Hinges provide a way of understanding Godel’s theorems without the need to justify certain basic beliefs within the system. I prefer to call these basic beliefs, non-propositional, which eliminates the need to refer to them as true or false. It seems to me that you can always ask of a true proposition, “How do you know it’s true?” – which evokes a justificatory response. Moreover, if you refer to them as true, can you also ask if they can be false? This seems to open a can of worms.

It seems to me that formal systems are held together by background beliefs, i.e., that you can’t create a formal system (epistemic or mathematical system) without the background. I’m specifically referring to the prelinguistic background that is even more fundamental than linguistic hinges. For example, the prelinguistic beliefs that occur as a result of engaging with the world, walking, running, touching, smelling, object and special awareness, etc. Even causal and simple logical relationships are probably part of these basic beliefs. So, the basic beliefs that are formed before linguistics play an important role in the more sophisticated linguistic beliefs (such as what it means to know) that come later.

Basic beliefs are important because they form the substructure that allows epistemic and mathematical systems to form without the need for justification, i.e., they are prior to our justificatory models. Basic beliefs, especially prelinguistic beliefs, are the scaffolding that allows our models of epistemology and mathematical systems to take root. These kinds of beliefs are necessarily prior to our world of justification.

The foundational axioms act as hinges in the Wittgensteinian sense. This would eliminate Godel’s requirement for the axioms to be proved within the system. — Sam26

I confess I don't have a background in mathematics, but I'm not sure I follow you here. As far as my understanding goes, Godel's incompleteness theorems do not show that a formal system of logic cannot prove its own axioms. That axioms cannot be proven by deductions from axioms is a foundational principle of mathematics and logic, and did not originate with Godel.

The incompleteness theorems show that formal systems of logic always produce truths that are not provable using only the axioms of the system. Even if those truths are adopted as axioms, further unprovable truths will still exist, generating an infinite list of axioms. If a given axiom ever makes that list 'complete', then the list can no longer be 'consistent', in the technical mathematical sense of those terms.

The truth-value of axioms in themselves is a question in the philosophy of mathematics (and philosophy generally). But your treatment of Wittgenstein's 'hinge-propositions' here effectively equates them with axioms – which is to say, claims that are accepted without being proven and on the basis of which formal logical reasoning depends. That is the basis of all formal reasoning, as far as I know, and not original to Wittgenstein or Godel.

Am I missing something? If Witt had meant that hinge-propositions were just like axioms, he would have said so. He had a good knowledge of mathematics, as I understand. But perhaps you are a mathematician too and you can explain where I'm going wrong here! Thanks in advance.

I’ll try to re-word my point to make it clearer. First, a summary of Godel’s incompleteness theorems. The first theorem states that in any formal system adequate for number theory there exists true mathematical propositions (including its negation) that cannot be proven within the system. The corollary (the second theorem) is that the consistency of the formal system capable of expressing arithmetic cannot be proved using only its axioms and rules.

My point is that if we think of the propositions in Godel’s theorem (the ones that cannot be proven within the system) in the same way Wittgenstein thinks of hinge propositions (basic beliefs), viz., that hinges are outside our epistemological framework, then there is no requirement to prove the propositions within the system. We could think of Godel’s unprovable statements as hinge-like. So, Godel’s unprovable statements are necessary for the formal system to operate, just as hinges are necessary for our epistemic practices. The systems are held fast by viewing certain statements as hinges. I’m assuming you understand Wittgenstein’s point about hinges in OC.

Thanks for this.

My point is that if we think of the propositions in Godel’s theorem (the ones that cannot be proven within the system) in the same way Wittgenstein thinks of hinge propositions (basic beliefs), viz., that hinges are outside our epistemological framework, then there is no requirement to prove the propositions within the system. We could think of Godel’s unprovable statements as hinge-like. So, Godel’s unprovable statements are necessary for the formal system to operate, just as hinges are necessary for our epistemic practices. — Sam26

But again, this formulation of 'unprovable statements' that are 'necessary for the formal system to operate' makes them sound more like axioms. How do they differ from axioms?

