Is deductive logic consistent and how does it influence other systems?

[caumaan]

I don't really know a lot about computers, so maybe I should just say this...

What is fuzzy logic and what are some of its applications?

 

[Jimmy]

http://www-2.cs.cmu.edu/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html

 

[Nephtys]

I always thought that fuzzy logic was essentially binary code carried to the next level, with each digit having a number of possibilities between 0 and 1. Isn't that right?

 

[Jimmy]

This is taken from the link I gave in my previous post:

Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth -- truth values between "completely true" and "completely false".

It seems you are correct, Nephtys.

Cauuman: What is fuzzy logic and what are some of its applications?

Fuzzy logic is used directly in very few applications. The Sony PalmTop apparently uses a fuzzy logic decision tree algorithm to perform handwritten (well, computer lightpen) Kanji character recognition.

 

[StarThrower]

Fuzzy logic is total nonsense.

Any statement is either true or false. There is no such thing as a statement being 1/3 true and 2/3 false, and that is exactly what fuzzy logic claims is possible.

Basically, of all possible logics that a reasoning agent can use, only one will be free from contradiction, and that is binary logic.


The following statement is true:

For any statement x, either X is true or X is false, and not (X is true and X is false.


We can condense this, using the 'exclusive or' operator XOR as follows:

For any statement X, |X|=0 XOR |X|=1

If you choose any logic other than binary logic, the statement above is false.

Thus, in any logic other than binary logic, there is at least one statement X such that not (|X|=0 XOR |X|=1), from which it follows that

not (|X|=0) if and only if not (|X|=1).

Now, if you do a temporal modal analysis of the meaning of the 'if and only if' operator, you will see that in an expression "A if and only if B" which is true, implies that the truth values of A,B are true simultaneously. And it is easy to show that

A if and only if B = not A if and only if not B

thus we have:

|X|=0 if and only if |X|=1

So what this tells us is that the statement

|X|=0

and the statement

|X|=1
are true simultaneously, therefore it follows that

0=|X| and |X|=1

And now using the transitive axiom of equality we can conclude that

0=1, which is false in all systems of logic.

 

[Hurkyl]

If you treat all the logical operators as if they were operating on binary logical values, then of course you're going to derive a contradiction if the values aren't restricted to be 0-1.



Fuzzy logic is one way to avoid hard decision making. For example, 5'9" is the average height for a male (or it was at one time), and 6'3" is tall. What do you call a person who is 5'11"? Average height? Tall? Does it even make sense to have a cutoff? Fuzzy logic let's you assign a number between 0 and 1 for both "average" and "tall", thus avoiding making this hard decision.

 

[Jeebus]

Originally posted by Hurkyl
If you treat all the logical operators as if they were operating on binary logical values, then of course you're going to derive a contradiction if the values aren't restricted to be 0-1.


Fuzzy logic is one way to avoid hard decision making. For example, 5'9" is the average height for a male (or it was at one time), and 6'3" is tall. What do you call a person who is 5'11"? Average height? Tall? Does it even make sense to have a cutoff? Fuzzy logic let's you assign a number between 0 and 1 for both "average" and "tall", thus avoiding making this hard decision.

Exactly.

{Part II}Or you can go with the Weisstein definition:

An extension of two-valued logic such that statements need not be true or false, but may have a degree of truth between 0 and 1.[/size]

Such a fuzzy logic systems can be extremely useful in designing controlled variables for real-world systems such as elevators. Same type of analogy Hurkyl gave above.

 

[phoenixthoth]

hurkyl, that's exactly the example i was going to give.

if every statement is either true or false, then tell me whether the following statements are true or false, ok?

1. i am tall (i'm 5'11'')
2. i am very tall
3. i am somewhat tall
4. i am roughly of average height

5. i am beautiful
6. i am one of the most beautiful people in the world

7. i always lie
8. this statement is false

9. the barber who shaves every man who doesn't shave himself shaves himself
10. the barber who shaves every man who doesn't shave himself does not shave himself

11. the set of all sets that are not members of itself is a member of itself.
12. the set of all sets that are not members of itself is not a member of itself.

13. this statement is true but not provable.
14. this statement is true but you can prove that it is false.

or did you just mean that all well formed formula in binary logic are either true or false?

this is the third thread with basically the same topic now.

 

[StarThrower]

Originally posted by phoenixthoth
hurkyl, that's exactly the example i was going to give.

if every statement is either true or false, then tell me whether the following statements are true or false, ok?

1. i am tall (i'm 5'11'')
2. i am very tall
3. i am somewhat tall
4. i am roughly of average height

5. i am beautiful
6. i am one of the most beautiful people in the world

7. i always lie
8. this statement is false

9. the barber who shaves every man who doesn't shave himself shaves himself
10. the barber who shaves every man who doesn't shave himself does not shave himself

11. the set of all sets that are not members of itself is a member of itself.
12. the set of all sets that are not members of itself is not a member of itself.

13. this statement is true but not provable.
14. this statement is true but you can prove that it is false.

or did you just mean that all well formed formula in binary logic are either true or false?

this is the third thread with basically the same topic now.

There is a difference between a sentence and a statement. A statement must either be true or false, and no statement can be true and false simultaneously. Now, in order to have truth value, a statement must have meaning. Consider the sentence, "I am tall". It is neither true nor false, hence it isn't a statement. Now, you could try to define 'tall' using ruler measurement, and say that such and such is tall if and only if its length is greater than the length of this ruler, but clearly if you feel these sentences are paradoxes of some kind, then that isn't what you mean by the adjective 'tall'. Thus, all your sentences which involve the word 'tall' are not statements. The same goes four the fourth sentence. The word 'beautiful' is also meaningless.
As for the sentence "I always lie", simple statement mechanics reveal things to you. Suppose the sentence is a statement. Next suppose it is true. Now, we need to know what it means. Suppose that it is assigned a meaning such that if it is true then it is false. It will now follow that the statement which the sentence denotes is false. Before you can assess the truth value though, the meaning of the sentence must be clear. If the meaning of the sentence isn't clear, then the sentence is meaningless, and hence cannot denote a statement.

X = this statement is false = X is a statement AND |X|=0

Suppose X is a statement.

