Peer Reviewed Bad ID Math

Mark,

Your blog is becoming one of my favorites with regards to shredding the shit that ID puts out. You get straight to the point, no fluff, no bother. Thank you.

For that very reason, they won't be hissing back at you -- because they'd actually have to show how you didn't refute their nonsense, which would actually require some semblance of intellectual rigor. Thus, no response will be forthcoming.

Thanks again.

Am I right that he's asserting that the relationship between symbolic and physical is always from symbolic to physical and never the other way? That we don't create symbols to describe the physical? It sounds a lot like "In the beginning was the Word..." done up in a tuxedo.

If he were a Wiccan, I'd think that he was trying to provide a mathematical basis for symbolic magic: Manipulate the symbol and you manipulate the thing.

I'm glad that you have the patience to wade through this stuff.

Hrm... What I find really funny about this argument is that even if you assume that only an intelligent designer is capable of creating a self-referential system, you cannot talk about such an intelligent designer in the same logical system. Doesn't that kinda make the whole idea of intelligent design as a "logical" explanation of life's origin impossible?

The first thing I thought when I read your first quote from Voie was how much it sounded like "quantum theory" arguments used to support health belief systems like homeopathy.

The second thing I thought, after reading more quotes, was that Voie is simply rehashing Berkeley's Dialogues in mathematical drag, with "symbolic representation" substituted for "perception."

The third thing I thought is that Voie seems to be arguing that reality is dependent on representation, which sounds really "vulgar postmodernist" to me.

The footnote on truth was added on January 5 of this year, so if the article was written before that (which seems likely), it could be that the author honestly doesn't understand Gödel's Theorem.

I started to read through the original paper. The author seems to have begun with the misconception that there is something very unusual or "artificial" about self-referential systems and then made the leap that such things are the product of human interpretation rather than a straightforward property of sufficiently powerful axiom systems. Take this statement, for instance:

The factor of human creativity in mathematical theories seems to have been overlooked in the history of science. Odd and controversial results might seem less odd when this factor is taken into account.

It's "overlooked" because the truth of Goedel's theorem does not depend on human creativity. It seems less odd and controversial once you become sufficiently familiar with the result, after which you realize that it is simply a theorem with a proof. The proof is not even all that difficult compared to many others (heck, even I understand it, but I'll probably never understand how Wiles proved Fermat's last theorem).

At this point, it seems "odd" to me that people ever supposed that every statement in first order logic would have a finite-length proof or disproof. Talk about optimism. As for "controversial", what controversy? Any good CS program requires a course in math logic where they teach you this as a theorem.

Self reference is all over the place. A lambda expression with a fixed point is not really any more amazing than a conventional numerical fixed point such as: x=1+rx (the equation for solving the convergent series 1+r+r^2+r^3+... for |r| less than 1). In both cases, you define a function for which x=f(x). Actually, both are pretty cool in their own ways, but there is nothing in either that is a product of the human mind in some mystical sense.

As for how this made it through peer review: probably a combination of reviewer laziness and lack of necessary background to evaluate.

Poor Kurt Gödel never deserved the kind of abuse he still receives, despite the valiant efforts of Cosma Shalizi and Torkel Franzen[1].
It's worth noting that Gödel is frequently invoked to "disprove" the possibility of evolution, as in this steamer from TCS:

[F]aith must precede reason, even if the faith is only in reason itself (as Gödel showed, reason cannot prove its own validity).

And let me add[2] in passing that this Left-Right struggle over science is not even nearly symmetric. This is only one small patch of anti-rational territory in the rapidly expanding empire that is Wingnuttia -- and conservatives really ought to be worried about it.

[1] See Cosma (near the bottom) for the link. I can't link again here without triggering SB's anti-spam sentry.
[2] Struggling, and failing, to suppress the political potshot.

I would stop short of any analysis and send back a paper that quotes Wikipedia in the first place. If you can't do it in a freshman history paper on the Battle of the Bulge, surely in a peer reviewed mathematics journal it should be totally out of bounds?

bmurray:

I do agree with you. I can see having a one-off citation of wikipedia solely in the context of "See what a common information source for the general public says about this"; or "here's a cute quotation about this subject". But to use it repeatedly as a major source for a real publication? The editors and the reviewers absolutely should not have permitted it.

