to have to be ones we could never even be aware of. It would have to be something more akin to the blindspot in the visual field (although even that example fails, since one can become aware of that with careful experiment).
Indeed. And it's perhaps somewhat of a historical question because plenty of thinkers prior to the heyday of foundationalist aspirations take it as somewhat obvious (to them) that some truths (and really the more important, "foundational" ones) cannot even be expressed in human language, let alone subjected to something like a mathematical proof. For example, Saint John of Damascus says this in matter of fact terms at the outset of the Exact Exposition, Plato inveighs against the inadequacy of words and justificatory dissertations in his seventh letter, and then there is Saint Paul's famous mention of being "caught up to the third heaven" and hearing "inexpressible words, which a human being is not allowed to speak."
I would guess the Cartesian dream of a world reducible to mathematics is a major impetus here. Timothy Shutt has an interesting lecture where he suggests that the decline in epics is in part due to the fact that they lost their place as authoritative sources to mathematics (and this is a problem Milton is grappling with as he tries to write a new Protestant epic in an environment where epic and scripture is losing this authority).
The problem I have with the essay is that it fails to distinguish between a notion of necessary truth as a relative, contingently stable structure of meaning (Wittgensteins hinges, forms of life and language games) and a notion of necessary truth as a platonic transcendental, which is how Godel views the necessary ground of mathematical axioms.
I think I had a similar thought. But the essay works in spite of that distinction. Functionally, there is a parallel. On both sides, you have the objects of discourse, and you have the unquestioned background. Both linguistic and mathematical discourses need both. And, what counts as object and background is relative in both cases: relative to the language game, and relative to the mathematical domain.
I see your point. So could you say that Wittgenstein's hinges can in some sense be situated, or understood in terms of lived existence and 'language games' whereas Godel's platonic transcendentals simply are, without any reference to context or situatedness?
That if I have six beers in the fridge, and you come and drink one, there will be five remaining, everything else being equal. But to provide more context for Thomas Nagel's expression, in particular, the paragraph from which it is taken was:
[quote=Thomas Nagel, The Last Word (New York: Oxford University Press, 1997), p 137 ]The only form that genuine reasoning can take consists in seeing the validity of the arguments, in virtue of what they say. As soon as one tries to step outside of such thoughts, one loses contact with their true content. And one cannot be outside and inside them at the same time: If one thinks in logic, one cannot simultaneously regard those thoughts as mere psychological dispositions, however caused or however biologically grounded. If one decides that some of one's psychological dispositions are, as a contingent matter of fact, reliable methods of reaching the truth (as one may with perception, for example), then in doing so one must rely on other thoughts that one actually thinks, without regarding them as mere dispositions. One cannot embed all one's reasoning in a psychological theory, including the reasonings that have led to that psychological theory. The epistemological buck must stop somewhere. By this I mean not that there must be some premises that are forever unrevisable but, rather, that in any process of reasoning or argument there must be some thoughts that one simply thinks from the inside--rather than thinking of them as biologically programmed dispositions.[/quote]
So he's saying that the attempt to rationalise what I think are being described as 'hinge propositions' is to view them from the outside - to evaluate them in some other terms, such as biologically-programmed dispositions. (We see a lot of that here.)
If only all philosophy writing was as clearly written as this essay.
How do axioms and hinges relate to knowledge?
Let the axiom be "the sun rises in the east". This axiom can never be proved true. If one day the sun rose in the west, then the axiom is false. The axiom "the sun rises in the east" is not knowledge, as it can never be proved true.
Let the hinge be "the sun rises in the east". This hinge is a certainty. If one day what we think is the sun rises in the west, then what we see cannot be the sun. The hinge "the sun rises in the east" is knowledge regardless of what we observe.
Whereas hinges enable knowledge, axioms don't serve a similar purpose, as they can never be knowledge
I did very much like the paper, but this statement of the thesis (which occurs a few times) actually strikes me as somewhat ambiguous.
The point could be either:
A. That actually, fully self-justifying, air tight foundational systems would be somehow deficient (e.g. if logicism re mathematics could be decisively demonstrated it would somehow actually undermine knowledge); or
B. Because a fully self-justifying system is impossible, ungrounded certainties are essential for knowledge.
I assume from the paper that B is meant though, since it does not give any indication of why A should be the case.
The problem I see, which @Joshs gets at, is that B seems to risk equivocating re many common and classical definitions of "knowledge." A critic could say that knowledge is about the possession of truth simpliciter. It is not about possession or assent to "what is true [I]given[/I] some foundational/hinge belief" (which itself may be true or untrue). This redefinition seems to open the door on "knowing" things that are false.
Hence, I think someone holding to a classical notion of knowledge as the possession of truth, and truth as "the adequacy of the intellect to being," might be inclined to say that the solution here is actually radically skeptical. All that is "known" is based on that which is not known. "Knowledge" ceases to be knowledge. Further, all demonstration from first principles would flow, ultimately, from premises that could be said to be less well known than their conclusions (making them bunk demonstrations from the Aristotleian point of view).
This would arguably be one of Kripke's "skeptical solutions" (as opposed to a straight solution), redefining "knowledge" in a fairly radical way (although perhaps not as radical as some moves, e.g. Quine).
An unrelated comment on that thesis statement: might the axiom not be more analogous to the hinge propositions than the unprovable statement? No doubt, the unprovable statement (as the existence of uncomputable or inexpressible statements) seems relevant, but I'm not sure if it fills the same role.