Suppose |X|=1. Therefore, |X|=0 and |X|=1, therefore 0=1, which is false. Hence not (|X|=1), hence |X|=0. Thus, not (X is a statement) or not (|X|=0). Hence, X isn't a statement or |X|=1. And we know that not (|X|=1), hence X isn't a statement.

All your other sentences are completely analyzable using binary logic. What must happen is this. The sentences which are statements are true or false but not both simultaneously, and the rest of the sentences arent statements.


Consider the sentence:

the barber who shaves every man who doesn't shave himself shaves himself

Suppose the barber exists.

Now, it is asserted that for any man in town, if that man does not shave himself then the barber shaves that man AND the barber shaves himself.

Let B denote the barber, let S denote the binary relation 'shaves' and let X denote an arbitrary man from the town. The sentence is asserting that:

For any X, if not (X-S-X) then (B-S-x)

The previous sentence is a first order logical statement, and the domain of discourse is the set of men in some town. Supppose that B is an element of the set of men in this town, then B is substitutable for X, which leads to the following statement:

if not (B-S-B) then (B-S-B), which can be shown to be tautologically equivalent to
B-S-B

Now, we can eliminate the assumption that the barber is a man in this town, and right the following absolutely true statement:

Either the barber is not a man in this town, or the baber shaves himself. It was also asserted that the barber shaves himself, and this does not contradict any of the reasoning.


In 10 however, we would reach a contradiction by getting the compounnd statement B-S-B and not(B-S-B) from which it follows that the barber is not a man in the town, still no paradox at all.

11 and 12 mimic 9,10

13,14 if statements, are compound statements which assert they are statements, and in that sense they are similar to sentence 8. The error in the Godel theorem is that the sentence isn't a statement, but is processed as a statement. The break in reasoning should occur by concluding that the sentence isn't a statement.

 

[Canute]

It doesn't matter whether Goedel's sentence was a sentence or a statement in this context. He did not suggest that some statements are neither true nor false. His proof was of the fact that any system of truths and falsities (formal axiomatic systems of a certain level of complexity etc.) must contain statements which are undecidable within the system.

As this is true for all such systems then there will always be statements which are undecidable. However this does not mean that they are neither true nor false in reality, just that they cannot be decided without extending the system.

But some people take this to mean that all such systems cannot produce truth, and infer from this that there are statements which are neither true not false in reality , just as non-dual philosophers assert.

As Lao Tsu said " Words that are strictly true seem to be paradoxical" (Tao Teh Ching). This relates directly to Goedel, in that the more true (provable) words become, the more inconsistent the system used to prove them appears. This relates also to something called the 'Quine-Duhem thesis', which states (by implication) that all axiomatically derived truths are only relatively true, and cannot ever be shown to be completely true.

Drawing conclusions from all this is tricky however, and academics argue incessantly about the real meaning of these proofs.

 

[StarThrower]

Originally posted by Canute
It doesn't matter whether Goedel's sentence was a sentence or a statement in this context. He did not suggest that some statements are neither true nor false. His proof was of the fact that any system of truths and falsities (formal axiomatic systems of a certain level of complexity etc.) must contain statements which are undecidable within the system.

As this is true for all such systems then there will always be statements which are undecidable. However this does not mean that they are neither true nor false in reality, just that they cannot be decided without extending the system.

But some people take this to mean that all such systems cannot produce truth, and infer from this that there are statements which are neither true not false in reality , just as non-dual philosophers assert.

As Lao Tsu said " Words that are strictly true seem to be paradoxical" (Tao Teh Ching). This relates directly to Goedel, in that the more true (provable) words become, the more inconsistent the system used to prove them appears. This relates also to something called the 'Quine-Duhem thesis', which states (by implication) that all axiomatically derived truths are only relatively true, and cannot ever be shown to be completely true.

Drawing conclusions from all this is tricky however, and academics argue incessantly about the real meaning of these proofs.

I've already completely analyzed the Godel sentence, and isolated an error in its analysis, as I now see it, there is nothing to argue about. It certainly does matter whether or not the Godel sentence is a statement, because there is no reason to use binary logic to try and process sentences which don't denote statements, no reason at all.

I can post my old work which isolates Godel's reasoning error if you wish.

 

[Canute]

Originally posted by StarThrower
I've already completely analyzed the Godel sentence, and isolated an error in its analysis, as I now see it, there is nothing to argue about. It certainly does matter whether or not the Godel sentence is a statement, because there is no reason to use binary logic to try and process sentences which don't denote statements, no reason at all.

I can post my old work which isolates Godel's reasoning error if you wish.

I think you're missing the significance of Goedel's theorems. Certainly not many people share your view that he was mistaken. Most people, even today, think he should have got a Nobel prize. Perhaps you know better.

The question is only whether Goedel's sentence was a true theorem within the mathematical system. Statements and sentences don't come into it. If you're saying that undecidable theorems are invalid as theorems because they are self-referential then you need to go back to the mathematics. Goedel wasn't a fool, and he dealt with this problem.

 

[phoenixthoth]

Fuzzy logic is total nonsense.

Any statement is either true or false. There is no such thing as a statement being 1/3 true and 2/3 false, and that is exactly what fuzzy logic claims is possible.

Basically, of all possible logics that a reasoning agent can use, only one will be free from contradiction, and that is binary logic.


The following statement is true:

For any statement x, either X is true or X is false, and not (X is true and X is false.

the correct term i think you mean is "well formed formula," not statement. x is tall is a statement and a sentence. is T(x) not a well formed formula? ;)

of course if you assume all well formed formulas are going to have precisely one truth value among {T,F}, then it follows that all wffs have a truth value among {T,F}. fuzzy logic could be interpreted as meaning that statements are 1/3 true and 2/3 false, but that interpretation would be coming from the human using it and not the system itself just like how something being true in binary logic may or may not be actually true. all it means is that you have a function with domain of all wffs to {T,F} (binary logic) or {T,M,F} (ternary logic) or [0,1] (fuzzy logic) that satisfies basic properties that make it a generalization of binary logic. that it is a function guarantees an unambiguity in the "truth value" so that no wff can have more than one truth value and it must have at least one. so an extended excluded middle is there.

other logics are free of contradiction though they contradict the assumption that all wffs must have truth value of T or F. however, they don't contradict their own set of assumptions that each wff must have one truth value which is a generalization.

for more info and to look and tell us where the contradictions are, check out the following articles:
http://plato.stanford.edu/entries/logic-manyvalued/
http://plato.stanford.edu/entries/logic-fuzzy/

and if you can publish a paper on where godel was wrong, maybe you can get a fields metal! good luck; it would be wonderful if there were no undecidable statements.