As far as I can see, there has never been a 'requirement to prove the propositions' that 'are necessary for the formal system to operate' either in Godel or elsewhere – those propositions are axiomatic and Godel did not try to prove them or to show that they could or could not be proven.

Godel showed that there would always be true but unprovable statements within any axiomatic logic system. If these statements are incorporated into the system as axioms (which are precisely those statements that are accepted as true without being proven), either those new axioms will contradict the existing ones, or they will result in the emergence of further true but unprovable statements. No system can ever fully incorporate all these true statements as axioms and remain consistent.

Again, I'm not a mathematician. But what you describe as 'hinge-propositions' sound a lot more like axioms (which form the basis of reasoning) than true but unprovable statements (which are not axiomatic).

I’m assuming you understand Wittgenstein’s point about hinges in OC. — Sam26

Thank you, but that's probably too charitable an assumption! To be honest, after reading 'On Certainty', I was surprised to find such widespread discussion of 'hinge-propositions' in the secondary literature. Wittgenstein only mentions hinges briefly and never seems to use the phrase 'hinge-propositions' at all (at least not in the Anscombe/Wright translation, unless I'm mistaken).

In section 655 he writes:

The mathematical proposition has, as it were officially, been given the stamp of incontestability. i.e.: "Dispute about other things; this is immovable – it is a hinge on which your dispute can turn."

He's talking about the proposition 12x12=144 here, which can be derived from basic axioms of arithmetic. I don't see how this relates to Godel's theorems but, again, no doubt there is much I don't understand here.

eta: I realise I've taken the above quotation out of context. But it comes from a section where Wittgenstein is teasing out certain epistemic similarities between mathematical propositions and empirical propositions. From 651:

... one cannot contrast mathematical certainty with the relative uncertainty of empirical propositions. For the mathematical proposition has been obtained by a series of actions that are in no way different from the actions of the rest of our lives, and are in the same degree liable to forgetfulness, oversight and illusion.

And 653:

If the proposition 12x12=144 is exempt from doubt, then so too must non-mathematical propositions be.

That's the context in which I understand the idea of 'incontestable', 'fossilised' or 'hinge-like' propositions in 'OC'.

Even causal and simple logical relationships are probably part of these basic beliefs. So, the basic beliefs that are formed before linguistics play an important role in the more sophisticated linguistic beliefs (such as what it means to know) that come later. — Sam26

As with everything 20th century tried to do away with by turning epistemology to language debates, it goes back to Kant :wink:

Godel showed that there would always be true but unprovable statements within any axiomatic logic system. If these statements are incorporated into the system as axioms (which are precisely those statements that are accepted as true without being proven), either those new axioms will contradict the existing ones, or they will result in the emergence of further true but unprovable statements. No system can ever fully incorporate all these true statements as axioms and remain consistent. — cherryorchard

An important difference between Gödel and Wittgenstein is that for the latter the synonymous concepts of hinge propositions, forms of life and language games are neither true nor false. They are outside all schemes of verification, since such schemes presuppose them.

The term 'hinge' occurs three times in On Certainty.

341. That is to say, 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.

343. But it isn't that the situation is like this: We just can't investigate everything, and for that reason we are forced to rest content with assumption. If I want the door to turn, the hinges must stay put.

655. The mathematical proposition has, as it were officially, been given the stamp of
incontestability. I.e.: "Dispute about other things; this is immovable - it is a hinge on which your
dispute can turn."

The third is the only example explicitly called a hinge. It is both a proposition, and true.

655. The mathematical proposition has, as it were officially, been given the stamp of
incontestability. I.e.: "Dispute about other things; this is immovable - it is a hinge on which your
dispute can turn."