Dan:

He doesn't get Gödel at all. His interpretation of it is that while pure formal systems can't cope with it, intelligent minds can formulate demonstrably true statements that escape the bounds of the formal system. So since he's allegedly an intelligent mind communicating with other intelligent minds about the existence of a supreme intelligent designer mind, he thinks that the argument is workable in the context of that kind of communication, even if it isn't in purely formal logic.

jre:

You can link as much as you want; it just might delay the appearance of your comment slightly. I try to check the spam queue every couple of hours, delete the spam, and publish everything legit.

Very nice exposure of ID babbling. I do have to take issue with a few statements, though:

3. Gödel's theorem says that incompleteness statements are true.(wrong)

The catch of course is that the statement is not really true. Incompleteness statements are neither true nor false. They are paradoxical.

This is nonsense. Assuming there actually is such a thing as the natural number system, every first-order statement about it is either true or false, and the Gödel sentence for a decidable set of axioms T (the sentence s which is equivalent in T to "s is not provable from T") is such a statement. (We assume that T is an extension of the basic arithmetic axioms, for example PA.)

Moreover, if the axioms of T are true, then s is itself true. To see this, note that it is provable in T (hence true) that s is equivalent to "s is not provable from T". So s is either provable from T and false (a contradiction), or true and not provable from T.

But wait: didn't we just prove s? Yes, but we didn't prove it in the axiom system T. We escape outright contradiction because this argument can't be formalized in first-order logic at all; in particular, there's no first order formula that says of a number n "n represents a true statement". (This is known as Tarski's Theorem.)

There are plenty of self-referential statements that don't trigger incompleteness. For example, in set theory, I can talk about "the set of all sets that contain themselves". I can prove that there are two sets that meet that description: one contains itself, the other doesn't. There's no paradox there; there's no incompleteness issue.

The main point here is correct, but I don't get your example at all. In every set theory I've heard of, sets are uniquely determined by their elements, so there would be at most one set of "all sets which are elements of themselves". In ZF or ZFC the empty set fits this description, as the foundation axiom forbids any set from being an element of itself, while in NF (for example) there would be no such set.

How does one cite wikipedia for a peer-reviewed article? I know it's generally pretty accurate, but it's inherently fluid, lacks accountability, and is not, not, not a primary source!

...it is published by Elsevier, a respectable publisher.

It's probably not all that uncommon that silly stuff like this gets published in mainstream journals. Last year I complained in this blog entry about an article in Elsevier's Theoretical Computer Science which contained similar abuses of logic. The article in question was written by Selmer Bringsjord, who is the head of the cognitive science program at RPI. Which I suppose goes to show that even people who ought to know better are in the habit of mangling Gödel to serve their purposes.

(By the way, perhaps I should add that Bringsjord is not trying to show the existence of a designer, but merely to show that human minds must be "hypercomputers". The logical argument grates just as loudly on one's sense of mathematical aesthetics, though.)

Have you considered contacting the editors of said journal regarding this tripe? It's possible they could be embarrassed into de-accepting it.

Chaos, Solitons and Fractals appears to be something of a vechicle for its editor, one M. Saladin El Naschie, to publish. This year alone, he has 21 papers in there. I think we can all agree that this is ridiculous. I'd be happy with that many in a decade or two.

I'm sure that good research is published in CSF, but it is an odd little journal.

... 3Gödel's theorem says that incompleteness statements are true.(wrong)
----
The catch of course is that the statement is not really true. Incompleteness statements are neither true nor false. They are paradoxical.
----
This is nonsense. Assuming there actually is such a thing as the natural number system, every first-order statement about it is either true or false,
---
I just want echo chad's earlier statement. The statement "This statement is false." is paradoxical, the statement "This statement can not be proved in system K." is not.

In fact,if the statement is statable in K we can say that it is true when referring to a consistent K and false otherwise. Proof of this is equivalent to Gödel's second incompleteness theorem.

