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
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
I’m assuming you understand Wittgenstein’s point about hinges in OC. — Sam26
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."
... 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.
If the proposition 12x12=144 is exempt from doubt, then so too must non-mathematical propositions be.
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
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
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."
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
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
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
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.
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
https://www.thecollector.com/ludwig-wittgenstein-on-certainty/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”.
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
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
Of course they are true or false. Wittgenstein isnt denying this. — Joshs
hinge propositions, forms of life and language games are neither true nor false. — Joshs
The riverbed is bedrock. — Joshs
The river-bank analogy refers to empirical propositions (96), Bedrock occurs once (498) and refers to what is beyond doubt. — Fooloso4
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
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
riverbed’s bedrock ( what is. beyond doubt) — Joshs
The riverbed is bedrock. — Joshs
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.
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
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
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."
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.