The third is the only example explicitly called a hinge. It is both a proposition, and true. — Fooloso4

Wittgenstein is arguing that we conventionally equate the fact there can be no dispute concerning the meaning of a mathematical proposition with its being true. This is not how Wittgenstein treats hinge propositions. His critique of Moore’s supposedly ‘true’ statement ‘this is my hand’ revolves around Moore’s confusing an empirical truth claim with the indisputability of a hinge proposition (form of life, language game).

Where does he make the claim that we do not dispute 12+12=144 but it is not true or false that 12+12=144?

Engineering calculations do not depend on lack of dispute.

It does not matter how many times Wittgenstein refers to this kind of proposition as "hinge," most philosophers use the term to refer to this kind of proposition (hinge, bedrock, foundational, basic, all mostly refer to the same thing).

Where does he make the claim that we do not dispute 12+12=144 but it is not true or false that 12+12=144?

Engineering calculations do not depend on lack of dispute. — Fooloso4

He is linking hinge propositions with forms of life and language games. They are all incontrovertible for the same reason. Not because they are true, but because they form a system of logic on the basis of which true and false statements are intelligible. It is not that the sum of proposition 12+12=144 is not either true or false, it is that the practices that allow us to know this are not themselves true or false. Before we can answer ether 12+12=144 is true or false, it has to be intelligible. Hinge propositions provide the bedrock of intelligibility.

657. The propositions of mathematics might be said to be fossilized. - The proposition "I am called...." is not. But it too is regarded as incontrovertible by those who, like myself, have overwhelming evidence for it. And this not out of thoughtlessness. For, the evidence's being overwhelming consists precisely in the fact that we do not need to give way before any contrary evidence. And so we have here a buttress similar to the one that makes the propositions of mathematics incontrovertible.

most philosophers use the term to refer to this kind of proposition (hinge, bedrock, foundational, basic, all mostly refer to the same thing). — Sam26

Your assumption that these are all terms referring to the same thing is questionable. The only thing that turns on bedrock, as Wittgenstein says, is the spade.

A hinge is not a foundation:

OC 152.

I do not explicitly learn the propositions that stand fast for me. I can discover them
subsequently like the axis around which a body rotates. This axis is not fixed in the sense that
anything holds it fast, but the movement around it determines its immobility.

It may be questionable for you but not me. If you want to interpret it another way that's fine, but I think it goes against Wittgenstein's general thinking.

Mathematics is certainly a part of our form of life and mathematics does have its language games, but this does not mean that mathematical propositions are neither true nor false. The bridge would collapse if the calculations are wrong. We would not have landed on the moon if the calculations were wrong. Building bridges and moon landings are part of our form of life, but unlike our form of life the mathematical propositions are not arbitrary or t.a matter of convention or agreement.

A hinge is not a foundation:

OC 152.
I do not explicitly learn the propositions that stand fast for me. I can discover them subsequently like the axis around which a body rotates. This axis is not fixed in the sense that anything holds it fast, but the movement around it determines its immobility. — Fooloso4

Wittgenstein’s “On Certainty” was a response to G. E. Moore’s essays which aimed to identify propositions that are beyond skepticism.Wittgenstein examined the idea that certain propositions serve as the bedrock or foundation for other empirical statements. He likened these foundational statements to a riverbed that must remain stable for the river to flow.For Wittgenstein, the certainty we feel about some propositions stems from their deep integration into our daily activities or “forms of life”.

https://www.thecollector.com/ludwig-wittgenstein-on-certainty/

Once again, you assume as answered what is in question. Whatever you might take his "general thinking" to be, he calls 12+12=144 a proposition and nowhere does he claim that it is neither true or false.

Where does this article discuss mathematical propositions?

The riverbed is not bedrock. It changes, sometimes slowly and other times rapidly. The axis around which a body rotates is not bedrock and is not held fast by bedrock.

