Happy Atheist Forum

I have oft heard the mention of "proof" as the inneffable source of mathematics, and , thus, of science. When I think about it, it makes sense. Science is a method of search for the truth, where the method of enquiry seeks to prove or disprove whatever statement you might think of about the world. Still, there is a twofold uneasiness between me and science, due to two rather strange observations:

1) it was Hume, I believe, who figured out, and proved mathematically, that there is nothing we can find in a cause that would lead us to deduce the resultant effect of that cause. This is troubling, because it calls into doubt the very validity of empericist thinking. Empiricism depends heavily on the idea that causes beget effects, and that the two are deductively linked. I am still considering this.

2) In two words, Kurt Godel. This is the man that also proved, mathematically, that there are an infinite number of theorems, statements one could make about the world that could not be proved in the negative, nor the positive. In this case, you should note that I am not talking about the statement "god exists", which can easily be proven given a proper definition of the term god. No, I am talking about contradiction.

Yes. Contradiction. If you were to write on a sheet of paper, "the statement on the other side of the paper is true", and then write on the other side, "the statement on the other side of this paper is false.", you would have produced two theorems about the world that could not be proven true nor false.

When Kurt Godel made this discovery, the entire mathematical world was in an uproar, because it effectively meant that a complete and internally consistent universe would require something beyond itself to solve these statements within itself. What a shock! I told this story to some religious friends of mine, and they immediately took it as a proof that God exists.

Indeed, the statement is unsettling. This thread is my response to Kurt Godel's discovery.

As I was pondering kurts argument, it struck me that all of these "incompleteness" theorems, so to speak, were always actually a pair of two or more concrete theorems. It strikes me that the statement, "the statement on the other side of this paper is true/false" depends on the statement on the other side of the paper for it's validity. What does the statement on the other side of the paper depend on for validity? Whatever it is talking about. So, the first observation I think is this:

1. Theorems make statements about other phenomena.
2. A theorem's validity is dependent upon the phenomena of which it speaks.

After some more thoughts, I realized that kurt godel's theorem relied upon the assumption that theorems are identical with phenomena. Think about it: Is a theorem, which speaks on phenomena, itself phenomena? I think we can agree that the defining characteristic of phenomena in general is existence, without which phenomena would not be phenomena. (it would be nothing). But a theorem on phenomena? Does it exist? Certainly, the concept of a theorem exists. Certainly, there can be little squiggles on a piece of paper that represent a theorem. But I am convinced that a statement about the world in general has no inherent existence.

Why? There was a clue in the first observation.  :D
To the ever expanding knowledge of the atheist mind!

Quote from: "Wechtlein Uns"1) it was Hume, I believe, who figured out, and proved mathematically, that there is nothing we can find in a cause that would lead us to deduce the resultant effect of that cause. This is troubling, because it calls into doubt the very validity of empericist thinking. Empiricism depends heavily on the idea that causes beget effects, and that the two are deductively linked. I am still considering this.

It was indeed Hume.

Quote from: "Wechtlein Uns"2) In two words, Kurt Godel. This is the man that also proved, mathematically, that there are an infinite number of theorems, statements one could make about the world that could not be proved in the negative, nor the positive. In this case, you should note that I am not talking about the statement "god exists", which can easily be proven given a proper definition of the term god. No, I am talking about contradiction.

...<snipped>...

All of Kurt Godel's "paradoxical theorems" are a collection of theorems that depend on the results of all the other theorems and nothing else. Because of this, the theorems go in one big circle, forever trying to get to the bottom of definite phenomena, but only finding something that is dependent on something else for it's validity.

This is where you are mistaken, sadly. Not all Gödel statements are statements about statements.

Axiom : The barber cuts the hair of exactly those people in the village who do not cut their own hair.
Gödel statement : The barber cuts his own hair.

Sorry.

If you're saying that not all of godels statements were statements about statements, then I agree. I didn't think I was claiming that, but it could have sounded like that.

What I was saying is simply this: Statements that are recursive(i.e. depend upon, ultimately, the determination of their validity to be valid) are equal to "Null". They are not dependent upon phenomena for their validity. Add to that my argument that they have no inherent phenomenal existence of their own, and what do you think?

Quote from: "Wechtlein Uns"If you're saying that not all of godels statements were statements about statements, then I agree. I didn't think I was claiming that, but it could have sounded like that.

