Fuzzy Logic [Archive] - Physics Forums

StarThrower

Dec24-03, 11:29 AM

Hurkyl

Dec24-03, 02:20 PM

If you treat all the logical operators as if they were operating on binary logical values, then of course you're gonna derive a contradiction if the values aren't restricted to be 0-1. [6)]

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 lets you assign a number between 0 and 1 for both "average" and "tall", thus avoiding making this hard decision.

Jeebus

Dec24-03, 02:37 PM

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 gonna derive a contradiction if the values aren't restricted to be 0-1. [6)]

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 lets 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.

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

Dec25-03, 01:55 AM

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

Dec29-03, 08:30 AM

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 isnt 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 statment.

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

Dec29-03, 09:00 AM

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 statments 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

Dec29-03, 01:14 PM

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 statments 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

Dec29-03, 03:02 PM

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

Dec29-03, 03:28 PM

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

Dec29-03, 03:32 PM

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 statments 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 realy doesn't apply. Philosophically, it's trecherous to use without abusing it.

At least that's my take on it.

phoenixthoth

Dec29-03, 05:20 PM

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

Dec29-03, 05:43 PM

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 realy 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

Dec29-03, 06:37 PM

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

Dec29-03, 11:51 PM

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

Dec30-03, 09:41 AM

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

Dec30-03, 10:56 AM

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

Dec30-03, 11:25 AM

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

Dec30-03, 12:38 PM

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

Dec30-03, 12:47 PM

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

Jan1-04, 07:44 PM

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

Jan2-04, 06:51 AM

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

Jan2-04, 07:29 AM

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

Jan2-04, 08:00 AM

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

Jan2-04, 09:52 AM

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

Jan2-04, 12:01 PM

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

Jan2-04, 03:15 PM

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 explaination was somewhat ambiguous. Yours is much better.

Canute

Jan3-04, 08:15 AM

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

Jan3-04, 09:48 AM

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

Jan3-04, 10:09 AM

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

Jan3-04, 11:18 AM

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

Jan3-04, 11:19 AM

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

Jan3-04, 11:35 AM

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

Jan3-04, 12:02 PM

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

Jan3-04, 12:12 PM

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

Jan3-04, 02:14 PM

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

Jan3-04, 04:32 PM

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

Jan3-04, 04:46 PM

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

Jan4-04, 06:50 AM

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 realise 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

Jan4-04, 09:26 AM

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

master_coda

Jan4-04, 09:29 AM

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

Jan4-04, 10:29 AM

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 Hawkings 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 refering 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

Jan4-04, 10:43 AM

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

Jan4-04, 11:26 AM

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 refering 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

Jan4-04, 02:51 PM

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 refering 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

Jan4-04, 03:16 PM

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

Jan4-04, 03:43 PM

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.

Hurkyl

Jan4-04, 04:17 PM

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.

I'm not entirely sure what you're trying to imply, so I'll take a stab in the dark; how about this axiom of real arithmetic:

\forall a,b \in \mathbb{R}: a + b = b + a

And as I mentioned before, any connection between reality and a mathematical system is not part of the mathematical system.

Bob3141592

Jan4-04, 05:10 PM

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.

Ah, I think this is where the discussion diverges.

A formal mathematical system means that the axioms are abstract statements containing various symbols that are assummed to be true. The sysmbols used in the axioms don't actually have any meaning. As such, they refer to nothing.

When a mathematician writes F = ma, he is making no statement beyond how vectors multiply together. But when you say force equals mass times accelleration, then you have added meaning to the symbols that are not part of the mathematics. Sure, they look the same, but even in this very simple equation of physics, you use a variable "m" to stand for something, mass, that in reality isn't really understood. The whole thing is more than just an approximation, it's a simplification that intentionally ignores major parts of reality in order to shoehorn it into a mathematical framework.