As I understand it, the "paradox" of a Godel statement is that we can see that the statement must be true (if the system is consistent), yet we cannot prove it.

Eh, this "paradox" business is just getting silly. The salient property of a Goedel statement is that no finite application of inference rules will suffice to deduce it or its inverse from the axioms. It is only surprising if we are predisposed to believe that every proposition or its inverse should have a finite proof. The lack of a finite-length proof should not really be a huge shock considering that the statement is universally quantified over an infinite domain. We're just spoiled by all the nice statements that have finite proofs.

Granted, we're accustomed to being able to prove lots of useful statements about infinite sets using a finite application of inference rules. If I say "the sum of two odd numbers is always even" I can prove that without writing an infinite enumeration of all pairs of odd numbers. There's a finite proof of the statement. But just because there's one in that case doesn't mean there always should be any more than we should expect every equation to have a closed form solution. (Heck, some ancient greeks may have found it a "paradox" that not all numerical solutions could be expressed as a ratio of integers; today's paradox is tomorrow's self-evident truth).

Very roughly Goedel's statement says: "For all finite-length proofs S, S is not a proof of this statement." (and his big insight was to show that you could represent such a statement within a system such as Peano's axioms). But there's no paradox. If you started to write each possible proof S, you could establish that each one you managed to test was, in fact, not a proof of Goedel's statement. Likewise, there is no disproof, because that would imply the statement is false and hence the existence of a proof, but you cannot have both in a consistent system.

Does that make the statement "true"? It's understandable that some might see it intuitively that way since you can sort of imagine enumerating all possible proofs. But you could add either the proposition or its inverse to the set of axioms and still have a consistent system. That is a very hazy shade of "true" in my book.

Thus, Goedel's statement is neither true, false, nor paradoxical. It is probably best characterized by its own wonderfully concise self-description.

Evan Murdoch,

The way to get around the fluidity of Wikipedia articles is to cite a specific revision in the article history. Wikipedia's administration only remove specific revisions if they contain legally actionable material, so they are stable for all practical purposes.

Whether or not you ought to cite Wikipedia is another question altogether. But I would like to point out that Wikipedia articles do not lack accountability, under either of the dictionary definitions of the word.

I started contributing to the Wikipedia two and a half years ago. In that time, I've made over five thousand individual edits and wrote five "Featured Articles", four of which have been displayed on the main page. And I still say that anybody who cites a Wikipedia article as a scholarly source is (to lift a phrase from John von Neumann) living in a state of sin.

In a funny way, editors on the Wikipedia are more accountable than people writing elsewhere in the Matrix, because the thousands of other people watching what they do take the policies of "verifiability", "reliable sources", "neutral point of view" and "no original research" pretty seriously. (While an individual revision of an article will last, for all practical purposes, forever, there is never a guarantee that said revision is a good one.) Every day, the Wikipedia becomes a better starting point for scholarship, but it will never be an end unto itself.

Re: Godel's statement

I was under the impression that the statement was assumed to be true.

Any statement must be true or false. If his statement is assumed to be false, it creates a contradiction, making the logical system it is embedded in inconsistent (because it contains a contradiction). If it is assumed to be true, then it is simply unprovable, making the logical system it is embedded in incomplete.

Mathematicians chose the latter, because an inconsistent system is 100% useless. An incomplete system is only useless for proving the specific unprovable statements - a decent system is still quite fine for everything else.

Am I mistaken?

I can hardly believe how you can bring it up to digest such an article and analyse it so clearly. Well done ...

From what you write about it, it seems to me that the author actually poses an interesting argument here. First, DNA is definitely symbolic in nature, so DNA does prove that symbols (language if you wish) appears in nature that man hasn't made. Second, such "natural symbols" do seem to be a good "whetstone of evolutionism": if you believe that such well-organized designs can only arise as the result of a conscious design effort, you are a creationist; if you believe the opposite, namely, that such a design is far too robust and intricate to possibly be the result of a conscious design effort, and gradual evolution is the only possible way for such things to come into being, you're an evolutionist.

