@TheMadFool, justification may be 100%, but justification may be not perfect, either.

If no counter-examples are found, and no proof is presented, then the justification is not perfect, but accepted as true.

For instance, if you substract a positive number from a larger positive number, you get a positive number. There is no proof for this, despite it ensuing from mathematical axioms. No proof, but then again, nobody can find a counter-example for it, either.

justification may be 100%, but justification may be not perfect, either.

If no counter-examples are found, and no proof is presented, then the justification is not perfect, but accepted as true.

For instance, if you substract a positive number from a larger positive number, you get a positive number. There is no proof for this, despite it ensuing from mathematical axioms. No proof, but then again, nobody can find a counter-example for it, either. — god must be atheist

a > b > 0
a > b
a - b > b - b
a - b > 0

Anyway

You made an interesting point here viz. but I want to ask you a question:

How do you know that a proposition P is true?

You need to justify P i.e. provide a logical argument to show that P is true.

How else are you doing philosophy or math or science or even just living?

This is at odds with Godel's claim that there exist propositions in math, say P, that are true but unprovable. How do you know P is true when you didn't prove it to be true? That's my question.

You provided a route out of what I think is a problem viz. that there are no counterexamples. As attractive and comforting as this sounds it's not the way I see mathematical theorems being proved. At best a theorem that doesn't have a counterexample is considered a conjecture that is likely to be true. It seems your method is central to scientific thinking (falsificationism) and you already know that scientific "truths" are provisional - assumed to be true but not for certain. However, math is different. I've never heard of theorems that are provisionally true and I'm reasonably sure that Godel's theorems are deductive in character and so excludes such a method as yours.

@TheMadFool How do you know you're going to die at one point? It is not proven that you'll die. Theoretically speaking, all living things are not proven to die at one point. Yet we know we'll die.

If nothing else, we'll die our hair when we get old enough. (-:

Truth or something being true can be accepted as such, even if not proven, if no counter-examples are given. In the above example, no counter-example is given to your being alive. Yet you know you will not live forever. Despite the lack of proof, and despite the lack of counter-examples. (Others' deaths are not yours; they are only people or things SIMILAR to you, but not quite you, so dead people's examples are, strictly speaking, not counter examples to your immortality.)

Nice proof, by the way, of my "theorem". Haha. You finished me off in a flash. But as a mathematician, using Google, you can surely find examples of stuff on the Internet where a theorem is not proven, but true... the only reason being for accepting it as true, is the lack of counter-examples that prove it wrong.

I can't say much else about this... you insist that for truth, the only valid criteria is a proof, esp. in mathematics. Some others think differently, and that's something I can't force you to accept, to think like them... or like us, since I am with "them" (whoever "they" are). And that, for the umpteenth time, is, that lack of counter-examples constitute a reason believe that something is true, or justified in belief.

I'd like to continue the discussion as you raised an important issue - counterexample as a method of proof.

I'm familiar with the term, counterexample, in the context of logic. It's used as a method to prove invalidity of arguments. Another area where counterexamples show up is in science - the disconfirming observation.

I must agree with you that counterexamples are used in math too but do notice the word itself - counterexample - indicating that it's an instance/example that opposes a claim or theorem/proposition.

So, given the meaning of the word, counterexamples obviously disprove some claim. I mean it's primary use is as a method of disproof and not proof. Although using counterexamples, specifically an absence of them, to prove a proposition, like you're claiming, looks quite attractive, I don't think it's the standard approach to mathematical proofs. If you disagree still, kindly furnish a counterexample that disproves that.

I think there's a solid reason why counterexamples aren't used in proofs. Consider a proposition P. P can either be true or false. Suppose no one has found a proof for P being true. Also suppose that P has no counterexample. The logical conclusion should be that P hasn't been proved either true or false - yes there is this option which you're missing. To assume P as true, which you're claiming should be done. would be committing the fallacy of argument from ignorance - P hasn't been disproved. Ergo, mistakenly, claiming P is true.

