The Book!
[quote=Wikipedia]Paul Erd?s (Hungarian: Erd?s Pál [??rdø?? ?pa?l]; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his social practice of mathematics (he engaged more than 500 collaborators) and for his eccentric lifestyle (Time magazine called him The Oddball's Oddball). He devoted his waking hours to mathematics, even into his later years—indeed, his death came only hours after he solved a geometry problem at a conference in Warsaw.[/quote]
[quote=Wikipedia]He had his own idiosyncratic vocabulary; although an agnostic atheist, he spoke of "The Book", a visualization of a book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, "You don't have to believe in God, but you should believe in The Book." He himself doubted the existence of God, whom he called the "Supreme Fascist" (SF).[/quote]
So, there's The Book, generalizing Erd?s' idea, which contains all the proofs, elegant or inelegant (I'm not as demanding as Erd?s), for each and every true proposition, mathematical or otherwise.
Consider now the statement G (Supreme fascist, lovingly shortened to SF) = God exists.
I pick up The Book and I scan it from beginning to end. No proof for G is found. Can I then conclude that ~G = God doesn't exist, is true?
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).
My question: Is it the case that there doesn't exist a proof for a proposition p imply that ~p, without the need for a proof of ~p?
P. S. The Book contains only proofs for true positive statements and not negative statements.
[quote=Wikipedia]He had his own idiosyncratic vocabulary; although an agnostic atheist, he spoke of "The Book", a visualization of a book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, "You don't have to believe in God, but you should believe in The Book." He himself doubted the existence of God, whom he called the "Supreme Fascist" (SF).[/quote]
So, there's The Book, generalizing Erd?s' idea, which contains all the proofs, elegant or inelegant (I'm not as demanding as Erd?s), for each and every true proposition, mathematical or otherwise.
Consider now the statement G (Supreme fascist, lovingly shortened to SF) = God exists.
I pick up The Book and I scan it from beginning to end. No proof for G is found. Can I then conclude that ~G = God doesn't exist, is true?
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).
My question: Is it the case that there doesn't exist a proof for a proposition p imply that ~p, without the need for a proof of ~p?
P. S. The Book contains only proofs for true positive statements and not negative statements.
Comments (43)
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?
Quoting 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.
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.
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.
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. Absence of proof is not proof of absence.
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 Erd?s' The Book.
Quoting 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 [math]\rightarrow[/math] 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.
Quoting 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.
Quoting jgill
:up: So your Erd?s 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?
Quoting 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.
Quoting 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.
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.
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?
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.
Yes, proud to say it's 0. I'm happy to shed no light on what you are trying to say. :cool:
:sad: I was hoping for more than that. Can't have it all, right?
:scream: You're Paul Erdös!
Wrong. He would not be seen dead in my image. :snicker:
You underestimate yourself! Perhaps I overestimate Erdös.
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:
Are you sure about that? :chin:
That's what I know about Gödel's incompleteness theorems.
Consider:
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.
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.
: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.
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.
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.
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:
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.
Thank you for your input. Arigato gozaimasu!
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?
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).
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.
Yes but that there is no reason to think p exists isn't a reason to think p does not exist. That's how things stand as of now I'm told. As one poster remarked, absence of proof is not proof of absence. To say p does not exist, we need proof that p does not exist.
Russell's celestial teapot is a response to some theists who take the stance that because science can't disprove god, god exists. It's basically an argumentum ad ignorantiam (argument from ignorance) fallacy to think this way. Note Russell stops short of claiming there is no celestial teapot in orbit around the sun. The reason: absence of proof is not proof of absence.
To make the long story short,
1. Despite claims that you can't prove a negative, you have to. You know what, this idea is probably a derivative of the legal principles innocent until proven guilty and onus probandi (burden of proof). I'm not sure about this though. However, sometimes it makes more sense to assume truth rather than falsity (Pascal's Wager). Like one poster remarked, it depends on the risk involved.
2. As for the main thrust of my OP (p can't be proven implies impossible that p is true i.e. in bivalent logic, p has to be false), it appears that, here too, the legal principle innocent until proven guilty applies but the argument for falsity/innocence is, to my reckoning, much, much stronger; after all, guilt/truth can't be proved (no proof exists).
Quoting Agent Smith
Pascal's Wager is nuts. What if you 'decide to believe' in the wrong God? It leads you no where. Also there's the problem, as I see it, that beliefs can't just be faked like this. You either believe something or you don't.
Main quest: If it's impossible to prove the proposition p is true , what truth value do we assign that proposition, p?
Side quest: Why can't, rather shouldn't, we prove a negative?
But why?
Given proposition p, p is either
1. True
or
2. False
For me to know about p, I need
3. Proof/justification for p
Suppose I looked thoroughly in The Book (of all proofs) and found p has no proof. Is this equivalent to it is impossible to prove p? So what if it is? More importantly, from this alone, what truth value should I assign p?
Note that there's a difference between I have no proof and there is no proof. If from the former, I say p is false (~p), I commit the argument from ignorance fallacy. From the latter, since it's not the same as the former, I reckon the aforementioned fallacy is not committed. Why? How? My questions.
I have a feeling that I'm getting mixed up between knowledge (JTB) and logic/truth. I'm still in a fog.