It's quite astounding that it seems like the complicating factors that were simplified away can be accounted for, and added back in by physicists. It takes trained physicists, because the equations get more and more involved, and more complicated, and most people go absolutely apoplectic when they see them. But no matter how elaborate those equations of physics become, they continue to make simplifying assumptions, and to leave out complicating factors tat make working with the equations just too complicated. Nobody even tries to write equations even for something as simple as an Estes model rocket using quantum mechanics. But if they don't, they can't pretend that the equations they use are an exact, meaningful represetation of reality. Most physicists don't even try to make that claim. It's only an approximation, and generally, they're quite happy that they get the aproximation good enough that it can be scaled up to throw bigger rockets at Mars. They're esctatic when their aim is true, but they don't think of it as Truth.

I suppose there is a class of physicists who are looking for Truth. t'Hooft wants to find the "ultimate" building blocks. Lederman wanted to find the "God particle" (though I understand it was more often refered to as the "Goddam particle." But it's presumptuous of us to asume we understand how those people really think philosophically. It's more that if they didn't write poeticaly, people like us wouldn't be able to understand anything they said.

Bob3141592

Jan4-04, 05:18 PM

Originally posted by Hurkyl
I'm not entirely sure what you're trying to imply, so I'll take a stab in the dark; how about this axiom of real arithmetic:

\forall a,b \in \mathbb{R}: a + b = b + a

And as I mentioned before, any connection between reality and a mathematical system is not part of the mathematical system.

Ironic, isn't it, that the example you gave to show mathematics isn't equivalenty to reality says "for everything in the Reals [:)]

Perhaps a more philosophically poetic example would involve the imaginary numbers instead of the reals.

And it remains ironic that physics requires imaginary operations to produce "real" results, though most casual philosophers I've talked to think imaginary numbers are entirely unreal and, well, imaginary.

Hurkyl

Jan4-04, 05:31 PM

Bah fine. How about the ring axiom:

\forall a, b: a + b = b + a

selfAdjoint

Jan4-04, 06:34 PM

Originally posted by Hurkyl
Bah fine. How about the ring axiom:

\forall a, b: a + b = b + a

That's a commutative ring axiom (or an Abeliean group axiom!)

I am not sure I understand what discussing individual axioms is getting us in this discussion?

Canute

Jan5-04, 03:21 AM

I wonder if this disagreement, my part of it anyway, is caused by my use of language, which may be technically sloppy, me not being a mathematician n'all.

All I'm defending is the idea that an axiom is an assumption, that by defintion it is not derived from the system for which it is an axiom, and that it therefore refers to something beyond that system. I'm still not clear how there can be any objection to this.

If I'm wrong then fine, my world-view doesn't depend on it. But I haven't seen anything I regard as a counter-argument yet. To be honest to me it seems inarguable.

I suppose, in Kantian terms, I'm arguing that systems with no axioms (or axioms that are provable within the system) are analytic, (self-referential) whereas systems with axioms are synthetic (refer beyond themselves).

Or, in logical positivist terms, that complete systems, in which the axioms are provable, are by definition tautological and thus trivial (in a formal sense).

Thus if an axiom is provable (all red roses are red) it can never give rise to knowledge of anything beyond the system. But if it is not (all roses are red) then it can, because it makes a claim about something outside the system.

Hurkyl

Jan5-04, 06:56 AM

As far as I can tell, your idea of an axiom is simply not the mathematical idea of an axiom.

Bob3141592

Jan5-04, 09:11 AM

Originally posted by Canute
I wonder if this disagreement, my part of it anyway, is caused by my use of language, which may be technically sloppy, me not being a mathematician n'all.

All I'm defending is the idea that an axiom is an assumption, that by defintion it is not derived from the system for which it is an axiom, and that it therefore refers to something beyond that system. I'm still not clear how there can be any objection to this.

If I'm wrong then fine, my world-view doesn't depend on it. But I haven't seen anything I regard as a counter-argument yet. To be honest to me it seems inarguable.

Well, the first part of that is right, an axiom is an assumption and it isn't derived from the system. The second part, the conclusion, is wrong. An axiom does not refer to something beyond the system. The axiom is one of the foundation parts of the system; there is no system without the axioms. And that's a big objection to what you're describing.

