Just asking for a bit of clarification. According to what you quoted, The Book involves only proofs of mathematical theorems. But you say that it includes proofs of every proposition, "mathematical or otherwise." Are you referring to an imaginary different book?

The rationale is simple: no proof at all for G is indistinguishable from there is proof of G but you haven't discovered it yet) — Agent Smith

Ah, I see. The fact that there's no proof at all of your version of The Book, that doesn't mean there is no proof of your version of The Book, right? Or the fact there is no proof at all of your version of The Book is indistinguishable from there is proof of your version of The Book but we haven't discovered it yet.

The Argumentum ad ignorantiam fallacy states that just because G hasn't been proved, we can't then conclude ~G. The rationale is simple: no proof at all for G is indistinguishable from there is proof of G but you haven't discovered it yet). — Agent Smith

Digression: The time to believe in something is when there is sufficient evidence. Sure, lack of evidence is not proof against a proportion, however that does not imply 'believe it anyway'. A responsible atheist would not say there is no god, they would say there is no good reason to believe in god. Just as there is no good reason to believe in goblins - even though goblins also can't be disproved. That said, reason is largely bypassed on this question anyway. Most tend to choose beliefs like these based on emotional grounds and dress them up with reasons.

Years ago, when I was giving a talk about one of my research projects, Erdős sat in the front row and listened intently for a few minutes, then lost interest and began doodling, then took a short nap.

He was a very unusual character, traveling around the world with a grocery bag of belongings, staying with fellow mathematicians. He never married and would live with his ageing mother in her apartment in Budapest when not going from place to place. He once stayed at the home of my advisor.

Have fun with The Book.

"God exists" is not a well-formed formula in mathematics. Hence it cannot be a theorem. The book contains only proofs of theorems. SO "God exists" could not be int he book.

So it's a muddled question.

More interesting is Gödel's work, which shows that the book can never be complete. There will always be missing theorems.

No proof for G is found. Can I then conclude that ~G = God doesn't exist, is true? — Agent Smith

No. Absence of proof is not proof of absence.

Just asking for a bit of clarification. According to what you quoted, The Book involves only proofs of mathematical theorems. But you say that it includes proofs of every proposition, "mathematical or otherwise." Are you referring to an imaginary different book? — Ciceronianus

Yes, I'm talking about a book, The Book that contains proofs of even nonmathematical true propositions from other disciplines. Consider my The Book to be an expanded version of Erdős' The Book.

Ah, I see. The fact that there's no proof at all of your version of The Book, that doesn't mean there is no proof of your version of The Book, right? Or the fact there is no proof at all of your version of The Book is indistinguishable from there is proof of your version of The Book but we haven't discovered it yet. — Ciceronianus

Kinda. I'm trying to see if there's a logical connection between

1. There's no proof at all of p (a proposition) in The Book

and

2. ~p

Does 1 imply 2 and vice versa?

Please note that

3. p hasn't been proven

doesn't imply that

4. ~p

To think 3 $\rightarrow$ 4 is a fallacy (argumentum ad ignorantiam/argument from ignorance).

Please pay attention to 1 and 3. They look similar but they're not. In the case of 1, the universe of proofs (The Book) is exhausted and no proof of p is found but in the case of 2, this isn't so. 2 and 4 are identical.

Digression: The time to believe in something is when there is sufficient evidence. Sure, lack of evidence is not proof against a proportion, however that does not imply 'believe it anyway'. A responsible atheist would not say there is no god, they would say there is no good reason to believe in god. Just as there is no good reason to believe in goblins - even though goblins also can't be disproved. That said, reason is largely bypassed on this question anyway. Most tend to choose beliefs like these based on emotional grounds and dress them up with reasons. — Tom Storm

To go from "there is no good reason to believe in god" and "there is no good reason to believe in goblins" to there is no god and there are no goblins is to commit the argumentum ad ignorantiam fallacy (vide infra).

1. There is no proof of p (there is/are a god/goblins)
Ergo,
2. ~p (there is/are no god/goblins)

Now consider the following:

Suppose p = there is/are a god/goblins.

3. p cannot be proven (there is no proof of p in The Book)

Can I now infer that,

4. ~p

???

"Most tend to choose beliefs like these on emotinal grounds" :up: I think so too but some do try and reason their position.

Have fun with The Book. — jgill

:up: So your Erdős number is low. I think someone with a mathematical background can shed light on what I'm trying to say.

There are some mathematical propositions that cannot be proven (say p) given a particular axiom set, call it A. Can I then conclude that in A, ~p?

"God exists" is not a well-formed formula in mathematics. Hence it cannot be a theorem. The book contains only proofs of theorems. SO "God exists" could not be int he book.

So it's a muddled question.

More interesting is Gödel's work, which shows that the book can never be complete. There will always be missing theorems. — Banno

You've not fully grasped what I'm trying to say. My fault entirely. Please read the replies vide supra for clarification.

Yes, Gödel's incompleteness theorems have significance to the discussion at hand.

As per Gödel, given an axiom set A there are mathematical statements (p) that can't be proven (no proof of them exist in The Book) BUT they're true).

Gödel's discovery contradicts the following argument:

1. p cannot be proven (no proof of p in The Book)
Ergo,
2. ~p (p is false)

However, the matter is quite complicated when it comes to Gödel: Is the sentence (Gödel sentence) "this statement is true but cannot be proven" a proof of itself? Other issues may exist but currently I'm unaware of what they are.

No. Absence of proof is not proof of absence. — Book273

Yep. Argumentum ad igonrantiam fallacy but what if for a proposition p, it's the case that p cannot be proven. Sounds similar to "absence of proof" but they describe entirely different scenarios. Vide supra (replies to other posters) for further clarification.

Thanks to all. Good day.

To go from "there is no good reason to believe in god" and "there is no good reason to believe in goblins" to there is no god and there are no goblins is to commit the argumentum ad ignorantiam fallacy (vide infra). — Agent Smith

You've missed the point. You keep skipping ahead. A responsible atheist does not say there is no god. S/he says there is no good reason to believe in god - the case has not been made. It's like a murder case in law. A person found not guilty is not innocent. They simply have not met the legal criteria for guilt.

Update

Suppose we start off with an axiom set A.

We write The Book containing every true proposition that are supported in A.

Take a proposition p.

It's discovered that The Book contains no proof of p. In other words, p cannot be proven in A.

Is it the case that ~p in A?

You've missed the point. You keep skipping ahead. A responsible atheist does not say there is no god. S/he says there is no good reason to believe in god - the case has not been made. It's like a murder case in law. A person found not guilty is not innocent. They simply have not met the legal criteria for guilt. — Tom Storm

Ah! So, "there is no good reason to believe in god" implies atheism but it doesn't imply "there is no god"? :chin: There are brands of atheism consistent with this line of reasoning. Could you elaborate on that. Thanks.

Have fun with The Book. — jgill


:up: So your Erdős number is low. I think someone with a mathematical background can shed light on what I'm trying to say. — Agent Smith

Yes, proud to say it's 0. I'm happy to shed no light on what you are trying to say. :cool:

Yes, proud to say it's 0. I'm happy to shed no light on what you are trying to say. :cool:
nowReplyOptions — jgill

:sad: I was hoping for more than that. Can't have it all, right?

Yes, proud to say it's 0 — jgill

:scream: You're Paul Erdös!

You're Paul Erdös! — Agent Smith

Wrong. He would not be seen dead in my image. :snicker:

Wrong. He would not be seen dead in my image. — jgill

You underestimate yourself! Perhaps I overestimate Erdös.

About Gödel

Gödel claims that given an axiom set A, there are true propositions (say p) that are true but not provable in A.

So, p is true but the question is, is p true in A or is p true in a different axiom set B?

Another way of saying that perhaps is, is p false in A but true in another axiom set B?

:chin:

Gödel claims that given an axiom set A, there are true propositions (say p) that are true but not provable in A. — Agent Smith

Are you sure about that? :chin:

Are you sure about that? :chin: — jgill

That's what I know about Gödel's incompleteness theorems.

Consider:

This statement does not appear in the book.

It is either false or not in the book, and hence the book is incomplete.

The book is either inconsistent or incomplete.

If it is inconsistent then one cannot rely on any proof that god exists.

If it is incomplete then one cannot conclude from god's not being mentioned that god exists or does not exist.

So the book is irrelevant.

Ah! So, "there is no good reason to believe in god" implies atheism but it doesn't imply "there is no god"? :chin: There are brands of atheism consistent with this line of reasoning. Could you elaborate on that. Thanks. — Agent Smith

Yes, however I'm no expert on atheism. Atheism is not a philosophy and it has no doctrines. Some atheists believe in the supernatural, for instance. And remember, most people who believe in a god, say, Allah, are atheists with regard to hundreds of other deities humans believe in. Most people are therefore atheists of a sort.

Not everyone agrees on categories - like any other area of belief. I am an agnostic atheist. This means I am atheist regarding belief - I am unable to believe in a god/s - and I am agnostic about whether knowledge of god/s is possible. I notice a similar view is held by American Atheists.

Consider:

This statement does not appear in the book.

It is either false or not in the book, and hence the book is incomplete.

The book is either inconsistent or incomplete.

If it is inconsistent then one cannot rely on any proof that god exists.

If it is incomplete then one cannot conclude from god's not being mentioned that god exists or does not exist.

So the book is irrelevant. — Banno

:chin:

T = This statement does not appear in The Book.

T can't be in The Book because then it would contradict itself. :ok:

If T is not in The Book then, you claim, The Book is incomplete. T could be false. If so ~T = This statement is in The Book. Nothing's amiss!

For the moment let's disallow self-referential sentences.