But this shifts focus on to why axioms are chosen. Certainly, it is sometimes "because they produce interesting results," particularly as mathematicians tinker with existing, established systems. Yet in general, they are selected because they are considered true, and indeed they are ideally indubitable. However, this is not "true given some prior axioms," but, hopefully, "true absolutely." For instance, Euclid's postulates held up so well because it seemed fair to dismiss someone who denied them as insane or acting in bad faith. "Take out paper, a pencil, ruler, and protractor and see for yourself." An ideal axiom hits that level, although obviously they do not always.
Does anyone suspect, as I do, that the linkage connecting Wittgenstein's hinges to Gödel's incompleteness theorems suggests some type of symmetry (and conservation of the possible scope of narrative elaboration (whether verbal or numerical) i.e., conservation of containable fundamentals within a system ) extending from verbal language to both numerical language and chains of reasoning?
That's close to the Protagorean view that truth is always relative to the perceiver or to the community's standards of justification ('man is the measure of all things"). There are only truths for us. But this view has profound implications, not least of which is that it undermines the possibility of truth as something we discover rather than merely decide. But If all truth is decided rather than discovered, then the proposition truth is what we decide must also be just a decision, not a truth, and one that we're under no rational obligation to accept. And if its presented as a universal fact, that stands in contradiction to relativism, as we're obliged to accept it.
Furthermore the whimsical example of the six-pack of beer made no reference to the absolute, but only to necessary facts - that given six of something, the subtraction of one will invariably leave five. I can't see how that can be a matter of controversy.
Quoting tim wood
And this cuts against your earlier claim - If some truths are so deeply embedded that theyre not decided but rather constitutive of meaning, then theyre not true because we say so. Theyre true as conditions of intelligibility - already a step away from relativism (and near in meaning to the 'hinge propositions' we're discussing.)
Modern discourse often shies away from talk of absolute truth its seen as naïve, dogmatic, or even authoritarian. But that taboo has become a dogma in its own right! It is true that articulating any notion of the absolute is difficult perhaps even impossible in a fully transparent or complete way but it is part of what philosophy is about.
By contrast, the idea that truth is what we decide it is, sounds superficially tolerant but collapses into incoherence if pushed. If we genuinely believe that all truths are relative to individual or social standards, we lose traction in anything beyond personal preference. Disagreement becomes either a clash of taste or a power struggle, not a pursuit of understanding.
So I think we have to ask ourselves not just in epistemology but across our culture whats lost when we treat truth as if it were merely a social construct. At the very least, philosophy ought to keep open the question of whether some truths are not of our own making, even if they are hard to articulate.
That is very much the thrust of Thomas Nagels The Last Word, where he defends the idea that reason has a kind of intrinsic authority that transcends subjective or cultural standpoints. He takes aim at the creeping relativism in contemporary thought that treats logic, objectivity, and justification as mere social conventions or evolutionary adaptations or instruments of power. Nagel argues that this position ends up undermining itself, because the relativist must rely on the very norms of truth and logic that theyre trying to dismiss. His point isnt that everything is absolutely true in some metaphysical sense, but that there are certain truths logical, mathematical, even ethical which are binding not because we agree on them, but because they compel assent through reason itself. Facts that reason compels us to accept.
This is a very relevant topic on TPF, so I think the submission is appropriate.
If ones notion of epistemic justification is dependent on other epistemically justified beliefs, then the infinite regress looms. Or as I've said , it "is a bit like a novice bricklayers idea that every brick needs to rest on two other bricks. But this leads to an infinite regress, for there must be a foundation which itself supports the lowest bricks."
The basic way to avoid this infinite regress is by positing more than one kind of justification. For example, the justification that attaches to foundational beliefs versus the justification that attaches to non-foundational beliefs, where the justification that attaches to foundational beliefs is not dependent on other epistemically justified beliefs.
Wittgensteins solution is apparently to make a distinction between different kinds of beliefs (or propositions) but to dismiss the idea that foundational beliefs require justification. So there are different kinds of beliefs but not different kinds of justification.
In other words, given the following argument, Wittgenstein would apparently accept 1, 2, and 3, but reject 4 and 5.
The rejection of 4 is a significant problem for Wittgenstein, but there is another problem. The justification/warrant of an argument's conclusion flows from the justification/warrant of the argument's premises, in much the same way that electricity travels from one end of a conductive surface to another. Yet Wittgenstein believes that he can begin with premises which possess no justification/warrant, and from them infer conclusions that possess justification/warrant. This is not coherent, and the same issue rears its head in reverse when we consider the fact that a modus tollens critique moves from conclusion to premises (or more precisely, from consequent to antecedent). It is irrational to try to divorce premises from conclusion qua justification.
Quoting Moliere
How should we respond to Wittgenstein here? Apparently by pointing out to him that there is a why, and that other people act differently than he does. As soon as two people who act in foundationally different ways come into contact with one another the "why" will become a question of interest.
But what about Gödel?
Quoting Moliere
This is simply a misunderstanding of Gödel. Mathematicians since Euclid knew that axioms could not be proved. This is nothing new. Gödel's contribution has to do with the completeness of formal systems, not the self-justification of formal systems. In fact most thinkers already believed that formal systems lacked completeness, but Gödel proved it and in the process destroyed the hopes of Wittgenstein's friends in the Vienna Circle.
I don't suspect that Gödel shared Wittgenstein's confusion in this matter, but perhaps someone who is familiar with Gödel's wider work could comment. I don't suspect that Gödel confused formal reasoning with natural reasoning. Formal logic has a very strong dichotomy between axioms and consequences, to the extent that there is a schizophrenic gulf between the two with regard to justification. Natural reasoning does not work that way.
(Note that Wittgenstein tends to shift haphazardly back and forth between psychological description and logical normativity, and this complicates but does not invalidate the picture I have drawn. Gödel does not do this. He is not arguing for the idea of unjustifiable premises.)