What I was saying is simply this: Statements that are recursive(i.e. depend upon, ultimately, the determination of their validity to be valid) are equal to "Null". They are not dependent upon phenomena for their validity. Add to that my argument that they have no inherent phenomenal existence of their own, and what do you think?

But the example of a Gödel statement I gave, was not a recursive one either.

Another example.

Axiom 1 : It doesn't rain or I use an umbrella
Axiom 2 : I don't use an umbrella or it rains
Gödel statement 3 : It rains.

BTW your proposal that Gödel statements are equal to null, would require another logic as it is a contradiction of the axiom of excluded middle.

Oh, I think I understand what your getting at.

I believe there are many different types of statements. But of all the statements that might or might not be experienced, It seems that they all require something other than themselves for their own validity. A statement that says, "it rains", depends upon whether there is a phenomenon that can be singled out as "raining", in my view.

Now, as far as science goes, the material is all that exists. But it's slowly turning about that this material isn't really different from other material. Rather, all that there is--is phenomena. Pretty much. I understand that there are abstract concepts like respect or failure that might seem like they are independent of the phenomenal world, but I see it otherwise. In looking at phenomena, one does not have to slice the universe up in an instant. Phenomena changes, and one can group changes in phenomena as a phenomenon in its own right. I don't want to get into all the ways one can slice up phenomena, but suffice it to say, any statement that has any meaningfulness at all can ultimately be traced back to directly experienced phenomena.

I feel that, when someone makes a statement that is a paradox, it is because the statement relies not on phenomena for it's validity, but on another statement. Since a statement can not have any inherent validity in itself, the second statement must dependo n phenomena for the first statement to become proven true or false. But what if the second statement depends on the first statement for its validity? This, in my opinion, is what I consider to be the structure of a paradoxical argument.

Now then, the reason I say that the two statements, taken as whole, must be equivalent to null is because if they were dependent on something that was not null, they would be able to be proven true or false. If the two statement system depends not on phenomena, but on what then? All that there is is phenomena. The only other thing that is not phenomenal is null.

I agree that these kinds of statements can not be proven true or false. However, I do not think that that means some inherent flaw in the mathematical system. And that was my original point. Kurt Godel seemed to prove that there are mathematical statements that can not be proven true or false with a complete and internally consistent mathematical system. My main point is that there is not just True or false among statements. True or false is to be used when the statement depends upon some mathematical or phenomenal result to see its validity. Whereas null is the result of statements that ultimately depend upon null for their validity. and it seems that paradoxical statemnts rely not on phenomena, but "not-phenomena", thus what else could they rely on, but null?

I think, however, that you might have misunderstood my argument. I believe that there are statements that can be proven true and false, and there are statements that can not. But the statements that can not be proven can still be evaluated within an internally consistent and complete architecture of symbols.

Really, all I'm try to say is that there can be an evaluation of a paradoxical statement.

Quote from: "bowmore"Axiom 1 : It doesn't rain or I use an umbrella
Axiom 2 : I don't use an umbrella or it rains
Gödel statement 3 : It rains.

.

At first glance I thought that this was a perfect example of what I was talking about, but It looks like this isn't. Followoing Godel statement three, it is obvious that "it rains", and "I use an umbrella" are chosen. What I would like to point out, however, is that this is not a paradoxical statement. The entire thing depends upon the phenomena of it raining. They don't depend on each other in a circle for their validity, and thus are not a paradox.

Soo...I don't see how this helps you at all. I all ready agreed that these types of statements exist, no?

Quote from: "Wechtlein Uns"

Quote from: "bowmore"Axiom 1 : It doesn't rain or I use an umbrella
Axiom 2 : I don't use an umbrella or it rains
Gödel statement 3 : It rains.

.

At first glance I thought that this was a perfect example of what I was talking about, but It looks like this isn't. Followoing Godel statement three, it is obvious that "it rains", and "I use an umbrella" are chosen. What I would like to point out, however, is that this is not a paradoxical statement. The entire thing depends upon the phenomena of it raining. They don't depend on each other in a circle for their validity, and thus are not a paradox.

Soo...I don't see how this helps you at all. I all ready agreed that these types of statements exist, no?

Whoops I messed up axiom 2.  