Is the following true:

1. If p cannot be proven (there is no proof of p in the axiomatic system A we're working in) then p is false (~p) in axiomatic system A.

What about undecidability?

Allow me to rephrase my question:

Is p cannot be proven true logically equivalent to it's impossible that p (is true)?

Theism/atheism is relevant to the discussion because

1. Neither is there proof that god exists, nor is there proof that god doesn't exist.

2. If god exists is unprovable, does it mean god doesn't exist?

Likewise,

3. If god doesn't exist is unprovable, does it mean go exists?

Please note the following difference:

Hasn't been proven vs. Can't be proven.

The difference is obvious:

There may be proof, we haven't found it vs. We looked, there is no proof at all (respectively).

That's all I have so far.

What about undecidability? — Agent Smith

I haven't found it hard to decide that I don't believe in gods and I have a high degree of confidence that the idea is false (based primarily on a familiarity of the classical arguments and the work of apologists, even presuppositional apologetics). But, importantly, one can't prove a negative. As you may have read elsewhere, the burden of proof is on the person making the claim about god or, for that matter, the Flying Spagetti Monster.

There are a lot of things I don't believe in and consider to be false but can't as yet be 'proven' to be false - Bigfoot; the Loch Ness Monster; alien abductions, leprechauns, Russell's infamous teapot. You could devise a very long list of such things.

Digression: The idea of god/s is so incoherent for me that the idea can only be accommodated through a perspective of mysticism (where reason is not involved) - and for which I have some sympathy. Certainly the least concrete, abusive and nasty accounts of theism seem to be those of the mystics, particularly in the Christian tradition, from Gregory of Nyssa to Thomas Merton.

But, importantly, one can't prove a negative. — Tom Storm

Yes, that's precisely what led me to this topic.

Why is the default truth value for a proposition false? Is it though? Atheism?

That seems to be linked to the present discussion on whether the fact that p can't be proven implies ~p.

JTB Theory of Knowledge & Gödel

For a proposition p,

1. If p is true then, there is a proof of p [justification is necessary for the truth of a proposition]

which means

2. If there is no proof of p then, p is false! :chin:

There is no proof of p = p can't be proven = there is no proof of p in The Book

However, Gödel claims that there are propositions (say p) that are true but unprovable. This basically means:

3. p is true & p has no proof

that means

4. ~(If p is true then, there is a proof of p) i.e. 1 is false.

:chin:

Why is the default truth value for a proposition false? Is it though? Atheism? — Agent Smith

I'm not a philosopher, so this question probably has a correct answer unknown to me.

I don't think the default is false - it has to do with the nature of the proposition. If you tell me that you have a pet dog at home I am not going to assume the proposition is false as dogs are an everyday thing we all know and can demonstrate to exist.

If, however, you tell me there is an Elf who lives in your pocket, I am going to need you to provide some proof as this is a claim of extraordinary nature. In this instance the burden of proof is upon you.

Mainly because people won't give a shit whether you have a dog at home or not. There is nothing at stake in this proposition. Different types of claims require a different approach.

:up: I seem to have made the rookie mistake even veteran philosophers (love to) make. Sweeping generalizations. Hilary Putnam says something to the effect that philosophers love to generalize, they always fail but that doesn't stop them. Oops! Putnam generalized.

Thank you for your input. Arigato gozaimasu!

:up: Nice chatting to you.

Questions:

1. Does the truth of a proposition p require a proof (of p)? In other words does p imply there is a proof of p? What is the purpose of a proof?

2. What is knowledge? JTB? If so is a Gödel sentence (true but no proof) knowledge? How does this relate to the Gettier problem?

3. What's the default truth value of a proposition p, given no proof that either p or ~p? Is it unknown (agnosticism) or is it ~p (atheism)?

4. If p cannot be proven does it mean that impossible that p?

So, there's The Book, generalizing Erdős' idea, which contains all the proofs, elegant or inelegant (I'm not as demanding as Erdős), for each and every true proposition, mathematical or otherwise. — Agent Smith

There are many books in the world. Let one of these books be The Book.

Some of these books have flawed proofs and untrue propositions, whilst The Book has true proofs of true propositions.

So how would it be possible to know which of the many books is The Book, in order to be able to say "there's The Book" ?

IE, even if we were looking at The Book directly in front of us, we wouldn't be able to recognise it as The Book (a bit like my posts, they hold the truth, yet tend to be ignored).

The Book is occupies the region between hypothetical & actual. Do we just have to figure everything out and compile the treatise or do we read off proofs from its imaginary pages?

By the way what is the purpose of a proof if it isn't sufficient to show that the proposition for which the proof is written is true?

My take would be if there is no proof of p, there's no reason to think p exists. Depending on what p is imagined to be, however, we may make inferences regarding the likelihood of p's existence. That we have no proof there is a planet-sized turtle orbiting the sun doesn't mean there could be one.