 

[Bob3141592]

It doesn't matter whether Goedel's sentence was a sentence or a statement in this context. He did not suggest that some statements are neither true nor false. His proof was of the fact that any system of truths and falsities (formal axiomatic systems of a certain level of complexity etc.) must contain statements which are undecidable within the system.

I believe that's quite correct.

If I understand Goedel (now that's a fuzzy statement!) he wasn't making a statement about the nature of absolute truth. He was making a statement about the nature of formal systems. Lots of people seem to miss this point. A formal system is one in which there a finite number of explicitly defined symbols, and a finite number of explicitely stated axioms, and where there are a finite number of explicitely stated operations on those symbols. The symbols themselves do not contain "meaning," and it is not necessary nor even desirable to have any interpretation of the symbols. A formal system can be reduced to a set of algorithmic actions on the symbols or the derived statements of the system.

Very little we deal with is really a formal system. I'm not sure that mathematics itself is a formal sstem, since I don't know for a fact that set theory can be completely axiomized.

And reality itself may not be a formal system. We can't make it a formal system unless we can both axiomize all of the principles by which reality operates, which is a dubious proposition, and define without any ambiguity all of the elements on which reality is built.

So while Goedel's Proof is very important logically, it's commonly overextended and used in domains where it really doesn't apply. Philosophically, it's trecherous to use without abusing it.

At least that's my take on it.

 

[phoenixthoth]

what it sounds like you're saying is that it is not logically impeccable to apply mathematical theorems to philosophy, and with that i would whole-heartedly agree.

 

[Canute]

Originally posted by Bob3141592

If I understand Goedel (now that's a fuzzy statement!) he wasn't making a statement about the nature of absolute truth. He was making a statement about the nature of formal systems. Lots of people seem to miss this point. A formal system is one in which there a finite number of explicitly defined symbols, and a finite number of explicitely stated axioms, and where there are a finite number of explicitely stated operations on those symbols. The symbols themselves do not contain "meaning," and it is not necessary nor even desirable to have any interpretation of the symbols. A formal system can be reduced to a set of algorithmic actions on the symbols or the derived statements of the system.

Very well put imo.

Very little we deal with is really a formal system. I'm not sure that mathematics itself is a formal sstem, since I don't know for a fact that set theory can be completely axiomized.

This is where I begin to disagree. I think I see what you're saying. In a sense you're right, there is no such thing as a formal system, since all such systems must be incomplete or inconsistent and therefore not strictly formal. But we treat systems as formal, in other words we act as if we deal with formal systems. We assume, for instance, that proofs can be derived from formal systems despite their lack of complete formality. This is ok on a superficial level, (2 + 2 does truly equal 4), but it goes horribly wrong when we use axiomatic systems to attempt to prove truths about reality, for it can't be done.

As nearly all our reasoning is formally axiomatic, and certainly all our systems of proof, I would say we do deal all the time with what we treat as being formal axiomatic systems, but that, as you say, we are kidding ourselves for no such thing exists, and these systems contain no non-contingent proofs or truths. (I'd be interested to know whether you agree with this point or not)

And reality itself may not be a formal system. We can't make it a formal system unless we can both axiomize all of the principles by which reality operates, which is a dubious proposition, and define without any ambiguity all of the elements on which reality is built.

I'd say that this is the heart of the issue. The non-dual view of reality is that it is built on something (/nothing) of which nothing completely true or false can be asserted. In this sense the totality of reality is seen as being beyond description by any formal system of truths and falsities. By this view Goedel's theorems are true as a natural consequence of the structure of reality, for all formally axiomatic systems used to describe it must leave out the most important part of it, and must therefore not be incomplete or inconsistent, for if fully developed they will be found to be wrong, only partially true.

So while Goedel's Proof is very important logically, it's commonly overextended and used in domains where it really doesn't apply. Philosophically, it's trecherous to use without abusing it.

I agree with that, but feel that the theorems are vitally significant to philosophy, so the dangers have to be faced. I like Roger Penrose's approach, but I know many people think he abused Goedel.

 

[master_coda]

I just feel that I should point out that for something to be a formal system, it does not have been be consistent and complete.

And even if consistency and completeness were a requirement for a system to be formal, that does not imply that there are no formal systems. There certainly are systems that are both consistent and complete.

 

[Bob3141592]

Originally posted by Canute
This is where I begin to disagree. I think I see what you're saying. In a sense you're right, there is no such thing as a formal system, since all such systems must be incomplete or inconsistent and therefore not strictly formal. But we treat systems as formal, in other words we act as if we deal with formal systems. We assume, for instance, that proofs can be derived from formal systems despite their lack of complete formality. This is ok on a superficial level, (2 + 2 does truly equal 4), but it goes horribly wrong when we use axiomatic systems to attempt to prove truths about reality, for it can't be done.

That wasn't quite what I said. No "nontrivial" formal system can be both complete and consistent (nontrivial in this case means powerful enough to do multiplication, but I'll use it in a differenct sense later). That's what Godel proved. The systems are formal. And if they're formal, they're limited by the constraint of Godel's proof. If, however, they're not formal, meaning they don't meet at least one of the essential criteria of a formal system, maybe they can be both complete and consistent. Is reality a formal system? If it isn't, then it can be both complete and consistent. It may not be completely comprehensible to us in that case, but in and of itself it can be complete and consistent.

Note that I'm not saying that it is here. That would be a statement of faith that I'm not prepared to make at this point (but if you ask me later, I'll say that it is, I hope).

As nearly all our reasoning is formally axiomatic, and certainly all our systems of proof, I would say we do deal all the time with what we treat as being formal axiomatic systems, but that, as you say, we are kidding ourselves for no such thing exists, and these systems contain no non-contingent proofs or truths. (I'd be interested to know whether you agree with this point or not)

I can't agree with that, because I don't think all of our reasoning is formally axiomatic at all. For example, I play chess, and I play moderately well. Most people think of chess as a formal system, that it involves nothing but it's own special form of reasoning. The rules about how the pieces move are finite, and the board and pieces don't contain any hidden meanings or interpretations that are relavant to how the game is played. those are part of what a formal system is. But I don't fully understand the axioms about what's good in chess, and how the rules operate on the board and pieces. I don't know that anyone fully knows or understands them. Despite the progress computers have made, I do not believe chess can be solved with a discrete algorithm. Generally, I play by an instinct I don't fully comprehend. Still, it manages to work enough that I'd play for money if you want.