Mathematics is certainly a part of our form of life and mathematics does have its language games, but this does not mean that mathematical propositions are neither true nor false. The bridge would collapse if the calculations are wrong. We would not have landed on the moon if the calculations were wrong. Building bridges and moon landings are part of our form of life, but unlike our form of life the mathematical propositions are not arbitrary or t.a matter of convention or agreement. — Fooloso4

Of course they are true or false. Wittgenstein isnt denying this. He is making a distinction between a micro and macro level of analysis. A particular qualitative system of interconnected logical elements ( the macro level). is implicitly used as a framework of intelligibility within which individual propositions can be true or false ( the micro level).

The riverbed is not bedrock. It changes, sometimes slowly and other times rapidly. The axis around which a body rotates is not bedrock and is not held fast by bedrock. — Fooloso4

The riverbed is bedrock. Bedrock changes slowly, because it itself is held in place by its relation to a slowly changing surround.

Of course they are true or false. Wittgenstein isnt denying this. — Joshs

Right, Wittgenstein is not, but you said:

hinge propositions, forms of life and language games are neither true nor false. — Joshs

If the only example he gives of a hinge propositions is true, then at least some hinge propositions are true.

The riverbed is bedrock. — Joshs

Bedrock is not made partly of sand:

OC 99. And the bank of that river consists partly of hard rock, subject to no alteration or only to an
imperceptible one, partly of sand ...

The spade may turned when digging in a river-bank, unless it hits a rock, but a rock is not bedrock.

The river-bank analogy refers to empirical propositions (96), Bedrock occurs once (498) and refers to what is beyond doubt.

The river-bank analogy refers to empirical propositions (96), Bedrock occurs once (498) and refers to what is beyond doubt. — Fooloso4

The river-bank analogy refers to the way that empirical
propositions can harden and change into conditions of possibility for empirical propositions. Wittgenstein distinguishes thoughout ‘On Certainty’ between empirical propositions and those propositions which we do not know through the test of experience, but which instead ground a way of interpreting experience. This is the distinction between the riverbed’s bedrock ( what is beyond doubt) and the shifting waters of the stream (empirical experience) that runs through it.

98. But if someone were to say "So logic too is an empirical science" he would be wrong. Yet this is right: the same proposition may get treated at one time as something to test by experience, at another as a rule of testing.

An important difference between Gödel and Wittgenstein is that for the latter the synonymous concepts of hinge propositions, forms of life and language games are neither true nor false. They are outside all schemes of verification, since such schemes presuppose them. — Joshs

Axioms in a formal logic system are also outside all schemes of verification, because they are presupposed and form the basis on which those schemes proceed. Whether or not axioms are 'true' is, as I understand, an open question in the philosophy of mathematics. From this thread, I gather that the truth-value of 'hinge-propositions' is an open question in philosophy too.

Again, I fail to see a distinction between the 'hinge-propositions' that are being discussed here and the simple concept of axioms – claims we accept without proof in order to begin reasoning in the first place. But I also admit I do not recognise Wittgenstein as having theorised anything called a 'hinge-proposition' in 'On Certainty'. I accept that he used a door hinge as a metaphor for the way we reason, in sections 341 and 343 and again in 655. But the metaphor was not very thoroughly pursued in any of these cases, and did not strike me as particularly crucial to his line of inquiry.

Of course, the academic consensus would strongly suggest I'm wrong – that 'hinge-propositions' do indeed form a key part of Wittgenstein's argument in 'On Certainty'. I just can't seem to make that out in the text itself.

I do not recognise Wittgenstein as having theorised anything called a 'hinge-proposition' in 'On Certainty'. I accept that he used a door hinge as a metaphor for the way we reason, in sections 341 and 343 and again in 655. But the metaphor was not very thoroughly pursued in any of these cases, and did not strike me as particularly crucial to his line of inquiry.