Axiom 2 : I don't use an umbrella or it doesn't rain

That should fix it right? i.e. now there is a paradox.

My point was that the Gödel statement was not reflexive. It doesn't refer to itself or another statement.
But you are obviously just referring to the fact that within this system the truth of (3) implies it being false.

I'd like to explore your null value approach.
So please answer me these (to establish the grammar involved with nulls) :

not null = ?

true and null = ?
true or null = ?

false and null = ?
false or null = ?

Not null = ... if it's apparent, it's not null.

In the null valua approach, it's not enough to say that a statement can be either true or false, and then tack on null, as in "true and null". It doesn't work that way, but rather a statement is either true, false, or null. A true statement mirrors the phenomena it represents. While a false statement invokes phenomena that is, for whatever reason, not apparent. or rather, phenomena that doesn't match up with apparent phenomena. And finally, a null statement would be a statement that does not depend on phenomena at all for it's validity, but either another statement which in turn depends upon the previous statement, or a statement that depends upon itself for validity. I don't think it's possible to make as tatement that depends upon itself for validity, BUT, it is possible to have a chain of statements, of whatever number, that ultimately depend on each other for validity in one big circle.

So, in response to your questions, just as you can not have a "true and false", you can not have a "true and null". However, you can have a "true or null".

Quote from: "bowmore"Whoops I messed up axiom 2.  :eek:

Quote from: "Wechtlein Uns"

Quote from: "bowmore"Whoops I messed up axiom 2.  :eek:

The axioms aren't trying to reflect actual reality.

Quote from: "Wechtlein Uns"So, in response to your questions, just as you can not have a "true and false", you can not have a "true and null". However, you can have a "true or null".

I'll try to clarify my question. Since you introduce a third possible value for 'booleans' the oprators in the algebra need to be able to cope with that new value.

the not() operator :

not(true) = false
not(false) = true
not(null) = ?

same for the 'and' and the 'or' operator.

Suppose a statement A is null (e.g. This statement is false)
And a statement B is not null (e.g. It rains)

What then is the value of the statement
not(A)
A and B
A or B

I wonder whether you are suggesting an alternate algebra to dodge Gödel's theorem. If so I'd like to explore that algebra to assess if you've been able to succesfully pull this off.
If you're just using null as a symbolic value to mark something as a Gödel statement, then this exploration isn't needed, but it'd also not be the great solution to Gödel's theorem.

I spent a lot of time intricately describing the system in detail, when all of a sudden, I press post, and lose the connection.  :(), I'll save it for next time. Better yet, try and figure out why it is so! What's life without a challenge, eh?

I would also like to add that in the tripartite boolean system, there is also this:

(True or False or Null) = (True or False or Null) Null (True or False or Null)

If B is True, then:

(Null) Null (True) = True. Ah, that's the simplest one.
We'll I'll be leaving you to think about this, bowmore. I hope you come up with some interesting stuff. It's been a pleasure talking with you.

Excuse me for saying so, but......

holy shit!

 :pop:

Quote from: "Wechtlein Uns"Or is like so:

(True or False) = (True or False) False (True or False). In the old algebra.
In the new: (True or False or Null) = (True or False or Null) False (True or False or Null)

It may be me, but I don't understand this notation, even in the old algebra.

What does (x)False(y) mean?

Quote from: "Wechtlein Uns"Therefore, if we are assuming that B is true(again, for simplicity), then:

(Null) False (True) Not(=) False, therefore (Null) False (True) = (Null) False[/i] (True) = True!

This was where I explained in detail why False correlates with or and True correlates with And, but after the lengthy description I allready gave(that was lost to a connection error :D

Well, I'm stuck until I can make sense of your syntax.
I'll play with it a little, perhaps it becomes clear.

Right. where to begin? I'm going to try to explain my syntax, though, if I'm being honest, I have yet to figure out the actual axiom for using true and false correlates with and (and) or. But I do have a very strong hunch that there's something solid there. I have a nose for sniffing out stuff like this.

First off, i think I need to clarify what we mean when we use the term "true" or "false". When a statement is true, all we're really saying is that the phenomena image evoked by that statement matches some phenomena somewhere in the universe. It's a matter of the Identity axiom, to say that Statement A = True. I believe that the And and Or operators operate in essentially the same way, except they are used with multiple statements to find one Truth or one Falsehood.