Redefining it in semantic terms is a deflationary or minimalist move. This aligns with the disquotational theory of truth (e.g. Snow is white is true if and only if snow is white), which claims that truth is not a substantive property, but merely a linguistic device for generalizing over propositions.
However, this doesnt rebut the charge of relativism it obscures it. If you say that no proposition is true in itself but only because we say it is T, then were right back to Protagoras:
"What is true is what we decide is true."
The sleight of hand here is that he avoids making a metaphysical claim about truth by shifting into a formal, semantic register but this move itself carries a metaphysical implication, namely that truth has no independent reality beyond the operation of language and consensus.
You ask for a definition of "truth" as though it's a settled term but even among philosophers, it remains contested. Some adopt deflationary or minimalist theories (like yours), others argue for correspondence, coherence, or pragmatic theories, and still others defend truth as a transcendental condition for meaning or knowledge. The Stanford Encyclopedia of Philosophy lists over a dozen major theories. So we can't assume it's a straightforward term reducible to a single semantic function.
If we reduce 'truth' to nothing but the property T of a proposition P and then define T solely in terms of human stipulation then we haven't solved anything; we've just defined truth out of existence, and replaced it with consensus or coherence within some human framework. But that doesnt answer the philosophical question. It dodges it.
What does "it is" mean? Does it mean "it is T"? Or "it is true"? Either way your definition is circular:
This is an indication that defining truth is more difficult than one might first expect. Truth is something which is characteristically resistant to univocal sequestering within the object language or meta language (which is why philosophers like Buridan explicitly rejected the notion that the two "languages" are separable).
I also liked the paper, and liked that it was clearly written. As Prof. Adrian Piper wrote in his article "Ten Commandments of Philosophical Writing" Thou shalt not obscure thy ideas with turgid prose.
Being clearly written then allows me to understand what the author is trying to say, even if I disagree with the author's premise that "ungrounded certainties enable knowledge", and even if I find parts of the author's essay ambiguous.
The author is standing their ground in being clear in what they are saying. This enables the reader to properly engage with their argument, even if the reader then disagrees with the author's argument. It is then up to the reader to explain why they disagree with the author's argument, thereby moving the philosophical debate forwards. Philosophy should be a dialogue, as Adrian Piper says in his article "Ten Commandments of Philosophical Writing".
A clearly written philosophical essay is the hinge upon which new philosophical knowledge may be gained.
===============================================================================
Quoting Count Timothy von Icarus
As you say, to say that hinges give knowledge is not generally how we understand the word knowledge, as being something that is universally true.
For the animalist, all things, including animals, plants and rocks, possess a distinct spiritual essence. For the animalist, their hinge proposition may be "this plant possesses a spiritual essence". This gives them the knowledge that this plant possesses a spiritual essence.
For the atheist, no thing possesses a distinct spiritual essence. For the atheist, their hinge proposition may be "this plant doesn't possesses a spiritual essence". This gives them the knowledge that this plant doesn't possesses a spiritual essence.
The animalist sees a plant and knowing that all plants possess a spiritual essence knows that the plant they are looking at possesses a spiritual essence. The atheist sees the same plant and knowing that no things possess a spiritual essence knows that the plant that they are looking at doesn't possess a spiritual essence.
As you say, this is not how we understand knowledge, being something universally true.
The hinge proposition imposes itself on the world. We then observe this world. This enables us to confirm that the hinge proposition is true. The hinge proposition confirms its own truth self-referentially.
Another example. Let my hinge proposition be "the sun always rises in the east". In the event that I observe the sun rising in the east, this confirms my hinge proposition. In the event that I observe what I think is the sun rising in the west, then it cannot be the sun, thereby again confirming my hinge proposition.
A hinge proposition such as "here is one hand" gives knowledge that here is one hand. But this is self-referential knowledge, which is not how we generally understand knowledge as being universally true, as you say.
I would say that it has something to do with the idea that truth is the water we swim in, and it is hard to identify that sort of thing. Probably only in the presence of two apparent and conflicting truths does the notion of 'truth' emerge more clearly.
At a much more general level, I would be very wary to discount a word/concept that is so ubiquitous throughout human civilization. Those sorts of words/concepts tend to have a meaning, even if the meaning is difficult to pin down.
Quoting tim wood
Suppose you said, "There are true [somethings], but there are no truths." I would just respond, "What are these [somethings] that are true?" The ontological problem attaches to propositions as much as it attaches to truths, or to whatever [something] the "adjective" modifies.
Doesn't the second claim contradict the first? How is the statement 'truth (in- and by-itself) does not exist' not a metaphysical claim?
Quoting tim wood
I see the logic of your position you're treating truth as nothing over and above the attribution of a property to a proposition, and I understand the deflationary intuition behind that. But the issue at stake, especially given the original context (Wittgenstein and Gödel), isnt just about semantics. I'm saying, that the issue is really about whether there is a domain of what we might call the unconditionally true truths not simply constructed or declared within discourse, but which ground discourse itself. Thats a metaphysical and ontological question, not simply a linguistic one.
Gödel, for example, was a mathematical Platonist. He believed that mathematical truths exist independently of our capacity to prove them that they are so, whether we grasp them or not. Wittgensteins hinges arent proven either, but theyre not arbitrary. They are taken to be true not because we say so, but because they constitute the background against which the very act of saying something becomes intelligible.
To deny that there is such a thing as truth beyond being just a general idea risks collapsing this structural distinction. If truth is only ever the local property of propositions as we use them, then you effectively deny the possibility of truths that are not contingent on our grasp or declaration. But isnt that exactly what Gödels theorems reveal? That some truths outstrip the systems we build?
This is just the kind of question Platos dialogues return to again and again what it means for something to be true or to be good in itself. The dialogues often end in aporia, yes but not as dismissals. Rather, they preserve the seriousness of the inquiry by refusing to reduce these questions to mere convention or definition or to provide a dogmatic solution. And we cant define truth in some final way, that may be a sign of its depth, not its non-existence. Likewise, Socrates' consistent refusal to declare that he knows any kind of final truth - he's not denying that there is, but inviting deep contemplation of the question. (Is this why Socrates was said to have sometimes fallen into a kind of trance, standing rooted to the spot for hours or days? That stillness might itself be a kind of answer: a living witness to the fact that some truths are not merely stated, but must be grappled with through a deep questioning.)
To consider whether anything is unconditionally true not merely 'true for us' we have to ask questions beyond usage and attribution. Were talking about the architecture of thought and language, of being itself. Those arent things you can show in an empirical way but neither are they merely artifacts of language. They belong to the domain of what Kant might call the (transcendental) conditions for the possibility of experience and understanding. And thats a philosophical question, not a semantic one.
So theres a deeper question here about the nature of truth and its one that cant be settled by appeal to semantics or usage alone. Certainly, truth doesn't exist as some abstract 'thing' out there in the world, waiting to be pointed to or depicted. But that doesnt mean its not real.
Classical philosophy speaks of 'intelligible objects' principles or forms that do not exist qua phenomena, but which nonetheless structure intelligibility (ref). Think, for instance, of the law of the excluded middle. Does it 'exist'? Not in the empirical sense. But is it real? It seems inescapably so not because we invented it, but because rational discourse depends on it.
So in that sense, the truths of reason logical principles, mathematical axioms, moral intelligibilities dont so much describe what exists as disclose the structure of intelligibility itself. They are not things among things, but conditions for thought, and for discourse.
This is why the denial of truth as a real though not empirical dimension is so radical. Its not just a semantic revision. It amounts to a dismantling of the very architecture of meaning. And thats why thinkers from Plato to Augustine (and indeed, Gödel and Wittgenstein in their own ways) were so attentive to this domain of the intelligible not as 'objects' in the modern sense, but as realities grasped by the intellect.
If we continue in your Analytic route we would simply say that a truth is a true proposition. Truth itself, apart from individual truths, could just be a general idea, sure. None of this seems problematic. We regularly appeal to general ideas, such as justice, mathematics, politics, sports, etc.
It's not that hard to give a definition. Truth is the adequacy of thought to being. Being a transcendental, "true" is "said many ways," as it is predicated analogously. For instance, we can think of an utterance in terms of it being a sign of truth in the intellect of the speaker (true versus false knowledge claims) or in terms of the utterance accurately reflecting the beliefs of a speaker ("telling the truth" versus lying).
In terms of logical truth:
https://www.newadvent.org/cathen/15073a.htm
See also: https://www.newadvent.org/summa/1016.htm
https://isidore.co/aquinas/QDdeVer1.htm
That is your description, written so as to support the point you're making. But it was not how Gödel understood it himself.
[quote=Rebecca Goldstein]Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptivenot of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception.[/quote]
Gödels view of mathematical intuition as a kind of perception echoes Platos claim that the soul has an eye suited to grasping the intelligible - the eye of reason. In the Republic, mathematics belongs to the level of dianoiaa faculty higher than belief, yet still dependent on symbols and hypotheses. Plato insists that geometrical and arithmetical truths do not belong to the world of becoming, but to a higher, stable, unchanging realm. Hence Gödels Platonism is not modern nostalgia, but a precise continuation of that classical outlook: mathematics is not invented, but discoveredseen by a faculty suited to such realities. And not a matter of belief, doxa or pistis, but insight into what is.
The truths of undecidable sentences are to be decided - not through formal deduction by using the system but externally of the system, by either the user of the formal system who makes a decision as to their truth value (in which case they become promoted to the status of non-logical axioms) or through external but presently unknown matters of fact or by future censensual agreement if the formal system is used as an open language. Undecidable sentences are a subclass of the more general undecided sentences, namely those sentences which are not yet decided, but might be settled either internally by appying the system, or externally by extending the system.
No, because it seems obvious that there is a difference between what is true and what merely appears to be true. Indeed, one cannot have a coherent appearance/reality distinction if there is "nothing but appearances." In that case, appearances just are reality, and we have something like Protagorean relativism.
I did?:chin:
Pretty sure that's the definition I gave. What exactly is the counterpoint, that thought cannot be adequate to being? Epistemic nihilism?
- I wasn't following his posts very carefully, but my hunch is that it might be unrelated to the discussion itself. Maybe he just felt that he was spending too much time on TPF and made a strong decision to leave.
I really do get that, and often consider it, although even if I decided to stop posting, I wouldn't have all my posts deleted.
I explained what happened in the Shoutbox.
Quoting Jamal
The whole story:
Quoting Jamal
If he comes to regret his departure he could always send an email to info@thephilosophyforum.com, and I could send him an invitation.
Quoting Deleted User
@Jamal - so sorry to hear of this dramatic turn of events.
All of this and the way it has been handled is most unfortunate.
I would like to add my thoughts but this is not the place.
Could the posts concerning Deleted User ( Tim Wood) be moved to a more appropriate spot. Not the Shoutbox where only a few enter!
Is the Feedback category accessible without signing in? The loss of a long-term member is not easy to take on board or process...the loss of the posts and shared thoughts, poetry and music. Ouch!