I might be wrong about a system as simple as chess. But certainly our reasoning as applied to psychology, politics and religion is not formally axiomatic. Not by a long shot. Yet we do reason about them. I don't event think the basic concepts people use to think about these things could be called axiomatic. We think based on concepts we've formed by extrapolations from our own idiosyncratic observations, and we change those concepts throughout our development. It's a rather recursive, even unstable operation. It's the very antithesis of a formal axiomatic system. In my opinion, philosophically, that may be our great strength, not a deficiency.

I'd say that this is the heart of the issue. The non-dual view of reality is that it is built on something (/nothing) of which nothing completely true or false can be asserted. In this sense the totality of reality is seen as being beyond description by any formal system of truths and falsities.

Now that sounds almost like a religious statement (is it acceptable to include religious notions in the discussions in these folders?). And if anything, it is religious beliefs that some, even many people believe to be a higher reality than the physical world. And about their religious beliefs, people often think of them as the absolute truth. It's everything else they're less confident about.

Myself, I'm rather areligious, since I think no concept of a non-trivial God can be consistent (in other words, makes any sense at all: and in this case, non-trivial means any active God, as opposed to the Deistic notion of a Creator who wills the universe into existence and then separates himself from it--that's what I call a trivial God, like the trivial solution to the harmonic equation). But we don't have to go there.

I take a more poetic view (poetry is another way around the restraints of a formal system, since the symbols it invokes have meaning but aren't unambiguously defined). I say the universe is nothing, just nothing that's unevenly distributed. Some places in the universe contains less nothing than other places, and some contain more nothing, so overall it balances out. And that's the truth, at least in a sense. Right?

... but feel that the theorems are vitally significant to philosophy, so the dangers have to be faced. I like Roger Penrose's approach, but I know many people think he abused Goedel.

I agree, the existence of Godel's theorem is significant even outside of the domain where it applies. It's useful in aguments by simile and by analogy, which are dangerous ways to philosophize themselves. But we have to use something to get a handle on things, and since we can't really define the symbols of our notions of the world to another, we have to do something. So trecherous or not, we forge ahead.

 

[selfAdjoint]

That wasn't quite what I said. No "nontrivial" formal system can be both complete and consistent (nontrivial in this case means powerful enough to do multiplication, but I'll use it in a differenct sense later). That's what Godel proved.

Not quite. What he proved is more like any system that contains second order propositional calculus is either incomplete or inconsistent.

As I keep posting, maybe it will take one of these days, Tarski proved that geometry is complete, and Hilbert and others proved it is consistent. And you can do multiplication in geometry (application of areas, or Eudoxian proportions). But the proofs don't use set theory, so they escape the flaws that lead to incompleteness.

 

[Canute]

Originally posted by master_coda
I just feel that I should point out that for something to be a formal system, it does not have been be consistent and complete.

And even if consistency and completeness were a requirement for a system to be formal, that does not imply that there are no formal systems. There certainly are systems that are both consistent and complete.

Yes, I was trying out an idea about systems to see if I got away with it or not. I didn't.

I was wondering whether a system could be accurately called 'formal' if it was inconsistent or incomplete. It's a odd way of looking at it I suppose. Such as system is formal in intent, but it cannot be proved to be entirely formal in fact. There will always be some doubt, with any system subject to Goedel's limits, as to whether the system is actually formal or not. After all if we know that there has to be a contradiction in it somewhere then we can define it as not being strictly formal.

Does that make any sense?

 

[master_coda]

Originally posted by Canute
Yes, I was trying out an idea about systems to see if I got away with it or not. I didn't.

I was wondering whether a system could be accurately called 'formal' if it was inconsistent or incomplete. It's a odd way of looking at it I suppose. Such as system is formal in intent, but it cannot be proved to be entirely formal in fact. There will always be some doubt, with any system subject to Goedel's limits, as to whether the system is actually formal or not. After all if we know that there has to be a contradiction in it somewhere then we can define it as not being strictly formal.

Does that make any sense?

I think I see what you are saying. But problems that happen "within" the system (such as inconsistancy and undecidability) don't actually affect formality, since a systems formality comes from how it is defined.

The truth-values of statements within the system don't really have anything to do with formality.

 

[Canute]

Originally posted by Bob3141592
That wasn't quite what I said. No "nontrivial" formal system can be both complete and consistent (nontrivial in this case means powerful enough to do multiplication, but I'll use it in a differenct sense later). That's what Godel proved. The systems are formal. And if they're formal, they're limited by the constraint of Godel's proof. If, however, they're not formal, meaning they don't meet at least one of the essential criteria of a formal system, maybe they can be both complete and consistent. Is reality a formal system? If it isn't, then it can be both complete and consistent. It may not be completely comprehensible to us in that case, but in and of itself it can be complete and consistent.

I agree with this. See also my reply to 'masta coda' above.

Note that I'm not saying that it is here. That would be a statement of faith that I'm not prepared to make at this point (but if you ask me later, I'll say that it is, I hope).

Interesting. I've convinced myself that it cannot be. The trouble is that if it cannot be, then there's no way of proving it cannot be. If there is a way that reality can be described by a non-trivial formal axiomatic system then I wonder what the fundamental axiom is, or was.

I can't agree with that, because I don't think all of our reasoning is formally axiomatic at all. For example, I play chess, and I play moderately well. Most people think of chess as a formal system, that it involves nothing but it's own special form of reasoning. The rules about... etc

I don't disagree with what you said about chess and so on, but I take it to mean something different.

If reality is logical, which most people assume it is, then to think rationally about it is to think strictly logically. In other words to form an understanding of it requires matching ones concepts and thoughts to their objects. I'd say that this entails the use of a formal axiomatic system of deduction. Only this could produce an explanation of reality that is isomorphic with the logic of what it is explaining, and thus be a true explanation.

If formal axiomatic systems are inevitably either inconsistent or incompletable then it seems much better to assume that this is also true of reality itself, rather than just take it to be a limit on our ability to reason. This is hardly a new idea, as I mentioned before.

Now that sounds almost like a religious statement (is it acceptable to include religious notions in the discussions in these folders?).

No, no, not religious. But you mentioned reality and metaphysics became unavoidable.

And if anything, it is religious beliefs that some, even many people believe to be a higher reality than the physical world.

Often it is yes. But quite often it isn't. There are some quite good non-religious reasons for believing in a 'higher plane' of some sort.

If you assume that you can describe the phenomenal universe by use of a formal axiomatic system of proofs, as I think you do, then you have the problem of the incompletenes theorems to overcome.

In this case the idea that there is a higher plane, a non-material meta-system that is axiomatic to existence is very useful. In principle it solves all the logical problems and it becomes possible, in theory at least, to complete the rest of the system consistently. This way you can have a completely consistent system of explanation of reality that can be as complete as you want it to be, with the one proviso that you can't prove that a higher plane or meta-system does exist. (Although you might know it non-systematically)

Myself, I'm rather areligious, since I think no concept of a non-trivial God can be consistent (in other words, makes any sense at all: and in this case, non-trivial means any active God, as opposed to the Deistic notion of a Creator who wills the universe into existence and then separates himself from it--that's what I call a trivial God, like the trivial solution to the harmonic equation). But we don't have to go there.

I'll just agree.

I take a more poetic view (poetry is another way around the restraints of a formal system, since the symbols it invokes have meaning but aren't unambiguously defined). I say the universe is nothing, just nothing that's unevenly distributed. Some places in the universe contains less nothing than other places, and some contain more nothing, so overall it balances out. And that's the truth, at least in a sense. Right?

No offense, but as an idea that doesn't seem any more consistent than the idea of God. Can one have more of nothing? I'm not sure.

I won't bore you with the details, but the Buddhist idea of 'emptiness' makes far more logical sense than this, although it's not really all that different in a way.

I agree, the existence of Godel's theorem is significant even outside of the domain where it applies. It's useful in aguments by simile and by analogy, which are dangerous ways to philosophize themselves. But we have to use something to get a handle on things, and since we can't really define the symbols of our notions of the world to another, we have to do something. So trecherous or not, we forge ahead. [/B]

Hmm. Sorry to write so much but this topic fascinates me.

Again I agree with what you say here, but I wonder if it's not a big mistake to forge on regardless of Goedel's proofs. To me they hold the key to understanding reality, but they're not being analysed properly because they are seen as obstacles to overcome rather than glaring clues to the fundamental nature of reality.

I think teams of professionals should be paid to work on why the incompleteness theorems apply to all non-trivial systems of truths and falsities in all possible universes in the infinite multiverse, if there is such a thing. I suspect that there is a reason.

Cheers
Canute

 

[Canute]

Originally posted by master_coda
I think I see what you are saying. But problems that happen "within" the system (such as inconsistancy and undecidability) don't actually affect formality, since a systems formality comes from how it is defined.

The truth-values of statements within the system don't really have anything to do with formality.

Yeah, I think there's two ways of looking at it. What you say is right but I still wonder whether a system known a priori to be either incomplete or inconsistent can properly be called formal. But I'm only being pedantic.

 

[Bob3141592]

Originally posted by selfAdjoint
As I keep posting, maybe it will take one of these days, Tarski proved that geometry is complete, and Hilbert and others proved it is consistent. And you can do multiplication in geometry (application of areas, or Eudoxian proportions). But the proofs don't use set theory, so they escape the flaws that lead to incompleteness.

He did? I'll have to check that out and see if I c an get anything out of it. I'm not a mathematician, though it is what my bachelor's is in. So I'm not completely ignorant of it, but I'm not that advanced either. If you've got any leads on where I can learn more about this at an intermediate level, I'd appreciate it. If not, I'll did around and see if I can muddle through. Thanks.

 

[Canute]

Originally posted by selfAdjoint
As I keep posting, maybe it will take one of these days, Tarski proved that geometry is complete, and Hilbert and others proved it is consistent. And you can do multiplication in geometry (application of areas, or Eudoxian proportions). But the proofs don't use set theory, so they escape the flaws that lead to incompleteness. [/B]

I'm with Bob on this. Are you sure of your facts here?

What do you mean by complete? Do you mean that the system proves its own axioms? Surely that would make the system trivial (unable to refer outside itself, and therefore tautological).

 

[master_coda]

Originally posted by Canute
I'm with Bob on this. Are you sure of your facts here?

What do you mean by complete? Do you mean that the system proves its own axioms? Surely that would make the system trivial (unable to refer outside itself, and therefore tautological).

This is quite correct. Alfred Tarski provided an axiomatization of Euclidean plane geometry that is complete.

By complete, I mean the traditional mathematical meaning. Given a geometrical statement G, either G or not G is true. In computiational terms, this means that there exists an algorithm which can take a statement in geometry and decide if it is true or not.

 

[Hurkyl]

For the sake of accuracy, it means that given a statement "G", either "G" or "not G" is deducible from the axioms. (A very fine distinction)

 

[Canute]

Masta coda

Still not quite clear about something. What exactly did Tarski do? Did he prove Euclid's axioms to be true within the system?

If so then surely his system is trivial, and therefore not subject Goedel's limitations, rather than an exception to them.

 

[selfAdjoint]

He took the whole modern set of axioms for geometry. This is an axiom system just as set theory has a system of axioms. Goedel proved the axioms of set theory, and anything that depended on them, to be incomplete. This was a proof in the meta theory, where the set theory axioms are content.

Tarski constructed a proof at the same level as Goedel's, in the meta theory of geometry, that showed that Geometry was complete. Tarski's proof is just as valid as Goedel's and is well known in the foundations community. Google on Tarski, complete.

Since you are so boulversee by this old news (Tarski's proof is from the 1940s), I won't mention to you the BSS machine and the vast spectrum of complete theories it has opened up.

 

[master_coda]

Originally posted by Hurkyl
For the sake of accuracy, it means that given a statement "G", either "G" or "not G" is deducible from the axioms. (A very fine distinction)

Looking at it again, my explanation was somewhat ambiguous. Yours is much better.

 

[Canute]

Originally posted by selfAdjoint
He took the whole modern set of axioms for geometry. This is an axiom system just as set theory has a system of axioms. Goedel proved the axioms of set theory, and anything that depended on them, to be incomplete. This was a proof in the meta theory, where the set theory axioms are content.

Tarski constructed a proof at the same level as Goedel's, in the meta theory of geometry, that showed that Geometry was complete. Tarski's proof is just as valid as Goedel's and is well known in the foundations community. Google on Tarski, complete.

My point was that if the axioms of geometry can be proved from within the formal system of theorems to which they give rise then that system is trivial in that it is tautological and cannot refer outside of itself. Such a system therefore is not subject to the contraints of the incompleteness theorems, and therefore Tarski doesn't seem relevant here.

I'm happy for you to explain why I'm wrong about this, if I am. It's something I've been taking for granted, but I'm no mathematician.

 

[selfAdjoint]

My point was that if the axioms of geometry can be proved from within the formal system of theorems to which they give rise then that system is trivial in that it is tautological and cannot refer outside of itself.

This is just not so. Geometry has content (points, lines) and isn't just some trivial a=> b, b=> a kind of tautology. In any case the Tarski proof can be extended to the real line.

This is all settled results. Rather than arguing in a vacuum, why don't you use google U. and find out what's been going on?

 

[master_coda]

Originally posted by Canute
My point was that if the axioms of geometry can be proved from within the formal system of theorems to which they give rise then that system is trivial in that it is tautological and cannot refer outside of itself. Such a system therefore is not subject to the contraints of the incompleteness theorems, and therefore Tarski doesn't seem relevant here.

I'm happy for you to explain why I'm wrong about this, if I am. It's something I've been taking for granted, but I'm no mathematician.

I just want to point something out here...no mathematical system refers to anything "outside of itself". The gist of the incompleteness theorem is that any system powerful enough to express the arithmetic of the natural numbers has theorems within the system that cannot be decided within the system.


Unless by "outside of itself" you mean "the system does not refer to anything undecidable". But in that case, you can't say that Tarski is irrelevant, since he was the person who showed that geometry "does not refer to anything undecidable".

 

[Bob3141592]

Originally posted by Canute, on my suggestion that reality might be expressible by a formal system
Interesting. I've convinced myself that it cannot be. The trouble is that if it cannot be, then there's no way of proving it cannot be. If there is a way that reality can be described by a non-trivial formal axiomatic system then I wonder what the fundamental axiom is, or was.[/B]

I've been thinking about this, and I've rather reversed my position (though it's quite possible that I'll change my mind about this again). I don't think it's possible to encapsulate the acausal nature of quantum mechanics into any formal system. Even if you could create a symbol to express that a quantum event occurs "for no reason except that the particle wants to" it'd be hard to convince me about that "wants to" part. And if acausality is replaced by a hidden variable theory, then in being hidden it would be outside the formal definition of the system

No offense, but as an idea that doesn't seem any more consistent than the idea of God. Can one have more of nothing? I'm not sure.

Like I said, I was being poetic. But the technical idea is that the BB was a quantum fluctuation event, with all positive components being balanced by a negative component. Perhaps the universe contains zero net energy (and zero net of everything else too), though I freely admit I don't understand what the physicists are talking about when they refer to dark or negative energy. That's what I was hinting about when I said the universe might be nothing, just unevenly distributed. But it's not worth serious discussion, though I rather like the way that sounds, and it is fun to say.

 

[Canute]

But surely an axiomatic system that proves its own axioms is not an axiomatic system. An axiom is by definition not derived from other theorems in the system.

 

[Bob3141592]

Originally posted by selfAdjoint
Tarski constructed a proof at the same level as Goedel's, in the meta theory of geometry, that showed that Geometry was complete. Tarski's proof is just as valid as Goedel's and is well known in the foundations community. Google on Tarski, complete.

Since you are so boulversee by this old news (Tarski's proof is from the 1940s), I won't mention to you the BSS machine and the vast spectrum of complete theories it has opened up.

Gee, if I knew what boulversee meant I could decide if I should feel insultd or not. The closest I could find was bouleversement which means "an overthrow; confusion; convulsion." and that certainly doesn't come across as flattering!

Like I said, I'm not a mathematician by any means, so I'm not in touch with the foundations community. I've read Nagal's book on Godel Proof, and Hofstadter's GED and Klne's The Loss of Certainty. Granted, I might not have fully understood them, and since they ain't textbooks, you might not think them worth much, but I do what I can.

Tarski... have I ever heard of him? Was he the one that proved you could take a sphere apart and then put it back together to make a sphere of twice the volume of the original? Is that included in his geometry?

 

[master_coda]

Originally posted by Canute
But surely an axiomatic system that proves its own axioms is not an axiomatic system. An axiom is by definition not derived from other theorems in the system.

That isn't quite accurate. An axiom is not defined by saying that you cannot derive it from other theorems in the system. Rather, an axiom is just a statement that you have assumed to be true.

You can often show that the axioms are true from the theorems. But since the theorems were derived by assuming the axioms are true, this isn't really meaningful...all you've done is show that if you assume the axioms are true, then you can prove the axioms are true.


What Tarski did was to provide a set of axioms for Euclidean geometry, and show that given any geometric statement G, you can either:
1. Derive G from the axioms.
2. Derive the negation of G from the axioms.

Obviously you can also derive the axioms from the axioms, since A => A is a tautology. But Tarski wasn't trying to show that his axioms were "true". He was trying to show that in Euclidean geometry, there are no questions that "cannot be answered".

 

[master_coda]

Originally posted by Bob3141592
Tarski... have I ever heard of him? Was he the one that proved you could take a sphere apart and then put it back together to make a sphere of twice the volume of the original? Is that included in his geometry?

This is the Banach-Tarski paradox. Given a ball in R^3, you can divide the ball into six pieces, and re-assemble them to form two balls of the same size as the original.

However, this paradox depends upon the axiom of choice. That is not an axiom of Euclidean geometry.

 

[Hurkyl]

None of the axioms of set theory, incidentally, are part of Euclidean geometry. (Though you can simulate some very simple set things via logic)

 

[selfAdjoint]

I didn't mean bouleversee in an insulting way, but only to jolly you along a bit, because you did seem upset that the Goedel theorem didn't have the universal scope you had ascribed to it.

