You are still conflating justification with truth, and consequently ascribing views to both of us that we did not state and do not hold. — aletheist

Do you think justification is necessary for truth?

I would say it's the toes.

But seriously speaking, other things are necessary than just the lack of known counterexamples. So I say this is a false assessement of my stand, please disregard the quote by TheMadFool that attributes this to me.

Now I am not confident that @aletheist claim is properly written, either.

So I refuse to make a statement on a potential Strawman. — god must be atheist

Sorry. Thanks.

You say that if a claim doesn't have a counterexample it can be taken as proof.

That's the fallacy of the argument from ignorance. There's a third option between true and false which is "I don't know".

Do you think justification is necessary for truth? — TheMadFool

No, justification is about why someone believes a proposition, while truth is about whether that proposition represents reality. Besides, if justification were necessary for truth, then only justified beliefs could be true, which is not the case.

You say that if a claim doesn't have a counterexample it can be taken as proof. — TheMadFool

No, he did not say that, either. He said that an absence of counterexamples can be taken as justification for a belief; proof is a stronger form of justification.

Besides, if justification were necessary for truth, then only justified beliefs could be true, which is not the case. — aletheist

Can you name one unjustified truth?

Can you name one unjustified truth? — TheMadFool

Lots of people believe things without justification that happen to be true. That is why the standard modern definition of knowledge is justified true belief, not merely true belief.

Surely Euclid's theorems were provisionally held to be as good as proven until we realised they are a special case of something bigger. Likewise Newton's vis-à-vis the quantum/relativity.

Basically, what we thought we "knew" turns out to be just part of a spectrum or continuum. Spectrums are not thin lines but more like a solid cheese with textures. Nuancing science is what mostly happens, rather than "revolutions" or even "falsifying" (unless the proposition had been very narrow).

So, knowing and believing are not as dodgy as we might fear, provided we are respectful of their complexity.

I think the subtlety and slight but not excessive looseness in all this is refreshingly hard-nosed. I've always intuited this, I think the family and my teachers always accepted something like this.

It's tragic how often the "great" and the "good" deny that the middle needn't be excluded when it doesn't contradict the law of non-contradiction.

I read in Knowles that Anselm (and I type from memory) translated "necessarius" as "admissible" and not "compelling". (However he spoiled everything by attempting a so-called "ontological proof" which I am told Descartes swallowed.)

0

Nuancing science is what mostly happens, rather than "revolutions" or even "falsifying" (unless the proposition had been very narrow). — Fine Doubter

This is truly a great observation although I was hoping the theories themselves constituted subtleties that are missed by the novice like me. However, you meant something else - the progress of science and how it's made of small drifts rather than major shifts. Am I correct? Kindly clarify.

I read in Knowles that Anselm (and I type from memory) translated "necessarius" as "admissible" and not "compelling". (However he spoiled everything by attempting a so-called "ontological proof" which I am told Descartes swallowed.) — Fine Doubter

Did Anselm use "necessarius" in a logical context? Did he make a mistake?

Lots of people believe things without justification that happen to be true. That is why the standard modern definition of knowledge is justified true belief, not merely true belief. — aletheist

:up: :ok: Thanks

The Mad Fool,
1 - I am sure all theories contain subtleties which however are not beyond our faculties to contemplate as we help each other along. The other thing is, "falsification" and "revolutions" in the rather weak sense might not happen for several generations. It looks silly for some people to crow that they have "ruled out" something that was the all-the-rage hypothesis a few years ago, when in 200 years time they will have arrived at a far more compendious halfway house. (I except the so called "many worlds" which in fact didn't make sense, all along, so I am not really excepting it - a thread of its own.)
2 - I haven't been reading Anselm, this was alluded to in passing by Knowles, 'The evolution of medieval thought'. But I don't think Anselm made a mistake - although some of his contemporaries perhaps gave it a little stronger meaning - and perhaps not much stronger (again my feeling from secondary sources). I think it is very important for us moderns to realise when we don't need to be quite so categorically categorical. I think of reality as one of those old ships that flexes and creaks - and sails. Not as a Titanic that cracks up (without us noticing). Principles are there for the apply-ing. Perhaps "necessarius" means "the rule" and a word for contingent would mean something like occasional and quasi-random. The two are of course on a continuum and might even sometimes coincide, in my opinion. I think it is vital to remember that there are two kinds of probability - both occurrences, and our knowledge of them. I think of reality as being on the non-approximate side of approximation.
3 - An interesting thing about Gödel indeed is that what he describes as "proof" can reputedly sometimes occur without help from an outside system, however it rather often does require input from outside. I really adore ifs and I really adore buts - I think they are so firm and reliable.