Perhaps what you mean isn't an "axiom" but an "observation." Models built to explain observations can use axiomatic systems in their method of explanation, but that isn't equivalwent to the model.

So perhaps it's all a matter of terminology.

I suppose, in Kantian terms, I'm arguing that systems with no axioms (or axioms that are provable within the system) are analytic, (self-referential) whereas systems with axioms are synthetic (refer beyond themselves).

What system has no axioms? That doesn't make any sense to me. Axioms are a crucial part of the definition of the system.

Or, in logical positivist terms, that complete systems, in which the axioms are provable, are by definition tautological and thus trivial (in a formal sense).

Axioms don't need to be proved. It's a given that they're true (at least, within the system they help define). If a system proves it's axioms false, then the system is contradictory.

I'm not certain about this, but I think even inconsistent systems don't prove their axioms false. An inconsistent system allows derived statements to be shown as both true and false. But that applies to derived statements, not the axioms. Is that right?

Thus if an axiom is provable (all red roses are red) it can never give rise to knowledge of anything beyond the system. But if it is not (all roses are red) then it can, because it makes a claim about something outside the system.

I don't think any formal system can ever give rise to knowledge of anything beyond the system. The symbols used in formal systems are in and of themselves meaningless, but the way you're talking it seems you assume they have a meaning that applies to other things.

If you're trying to build a model of other things based on observations, and the model develops inconsistencies or contradictions, then the first place to look for problems would be in the observation or the interpretation of the observation. There are countless more ways to go wrong there. You can't asume the observation is perfect and that problems come about because the axiomatic system used to model the observation is defective.

Or do I misunderstand the intent of your argument?

Canute

Jan5-04, 11:51 AM

Originally posted by Bob3141592
[B]Well, the first part of that is right, an axiom is an assumption and it isn't derived from the system. The second part, the conclusion, is wrong. An axiom does not refer to something beyond the system. The axiom is one of the foundation parts of the system; there is no system without the axioms. And that's a big objection to what you're describing.
I think you misunderstand what I mean. Of course there is no system without the axioms. But the axioms refer beyond the system. Unlike all the other theorems of the system they deal with the outside world, they refer directly to it. That is why they are axioms. If they didn't do this they couldn't be axioms. Instead they would be theorems derived from the system.

Perhaps what you mean isn't an "axiom" but an "observation." Models built to explain observations can use axiomatic systems in their method of explanation, but that isn't equivalwent to the model.
No, that isn't it.

What system has no axioms? That doesn't make any sense to me. Axioms are a crucial part of the definition of the system.
If an axiom is proved within the system then it is not an axiom. If a system proves its own axioms it ceases to be a non-trivial axiomatic system. It becomes entirely self-referential. A system that says that a=b, b=c, c=a is trivial in a mathematical sense, and by one point of view does not even contain an axiom.

Axioms don't need to be proved. It's a given that they're true (at least, within the system they help define). If a system proves it's axioms false, then the system is contradictory.
OK.

I'm not certain about this, but I think even inconsistent systems don't prove their axioms false. An inconsistent system allows derived statements to be shown as both true and false. But that applies to derived statements, not the axioms. Is that right?
Hmm. I think maybe you're right in 'local' mathematical terms, but otherwise I'm arguing that Goedel is right precisely because all axiomatic systems do have false or inconsistent axioms inasmuch as they refer to reality. That's a personal view though, and based on metaphysics more than mathematics.

I don't think any formal system can ever give rise to knowledge of anything beyond the system. The symbols used in formal systems are in and of themselves meaningless, but the way you're talking it seems you assume they have a meaning that applies to other things.
No, I agree with you as far as mathematics goes. But in theorising about reality we have to give those symbols meaning, and interpret what the mathematical behaviour of the symbols might mean, as a physicist would.