There is active work going on now to see how much of the computations with real numbers can be incorporated into a complete system. The BSS machine I mentioned is one of the (purely abstract) tools for doing this.

By the way, I have read Nagel and Hoffsteader too. I can never decide if I like the Nagel book or hate it. About Hoffsteader I am ravingly enthusiastic.

 

[Bob3141592]

Originally posted by selfAdjoint
I didn't mean bouleversee in an insulting way, but only to jolly you along a bit, because you did seem upset that the Goedel theorem didn't have the universal scope you had ascribed to it.

No, not upset. I'd rather learn somethin than go on being wrong. I know just enough about these things to be dangerous, so I try to never trust what I think I know.

By the way, I have read Nagel and Hoffsteader too. I can never decide if I like the Nagel book or hate it.

Sounds ironically appropriate, doesn't it?

About Hoffsteader I am ravingly enthusiastic.

I've bought four copies of Hofstadter's book, since I'm always loaning it out and don't always see it returned. Not often you find a mathematics book with real wit and humor in it.

I tell people even if they find the body of the sections to be too much, just read the dialogues. Generally, if someone actually goes that far, they ask to keep the book longer so they can read the whole thing.

By the way, I knew my comment about Tarski wasn't legit. I was trying to be snide back, in response to the perceived slight. Sorry about that.

 

[Canute]

I suspect we're a bit at cross purposes here. Bob brought up the issue of whether reality itself forms an axiomatic system (which I take to mean - can be fully described by such a system). A system describing reality, whether it's metaphysical, scientific, theistic or anything else, must refer outside itself. It must therefore, according to Goedel (according to my understanding of Goedel) be either incomplete or inconsistent. This raises a few interesting question about reality and our ability to reason our way to the truth about it.

Whatever Tarski proved is not relevant here, because Tarski's axiomatisation of geometry works precisely because his system is entirely self-referential, it does not refer to anything real. As Einstein said somewhere "So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality”

This is what I meant by saying that Goedel's theoerems are not in any way limited in their applicability by Tarski's work. They are not dealing with the same kind of systems. Goedel's theorems simply don't apply to systems that are completely circular.

I'd like to discuss this because I've been trying to sort out some issues for quite a while but this is the first time I've found myself talking to people who seem to know something about these things.

My interest is in the implications of Goedel for science and metaphysics rather than for mathematics. Dangerous territory but fun to explore.

Hawkings says this:

"What is the relation between Goedels theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted….( )

In the standard positivist approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models. That is, a model can be arbitrarily detailed, and can contain an arbitrary amount of information, without affecting the universes they describe. But we are not angels, who view the universe from the outside. Instead we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Goedels theorem. One might therefore expect it to be either inconsistent, or incomplete. The theories we have so far, are ~both inconsistent, and incomplete."

(Stephen Hawking – Goedel and The End of Physics – net article (http://www.damtp.cam.ac.uk/strtst/dirac/hawking/ )

The question for me is whether the incompleteness theorems are just a quirk of epistemology or whether they tell us something important about reality. I feel it is the latter. They apply in all possible universes, and seem to place limits on what can be known by any system of proofs. Even if we take our perceptions as axiomatic (I think therefore I am) the same problems arise. (A fact I believe Plato saw, hence the shadows on the cave wall and our inability to reason our way out of the cave).

I feel that this relates directly to the use of undefined terms.

“…since every word in a dictionary is defined in terms of another word…The only way to avoid circular reasoning is a finite language would be to include some undefined terms in the dictionary. Today we must realize that mathematical systems too, must include undefined terms, and seek to include the minimum number necessary for the system to make sense.”

Leonard Mlodinow – Euclid’s Window p144

All this suggests that to explain reality one must have, at minimum, three ingredients in the explanation, namely an axiom, an undecidable question and an undefined term. Then one has to circumvent Goedel.

I'd like to hear your comments on this because it's such a slippery topic, expecially for a non-mathematician, that know I might be misunderstanding some of the issues.

I feel that the incompleteness theorems can be understood as arising from the nature of reality, if one assumes that it arises from a 'non-dual' ontology. (Something Hofstaedter just missed seeing in GEB, although I don't know how, he was so close). But before trying that one out I'll wait for a response to this bit.

 

[Bob3141592]

Originally posted by Canute
I suspect we're a bit at cross purposes here. Bob brought up the issue of whether reality itself forms an axiomatic system (which I take to mean - can be fully described by such a system). A system describing reality, whether it's metaphysical, scientific, theistic or anything else, must refer outside itself. It must therefore, according to Goedel (according to my understanding of Goedel) be either incomplete or inconsistent. This raises a few interesting question about reality and our ability to reason our way to the truth about it.

I'm not sure that necessarilly follows.

Perhaps much of the turmoil here is caused by a subtly of what Goedel's Proof is really about. The focus of the proof concerns only second order predicate logic. I'm not sure of the details of the distinctions, so it's time to do a bit more work. Here's one place to start -- http://en.wikipedia.org/wiki/First-order_predicate_calculus

Now, back to your original point. Let's say we have a complete and perfect model of reality at the atomic level. Just because the model is abstract and not made of the same "stuff" as the thing it models doesn't mean it's referring to something outside itself. That was never a formal requirement, was it? So I'm not sure this kind of argument applies. Besides, we don't know what the stuff of the universe is actually made of. Perhaps by time they work out all the details they'll find that strings or membranes or whatever is really vibrating in those higher dimensions are really just numbers after all. So, philosophically speaking, even if it mattered, we don't know that the model has to refer to something outside of itself.

Do I think that's really the case? No, not really, but I don't know, and it's not a legitimate assumption one way or another.

Anyway, got to go fix breakfast, and in between chores read up on predicate calculus of various orders.

 

[master_coda]

Originally posted by Canute
A system describing reality, whether it's metaphysical, scientific, theistic or anything else, must refer outside itself. It must therefore, according to Goedel (according to my understanding of Goedel) be either incomplete or inconsistent.

I just think I should mention this again. The incompleteness theorem does not say anything about systems that refer to things outside themselves. It talks mathematical systems, and no mathematical system refers to anything outside of itself.

If you insist that a system describing reality must refer outside of itself, then you have already shown that we cannot use a mathematical system to describe reality.