When I say: (True or False) = x, the True or false on the left are the possible answers to the statement on the right, namely, "X". The thing is, however, that "X" in this case, consists of more than one statement. So how are we to determine whether X is true or false? Just like with And and Or, we can require that all Statements be either true or false, or we can require that any statements be true or false. I would like to point out here, however, that when we say "all statements" we are, in essence, evoking a "True" type operator. Put it this way:

If A = B then (a and b) = True.

If A = True
And B = True,
the (A = B) = (True = True) = (True). What I've noticed here, is that in order for the statement to be True, Both statements have to inherently match up. And what do you call a statement that matches up? You call it True. That's why I equate "True" with And.

On the flip side, and this might be easier to explain, falsehood comes up because there is no match between a statement and percieved phenomena. The Or operator operates, on a basic level, upon the non-matching up of the multiple statements! If both A and B are true or false, then yes the statement is True or False accordingly, but we're actually calling upon an AND statement to do that. Whereas, if A is true and B is falses, then it is Or that does it's work. Or depends upon the two statements Truthfullnes and Falsity to not match up. And what do you call a statement that doesn't match up? You call it False. That is why I use False in my syntax in replace of or.

So: In the old algebra:

(true or False) is a basic or statement. It means the statement is either True OR false! However, it could be re-written as: (True) False (False) and still retain it's meaning.

The next true or false has to do with the truthhood or falsity of the first statement. It could also be re-written as (True) False (False). The truth is, "And" operators and "Or" operators can both be written in completely boolean logic. This has been done in Computers, where A string of Trues(1's) and Falses(0's) Represent And and Or. This is nothing new, and has been done in computer science before.

So:

(True or False) = (True or False) True (True or False) is simply the computer scientists way of representing an And statement with 0's and 1's. My above new operators, with (True or False or Null) uses a Tri-partite system. In this system, because there are three types of logic, there is actually an extra operator! I haven't named this operator, but it stems from the Null boolean.

anyways, I hope that clears things up. I'm not very good at putting my ideas forth, but if you still have questions, I should be able to answer them.

Quote from: "Wechtlein Uns"If A = B then (a and b) = True.

If A = True
And B = True,
the (A = B) = (True = True) = (True). What I've noticed here, is that in order for the statement to be True, Both statements have to inherently match up. And what do you call a statement that matches up? You call it True. That's why I equate "True" with And.

Equating the operators with the values doesn't add clarity. Frankly I wonder why you do it at all. What's worse it looks like you're actually changing the old algebra in the process.
I'm also wondering why you think operators must match a value. With what value does the "not" operator corespond or the "implication" and "equivalence" operators?

On top of that you seem to reach different results with one notation vs. the other :

(I've underlined the difference)

Quote from: "Wechtlein Uns"(Null) True (True) Not(=) False, therefore, (Null) True (True) = (Null) or (True) = False!

Quote from: "Wechtlein Uns"(Null) False (True) Not(=) False, therefore (Null) False (True) = (Null) False[/i] (True) = True!

At this point I'm not very convinced you're on to something at all.

ok. I've thought about this for a good long time.

The problem of Godel statements, or actually, what I had percieved as godel statements, but that were actually reflexive statements, originally struck me as a fitting challenge. I set out to define what was meant when one uses the term statement. But I have found that you can not know what is meant, and yet at the same time, you can grasp it's underlying structure.

A statement is not a thing. It is a pointer. In computer science, we have things known as pointers, which do not contain values in bits, but contain the address location of another location in memory. I think this is an elegant metaphor, because statements do not contain data in themselves. They point to data. When you read a book, and your eyes look over myriad statements, it becomes crystal clear that the word "tree" does not made a giant block of wood grow out of the page and into your eye. But rather, it directs the attention[/i] to the phenomena in question.

I believe this is why, when someone asks, "what is the meaning of meaning?", we get an error. Meaning, is, in essence, a translation mechanism. It is nature's way of switching "formats" so to speak. For example, If I were to make a statement that utilized large, obtuse sounding words to indicate that my keyboard is black, there is a chance that you will not understand the words and ask me for their meaning. If I say, "such and such means", I am utilizing the axiom prinicple. The two statements are equivalent, but not in the way you might think.

Statements that are equivalent to each other are not so because they are identical things[/b]. They are equivalent because they point[/i] to the same thing. The "meaning" I am speaking of, is simply a translation tool, to switch the data from one format to another, so that another person might understand.