If you're trying to build a model of other things based on observations, and the model develops inconsistencies or contradictions, then the first place to look for problems would be in the observation or the interpretation of the observation. There are countless more ways to go wrong there. You can't asume the observation is perfect and that problems come about because the axiomatic system used to model the observation is defective.
Agreed

Or do I misunderstand the intent of your argument?
I think you do, but I can't tell whose fault it is. I may be expressing myself badly, but I'm in the worst position to judge. [:D]

phoenixthoth

Jan6-04, 01:50 PM

a bit of delicacy. some axioms in set theory can be proven from others and they do not need to be called axioms; they could be called theorems. however, there do seem to be axioms that one can prove are "independent" and my understanding is that the gist of that means that one cannot prove the other.

part of what you wrote is that axioms are assumptions and this is my understanding of axioms. i hope this isn't what hurkyl was objecting to because that means i'm wrong. that's what i think the kernel of an axiom is. in addition, it's also a statement along the lines of "this is what our system depends on" and "if you accept this then these must follow."

so instead of being like a=b, b=c, and c=a for those axioms which can prove each other, it's more like a=b+d, b=c, d=c-b. you still have a=b=c yet 1. there is this independent d and 2. these "equations" have more "information" than a=b=c. i'm probably not explaining this right.

Canute

Jan6-04, 02:50 PM

Yes, that's about what I was getting at. An axiom is an assumption about something outside the system to which it gives rise.

phoenixthoth

Jan6-04, 02:57 PM

i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

hmm... what are we really talking about here? read that one more time and you'll get the allusion.

master_coda

Jan6-04, 04:12 PM

Originally posted by Canute
Yes, that's about what I was getting at. An axiom is an assumption about something outside the system to which it gives rise.

What outside of the system does that axiom refer to?

phoenixthoth

Jan6-04, 04:33 PM

math is replete with examples.

eg: parallel line axiom is related to what is observed in reality.

eg: the empty set is related to the concept of zero, nothingness, and emptiness eg one ten and zero ones is 10.

eg: the peano axioms are in relation to numbers which relate to measurement of things in the real world.

eg: field axioms relate to real numbers which relate to geometry which relates to the real world. e, for example, is used in compound interest (in reality) and solving a huge chunk of differential equations which model reality not to mention sine and cosine, which depend on e, which model periodic behavior in reality.

i repeat: i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

master_coda

Jan6-04, 05:17 PM

Originally posted by phoenixthoth
math is replete with examples.

eg: parallel line axiom is related to what is observed in reality.

eg: the empty set is related to the concept of zero, nothingness, and emptiness eg one ten and zero ones is 10.

eg: the peano axioms are in relation to numbers which relate to measurement of things in the real world.

eg: field axioms relate to real numbers which relate to geometry which relates to the real world. e, for example, is used in compound interest (in reality) and solving a huge chunk of differential equations which model reality not to mention sine and cosine, which depend on e, which model periodic behavior in reality.

i repeat: i agree but i don't think it is wrong to say that an axiom only refers to the system it is part of the foundation of. that's what it refers to but that is not why it exists ie, why it was created.

You cited a number of examples of axioms that "correspond" to what we see the real world. But it is in no way necessary for axioms to manifest themselves some way in reality.

For example, non-Euclidean geometry generally replaces the parallel lines axiom with an axiom that is contradictory to the parallel lines axioms. And non-Euclidean geometry is just as much of a valid system as Euclidean geometry.

Most of the other examples are very circular...they're concepts that relate to the real world because humans use them...and we use them because they relate to the real world. Perhaps they only seem to be "intrisically true" to us because we were raised to believe that.

Whether or not the axioms are "true" outside the system is irrelevant. It's usually best to start with axioms that intuitively seem to be true, but we can just as easily do the opposite, and the math is just as valid.