. I really adore ifs and I really adore buts - I think they are so firm and reliable. — Fine Doubter

Me too. :joke:

Isn't true but unprovable a contradiction? — TheMadFool

There is something very syntactic about the proof. The starting point is the diagonal lemma, which says:

• °x is the numeric encoding of number x
• #(θ) is the numeric encoding of formula θ

Lemma:

For each formula F with one free variable in the theory T, there exists a sentence ψ such that:

ψ ↔ F(°#(ψ))

is provable in T.

If we now look at the provability function B(x) which accepts the encoding of any sentence and returns yes/no if it is provable, we can see that it has a strange "corner case". By choosing F(x)=~B(x), we obtain the following proposition:

ψ ↔ ~B(°#(ψ)) is provable in T

The sentence ψ is not provable in T, because that is exactly what the expression says. So, this "corner case" sentence is true. This issue is caused by the diagonal lemma, which insists that each one-variable formula in the theory must have a fixed point.

The proof for the diagonal lemma itself is also purely syntactic. It is the result of just symbol manipulation. The final result may give a semantic impression, but Gödel's theorem is also purely syntactic.

The proof for the diagonal lemma itself is also purely syntactic. — alcontali

Like "Colorless green ideas sleep furiously"? No meaning? I though proofs need to be meaningful.

Like "Colorless green ideas sleep furiously"? No meaning? I though proofs need to be meaningful. — TheMadFool

The justification has to be purely syntactic. I am currently still looking into what it means for alternative functions F(x):

ψ ↔ F(°#(ψ)) is provable in T

Since it should work for any function F(x), I think that the secret to a proper interpretation of this lemma can be found by looking at alternative choices for F(x). The following stackexchange question also asks for alternative examples, but I find the answers not particularly satisfactory. Maybe there are better links for this?

:up: :ok:

Strictly a meta-comment here: this discussion is a good example or why I come out to this forum. A great exchange of ideas which leaves me more informed than before. And on top of that, no insults! :smile:

True BUT unprovable? How?

That a proposition is true means you have a proof that demonstrates that proposition to be true and unprovable means you don't have a proof. :confused: :chin:

Isn't true but unprovable a contradiction? — TheMadFool

No. Consider Goldbach's conjecture(GC):Every even integer greater than 2 can be expressed as the sum of two primes.

GC may be true even if it cannot be proven true. If true, and unprovable, it is unknowable.

may be — Relativist

Important clause.

But it demonstrates that conceptually, propositions can be true but unprovable.

But it demonstrates that conceptually, propositions can be true but unprovable. — Relativist

There's a difference between "can be true" and "is true". Godel statements are not of the "can be true" type, they're of the "is true" type. The transition from "can be true" to "is true" is done via justification aka proof. That's the usual procedure anyway.

Unprovable in the system, but provable outside the system.

Godel's G says that G is unprovable. And within G's system, basically arithmetic, G cannot be proved. Nor it's negation, ~G. But we can think about what that means, and that thinking shows that G is true.

It's an exercise to work it through.

Fair point, but this leads to the question of what constitutes reasonable justification? Most beliefs are not provable by deduction - we make inductive inferences all the time. You can't prove the sun will rise tomorrow, but it's reasonable to believe it will. Science advances by abductive reasoning, not deduction. To deny justifications other than deduction leads to extreme philosophical skepticism, which is paralyzing.

https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714

A fairly decent layman's guide I think, if anyone is interested.

@Relativist

Unprovable in the system, but provable outside the system. — tim wood

The problem is this: if I can't prove a proposition in a given system x, that proposition can't belong to that system. Proof of a proposition is a precondition for inclusion in a system. Ergo, no Godel sentence for system x, since it can't be proved in x, can be part of the system x. For instance I can't prove anything about quantum physics in the system of theism, so, how can anything about quantum physics be part of theism?

I can form the grammatically correct sentence "the uncertainty principle is true" with words that occur in theism in statements like " it is true there's no uncertainty regarding god's existence", "god commands us to live by the moral principles he laid out". The uncertainty principle is a fact in quantum physics, a totally different system to theism and insofar as theism is concerned, the sentence "the uncertainty principle is true" is meaningless.