I tend to think that incompleteness is more of a property of logic than the universe. Of course, if logic is an intrinsic part of the universe, then incompleteness is also a property of the univserse that we cannot escape. However, I'm not familiar with the philosophical arguments of that form...I'm just a mathematician.

 

[Canute]

Originally posted by Bob3141592
I'm not sure that necessarilly follows.

Perhaps much of the turmoil here is caused by a subtly of what Goedel's Proof is really about. The focus of the proof concerns only second order predicate logic. I'm not sure of the details of the distinctions, so it's time to do a bit more work. Here's one place to start -- http://en.wikipedia.org/wiki/First-order_predicate_calculus

I think it is generally accepted that Goedel's theorems have implications well beyond mathematics. In this I have Roger Penrose and Stephem Hawking on my side.

Now, back to your original point. Let's say we have a complete and perfect model of reality at the atomic level. Just because the model is abstract and not made of the same "stuff" as the thing it models doesn't mean it's referring to something outside itself.

If the model is a model of something else how can it not be referring to that something else. A model refers to something else by defintion. (Are we using 'refer' in different ways?)

So, philosophically speaking, even if it mattered, we don't know that the model has to refer to something outside of itself.

I think we do. What's the point of a model of reality that doesn't refer to anything?

An axiomatic system can be defined as a system that refers outside of itself, since its fundamental axiom is an theorem about something other than the system. It points outwards rather than inwards. (E'g' I think therefore I am, there is a line such that ... , God exists, etc).

 

[Canute]

Originally posted by master_coda
I just think I should mention this again. The incompleteness theorem does not say anything about systems that refer to things outside themselves. It talks mathematical systems, and no mathematical system refers to anything outside of itself.

I don't agree I'm afraid. The incompleteness theorems apply to axiomatic systems. Axioms refer outside the system by definition.

Also mathematical systems can refer ourside themslves. But, as Einstein said:

“ So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality”.

If you insist that a system describing reality must refer outside of itself, then you have already shown that we cannot use a mathematical system to describe reality.

I think that there's a get out clause.

I tend to think that incompleteness is more of a property of logic than the universe. Of course, if logic is an intrinsic part of the universe, then incompleteness is also a property of the univserse that we cannot escape. However, I'm not familiar with the philosophical arguments of that form...I'm just a mathematician. [/B]

I'm pleased to be talking to a mathematician about this, since it keeps some rigour in the discussion. My maths is rubbish I'm afraid, but to me Goedel has far more significance outside mathematics than inside it. If you're interested in the philosophical side and haven't already read it Penroses's stuff on this is brilliant, (since he can cope with the maths!) although many believe he went too far with his metaphysics in almost proving that God exists.

 

[master_coda]

Originally posted by Canute
I don't agree I'm afraid. The incompleteness theorems apply to axiomatic systems. Axioms refer outside the system by definition.

Also mathematical systems can refer ourside themslves. But, as Einstein said:

“ So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality”.

How do axioms refer to something outside of the system?

For example, in set theory one of the standard axioms is the axiom of the empty set. The axiom asserts that there exists a set that contains no elements.

But that does not refer to anything outside of the system (set theory). It is considered to be an axiom because it is asserted to be true instead of being deduced to be true.


The Einstein quote is a common one...it's usually used to try and put mathematical physicists in their place. But although the quote says "laws of mathematics refer to reality", that isn't really accurate. The math itself does not refer to reality in any way.

The quote is actually referring to the practice of taking mathematics and trying to derive conclusions about reality from math. Just because one can show that Euclidean geometry is consistent, it doesn't mean the nature follows Euclidean geometry.

 

[Canute]

Originally posted by master_coda
How do axioms refer to something outside of the system?

For example, in set theory one of the standard axioms is the axiom of the empty set. The axiom asserts that there exists a set that contains no elements.

But that does not refer to anything outside of the system (set theory). It is considered to be an axiom because it is asserted to be true instead of being deduced to be true.

But doesn't it assert something about reality that is not part of the system? Isn't that precisely what makes it an axiom?

The Einstein quote is a common one...it's usually used to try and put mathematical physicists in their place. But although the quote says "laws of mathematics refer to reality", that isn't really accurate. The math itself does not refer to reality in any way.

As a mathematician you may be forgetting that people use mathematics to count their change, not just for developing heuristic systems of proof.

The quote is actually referring to the practice of taking mathematics and trying to derive conclusions about reality from math. Just because one can show that Euclidean geometry is consistent, it doesn't mean the nature follows Euclidean geometry. [/B]

I agree. If one can show that Euclidean geometry is consistent then one can know it says nothing about reality, as Einstein suggests.

If it is to say something about reality, to prove something about it, a system cannot be made entirely self-referential. It seems to me that at some point it has to make an assertion that cannot be deduced but must be checked against the external facts. Is this not so?

 

[Hurkyl]

But doesn't it assert something about reality that is not part of the system? Isn't that precisely what makes it an axiom?

No. It's an axiom merely because it was selected to be an axiom.



You're missing a major piece of the puzzle here. It is not the job of a mathematical system to say anything about reality. It is someone else's job (e.g. a physicist) to devise a correspondence between reality and a mathematical system. It is through this correspondence (which is not part of the system) that the system says anything about reality.

 

[Canute]

Originally posted by Hurkyl
No. It's an axiom merely because it was selected to be an axiom.

I doubt that you can give an example of a fundamental axiom which refers only to the the system of theorems which can be derived from it.

Here's Anton Setzer from http://www-logic.stanford.edu/proofsurvey.html

"I think proof theory is mainly dealing with foundations, and after some mathematical reduction steps we will always end up with some principles, which can only be validated by philosophical arguments. Here interaction with philosophy is required.

You're missing a major piece of the puzzle here. It is not the job of a mathematical system to say anything about reality. It is someone else's job (e.g. a physicist) to devise a correspondence between reality and a mathematical system. It is through this correspondence (which is not part of the system) that the system says anything about reality. [/B]

I don't think it matters whose job is what. A mathematical system must be axiomatic in structure (based on or containing an underived/unproved theorem) in order to refer beyond itself, just as a dictionary must contain an undefined term in order to do so. That doesn't mean mathematical systems have to refer to anything, many of them don't, as you say.