Of course, using the mathematical definitions of what a formal system is and what axioms are tends to rob those concepts of any philosophical meaning...but if you want to use the incompleteness theorems then you have to use the mathematical definitions. You can't just discard those definitions because you prefer more meaningful ones and keep the theorems at the same time.

phoenixthoth

Jan6-04, 07:10 PM

part of what i said earlier was:
an axiom only refers to the system it is part of the foundation of (this is what you said in more words). that's what it refers to but that is [edit: usually] not why it exists, ie, why it was created.

as you said, the new axioms from post 1850 or so had no correspondance to reality except that what we imagine is real to an extent in that we really imagine. so those axioms still corresponded to or were invented by an observation of an aspect of reality: the reality inside our minds which of course is not real in the usual sense. but then it turned out that certain axiomatic adjustments lead to reality anyway, which is kind of interesting. i wonder what more we can piece together by adjusting the way we think.

this leads back to max tegmark's TOE article where existence is equivalent to freedom from contradiction.

Canute

Jan7-04, 06:41 AM

But a non-trivial axiomatic system cannot prove its own axioms. They are only partly part of the system.

(On the problem of explaining the cosmos) – “Every proof must proceed from premisses; the proof as such, that is to say the derivation from the premisses, can therefore never finally prove the truth of any conclusion, but only show that the conclusion must be true provided the premisses are true. If we were to demand that the premisses should be proved in their turn, the question of truth would only be shifted back by another step to a new set of premisses, and so on to infinity.”

Karl Popper – The Problem of Induction (1953, 1974) from ( http://www.dieoff.org/page126.htm)

Aristotle got around this by saying that there are earlier premisses which are indubitably true, and which do not need any proof. He called these ‘basic premisses’.

For both Aristotle and Pooper the problem was the in principle impossibility of proving ones axioms, or basic premisses, without rendering the system trivial.

Inasmuch as human knowledge is based on axiomatic reasoning and proofs Popper also writes this:

“What we should do, I suggest, is give up the idea of ultimate sources of knowledge. And admit that all human knowledge is human: that it is mixed with our errors, our prejudices, our dreams, and our hopes: that all we can do is grope for truth evn though it be beyond our reach. We may admit that our groping is often inspired, but we must be on our guard against the belief, however deeply felt, that our inspiration carries any authority, divine or otherwise. If we thus admit that there is no authority beyond the reach of criticism to be found within the whole province of our knowledge, however far it may have penetrated into the unknown, then we can retain, without danger, the idea that truth is beyond human authority. And we must retain it. For without this idea there can be no objective standards of enquiry; no criticism of our conjectures; no groping for the unknown; no quest for knowledge.”

Karl Popper – ibid.

I believe he was right given his scientific definition of knowledge, but wrong because he defined knowledge too narrowly.

Descartes recognised the problem, and thus chose a fundamental axiom that was in principle impossible to prove.

master_coda

Jan7-04, 04:18 PM

Of course we can prove the axioms of a non-trivial system. They're true by definition.

I'm well aware of the fact that you can't prove that the axioms are true "in reality". That's why mathematical axioms don't refer to reality. If they did, we would have to justify the axioms (we would have to show that they in fact do refer to the real reality). But since they don't we can just say "define these axioms as true" and they are.

phoenixthoth

Jan7-04, 04:24 PM

one can even define contradictory axioms to both be true but the result is a trivial system in which one can prove everything is true and false. so a major goal in any system should be consistency, i.e., proving that no theorem can be proved true and proved false. does the method called "forcing" do this in set theory; what is forcing for?

master_coda

Jan7-04, 05:38 PM

Originally posted by phoenixthoth
one can even define contradictory axioms to both be true but the result is a trivial system in which one can prove everything is true and false. so a major goal in any system should be consistency, i.e., proving that no theorem can be proved true and proved false. does the method called "forcing" do this in set theory; what is forcing for?

I'm not very familiar with the details of forcing. But it isn't for proving consistency, it's for proving relative consistency.

Paul Cohen originally used it to show that if ZF set theory is consistent, then ZF set theory + the negation of the axiom of choice is consistent.

Canute

Jan8-04, 05:34 AM

Originally posted by master_coda
Of course we can prove the axioms of a non-trivial system. They're true by definition.