Please note that the counterexample method is used in science where all claims are provisional, subject to change/deletion as soon as a counterexample occurs. However, don't forget that in science we need confirmatory evidence too. For example the theory of relativity is true not just because there are no counterexamples but also because it's been confirmed by many other observations. As you can see counterexamples are used mainly for disproving and not proving.

Thanks for your valuable comments.

How is it then that a statement like (refer highlighted section of quote above): "true, but that are unprovable" occur in Godel's incompleteness theorem. — TheMadFool

Justified True Belief (JTB) : Knowledge of proposition P = P is true, P is justified and you believe P. — TheMadFool

An important condition to incompleteness theorem is the axiomatic system is strong enough to do arithmetic.

The concept of complete is that every true statement that can be expressed in the system can be provable. The concept of consistent is simply that there are no contradictions.

The axiom of P = P forms a system of a single axiom where the only statement understandable to the system is P = P which doesn't contradict P = P, so it's complete and consistent.

Incompleteness theorem doesn't create doubts about things like P = P, and even adding a few more rules about P doesn't necessarily run into incompleteness theorem.

Enough rules to do arithmatic are needed.

Once arithmatic is possible, statements understandable to the system can also be statements about the system.

How this is possible is because all statements can be encoded as a number and rules about arithmatic describe numbers and what can be done with them.

Armed with this, it's then possible (in a completely proper mathematical way, not just conceptually) to make statements such as "This theorem is unprovable" and then show we can't prove that theorem with the axiom. But it's a true statements!!! We know it's true because we proved that it's unprovable, there's just no axiomatic way to prove it.

Without getting technical, I hope this gives some key intuitions as to what's going on.

Why this theorem was so important, is because before incompleteness theorem, mathematicians were looking for the "one true system", a system of rules about numbers from which all true statements about numbers can be proven and no contradictions arise.

The "no system of arithmetic can prove it's own consistency" part is, for me anyways, much more abstract relevance, I'm not sure there's some key intuitions about it.

Hence, mathematics are justified beliefs, but not justified true beliefs. — alcontali

Using a clever hack, Gödel manages to create a theorem that is algebraically "true" in the abstract, Platonic world of number theory, but which by simply asserting its own unprovability, is not provable from the construction logic of that world. Hence, true but not provable. — alcontali

Indeed. I sensed that there wasn't the required level of correspondence between Godel's incompleteness theorems (GIT) and the justified true belief (JTB) of philosophy. The shared characteristic between the two I was hoping to emphasize was the need for proof to establish truth. As you can see I'm under the impression that the only way to claim truth in mathematics is by proof. Hence my puzzlement when "true, but unprovable" cropped up in GIT.

Can you tell me more about

algebraically "true" — alcontali

?

My logic:

If a proposition P is true then necessarily that a proof must exist for P being true.

Mx = x is a mathematical proposition
Px = x has a proof
t = mathematical proposition that Godel says is true BUT not provable.

Mt = t is a mathematical theorem
~Pt = not the case that t has a proof
1. Ax(Mx > Px)....premise
2. Mt.......premise
3. ~Pt......premise
4. Mt > Pt.......1 UI
5. Pt...........2, 4 MP
6. Pt & ~Pt....contradiction

As you can see I have to reject one or all of the three propositions 1, 2, or 3.

If a proposition P is true then necessarily that a proof must exist for P being true. — TheMadFool

This is the premise you need to abandon.

There can be true statements that have no proof. Incompleteness shows us an example.

There can also be just "true facts" about numbers and arithmetic that are true and there's simply no proof possible.

For instance, the Collatz conjecture we may simply never be able to prove is true, false, or even undecidable, it just stays unknown (beyond what we can check through computer calculation, which wikipedia says we've done to 87 * 2^60 which seems impressible is minuscule compared to "all numbers"). I.e. it can be "true" but also true that no proof nor proving it's undecidable is possible; some things that "resist refutation" can potentially just stay a big question mark indefinitely. The halting problem is a related concept.