Also, I think Tim might find it helpful to read...even if he might be needing a really, long break.
Yes, feel free to start a thread there asking for clarification, and I'll respond with information I've so far scattered across various threads.
Done. Perhaps all posts, irrelevant to the essay, can now be moved there, or deleted. Thanks.
https://thephilosophyforum.com/discussion/16007/deleted-user
Quoting Moliere
Does this systemic inquiry serve a practical purpose? Or is it more like Sudoku?
Quoting Moliere
That's from experience. Chairs were made by people for people to sit on and pencils were made by people for people to write with: we've known these things from early childhood. A tentacled alien would not guess how to use them. As for questioning the existence of such mundane objects, Virtual Reality and holography have brought doubt back into play.
Humans made language long before they made philosophy and acquire language before the age of questioning. We have been living on this planet and using the things it provides since long before reasoning or justification. Couldn't do any reasoning or justifying without gravity, oxygen, and all that other necessary stuff we don't think about until they're absent.
Quoting Moliere
And yet, we can be wrong about those things. When the pencil point breaks, we fail to write; when a chair breaks, we fall down with a painful thump; when the ground is quicksand, we sink and suffocate to death. Cavemen, whose language consisted of gestures and vocalizations knew enough to test the reliability of physical objects by simple physical means.
Quoting Moliere
And that makes them dangerous, because of the exceptions, gaps, biases, delusion, misinterpretation and incorrect information. Much worse, our "language games" include fiction, deception, mis- and dis-information, which are all too easily internalized as foundational. Thus cultures become interwoven with false certainties and civilizations collapse.
I'm skipping the section on mathematics, by reason of skeptical ignorance.
Quoting Moliere
Bedrock is not tested with logic; it's tested with a drill. It doesn't need justification, it just needs to be hard. Only direct testing can justify non-linguistic certainty and only practice can justify linguistic certainty.
Quoting Moliere
I think acknowledging its failures and limitations improves cohesion of thought.
Quoting Moliere
To what end? This is a sincere question: What is it you hope to learn or achieve?
Part 1
This essay is heavy with dry theory. The author attempts a comparative analysis of foundational certainties. Structural parallels are explored: between Wittgensteins concept of hinges in On Certainty (OC) and Gödels incompleteness theorems.
According to the author:
.
As a reader, I want to know what this means, how true it is and why it matters.
The author tells us it has:
About theories:
Ray Monk quotes Wittgenstein:
From: https://www.prospectmagazine.co.uk/regulars/55561/wittgensteins-forgotten-lesson
The author writes:
With this distinction, the author goes further than Wittgenstein. There is no mention of non-linguistic hinges in OC. More of that later. Back to bedrock.
Bedrock is not the ground or foundation of belief.
From On Certainty:
From the author:
What happens when we reach bedrock?
In an article, Wittgenstein on Faith and Reason, Duncan Pritchard says:
: https://www.academia.edu/19857441/Wittgenstein_on_Faith_and_Reason_The_Influence_of_Newman
In On Certainty: https://prawfsblawg.blogs.com/files/wittgenstein-on-certainty.pdf
Wittgenstein writes:
In this essay, the author writes that we often perform actions without hesitation. Examples are given of sitting on a chair or picking up a pencil. The unthinking action illustrates Wittgensteins concept of a hinge proposition.
I find this problematic. It is an example of the authors theoretical non-linguistic hinge.
So many things will count as hinges. There is no doubt that there is a chair when I sit on it. But what is the hinge? The existence of chairs, this chair, that it will hold me, that the floor will hold the chair and the weight of me?
Even if we agree to everything that the author concludes. What of it?
Would we understand ourselves any better?
Non-theoretical understanding. Isnt that what Wittgenstein pursued?
Part 2
According to Ray Monk, Wittgenstein saw philosophy as the understanding that consists in seeing connections. For him his philosophy is an activity not a body of doctrine.
From PI 122:
A worldview. That of an individual or a collective? Both? Interacting. Linking. It is how we come to know by sharing thoughts and images.
As a collection of notes, On Certainty is thought-provoking with an imaginative use of metaphors. Some rigid pictures are painted, like hard hinges, structures. Others are soft and fluctuating like rivers.
Wittgenstein calls his/our picture of the world its inherited background. We dont get this by looking at its correctness or because we are satisfied with it. (OC 94).
This is one example of how Wittgenstein replaces the need for grounds and foundations. Others can be found with movement around an axis (OC 152), community (OC 298) and system (OC142).
The inherited background is like the river-bed. However, cultural concepts change. Different generations see things differently but some things stay the same. The river-bed may shift (OC 97).
This bedrock is not about justification of knowledge. Justification comes to an end. The river-bed channels thought.
The movement of the water (day-to-day thinking)) interacts with the river-bed structures. The flow of thoughts can change the shape of the rivers bank and course. There is not a sharp division (OC 97). We can change the world or our picture of it and vice versa.
Ungrounded grounds and ungrounded foundations are not a necessary condition.
Instead of grounds, ungrounded or otherwise, Wittgenstein points to 152, 94, 298, 142 and probably other passages. No mention of foundations. He rejects that model.
The author is correct regarding Wittgenstein's break with traditional epistemology. He moved away from old views about establishing foundations, a solid base on which to build knowledge. Wittgenstein believes that our understanding of knowledge is found in how we live, our beliefs and practices. Our certainty is practical rather than theoretical.
It is philosophy as a way of life. Not just for academics but for all.
The essay addresses the important philosophical questions of knowledge, certainty and foundations. However, the author's theoretical interpretation of W. is open to doubt. Some might say incorrect. There are several things that have been pointed out that are contrary to what is claimed in the essay.
Finally, what matters lies in the connection and activity. Sharing the challenge to improve understanding.
Quoting Vera Mont
The paper explores why we can know things at all by connecting two big ideas: Wittgensteins notion that our knowledge rests on unquestioned "hinges" (like assuming the ground will hold when we walk) and Gödels discovery that even math has true statements it cant prove within its own rules. The goal isnt to solve a practical problem like building a bridge, but to understand the foundations of how we think and reason, whether in everyday life or in fields like math and science. By demonstrating that both knowledge and mathematics depend on unprovable starting points, the paper reveals a universal idea, viz., that our systems of understanding require ungrounded foundations to function. This matters because it helps us appreciate the limits and strengths of human reasoning, encouraging a bit of humility about what we can justify and confidence in the systems despite these limits. Its like mapping the bedrock of thought, not to change how we live day-to-day, but to deepen our grasp of what makes knowledge possible, which can inspire clearer thinking in any field, from philosophy to science to ethics.
Thanks,
Sam26
Thank you for your response and for highlighting Gödels Platonism, which is a crucial aspect of his philosophical framework. I agree that Gödel, as a mathematical Platonist, believed in the absolute, objective reality of mathematical truths, existing independently of human constructions or formal systems, as Penrose notes. This view contrasts with a more contingent, practice-enabled nature of Wittgensteins hinges, which are grounded in our form of life and seem to carry a relativistic or pragmatist flavor. However, the parallel I propose between Gödels unprovable statements and Wittgensteins hinges focuses on the structural similarity as opposed to a complete philosophical alignment.
The key connection is how both thinkers reveal the necessity of ungrounded foundations within their respective systems. Gödels incompleteness theorems show that any consistent formal system of arithmetic requires axioms or truths that cannot be proven within that system, pointing to a limit of internal justification. Similarly, Wittgensteins hinges are basic beliefs that lie beyond justification or doubt, enabling our epistemic practices. While Gödel might have resisted the idea that these mathematical grounds are merely pragmatic or contingent (as hinges might appear in Wittgensteins framework), the structural parallel holds: both systems depend on foundational elements that are not internally justifiable but are necessary for the system to function.
Gödels Platonism suggests that unprovable truths, like the consistency of a system, exist in a realm of absolute mathematical reality, accessible perhaps through intuition or external perspectives (e.g., a stronger system). Wittgenstein, by contrast, sees hinges as embedded in our lived practices, not as absolute truths but as practical certainties that make inquiry possible. The paper doesnt claim Gödel would endorse Wittgensteins pragmatism but argues that the incompleteness theorems and hinge propositions both expose a universal feature of systematic thought: the need for ungrounded starting points. This structural necessity persists whether one views those foundations as God-given (Gödel) or as contingent features of human practice (Wittgenstein).
In short, while Gödels Platonism might make him somewhat wary of relativistic or pragmatist readings, the parallel with Wittgensteins hinges highlights a shared insight into the limits of formal and epistemic systems. This comparison enriches our understanding of foundational certainties, showing how they function across domains, even if their metaphysical status differs.
This response acknowledges Gödels Platonism and the critics concern, clarifying that the papers parallel is structural, not ontological. It defends the comparison by emphasizing the shared necessity of ungrounded foundations, while respecting the philosophical differences between Gödel and Wittgenstein, thus engaging the critique constructively without conceding the papers core argument.
Thanks for your response @Josh
Yes, that's the part I didn't get: 'ungrounded foundations' seemed to me a contradiction in terms. I assumed unquestioned assumptions were formed either through empirical testing or specialized faith.
But if it's a study in how humans think about lofty matters, I can see a purpose to the exercise.
Thank you for the thought-provoking response. Your subjective-objective distinction, self-referentiality, and the market pricing example adds to the discussion, and they challenge my claim about ungrounded foundations. Let me see if I can clarify my argument and explore the points you raise.
You rightly emphasize the subjective-objective distinction in the context of Wittgensteins hinges and Gödels incompleteness theorems, framing subjectivity as tied to self-referentiality and objectivity as a view without a viewpoint. I find this interesting, particularly your point that objective truths in logic and math are typically computable and provable, while subjective ones involve self-reference, evading such formalization. Your reference to Wittgensteins Tractatus (3.3323.333) and his solution to Russells paradox is spot-on: Wittgenstein identifies self-referentiality as a source of logical trouble, arguing that propositions or functions cannot contain themselves. This insight resonates with Gödels incompleteness theorems, which, as you note, cleverly navigate self-referentiality (e.g., the statement This statement is unprovable in the system) without falling into the traps of Russells paradox.
Quoting ssu
Your market pricing example is an interesting example of how self-referentiality complicates objective modeling. When a model of market prices incorporates the actions of everyone, including the modelers own decisions based on the model, it becomes self-referential, undermining objectivity. This aligns with Wittgensteins thinking in that certain propositions (or models) cannot contain themselves without losing their coherence. It also echoes my papers broader point: systems of thought, whether epistemic or mathematical, often rely on foundational elements that resist internal justification. In your market e.g., the hinge might be the assumption that prices reflect aggregate behavior, but using the model to act within the market introduces a self-referential loop that defies objective grounding (if I understand what you're saying), which is akin to the unprovable truths in Gödels systems or the unquestioned certainties in Wittgensteins hinges.