I'm not the best person to talk to as logic doesn't engage me. The logical absolutes are tautologies.
No one denies that we can say that God either exists or does not exist. There's your Aristotle. But in the real world it is not always possible to know what is the case.
Justified true belief is not logic. It simply means that a belief held is resting on good evidence. And of course justified true belief is not always correct.
My own view is that is no such thing as absolute truth and that truth is not found outside us but made by humans. No doubt many will disagree.
Tom Storm has stated something very important by proferring a "truth value" over & above the classical Aristotelian true and false viz. unknown.
This is interesting, very interesting in fact, because unknown pertains to, in my universe, knowledge and it's about us, our ignorance to be precise and not, presumably, about a/the propsoition and the state of affairs it talks about. In short, unknown isn't/can't be (?) be a truth value.
As to how all this relates to my OP, all I can say is unknown, if posited as a truth value. is N/A.
Quoting Agent Smith
Quoting Agent Smith
P has a truth value, even though we may never know what it is.
Quoting Agent Smith
No statement can ever be proved
The phrase "to say p does not exist" is incorrect. The question is whether the proposition is true or false, not whether the proposition exists or doesn't exist.
If I say that p is false, then I am saying that Shakespeare did not drink a cup of coffee on the morning of 15 July 1584. If I make such a statement, then it is only reasonable that I justify myself, giving sensible reasons why I believe that my statement is true. But I can never prove my statement. Even science only deals in probabilities, not proofs. No statement can ever be proved with 100% certainty.
Consider an example of deductive reasoning: “All men are mortal. Harold is a man. Therefore, Harold is mortal". It is true that there is a sequence of logical statements, one implying another, and giving an explanation of why a given statement is true. But the starting point is always axioms, accepted "rules", statements or propositions which are regarded as being established, accepted, or self-evidently true, ie, unproved.
IE, for me to say that Shakespeare did not drink a cup of coffee on the morning of 15 July 1584, it is reasonable that I am required to justify my belief giving sensible reasons, but it would be impossible for me to prove my statement, in that it is impossible to prove any statement.
As a wannabe skeptic, that's music to my ears but then deduction?
Quoting RussellA
Nice!
Is there is no proof equivalent to impossible to prove p?
In a legal context, the concept of burden of proof would be more applicable than innocent until proven guilty, I think, as the latter may be said to represent a high burden of proof. Generally, those making a claim have the burden to establish it, or at least make what we love to call a prima facie case, in legal Latin. Then, the burden may shift to the opponent of the claim, rather than the proponent. In some instances, the significance of the public policy behind a law is said to place the burden on the opponent rather than the proponent of a claim.
If the analogy holds, those making the claim that there is a God would have the burden to establish that's the case.
How to prove that the statement "no true statement can ever be proved true" is true.
Analytic statements
Quine in The Two Dogmas of Empiricism distinguished between two kinds of analytic claims - i) logical truths, true no matter how we interpret the non-logical parts in the statement, such as "no not-X is X" - and ii) synonymous truths, such as comparing bachelor with unmarried man.
As regards logical truths, logical truths are necessary truths, and necessary truths are beyond proof, in that no argument can be found to establish the truth of a necessary truth, as the nature of a necessary truth is to be true.
As regards synonymous truths, consider the statement "the sun rises in the east". By definition, the "sun" is something that rises in the east. So, the statement "the sun rises in the east" is analytically true. IE, if one morning something rose in the west, rather than the east, then that something wouldn't be the "sun", it would be something else.
IE, logical truths are beyond proof, and synonymous truths don't need to be proved as they are true by definition.
Synthetic statements
As regards whether the statement "the sun rises in the east" is synthetically true, one needs to prove that the sun rises in the east rather than the west.
If Idealism is true, and there is no external reality and the world only consists of ideas, then there is no sun, and the question as to where it rises becomes irrelevant.
If Indirect Realism is true, and our conscious experience is not of the real world itself but of an internal representation, then impossible to prove something outside that which we have direct knowledge of.
If Direct Realism is true, and the senses provide us with direct awareness of the external world, then it would be possible to prove that the sun rises in the east.
But how do we prove which of Idealism, Indirect Realism or Direct Realism is true, something philosophers have debated for thousands of years. As my perception of the sun would be identical, regardless of whether Idealism, Indirect Realism or Realism was true, it would therefore be impossible to prove which of these is in fact the case.
IE, synthetic statements cannot be proved.
Summary
Given the statement S "no true statement can ever be proved true", if S is true, proving it true would result in a contradiction.
IE, some true statements cannot be proved true. Though, in practice, proof is secondary to what pragmatically works.
1. Positive statement: G = God exists. To assert that something is a fact (a part of reality). It is possible that one believes something to be a fact when it isn't.
2. Negative statement: ~G = God doesn't exist. To aver that something is not a fact (not a part of reality). It is possible that one believes something to be not a fact when it is
If one believes something to be a fact, all one needs to do is show it is so in a particular corner of reality while to prove that something is not a fact, one has to exhaust all reality itself (a corner just won't do). It's quite obvious that the former is orders of magnitude easier than the latter. Ergo, one can't prove a negative.