Armed with this, it's then possible (in a completely proper mathematical way, not just conceptually) to make statements such as "This theorem is unprovable" and then show we can't prove that theorem with the axiom. But it's a true statements!!! We know it's true because we proved that it's unprovable, there's just no axiomatic way to prove it. — boethius

Thanks a lot. I remember quite vaguely that self-reference or the liar paradox has something to do with Godel's theorem.

Please note the underlined part of your quote.


If "this theorem T is unprovable" is proof, as you say, then doesn't that mean it's provable after all and that too within the axiomatic system?

How do you go from "this theorem is unprovable" to "this theorem is true but unprovable"?

How did Godel know that "this" theorem was true?

If "this theorem T is unprovable" is proof, as you say, then doesn't that mean it's provable after all and that too within the axiomatic system? — TheMadFool

Yes (that's a good question) and why the phrases "inside and outside the system" come up.

"Inside the system" the theorem isn't provable, there's no problem; the axioms are content to just leave it at that. It's only us outside the system that we realize that if the system can't deal with that statement, then that statement is actually true.

It's basically the liar paradox but there are two different systems to evaluate the theorem, whereas the liar paradox is fully "in our minds" and we can't look at it "outside the system it's expressed in".

So, "within the system" we can follow Godels axioms and statements to arrive at a completely proper conclusion. We need to use reasoning "outside the system" (that are not based on the axioms of the system) to arrive at the conclusion the statement is actually true. We can't do this with the liar paradox and so cannot do a similar thing to conclude it's actually true, as if it's true then it's false; incompleteness is a version of this idea that somehow works out due to these nuances of building it in a system that is smaller and weaker than our own minds and these nuances of "unprovable" doesn't necessarily mean "false"; so saying that statement is "actually true" doesn't make a contradiction with the truth value within the system which just says "unknown (as far as these axioms are concerned)"; i.e. if I say "I don't know if it's raining outside" isn't contradicted by you coming and saying "it's raining".

True things in a deductive system being unprovable.

True things without justification.

Kinda the same thing.

Yes (that's a good question) and why the phrases "inside and outside the system" come up.

"Inside the system" the theorem isn't provable, there's no problem; the axioms are content to just leave it at that. It's only us outside the system that we realize that if the system can't deal with that statement, then that statement is actually true.

It's basically the liar paradox but there are two different systems to evaluate the theorem, whereas the liar paradox is fully "in our minds" and we can't look at it "outside the system it's expressed in".

So, "within the system" we can follow Godels axioms and statements to arrive at a completely proper conclusion. We need to use reasoning "outside the system" (that are not based on the axioms of the system) to arrive at the conclusion the statement is actually true. We can't do this with the liar paradox and so cannot do a similar thing to conclude it's actually true, as if it's true then it's false; incompleteness is a version of this idea that somehow works out due to these nuances of building it in a system that is smaller and weaker than our own minds and these nuances of "unprovable" doesn't necessarily mean "false"; so saying that statement is "actually true" doesn't make a contradiction with the truth value within the system which just says "unknown (as far as these axioms are concerned)"; i.e. if I say "I don't know if it's raining outside" isn't contradicted by you coming and saying "it's raining". — boethius

I watched a video on Godel's theorem. A very simple argument and I think you're on the right track with your explanation.

Basically, every new theorem "proved" by Godel's method may be added to the original axioms creating a new axiomatic system. This new system itself is susceptible to the same problem and we have an infinite regress.

In the Godel statement "This statement is true but unprovable what does "this" refer to? To me "this" looks like a placeholder for theorems that satisfy the Godel condition of true but unprovable or is it that "this" refers to the statment "this statement is true but unprovable"?

If "this" refers to a particular theorem T then we get "Theorem T is true but unprovabl" which is a proof and still inside the axiomatic system that generated the statement.

IF "this" refers to the statement itself: "this statement is true but unprovable" then it doesn't make sense because "this statement is true but unprovable" is NOT a mathematical proposition.

Thanks again.

"This statement is true but unprovable what does "this" refer to? — TheMadFool

Answer: itself. Godel's sentence is an expression of mathematics that can be expressed - roughly - in English. In Godel's paper the sentence reads "17 gen r." What does 17 gen r say? That 17 gen r is not provable. See, no problem with "this" at all.

It's useless to go into this stuff without some knowledge of it - but that is not all that hard to come by, if you're actually interested.

Answer: itself. Godel's sentence is an expression of mathematics that can be expressed - roughly - in English. In Godel's paper the sentence reads "17 gen r." What does 17 gen r say? That 17 gen r is not provable. See, no problem with "this" at all.

It's useless to go into this stuff without some knowledge of it - but that is not all that hard to come by, if you're actually interested. — tim wood

I'll have to delve deeper to understand Godel. Thanks for good advice :rofl:

I sensed that there wasn't the required level of correspondence between Godel's incompleteness theorems (GIT) and the justified true belief (JTB) of philosophy. The shared characteristic between the two I was hoping to emphasize was the need for proof to establish truth. — TheMadFool

Proof does not establish truth, it establishes justification. However, since mathematics is the science of drawing necessary conclusions about hypothetical states of affairs (Peirce), there is a sense in which mathematical justification is equivalent to mathematical truth. A sentence is "true" within a consistent formal system as long as it does not contradict the underlying assumptions (axioms). A sentence is "undecidable" within that same system if it can neither be proved nor disproved on the basis of those particular axioms.

What Gödel proved is that it is possible to formulate an "undecidable" sentence within any sufficiently powerful formal system. However, there are still many formal systems--include standard first-order propositional and predicate logic--that are both consistent and complete, such that it is not possible to formulate an "undecidable" sentence within them.

As I have pointed out in other threads, classical logic does not require proof to establish truth, which is why "proof" by contradiction (reductio ad absurdum) is allowed. By contrast, constructive systems such as intuitionistic logic do require positive proof, such that the Law of Excluded Middle (every proposition is either true or false) does not apply.

@TheMadFool (et al) this is a theorem by Godel, not a proof.

So it stands to be proven or to find counter-examples that prove it wrong.

Therefore your objection or criticism (how can you know what's not proven) hinges upon the truth of Godel's theorem, which FIRST needs to be proven or else counter examples need to be found to disprove it, BEFORE your criticism could find validation.

In a way, this theorem by Godel is recursive, or self-reiterating.

"I can't prove this, but you can accept it as truth, that things can be accepted as truth without a supporting proof."

There, you have it. If you accept it, you accept all similarly stated truths; if you don't accept it, then it is self-evident that it isn't true. But acceptance and non- both depend on an individual's autonomous choice, as the theorem is still a theorem, no proof or counter-example has been presented yet.

@TheMadFool (et al) this is a theorem by Godel, not a proof. — god must be atheist

No, it is called a theorem because Gödel provided a proof; otherwise, it would be called a hypothesis or conjecture. Fermat's conjecture came to be known as a theorem because he claimed to have a proof, which no one ever found; Andrew Wiles finally came up with one in 1994.

No, it is called a theorem because Gödel provided a proof — aletheist

Oops, I screwed up big time. I was sure theorems were unproven theories. My ignorance. Totally.

But now I KNOW the meaning of the expression "Theorem".

Thanks, @aletheist.

By-the-by: what does your moniker mean?

By-the-by: what does your moniker mean? — god must be atheist

Thanks for asking. The Greek word for "truth" is transliterated aletheia, so I call myself "aletheist" because I believe that there is such a thing as (absolute) truth.

Transliterated or translated?

Noble conviction you have.

Now, go out and find it!!!

(Hehehe)

Transliterated or translated? — god must be atheist

Transliterated from the actual Greek work αλήθεια.

Noble conviction you have. — god must be atheist

Thanks; but then, the denial that there is such a thing as (absolute) truth is self-refuting.

You made me think. Truth as such can take some differing forms: 1. logical truhts, math truths, 2. empirical truths (finding real material world as it is), and there may be some others that are so true I don't even know abou them.

Funny thing is humans first envisioned they have the truth about the material world, but solipsism and many of its sub branches destroyed that hypothesis. So the first one we took for granted was the first one we took as unprovable.

The second thing we envisioned as being true are tautisms, logical proofs, math proofs. Then we realized they rest on axioms and have nothing to do with the physical world, despite math's and logic's original uses, that made the physical world be understood better.

Then came quantum mechanics that destroyed by stating counter examples to them, our faith in the tenets of intuitive logic.

So now we are here, not trusting our senses, not trusting our sensibilities, not trusting our intuition.

Transliterated from the actual Greek work αλήθεια. — aletheist

Whew! What a relief. Thanks. You have a nice n superb command of English. And you know how to use a computer for more things than the mainstream. I mean, to get to the Greek keyboard and type actual words is a bit more complex than sending cute cat pictures on the Internet (by females) or sending magnificent pictures of people's own genitals (by males).

The Greek word for "truth" is transliterated aletheia, — aletheist

With any of old Greek, translation and meaning is not a simple topic. The "a" is privative, meaning "without" or "absent." But without or absent what?

"Lethe was also the name of the Greek spirit of forgetfulness and oblivion, with whom the river was often identified.
In Classical Greek, the word lethe (λήθη) literally means "oblivion", "forgetfulness", or "concealment". It is related to the Greek word for "truth", aletheia (ἀλήθεια), which through the privative alpha literally means "un-forgetfulness" or "un-concealment"." (Wiki: "lethe.")

So the Greek ἀλήθεια passes for truth, but isn't. We might go so far as to say that for the ancient Greek aletheia meant the thing was apprehensible, and that he was (had better be) alert and aware and paying attention.

And this seems trivial and too academic, but instead it floats on a deep of buried meaning and significance. In a word or two, for the Greek, his "truth" - aletheia - is a matter of functionality within the community of his concerns (better-worse, bigger-smaller, etc.). For us, most of the time, truth is an abstract quality having to do with demonstrations of abstract concerns to earn a disinterested - uninvested - assent (mainly and mostly, is/is not the case).

E.g., for our Greek, his truth might be, "this ax is a good ax, and sharp. (Or not.)" On the other hand, almost anyone today buying an ax goes to a hardware store, is directed to the section where axes are sold, and buys one - without even the clue that he ought to be concerned with everything about the ax from the direction of the wood-grain of the handle to the steel to the shape, size, and grind of the blade. Or another way, for the Greek, under his aletheia, the truth of his ax lies in its functionality as an ax. Most of us, on the other hand, buy uncritically the idea of an ax, an ax-like object.

Confusion on this means that while you can "translate" ancient Greek, from Homeric/Attic/Classical and New Testament Koine to English all day long and feel good about yourself doing it, you will have at the same time buried both meaning and the possibility of meaning, as well as the possibility of experiencing and understanding that world and how and why it matters. When it comes to biblical "translation," this is fraught territory, filled with ignorance and fraud.

delete

aletheist claims that proofs are sufficient but not necessary for truth.

god must be atheist claims that that truth can be based on the absence of counterexamples.

Where do the two of you stand on each other's claims? Thanks.

aletheist claims that proofs are sufficient but not necessary for truth.

god must be atheist claims that that truth can be based on the absence of counterexamples.

Where do the two of you stand on each other's claims? Thanks. — TheMadFool

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.

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

Proof does not establish truth, it establishes justification. — aletheist

Proofs are sufficient, but not necessary, for justification.

... lack of counter-examples constitute a reason believe that something is true, or justified in belief. — god must be atheist

Justification can be based on the absence of counterexamples.

I tentatively agree with aletheist on what he said in this last post above this one of mine.

Excellent, and I agree with you that justification can be based on the absence of counterexamples. It is what Charles Sanders Peirce called "crude induction," describing it as the "weakest kind of inductive reasoning" and "the only kind of induction that is capable of inferring the truth of what, in logic, is termed a universal proposition." Its chief virtue is that it is readily self-correcting, because it only takes one counterexample. The paradigm case is "all swans are white," which was a justified belief of Europeans for centuries, but turned out not to be true.