However, I should clarify the papers claim about ungrounded foundations in light of your critique that not all systematic thought lacks grounded foundations. My paper argues that both Wittgensteins hinges and Gödels incompleteness reveal a structural necessity: systematic thought (in sufficiently complex epistemic or mathematical systems) requires foundational elements that cannot be justified within the system itself. This doesnt mean all foundations are ungrounded in an absolute sense, but that their grounding lies outside the systems internal justificatory framework. For Wittgenstein, hinges are grounded in our form of life, i.e., in our shared practices and interactions with reality, but they resist justification through argument or evidence within the epistemic system they support. For Gödel, axioms (like those of Peano arithmetic) are grounded in their mathematical fruitfulness or intuitive plausibility, but they cannot be proven within the system they define. The ungrounded part refers to this internal limit, not a denial of external grounding (e.g., in practice, intuition, or objective reality for Gödels Platonism).
Quoting ssu
Your point, that not all systematic thought can be brought back to grounded foundations, is a helpful perspective, but Id argue it complements rather than contradicts the my claim. The paper doesnt assert that all thought lacks grounded foundations, but that sufficiently complex systems (epistemic or mathematical) require ungrounded foundations within their own justificatory scope. Simpler systems, like those covered by Gödels completeness theorem or basic linguistic practices, may achieve internal grounding, but that the parallel with Wittgenstein and Gödel emerges in domains where complexity has limits, necessitating external or unprovable foundations.
Your market example actually strengthens my point. The self-referential nature of the pricing model mirrors the way hinges and unprovable statements function as enabling conditions that cannot be fully justified within the system. Just as a market participants actions disrupt the objectivity of the pricing function, hinges and axioms enable systematic thought by standing outside the systems justificatory reach. This suggests that the subjective-objective interplay you highlight is not just a logical issue but a structural feature of how systems, whether markets, math, or knowledge, must be organized.
Finally, your insights about self-referentiality and the subjective-objective distinction enrich the papers framework, and your market example vividly illustrates the challenges of grounding complex systems. While Gödels completeness theorem reminds us that not all systems face incompleteness, the parallel with Wittgensteins hinges holds for systems where internal justification hits a limit, revealing the necessity of ungrounded foundations.
Quoting Sam26
There's one Holy Grail there if one could make it a true mathematical theorem: if that "objective truths in logic and math are typically computable and provable, while subjective ones involve self-reference, evading such formalization" could be made into "objective truths in logic and math are all computable and provable, if there isn't self-reference that leads to subjectivity". Or something like that.
This leads to understanding that there's also true but uncomputable math and we cannot just assume objectivity to compute them. And that we do have to understand that in some occasions, the best models would be uncomputable.
Because look at just what we have now for a definition of computation: the Church-Turing thesis. And what does that basically tell us? Basically (and not rigorously defined) that computation is something that a Turing Machine can do. Which means that something that is uncomputable is something that a Turing Machine cannot do. And not that this isn't a theorem, just a thesis. The Church-Turing thesis is said to be unprovable or basically undecidable. And this is because a direct proof and computation are so close to each other.
The dichotomy of the subjective and the objective and Wittgenstein's remarks could really here help. It's worth mentioning that when Alan Turing and Wittgenstein met, they simply didn't understand each other. Wittgenstein say the paradox in Alan Turings undecidability result, yet as you noted that just like with Gödel's Incompleteness theorems, the example of the Turing Machine doesn't end up in a paradox. However, Wittgenstein does have an important point.
Quoting Sam26
Yes, once you are an acting part of a universe you are trying to model, the problem arises. Many times when you don't notice the problem, you get to a problem of infinite regress. Yet do notice that self-referential loops can get to a "objective grounding". If we have something like a self-fulfilling prophecy, that can indeed be modeled and computed.
Quoting Sam26
I agree. The uncomputable are really special occasions to the norm. At least when we try to make objective scientific models.
Quoting Sam26
Yes, exactly. There isn't any problem with having Gödel's completeness theorem and incompleteness theorems being true at the same time.
But let's think about just what is meant by "ungrounded foundations". Just what do we mean by this is important. In my opinion, with grounded foundations we go back to the way that an algorithm works: follow these foundations, and you can make correct model / compute the correct answer. Yet if in the foundations there is the aspect of subjectivity, all hell is loose. If the order or step would be "Here you decide what ice cream you like" it's not anymore an objective truth as it needs that subjective decision. Or the classic instruction of "Do something else not written in these instructions", which is a command that a computer cannot follow as it isn't itself a subject capable of making subjective decisions.
I think the objectivity/subjectivity dichotomy would be an interesting way to look at this problem. I remember last year we had a good thread about , where people went through professor Noson S. Yanofsky's interesting paper True but Unprovable and the PF thread was Mathematical truth is not orderly but highly chaotic. Perhaps then the subjective / objective issue wasn't at the center stage, but it really puts the issue back to simple logic.
Anyway, I hope these have been useful comments to you.
Thanks @ssu for the compliment. There are some really interesting ideas to pursue in these posts, especially as they relate to my interest in epistemology. I'm not a mathematician, but I did manage to see a connection between Wittgenstein and Godel. I've been trying to find other writings that've made a similar connection, but I haven't been able to find anything.
I have to move on to answering some of the other replies to my paper, but your responses were interesting.
I can see the structural parallel. There's a part of me that still wonders: Why this particular set of parallels? My first guess is that in two disciplines in which complicated thought is required we find a common between Godel and Wittgenstein, and that particular combination is persuasive of a larger structure in thinking that must be -- namely that there will be truths that are not grounded at the same level within any sufficiently "complicated"* body of -- knowledge?
*Whatever that is cached out as
I can see the analogy, but it's the part that I think could really sell the argument home -- not just a strong analogy, but even a reason to bring these people together due to the structure of thought, or something like that. Somehow strengthening the tie between the two examples.
Still, I say that in an attempt to be helpful, and your essay far surpasses my little comments on it. Thanks for your submission!