But I do not agree with your point 3. Some 90 years ago there was a big discussion among mathematicians on whether mathematics was about reality or about formal constructs. Does Euclidean geometry merely describe the formal consequences of a particular set of formal axioms and a particular formal theorem derivation system, and is its practical applicatability a mere fortunate coincidence? In that case, no mathematical statement about geometry can possibly be true: truth doesn't apply to them. Or is it the other way round, does Euclidean geometry describe the fundamental properties of land measurement, and are the axiomatization and formal reasoning system behind it only formulations of these properties, that might just as well be replaced with any other formalism that captures the same properties? In that case, all mathematical statements are true or false, while their provablility is just a fairly arbitrary property of their formalization.

You can argue that the difference is only terminological, but I think it goes too far to imply without argument that truth doesn't apply to mathematical statements at all, as you seem to do.

Xanthir:

Any statement must be true or false. If his statement is assumed to be false, it creates a contradiction, making the logical system it is embedded in inconsistent (because it contains a contradiction).

Unless I'm really confused, if you take a consistent axiom set and a Goedel sentence (G) for it, and add the inverse of that sentence (~G) as a new axiom, then you get a consistent set of axioms as a result.

Consider that a very standard way to prove a proposition is to start with the inverse of that proposition and derive a contradiction from it. We know that there is no proof of G in the original axiom set, so it follows that if we start by assuming ~G then we will not be able to derive a contradiction. Thus, adding ~G to the axiom set will not result in an inconsistency assuming the original axiom set was consistent.

OK, it's been years since I studied it, but am I wrong that all three axiom sets are consistent? I.e., The original, the original with G added as an axiom, and the original with ~G added as an axiom.

PaulC:

Yes, that's the way I understand Gödel as well: true statements + G is valid; true statements with not G is valid; and true statements without either is valid. That's what I meant by paradox: it makes no sense that both G and not G can be valid.

rp:

I think you misunderstood me. I was asserting that the statement that the paradoxical Gödel statements are true is not a valid statement. There's some philosophical interpretation involved in that statement as well; some people would argue that a Gödel statement G, *and* its negation, not G, are both true. I'd say that neither is really true; they're intrinsically inconsistent.

But no matter how you want to play the philosophy, asserting deep meaning from the "truth" of a Gödel statement is silly, since both the statement and its negation are equally true.

PC:OK, it's been years since I studied it, but am I wrong that all three axiom sets are consistent? I.e., The original, the original with G added as an axiom, and the original with ~G added as an axiom.

IIRC, yes, assuming the original system S is consistent. However (S and {~G}) is only consistent in that "(there exists)x such that P(x) holds" can not be shown to be inconsistent (under the rules of the game for some specific Ps) even when no such x exists.

Outside of the game,despite the Godel statement being one of these Ps, we can understand that if an explicit x that proved ~G existed then S would be inconsistent and hence S and {~G} would also be inconsistent.

Which leaves us with (S and {~G}) is consistent iff ~G is "false" i.e. there is not one theorem in S that implies G.

Phew. *IIRC disclaimer*

Does this song ring a bell?

"Don't let the sound of your own wheels,
drive you crazy."

Life as an abstract? Gee, when I woke up this morning, I thought I was real,,,

Mark CC:

I think you misunderstood me. I was asserting that the statement that the paradoxical Gödel statements are true is not a valid statement. There's some philosophical interpretation involved in that statement as well; some people would argue that a Gödel statement G, *and* its negation, not G, are both true. I'd say that neither is really true; they're intrinsically inconsistent.

But no matter how you want to play the philosophy, asserting deep meaning from the "truth" of a Gödel statement is silly, since both the statement and its negation are equally true.

So it's your position that there is no system of natural numbers? Because otherwise this makes no sense, as I argued above.

PaulC:

OK, it's been years since I studied it, but am I wrong that all three axiom sets are consistent? I.e., The original, the original with G added as an axiom, and the original with ~G added as an axiom.