I'm well aware of the fact that you can't prove that the axioms are true "in reality". That's why mathematical axioms don't refer to reality. If they did, we would have to justify the axioms (we would have to show that they in fact do refer to the real reality). But since they don't we can just say "define these axioms as true" and they are.
Defining an axiom as true is not quite the sane thing as proving that they are true. Anything at all can be defined as true. Don't forget that the first axiom in a system is adopted before there is any other theorem to contradict it. You can adopt any old axiom if you take no account of 'reality'.

A system that proves its own axioms must be trivial, (contain only trivial theorems) for the same reason that all analytic propositions are trivial (in a scientific sense).

Thus

All men are mortal
Socrates is a man
Socrates is mortal

is entirely trivial unless one assumes that the first two propositions refer beyond the system to real men and the real Socrates. If they do not then the syllogism is tautological and entirely trivial.

Canute

Jan8-04, 05:35 AM

Originally posted by master_coda
Paul Cohen originally used it to show that if ZF set theory is consistent, then ZF set theory + the negation of the axiom of choice is consistent. [/B]
Can you expand on that a bit, or give a link.

master_coda

Jan8-04, 07:22 AM

Originally posted by Canute
Defining an axiom as true is not quite the sane thing as proving that they are true. Anything at all can be defined as true. Don't forget that the first axiom in a system is adopted before there is any other theorem to contradict it. You can adopt any old axiom if you take no account of 'reality'.

A system that proves its own axioms must be trivial, (contain only trivial theorems) for the same reason that all analytic propositions are trivial (in a scientific sense).

Thus

All men are mortal
Socrates is a man
Socrates is mortal

is entirely trivial unless one assumes that the first two propositions refer beyond the system to real men and the real Socrates. If they do not then the syllogism is tautological and entirely trivial.

Yes, you can adopt any old axiom.

Of course, if you consider any tautology to be trivial, then you must consider all of mathematics to be trivial. After all, math is really nothing but a bunch of tautologies. However, that is the strength of mathematics. We always know that the conclusions necessarily follow from the premises.

Of course there is a great deal of science that needs more than math can provide. Thus we can "extend" math and try and make it refer to reality. For example, one can attempt to apply geometry to the universe.

However, when you try to apply math to "reality", you can't change the math at will, or apply it completely beyond its scope. Or at least, if you're going to do that, don't try and pertend you are making use of math.

For exapmle, you can't change the definition of a line to make it match reality more closely, and then apply geometry. The results of geometry come from the properties of the original definition, and changing the definition changes the results.

Similarly, incompleteness applies to a specific definition of what a formal system is, and it result is about a specific definiton of completness (or lack thereof).

Originally posted by Canute
Can you expand on that a bit, or give a link.

Well, here's a page that mentions what Cohen used forcing to prove:

http://mathworld.wolfram.com/Forcing.html

And here's a page that gives a bit of an overview of forcing.

http://planetmath.org/encyclopedia/Forcing.html

Canute

Jan8-04, 10:05 AM

master-coda

Perhaps the problem is the word 'trivial'. When I use it I don't mean 'of no importance' or 'of no utility'. I mean it in the scientific sense of 'trivially provable'.

This is synonymous with its mathematical use as (of the solutions of a set of homogenous equations) 'having zero values for all the variables' (Collin's Dictionary).

So when I say that a self-proving system is trivial I don't mean that most of mathematics is not useful, I just mean that it is trivial in a scientific sense, it makes no assertions about anything beyond the systems employed (insofar as they are tautological).

I tried your links but they're too technical for me. Is Cohen saying that for every consistent system there is an equally consistent system that can be derived from the negation of its axioms?

master_coda

Jan8-04, 01:45 PM

The link is somewhat too technical for me as well. I only have a general idea of the method, this isn't my field of expertise.

The basic idea is that, under certain conditions, you can add an extra set to set theory without breaking the consistency of theory. The idea is this:

1. Take some model of set theory A.
2. Take some set B that is not part of set theory A.
3. Create a new set theory C that models A but also contains B.
4. If B satisifes certain conditions, we know that the model C is relativly consistent with the model A.