This is important, I believe, because there are statements that do not point to the same phenomena, and these are not equivalent. But even more importantly, there are statements that point to statements, just as in computer science there are pointers that point to pointers. But, my point(pardon the pun) in all this, is the one underlying principle that Statements are not Inherently existent. They do not exist. In computer science, a pointer doesn't exist. It simply points. This is proven when in programming, it is revealed that "no data can be extracted from a pointer, that is itself not a pointer."

In this challenge, I have thought to create a third bit that would resolve the issue, but only now do I see, that I was, in essence, simply labeling the situation. There can be no third bit, or third value, because a bit or value is just that: data, value. In reality, however, when you come across a reflexive situation, there is no data that can be honestly extracted. And yet, there is data in this bizzarre situation:

The concept of Zero. In computer science, Zero is not equal to Null. It is an actual value. And yet, the two ideas share the same value. Truth is, zero is the concept of absolutely nothing. No data. No value. It is here that I believe the answer to reflexive statements may be found. A reflexive statement, by itself, points to another statement. But the system of reflixive statements as a whole, holds no data, and thus it's data is zero. It's data is null.

But Null is very different from true and false. However, I think that the statement "something is true" is also a statement about a statement. Only this time it is a statement about a statement with it's relationship to some data. In the same sense, to say "something is null" is to make a statement about that statement with it's relationship to zero. This relationship is, in fact, the very thing we have started with to define true. It is the axiom of identity. Null = Null. To say something is Null, is, if it is null, then it is True. If it is not, then it is false.

But what of the original system of reflexive statements? Are they true, or are they false? To this I have realized that the terms True, and False, only apply when there is something. Some data. I have all ready shown how to make a true or false with the systems relationship to zero, but when you take that relationship away, you are left only with zero.

And the only comfort that can be sought is that in computer science, Zero is false.
What is false? It is, of course, the idea that the statement and reality don't match. In other words, the statement, and the phenomena it points to, do not exixt. This might lead to the idea that Reflexive statements do point to data. But if they did, that phenomena would not exist. Thus, as a whole, they would be false.

Quote from: "Wechtlein Uns"In this challenge, I have thought to create a third bit that would resolve the issue, but only now do I see, that I was, in essence, simply labeling the situation. There can be no third bit, or third value, because a bit or value is just that: data, value. In reality, however, when you come across a reflexive situation, there is no data that can be honestly extracted.

So no more algebra  

Quote from: "Wechtlein Uns"And the only comfort that can be sought is that in computer science, Zero is false.
What is false? It is, of course, the idea that the statement and reality don't match. In other words, the statement, and the phenomena it points to, do not exixt. This might lead to the idea that Reflexive statements do point to data. But if they did, that phenomena would not exist. Thus, as a whole, they would be false.

I on the other hand can live with the fact that human cognition is limited, and by extension that the systems we build to describe reality are in themselves limited, which is ultimately what Gödel's incompleteness theorems prove.

BTW zero is false is not a universal in computer science. In Java for instance '0' is not the same as 'false', and both are not equal to 'null'.

Quote from: "bowmore"I on the other hand can live with the fact that human cognition is limited, and by extension that the systems we build to describe reality are in themselves limited, which is ultimately what Gödel's incompleteness theorems prove.

Heh. Yeah. I am still young and arrogant, and I feel a call to prove some major theorem or teard down the established walls. There's revolutionary blood in my viens. It's just of the scholarly type.  :blush:

Thank you for hanging with me through this, bowmore. You're a great guy to give me an ear when I wanted to talk to someone about my crazy ideas. Thanks a lot.

Quote from: "Wechtlein Uns"Heh. Yeah. I am still young and arrogant, and I feel a call to prove some major theorem or teard down the established walls. There's revolutionary blood in my viens. It's just of the scholarly type.  :blush:

np

Quote from: "Wechtlein Uns"Thank you for hanging with me through this, bowmore. You're a great guy to give me an ear when I wanted to talk to someone about my crazy ideas. Thanks a lot.

You're welcome.
I was curious for that algebra, as it would constitute a new formal system, and it would in fact be subject to Gödel's incompleteness theorem again.  
You concluded the algebra thing wasn't going to work before I could show that, so kudos to you.