Essentially, yes, but it's contingent on the truth notions above; that is, you have to start with a theory T that's true on the natural numbers (call them N). Also it has to be a decidable extension, etc. Then, if g is a Gödel sentence of T, we have T and (T + g) consistent (since they hold on N), and (T + ~g) consistent (by Gödel's Theorem).

But now, take the theory T1 = T + ~Con(T), where Con(T) is shorthand for "T is consistent". This is consistent (by Gödel's 2nd Theorem). It is consistent, etc., so it has a Gödel sentence g1, and Gödel's Theorem tells us that T1 + ~g1 is consistent. But T1 + g1 is not consistent, because one of its consequences is "T is not consistent, but T1 (which extends T) is". This is plainly impossible; if you can prove a contradiction in T, you can prove it in any extension.

But no matter how you want to play the philosophy, asserting deep meaning from the "truth" of a Gödel statement is silly, since both the statement and its negation are equally true.

It's been a while since I studied this, but I remember it this way...

One reading of "G" is: "Given a predefined mapping of numbers to proofs, there is no number that maps onto a proof of G".

So "not-G" is "There is a number X that maps onto a proof of G."

If not-G is true, and if we can determine X, then we can convert X into a proof of G and so the system is inconsistent. If we want to add not-G as an axiom and keep the system consistent, then we have to say that X is unknowable...it's infinite or otherwise off the number line, and we can't determine the proof that it maps to.

So adding not-G as an axiom is internally consistent, but in order for the system to work we need to change the definition of "number". Adding G as an axiom lets us keep the standard definition. (I think.)

chaosengineer:

So "not-G" is "There is a number X that maps onto a proof of G."
If not-G is true, and if we can determine X, then we can convert X into a proof of G and so the system is inconsistent. If we want to add not-G as an axiom and keep the system consistent, then we have to say that X is unknowable...it's infinite or otherwise off the number line, and we can't determine the proof that it maps to.

I think it's simpler than this. If you give me such a number X, that's a constructive proof of not-G and a refutation of G, which is what you'd expect to have in the extended axiom system. Not only that, it's easy to find such a number X. Just take the Goedel number for the axiom not-G. It's not the only proof of not-G. Actually, there are lots of them, although they all include the new axiom somewhere, since as we know, you cannot prove the assertion without it.

I think the only "paradox" is the fact that people historically conflated the notion of "truth" in a potentially infinite model with having a proof of finite length to demonstrate it. Goedel showed that this conflation is incorrect. Once you adapt your intuition, there is nothing that weird about it. Of course, that is what people often mean by paradoxes--not something truly contradictory but something counterintuitive.

On second thought, I probably spoke too hastily. G would have to encode the original axiom system to represent the predicate "X proves G" so an encoding of the new axiom would not satisfy the predicate. OK, I admit I have to think about this for a while.

I just dug up my copy of Hofstader's "Godel, Escher, Bach" to refresh my memory. He talks about adding "not-G" as an axiom in chapter 14..."X" winds up being what he calls a "supernatural number", and apparently there's a branch of math dedicated to studying them.

He also notes that it's theoretically possible that "not-G" could be proven without adding it as an axiom. In that case we'd have to accept supernatural numbers as a natural consequence of the system. (The same way that imaginary numbers are a natural consequence of quadratic equations.)

Good show! This will be an excellent reference in the future.

Two notes on "sign systems".

First: The reference is to a paper in the "Journal of Semiotics", and according to Wikipedia "Biosemiotics (bios=life & semion=sign) is a growing field that studies the production, action and interpretation of signs in the physical and biologic realms in an attempt to integrate the findings of scientific biology and semiotics to form a new view of life and meaning as immanent features of the natural world. The term "biosemiotic" was first used by F.S. Rothschild in 1962,"

I don't know the status of semiotics and especially biosemiotics. The SEED magazine looks respectable on the surface.