Cohen used this to add a new set to ZFC without breaking the consistency of ZFC. In the extended version of ZFC with his set, he was able to prove that the continuum hypothesis is false. Since his new set theory is at least as consistent as ZFC, we know that we can add the negation of the continuum hypothesis to ZFC without breaking consistency.

Cohen used a similar technique to show that we can add the negation of the axiom of choice to ZF set theory without breaking consistency.

Since Godel earlier proved that we can add the axiom of choice to ZF or the continuum hypothesis to ZFC without breaking consistency, we know that if ZF is consistent, then both ZF+axiom of choice and ZF+not axiom of choice are consistent. The same goes for ZFC and the continuum hypothesis...we can add it or its negation without breaking consistency.

When you want to apply math to reality, you have to be careful how you do it. Obviously when we start extending math to reality, we're only getting an approximate model. For example, lines drawn on a piece of paper aren't the same as lines in geometry. Lines in geometry have no width, and lines you draw do.

But the geometric lines are a very good approximation in most cases. So it's OK to apply geometry to drawings as long as we recognize that sometimes there'll be a conflict between the math and reality.

Yet you have to be careful to use at least a reasonable approximation. And when you aren't using an entire system, but only a single theorem, then you have to be even more careful, since violating even a single condition of the theorem will generally break it.

Canute

Jan8-04, 02:16 PM

Thanks for that. Now all I need to know is what the axiom of choice and continuum hypothesis are. [:D]

I'll have a look around and see if I can find an explanation simple enough for me.

selfAdjoint

Jan8-04, 02:30 PM

The axiom of choice says that if you have a (possibly infinite) collection of (possibly infinite) sets - any such collection - you can form a new set containing one element from each of the sets in the collection.

master_coda

Jan8-04, 02:37 PM

The continuum hypothesis is the hypothesis that there isn't any set larger than the set of natural numbers and smaller than the set of real numbers.

Canute

Jan8-04, 04:07 PM

Wow, I expected pages of mathematics. Thanks.

So which of these is right and wrong?

1. It been proved that ZFC is consistent whether or not these two propositions are taken as true.

2. These propositions are undecidable within ZFC, in some sense they are 'Goedel sentences'.

3. If the axiom of choice proposition is true then in ZFC there are an infinite number of possible sets.

4. If the continuum hypothesis is true then there are an infinite number of natural numbers.

5. ZFC is consistent if both propositions are taken as true, or both as false, but not if one is assumed false and the other true.

6. Neither proposition is a theorem in ZFC.

What do you reckon?

I don't get the bit about 'smaller than the set of real numbers'. (I always get confused by all these different sorts of numbers. I always forget which is what).

Cheers
Canute

Hurkyl

Jan8-04, 04:45 PM

(p.s. ZFC means "ZF + the axiom of choice"; I think you meant to say ZF everywhere you said ZFC in this post)

1. It been proved that ZFC is consistent whether or not these two propositions are taken as true.

False; only relative consistency may be proven mathematically.

2. These propositions are undecidable within ZFC, in some sense they are 'Goedel sentences'.

The first half is correct; I'm not sure I'd call them Goedel sentences, though.

3. If the axiom of choice proposition is true then in ZFC there are an infinite number of possible sets.

Correct... but the hypothesis is irrelevant; the axiom of infinity in ZF guarantees an infinite number of sets.

4. If the continuum hypothesis is true then there are an infinite number of natural numbers.

Again, this is guaranteed by the axiom of infinity.

5. ZFC is consistent if both propositions are taken as true, or both as false, but not if one is assumed false and the other true.

I *think* that ZF + any combinations of accepting or denying the axiom of choice and continuum hypothesis is relatively consistent to ZF.

6. Neither proposition is a theorem in ZFC.

Because the axiom of choice is an axiom of ZFC, it is also theorem in ZFC, via the trivial proof:

The axiom of choice.
Therefore, the axiom of choice.

However, it is correct that the axiom of choice is not a theorem in ZF.