Both Wittgenstein and Gödel were investigating questions related to completeness. Wittgenstein was asking whether epistemic practices could be completely self-justifying, and Gödel was asking whether formal systems could be completely self-proving. Both men discovered that the answer was no, and both showed that this isnt a defect to be solved but a structural necessity. These ideas solve the problem of infinite regression and the problem of circularity.
The "why these two" question has a deeper answer, viz., they represent the most rigorous investigations into foundational questions in their respective domains, and its during the same historical period. Wittgenstein was examining the foundations of ordinary knowledge and language, while Gödel was examining the foundations of the most rigorous knowledge we possess (mathematics). That they independently discovered analogous structural limits suggests this isn't domain-specific but reveals something about the structure of systematic thought itself.
As far as I can tell, no one else has made this connection, but who knows? The paper demonstrates this isnt just about complicated systems needing ungrounded elements, but about the logical structure of any system that strives for internal coherence. Even simple systems, if theyre to be complete and self-justifying, will encounter such limits.
The Wittgenstein-Gödel parallel reveals a universal logical constraint, viz., any systematic framework that attempts to ground all its statements internally will necessarily rely on elements that lie beyond its internal capacity for justification. The parallel is compelling precisely because it shows this constraint operating across the most fundamental domains of human understanding.
As far as I am concerned, no comments on this submission are 'little'. The essay is undoubtedly substantial and received a great deal of interest from certain posters. Not surprising given the attraction, sometimes an obsession, with the name 'Wittgenstein', his importance and influence in 20th century, Western Philosophy.
It stood out as the thickest text and, yes, it was indeed an academic paper. Specialist and exclusive.
It stands at the extreme end of the spectrum of 'philosophy writing'.
The challenge being met for this June event included 13 pieces of diverse creations. Thought-provoking.
Comments, replies and responses varied accordingly. To the sense and sensitivity of the subject.
As far as able to read and understand by questioning. Or enjoying the seeming simplicity.
For this essay, both Vera and myself offered responses:
Vera: (20 days ago)
https://thephilosophyforum.com/discussion/comment/993677
Mine: (in 2 parts, 16 days ago)
https://thephilosophyforum.com/discussion/comment/994486
https://thephilosophyforum.com/discussion/comment/994488
Sam's response to Vera engaged only with her final question. The response pertinent to his aims:
https://thephilosophyforum.com/discussion/comment/994973
Quoting Sam26
As someone who only studied W. briefly, a long time ago, I spent time and energy attempting a response. Given that I could see only a wall of incomprehensible gobble-de-gook, I did my best to connect. It was exhausting. However, so far, no response.
In my original reply, I didn't include this. From the penultimate paragraph of the conclusion:
Quoting Moliere
In my notes, exclamation and question marks surround this. For me, it sounds more like a machine than a human. As do many of the detached repetitions.
14 days ago:
Quoting Sam26
No further answers to the other replies were given. Instead:
13 days ago: https://thephilosophyforum.com/discussion/comment/995416
The thread, started by Sam26, 5 years ago - An Analysis of "On Certainty".
https://thephilosophyforum.com/discussion/8443/an-analysis-of-on-certainty/p1
The essay, then, is based on a body of lengthy, analytical work. Theory-driven.
Unlike many of the others who are arguably more refreshing and human. Unique with a spirit of creativity and engaging. Not exclusive. Or sounding like AI with its dry, authoritative tone.
Quoting Sam26
The connection could easily have been made by AI. Indeed, any author can now claim a 'new' find after using a prompt and conversing with AI.
This is not to suggest that the author relied on such. It is clear that he has given his all to the project.
This is what many think philosophy is all about.
I hope that this event has shown otherwise. It can be this and much, much more.
I don't think AI could have made such a connection. I made this connection more than a year ago, possibly longer, and the AI available at the time surely couldn't have made the connection between Wittgenstein and Godel.
It's worth noting that Wittgenstein disagreed with Gödel's incompleteness theorem (although apparently he misunderstood it).
From Remarks on the Foundations of Mathematics:
Wittgenstein was skeptical of Gödels theorem, and it seems that the disagreement, as far as I understand, was largely due to what appears to be a misunderstanding of the distinction between provability within a system and truth in the standard interpretation.
However, the structural parallel is valid regardless of Wittgensteins views of Gödel. Im not claiming that Wittgenstein endorsed this connection. Im just making an independent philosophical observation about structural similarities in foundational limits.
One could say that Wittgensteins objection supports my broader point, viz., his insistence that we must ask True in what system? and Provable in what system? which seems to support the idea that systematic frameworks require external grounding. He argues that truth and provability are always relative to a particular system, and this aligns with my argument about the necessity of external foundations.
Regardless of how this is interpreted, my paper shows that they were both concerned with the same foundational question, i.e., how do systematic frameworks relate to what lies outside them?
Thanks for the reply.
Thank you for your thoughts re the use of AI.
I didn't realise that you had previously made the connection, and argued for this before.
Did you publish anywhere? There's no reference to this effect.
The essays I thought were to be freshly spun.
6) Entries must not be previously available online, or in print form.
The legalities don't matter much to me. You made me think...
I don't always respond to every challenge or question because I just don't have the time. Right now, I'm working on a book on NDEs, so that occupies my time.
Ah, OK. I'm not that concerned about the first mention. :smile:
Quoting Sam26
Understood.
:up: That satisfies me, at least. Similar time periods, different areas of inquiry, similar conclusions indicate that we're dealing with something deeper than just a single thinker. Especially as the thinkers, as others have said here, don't see eye-to-eye elsewhere.