However, Rotschild was a religious crank. "Kull dubbed Rothschild an "endemic semiotician", as Rothschild was quite aware that semiotics grounded and synthesized his own work in psychology, psychotherapy, psychoanalysis, embryology, neurobiology, theoretical biology, and philosophy (and theology!), although his most intense interactive discourse community must have sometimes been limited to himself alone." "Keep in mind that, while he construes biosemiotics almost anthropocentrically, Rothschild's overarching semiotics is ecumenical, and recognizes sign behavior in inert as well as in living realms, even in psychokinesis and in telepathy." ( http://64.233.183.104/search?q=cache:Ipi6kopCKdYJ:www.ut.ee/SOSE/sss/an… )

Second: Information in nucleic acid chains is dependent on physical structure. For example, RNA stem loops and other secondary and tertiary structures work with the specific molecule in mind. ( http://www.pandasthumb.org/archives/2006/07/the_technogeek.html )

I liked that you also pointed out the inherent dualism in the paper. There is also the usual quantum babble whenever crank science touches the mind.

"It is interesting to see how widespread the belief in randomness is as a scientific explanation among scientists."

Sure, quantum mechanics is quite powerful.

"However, the problem of quantum measurements could indeed be related to the same kind of logic as in the origin of life; the problem of "self measurement"."

No respectable quantum interpretation suggests that dualism is an explanation.

A question though:
"For example, in set theory, I can talk about "the set of all sets that contain themselves". I can prove that there are two sets that meet that description: one contains itself, the other doesn't. There's no paradox there; there's no incompleteness issue. "

I was under the impression that this leads to the paradox of contradiction (Russell's Paradox) and thus new set theories were invented for this reason alone.

ebohlman says:

"The third thing I thought is that Voie seems to be arguing that reality is dependent on representation, which sounds really "vulgar postmodernist" to me."

The correct mapping is that more that one representation map to the same reality. The holographic principle, AdS/CFT, and other dualistic representations are examples.

Torbjörn:

Two quick things:

You say "The SEED magazine looks respectable on the surface." I sure hope so; they're the folks who are hosting this blog :-) I know from my contact with them that they're certainly trying to be a respectable layman's science mag.

And the question you asked about self-referential statements... It's actually a slightly complicated question. But the real statement that led to a lot of problems isn't "The set of sets that contain themselves"; it's sort of the inverse of that: the problem is the set of all sets that do *not* contain themselves.

The set of sets that contain themselves is slightly problematic, because what looks like a valid description of a set turns out to not uniquely describe one set; there are two sets that meet the description. You need some additional way to differentiate them. But that's not a huge problem, really - it just means that the basic statement "The set of all sets that contain themselves" is not a statement that specifies one set.

On the other hand... The set of all sets that *don't* contain themselves; that's a bugger. *That's* the one that took a ton of work in set theory and type theory to try to work around.

The set of sets that contain themselves is slightly problematic, because what looks like a valid description of a set turns out to not uniquely describe one set; there are two sets that meet the description. You need some additional way to differentiate them. But that's not a huge problem, really - it just means that the basic statement "The set of all sets that contain themselves" is not a statement that specifies one set.

I've already pointed out that this is false. By identifying the members of the set, you uniquely identify the set, if indeed there is one.

Chad:

And as I said: the only problem with "the set of all sets that contain themselves" is that the description *does not uniquely identify all of the members of the set*. It is an ambiguous description, which can describe two different sets. You need to write the definition more precisely to be a valid description of a set.

And as I said: the only problem with "the set of all sets that contain themselves" is that the description *does not uniquely identify all of the members of the set*.

Except that it does; it says that x is in A (where A is the set in question) iff x is in x. Let's walk through it; say A and B are two sets of "all sets which contain themselves". Then for all x, x is in A iff x is in x, which occurs iff x is in B. Thus x is in A iff x is in B, which by extensionality implies A = B. (Are you using a set theory which doesn't assume extensionality? If so, what is it?)

Now, if you get such a set A, it's not *immediately clear* whether A is an element of itself or not. In ZF, A exists, but is empty, so A is not an element of A.

Mark,

I'm as irritated as you are at people trying to use Godel's theorem to prove their pet metaphysical theory. The great physicist/mathematician Roger Penrose is guilty, as well, (in "The Emperor's New Mind", he argues that Godel's theorem implies the impossibility of artificial intelligence).

