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: — ssu

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 Wittgenstein’s hinges and Gödel’s 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 Wittgenstein’s Tractatus (3.332–3.333) and his solution to Russell’s 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 Gödel’s 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 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. — 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 modeler’s own decisions based on the model, it becomes self-referential, undermining objectivity. This aligns with Wittgenstein’s thinking in that certain propositions (or models) cannot contain themselves without losing their coherence. It also echoes my paper’s 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 Gödel’s systems or the unquestioned certainties in Wittgenstein’s hinges.

However, I should clarify the paper’s claim about “ungrounded foundations” in light of your critique that not all systematic thought lacks grounded foundations. My paper argues that both Wittgenstein’s hinges and Gödel’s 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 doesn’t mean all foundations are ungrounded in an absolute sense, but that their grounding lies outside the system’s 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 Gödel’s Platonism).

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 always.

Just as there is also Gödel's completeness theorem, that theorem doesn't collide with the two incompleteness theorems. — ssu

Your point, that “not all systematic thought can be brought back to grounded foundations,” is a helpful perspective, but I’d argue it complements rather than contradicts the my claim. The paper doesn’t 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 Gödel’s 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 participant’s actions disrupt the objectivity of the pricing function, hinges and axioms enable systematic thought by standing outside the system’s 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 paper’s framework, and your market example vividly illustrates the challenges of grounding complex systems. While Gödel’s completeness theorem reminds us that not all systems face incompleteness, the parallel with Wittgenstein’s hinges holds for systems where internal justification hits a limit, revealing the necessity of ungrounded foundations.

Thank you for one of the best replies I've ever gotten in this Forum. It's really great when somebody understands my points. Here are some comments that hopefully forward this good discussion.

You rightly emphasize the subjective-objective distinction in the context of Wittgenstein’s hinges and Gödel’s 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 Wittgenstein’s Tractatus (3.332–3.333) and his solution to Russell’s 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 Gödel’s 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 Russell’s paradox. — 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.

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 Gödel’s systems or the unquestioned certainties in Wittgenstein’s hinges. — 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.

Your point, that “not all systematic thought can be brought back to grounded foundations,” is a helpful perspective, but I’d argue it complements rather than contradicts the my claim. — Sam26

I agree. The uncomputable are really special occasions to the norm. At least when we try to make objective scientific models.

The paper doesn’t 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 Gödel’s 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. — 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.

Thank you for one of the best replies I've ever gotten in this Forum. It's really great when somebody understands my points. Here are some comments that hopefully forward this good discussion. — ssu

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.