Of course, the academic consensus would strongly suggest I'm wrong – that 'hinge-propositions' do indeed form a key part of Wittgenstein's argument in 'On Certainty'. I just can't seem to make that out in the text itself. — cherryorchard

I consider the notion of hinge proposition to be redundant; it’s just another way for Wittgenstein to talk about language game and forms of life, as a hinge on the basis of which we organize so many empirical claims that it makes no sense to subject it to doubt.

It's not a good idea to consider hinges to be "outside our epistemological framework", anymore than it would be a good idea for a hydrologist not to consider the riverbed. They are rather the foundation on which an "epistemological framework" rests.

Nor is it a good idea to think of hinges as not propositional. if they are not propositional then they cannot fulfil the task set them, which is to show that other propositions are true. They cannot act as a hinge unless they are true.

It's a bit like saying that the hinges of a door must be either part of the door or part of the door frame, and so failing to recognise that they are neither and both.

Nor is it a good idea to think of Gödel's unproven sentences as "outside the system" - they are very much a part of the system.

riverbed’s bedrock ( what is. beyond doubt) — Joshs

Saying the riverbed's bedrock is not the same things as saying:

The riverbed is bedrock. — Joshs

I suspect that his use of the river analogy intentionally points back to Heraclitus. He says, for example::

97. The mythology may change back into a state of flux, the river-bed of thoughts may shift. But I distinguish between the movement of the waters on the river-bed and the shift of the bed itself;
though there is not a sharp division of the one from the other.

The mythology is our world picture (95). That the riverbed of thought can change back into a state of flux means that it is not entirely stable or unchanging. It may not be doubted at some given point in time, but consider his example of being on the moon. It was not too long ago that the proposition: Man has never been on the moon, was beyond doubt. Although there are still some who doubt it, it is part of our scientific world picture that man has been on the moon. It is beyond doubt that we have been there. As before it was beyond doubt that we were not.

The riverbed is bedrock. — Joshs

Nuh. The river bed is silt, sand and rocks. It stays relatively fixed while the river flows past. If it didn't, we wouldn't have a river - we'd have a swamp or a delta or some such.

It was not too long ago that the proposition: Man has never been on the moon, was beyond doubt. Although there are still some who doubt it, it is part of our scientific world picture that man has been on the moon. It is beyond doubt that we have been there. As before it was beyond doubt that we were not — Fooloso4

So what do you think it means to say that some proposition is part of our world picture? Is a world picture simply a set of facts that we believe are verifiably true? Or is a world picture a system of relations that include certain possibilities and exclude others? Is it the fact that man has been on the moon that alone constitutes the ground for its indubitably, or is it a system of grounds underlying this fact which make the fact indubitable (our awareness of the the science of space flight and our trust of media)?

108. "But is there then no objective truth? Isn't it true, or false, that someone has been on the moon?" If we are thinking within our system, then it is certain that no one has ever been on the moon. Not merely is nothing of the sort ever seriously reported to us by reasonable people, but our
whole system of physics forbids us to believe it. For this demands answers to the questions "How did he overcome the force of gravity?" "How could he live without an atmosphere?" and a thousand others which could not be answered. But suppose that instead of all these answers we met the reply:
"We don't know how one gets to the moon, but those who get there know at once that they are there; and even you can't explain everything." We should feel ourselves intellectually very distant from someone who said this.

Nuh. The river bed is silt, sand and rocks. It stays relatively fixed while the river flows past. If it didn't, we wouldn't have a river - we'd have a swamp or a delta or some such. — Banno

When I have exhausted my justifications, I come
face to face with the limits that define the boundaries of a language game. This bedrock is fixed, but only relatively so.

If I have exhausted the justifications I have reached bedrock, and my spade is turned. Then I am inclined to say: "This is simply what I do."

Cognitive science and human language development are outside the scope of the propositions themselves, yet they explain a brain that can create propositional statements about the world.

Yep. And I also agree that hinges get too much attention.