Could all Godel sentences be of such kind, syntactically sound but semantically empty? If yes then Godel's incompleteness is "harmless" to mathematics because Godel sentences are meaningless gibberish spoken with perfect grammar.

Take a look at Un's pot just above yours and follow the link.

if I can't prove a proposition in a given system x, that proposition can't belong to that system. — TheMadFool

That's not true.Goldbach's Conjecture is not "semantically empty": all the terms and relations are well defined (prime number, even number, sum). If true and unprovable, it is because the formal system of peano arithmetic is incomplete. That's what Godel's incompleteness theorem is all about: incompleteness = there are unprovable true statements in the formal system.

I can form the grammatically correct sentence "the uncertainty principle is true" with words that occur in theism in statements like " it is true there's no uncertainty regarding god's existence", "god commands us to live by the moral principles he laid out". The uncertainty principle is a fact in quantum physics, a totally different system to theism and insofar as theism is concerned, the sentence "the uncertainty principle is true" is meaningless. — TheMadFool

Godel's Incompleteness theorem is about formal mathematical systems. It has no metaphysical implications. Similary, the uncertainty principle of Quantum Mechanics has no relation to the uncertainty various metaphysical claims.

@unenlightened

Take a look at Un's pot just above yours and follow the link. — tim wood

I must confess that I didn't understand the proof in the link but one thing stood out: the proof in the link made the following contradictory claims:

The mapping works because no two formulas will ever end up with the same Gödel number. — Quanta article

and

“The formula with Gödel number sub(y, y, 17) cannot be proved” — is sure to translate into a formula with a unique Gödel number. Let’s call it n.

Now, one last round of substitution: Gödel creates a new formula by substituting the number n anywhere there’s a y in the previous formula. His new formula reads, “The formula with Gödel number sub(n, n, 17) cannot be proved.” Let’s call this new formula G.

Naturally, G has a Gödel number. What’s its value? Lo and behold, it must be sub(n, n, 17). By definition, sub(n, n, 17) is the Gödel number of the formula that results from taking the formula with Gödel number n and substituting n anywhere there’s a symbol with Gödel number 17. And G is exactly this formula! Because of the uniqueness of prime factorization, we now see that the formula G is talking about is none other than G itself. — Quanta article

1. There's a formula with Godel number sub(n, n, 17)

AND

2. The Godel number for "the formula with Godel number sub(n, n, 17) cannot be proved" is also sub(n, n, 17)

It's like the Godel number critical to the proof is ambiguous and, from the rudimentary math I know, Godel numbering is then not a function. How this impacts Godel's proof? I have no idea.

:chin:

So what do you think is happening here? Do you think that Godel's argument fails? Or do you think that you need to do more to understand it?

One choice leads to understanding. The other, to psychoceramics.

What's your point Madfool? Consider sentence S, "This sentence is unprovable." Three options: it does not meet the criteria for a meaningful English sentence, or it's true, or it's false.

1) It's meaningless. Well, that would be the end of it, Except that S is not meaningless and it is a well-formed and meaningful English sentence.

2) It's true. In that case there are true sentences that are not provable.

3) It's false. If it's false then it must be provable, which means that you have proved a false proposition. In most systems proving P and ~P is a catastrophe for the system.

Getting all this into arithmetic is more than a little tricky, but @unenlightened's link above is as he says "a fairly decent layman's guide." Go back to it and stay with it until you've got it.

So what do you think is happening here? Do you think that Godel's argument fails? Or do you think that you need to do more to understand it?

One choice leads to understanding. The other, to psychoceramics. — Banno

What's your point Madfool? — tim wood

The issue is simple: Godel claims any given axiomatic mathematical system is incomplete because there's always going to be a mathematical proposition that's both

1. true

AND

2. unprovable in the system.

I want to know how a proposition can be true sans any justification. The normal, standard procedure when it comes to assigning truth values is to offer proof but Godel is quite clear on the unprovable nature of Godel sentences.

Then tim wood said:

Unprovable in the system, but provable outside the system. — tim wood

This, however, is incorrect. All articles on Godel's theorems only mention adding Godel sentences as axioms to the axiomatic system they render incomplete. Nowhere is it mentioned that they're "provable outside the system".

:chin: