Reading Wittgenstein's On Certainty as a Whole: An Interpretive Picture

Metaphysics & Epistemology
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Wittgenstein’s discussion of Gödel in Remarks on the Foundations of Mathematics makes the argument that if someone says of the Gödel sentence, “This proposition is true but unprovable in the system,” we have to ask what exactly is meant by “true.” If “true” simply means “provable in the system,” then calling the sentence true is incorrect, because by construction it is not provable there. But if “true” means something like “correctly describing its own unprovability when we view the system from outside,” then we have already shifted into a different language-game, a meta-level description of the calculus. Wittgenstein’s point is that the philosophical drama arises precisely from sliding between these two uses of “true.”

The key point is that if the Gödel sentence is interpreted as saying “This statement is not provable in the system,” and we can prove that this is indeed the case, then the statement has effectively been proved, but only in a different system. The natural mathematical response is simply to extend the calculus. What philosophers then describe as an eternal “true but unprovable statement” is, from Wittgenstein’s perspective, just a feature of how we choose to organize our formal systems.

So Godel’s comparison between ‘true within the system’ and ‘true but unprovable’ is not like the relation between the provable rules within a language game and the ‘true but unprovable’ hinge making that game intelligible. Rather, it is a relation between two different language games. Assuming the former is to confuse an epistemological and a non-epistemological grammar of truth, which is exactly what Moore did.

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The reading from the RFM is accurate as far as it goes. Witt did resist the philosophical drama surrounding “true but unprovable,” and he was right to point out the slide between uses of “true.” But I think you’re drawing the wrong conclusion from this for the parallel I’m making.

You say the Godel result is just a relation between two different language-games, one system and its meta-level extension. And that the natural mathematical response is simply to extend the calculus. But that’s the point. The extended system then has its own Godel sentence. And the further extension has another. The result isn’t a one-time problem solved by moving to a meta-level. It’s a structural feature that recurs at every level. No matter how many times you extend the calculus, the new system can’t fully justify itself from within. That’s not a feature of how we choose to organize formal systems, as though a better choice would avoid it. It’s an unavoidable feature of any sufficiently complex formal system (And not just mathematics).

Witt’s critique in the RFM is about a particular philosophical interpretation of Godel, the one that treats “true but unprovable” as a mysterious metaphysical discovery. I agree with that critique. But the math result itself survives Witt’s deflationary reading. Even after you strip away the philosophical drama, you’re still left with the fact that formal systems of sufficient complexity cannot be self-contained. That’s the parallel I’m drawing.

You say I’m confusing an epistemological and a non-epistemological grammar of truth, which is exactly what Moore did. But the parallel doesn’t depend on treating the Godel sentence as “true” in some extra-systemic sense. It depends on the structural fact that systems of justification, whether formal or epistemic, necessarily rely on something they can’t justify from within. Godel showed this for formal systems. Witt showed it for epistemic language. The methods are entirely different. The conclusion converges.

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One of the aims of this thread is to follow the argument as it develops remark by remark, since no existing book-length treatment of OC does this. It would probably take more than a year to complete.

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You argue that every sufficiently strong formal system has a Gödel sentence that it cannot prove. When we extend the system to prove that sentence, a new Gödel sentence appears. Therefore no such system can justify itself completely from within, and this leads to the structural conclusion concerning justification that systems must rely on something they cannot justify internally.

But this conclusion still depends on how the word “true” is functioning in the Gödel discussion. Inside a formal calculus, the relevant notion of truth is provability according to the rules. That is the only internal standard. The notion of truth as justification of the rules of a system is not a notion internal to the calculus. A formal system doesnt attempt to justify itself. It simply defines a rule-governed notion of proof. To move from from the internal notion as probability to the notion of justification of a system from outside of it involves a change in language game and a corresponding shift in the sense of truth being used.

The step from “every system has an undecidable Gödel sentence” to “no system can justify itself from within” requires such a shift in philosophical interpretation. These simply aren’t the same structural sense of what it means to be ‘true’, and therefore it amounts to a confusion of two grammars of truth to claim that systems of justification must rest on something they cannot justify internally.

A move in chess is justified or not based on the rules. But the rules don’t need to justify themselves at all. They are made intelligible from within a hinge, form of life, etc. Notice that in shifting our attention from the rules of chess to hinges and forms of life we are not moving from one language game to another, and we could say that a kind of structure of inclusion is involved. By contrast, when Godel shifts from the sense of ‘truth’ as provability within the rules of a formal calculus to ‘truth’ as what cannot be proven within any system, Wittgenstein says that this does amount to a change in language games.

As a result, the inclusive structural logic that obtains between moves within chess, the language game of chess itself and the hinges and forms of life that make such games intelligible doesn’t apply to what seems superficially to be a similar ‘inside-outside’ dynamic in Gödel. If what is involved here are two different language games then there is no relation of ‘reliance’ connecting the intelligiblity of the rules of an axiomatic system with an extra-systemic formal truth. Language games form a family of resemblances, not an inclusive logic.

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It is not you who are confused but those you do not understand the place and function of hinges who are confused. They insert a pseudo-problem of doubt at the foundation of our lives.

  1. The child learns by believing the adult. Doubt comes after belief.

And hinges come even later. Hinges are established late in the game, they are not a condition for playing.

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The claim that hinges are established late and aren’t a condition for playing is again, flatly wrong. OC 341-343 says “The questions that we raise and our doubts depend on the fact that some propositions are exempt from doubt, are as it were like hinges on which those turn.” That’s not describing something that shows up after the game is underway. It’s describing what makes the game possible. “If I want the door to turn, the hinges must stay put.” The hinges are there before the door moves, not after.

You cite OC 160, the child learns by believing the adult. Exactly. And what makes that learning possible? That certain things already stand fast. The child doesn’t first learn to speak and calculate and then later acquire hinges. The child can learn at all because object permanence, the reliability of her senses, the existence of other people are already in place, functioning as conditions of the training. OC 34 explicitly says trust in the teacher isn’t a separate lesson. It’s embedded in the learning itself. That trust is a hinge, and it’s there from the beginning, not established late.

What comes late is the articulation of hinges, not their functioning. We can name and describe hinges only after we have language and philosophical reflection. But a child acts with object-certainty long before anyone formulates “there are physical objects.” The hinge is operative. It just hasn’t been stated. Confusing the naming of something with its functioning is exactly the kind of error Witt is working against throughout OC.

Your reading has hinges as products of developed practices, which conveniently supports your claim that hinges are scientifically shaped. But Witt’s picture runs the other direction. Hinges are what developed practices rest on, not what they produce. OC 205 says “If the true is what is grounded, then the ground is not true, nor yet false.” The ground comes first. That’s the whole point.

You should start your own thread explaining your view. We get it, you disagree.

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Right.

[quote=“Sam26, post:82, topic:129”]
And what makes that learning possible? That certain things already stand fast.

What stands fast is not propositions. What stands fast is the uniformity of nature.

All of that is correct, but that objects do not appear and disappear is a fact of nature, not a proposition. It can be made into a proposition, the principle of the uniformity of nature, but the child does not learn this principle. Experience tells him that things do not appear and disappear, and that is why he is puzzled if something does seem to just appear or disappear.

[quote=“Sam26, post:82, topic:129”]
That trust is a hinge[/quote]

The child learns to trust when she learns to distrust. You are inserting doubt where it does not belong.

This is close but certainty is out of place here. There is no certainty without there being uncertainty. The child does not act with certainty, he simply acts.

It is not a matter of convenience they are part of the same issue.

You are making the mistake that Ludwig pointed to, although he is perhaps too polite or circumspect to call it a mistake,
using ground and hinge as if they are the same thing.

I thing Wittgenstein would not agree:

  1. You must bear in mind that the language-game is so to say something unpredictable. I mean: it
    is not based on grounds. It is not reasonable (or unreasonable).
    It is there - like our life.

I was responding to Ludwig. I think there are already enough topics on Wittgenstein. There may be others who are not part of your “we” who are interested in reading the text and considering different perspectives on it.

This is, after all, a discussion forum, not somewhere for you to give an interpretation without that interpretation being challenged.

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It’s just a suggestion.

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Do you have any textual support for this? Wittgenstein indicates the opposite:

  1. As children we learn facts; e.g., that every human being has a brain, and we take them on trust. I believe that there is an island, Australia, of such-and-such a shape, and so on and so on; I believe that I had great-grandparents, that the people who gave themselves out as my parents really were my parents, etc. This belief may never have been expressed; even the thought that it was so, never thought.

  2. The child learns by believing the adult. Doubt comes after belief.

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That objects do not appear and disappear is the way things appear to us, and not necessarily “a fact of nature”. This is what is at the base of the type of doubt which Wittgenstein is talking about. Many philosophers recognize that the way things appear to us is not necessarily the way that things actually are. So doubt is warranted.

For example, some propose that the continuity of existence which we observe, is really more like a succession of still frames. If this is the case, then things actually do come and go, but the change is so fast its invisible to us. Then we have the question of what happens in between, when the things are not there.

It is not the uniformity of nature which stands fast for us, it is how nature appears to us which stands fast for us. And since the philosopher understands that it is the way things appear which stands fast, the philosopher doubts whether the appearance is an adequate representation. This doubt enables the eureka moment, when someone suddenly realizes “that’s the way things really are, I knew we were seeing it wrong all this time”. For example, when it was realized that the earth revolves around the sun, not the other way around.

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How about sticking with the thrust of the thread. The thread isn’t just about hinges. I’m trying to give an overall picture line-by-line of OC.

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I believe Wittgenstein is wrong on this point. “We cannot have miscalculated” adds a subjective certainty, just like Moore’s “I know this is a hand” adds a subjective certainty to “this is a hand”. Therefore we cannot collapse it into the objective “12x12=144”, because this would be to ignore the subjective aspect being expressed.

To understand this better we can contrast “We cannot have miscalculated in 12x12=144” with “I think 12x12=144”. The former expresses subjective certainty, the latter subjective uncertainty (perhaps someone just learning mathematics). Since these two express a person’s subjective attitude toward 12x12=144, they cannot be collapsed to simply 12x12=144 without altering the meaning. Clearly, we could not collapse the latter expression of uncertainty into the objective statement, as it might be “I think12x12=142”. Nor can we collapse the expression of subjective certainty into having the same meaning as the objective expression.

I believe this is important, because Wittgenstein’s MO here is to deny the importance of the “inner process” in mathematical calculations (38). But in reality it is the inner process which actually determines the validity of the specific calculation, i.e. produces the rules. So the inner process actually remains more important, as logically prior to the calculation which could be represented as external.

For example, we can assign all the calculations to a computer, making them completely external. However, deciding and assigning algorithms is an internal process which is logically prior to having the computer carryout the calculations, and these internal processes are no less important.

Therefore Wittgenstein is proceeding from a false premise when he states that the internal is unimportant, as it is clearly just as important, if not more important, as logically prior. Then he wrongly collapses “We cannot have miscalculated in 12x12=144”, into “12x12=144”, as support for the false premise. This confusion is clearly expressed at OC 41, which is simply wrong, and a contortion designed to support the preceding false premise and lay the grounds for the falsity which follows.

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Now, he has successfully disproven his faulty premise at 43. He has shown that “We cannot have miscalculated in 12x12=144” really does have meaning over and above the simple objective “12x12=144”. It has the added certainty of “I know because I’ve checked.” That is the subjective certainty, the “inner process”, which actually is very important.

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Ultimately, Wittgenstein demonstrates that the “inner process” which he proposed to dismiss as unimportant at 38, is actually of the utmost importance. This is what he calls “learning to calculate” at 45.

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Yes, we don’t need an extra rule to justify the rule internal to calculation. But what does it mean to say that we calculate according to this internal rule? It doesn’t mean that rules function as hidden engines guiding practice. Rules do not stand behind the activity giving it authority. Any regularities in what practitioners have previously done does not have any authority to bind subsequent performances to the same regularities. We have to wait and see what we are actually doing with the rule each time we calculate ‘according’ to it in order to know what the according to means. This is what it means to say that the rule is expressed in the way the practice unfolds. Wittgenstein said that the practices behind a rule like 2+2=4 have a stability akin to a hinge, which means that once they have been learned we would expect little disagreement over what constitutes a correct answer.

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The question is the relationship between doubt and hinges:

[quote=“Fooloso4, post:81, topic:129”]
They insert a pseudo-problem of doubt at the foundation of our lives.[/quote]

Some propositions are exempt from doubt. (341) The child believing is not exempt from doubts because there is no doubt there to be exempted from. Hinges are what doubts turns on. There are no hinges where there is not yet doubt.

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This puts you in the hinges come later camp.

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I’m in the hinges are a false proposition camp.

I don’t see any mention of a regress problem here, or something which “stands fast”, or anything like that.

What Wittgenstein says is that even when we know very well how to follow a rule, like a calculation, we might still make a mistake in applying that calculation. And if we try to make a rule about how to apply the calculation, the best we can do is to say “in normal circumstances”. And that serves nothing because we can’t even say how to recognize normal circumstances.

This is what I explained earlier with the example of the difference between valid logic, and true propositions. We can apply logic validly (correctly calculate) but if a premise is false (we incorrectly apply the calculation), it is still a mistake, but a different type of mistake. This is two distinct types of mistake which Wittgenstein is pointing to OC 26: “What use is a rule to us here? Mightn’t we (in turn) go wrong in applying it?”

There is no regress talked about here. Nor is there a mention of something that stands fast. What is said is that even if we know the rule, and make no mistake in following it, we might still make a mistake in applying it.

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You’re right that Witt doesn’t use the word “regress” in OC 25-27, and I should have been clearer that I’m naming the structure rather than quoting him. But I think the structure is present in the logic of the passage. OC 26 asks “What use is a rule to us here? Mightn’t we go wrong in applying it?” If every rule for excluding mistakes is itself subject to misapplication, and you think that possibility needs to be covered by a further rule, you have a regress. Witt doesn’t spell out the infinite chain, but the question he’s asking is what generates one if you insist that every possibility of error needs a rule to cover it.

Your distinction between following a rule correctly and applying it to the wrong case is valid and is part of what Witt is pointing to. But that distinction doesn’t eliminate the regress problem. It deepens it. If correct rule-following can still produce the wrong result because the application goes astray, then no rule for following rules will guarantee correctness. That’s the regress. And Witt’s response isn’t to find a rule that finally stops it. It’s to show that the demand for such a rule is misguided. At some point we act within the practice without further rules to back us up.

You’re also right that “stands fast” doesn’t appear in these specific remarks. I’m reading forward, connecting OC 25-27 to what Witt develops later. In a sequential reading that tracks how the argument builds, that’s legitimate, but I should have flagged that the language is mine at this stage, not his.

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Are you claiming that this was Wittgenstein’s position? If so, based on what?