However, I think that there is a real sense in which the Godel statement G for Peano Arithmetic is in fact true. It is not correct to say that it is neither true nor false.

Let me take a famous example of a statement that is not known to be true or false: Goldbach's conjecture. GC is the claim that every even number greater than 2 is equal to the sum of two prime numbers. For example:

4 = 2+2
6 = 3+3
8 = 5+3
10 = 5+5
12 = 5+7
14 = 7+7
etc.

This is an empirical observation; it's true for every example that anyone knows of, but there is no proof that it's always true. Is it possible that GC is neither true nor false? I claim, that no, it is not possible.

Why not? Well, if there is a counterexample to GC, then GC is disprovable (and therefore false). Turning that statement around, we have:

If GC is not disprovable, then there is no counterexample to GC.

But what does it mean to say "There is no counterexample to GC"? It means "There is no even number that is not the sum of two prime numbers", which is another way of saying "Every even number is the sum of two prime numbers". But that's just GC itself. So what we have is

If GC is not disprovable, then GC.

So, if we assume that GC is neither provable nore disprovable, then it follows that GC is true.

The same holds for Godel's sentence G. If G is not disprovable, then it is true. There is no symmetry between G and not-G here, because G has the property

If G is not disprovable, then it is true.

while not-G has the property

If not-G is not provable, then it is false.

Now, it is correct to say that we can consistently
add not-G as an axiom. But if we do that, we are creating a new theory that is not a theory of the natural numbers. It is, instead, a theory of the so-called "nonstandard naturals", which are like the naturals, but they have additional elements corresponding to "infinite" numbers (bigger than any natural number).

I agree with Daryl here. More to the point, if we're admitting to the truth or falsity of sentences (whether in the "real world" or in some model), a sentence S and its negation are never "equally true"; one is true and the other is false.

Daryl McCullough's explanation sounds consistent to me, but it doesn't strike me as the only intuitive way of looking at it. You can say that an undecidable proposition is "true" because it has no counterexample, but I don't see what good that does. Absent a proof, you don't really know that it's true, so you're still left with the fact that you don't know if it is true. The proposition G has the property that one can at least prove that G is undecidable, but that is not true of all undecidable propositions.

Goldbach's conjecture may or may not have a counterexample, but if it has neither a proof, nor a counterexample, nor a proof of its undecidability, then we may never know its actual status with respect to the theory of the natural numbers. You can say "well in that case it's actually true" but this is indistinguishable to us from GC's being false but we just haven't been clever enough to find the counterexample.

The term "true" is a label that was part of natural language before people understood formal logic. Thus, it's not clear to me why one should be able to settle on a single definition of what it means in light of Goedel's theorem. You could attach it to all propositions that lack a counterexample or reserve it for those that actually have a proof. I don't see why either is obviously preferable.

Now, it is correct to say that we can consistently
add not-G as an axiom. But if we do that, we are creating a new theory that is not a theory of the natural numbers. It is, instead, a theory of the so-called "nonstandard naturals", which are like the naturals, but they have additional elements corresponding to "infinite" numbers (bigger than any natural number).

Likewise, given that both theories define all the useful properties of the natural numbers, it's not obvious to me why one is "not a theory of the natural numbers." Why can't they both be? I.e., the only way to nail down what you mean by natural numbers is to write down a set of axioms for it. There is no way to state definitely that either one is somehow closer to the fuzzy notion of natural numbers that we have in our brains. Granted, there might be a sense that one is a minimal theory, but I'm way out of my league in speculating.

Actually, my view of Goedel's theorem is largely pragmatic: In mathematics, we often make statements about all members of some infinite set. It should come as little surprise that not all of these statements can be settled one way or the other by an argument composed of finitely many inference steps. There's nothing spooky going on. Actually, it would be a lot spookier if every statement about infinities could be reduced to a finite argument.

PaulC writes: You can say that an undecidable proposition is "true" because it has no counterexample

No, not every statement is true just because it lacks a counterexample. Only statements of the form "There are no counterexamples to ....". If there are no counterexamples, then what it says is literally the truth.

but I don't see what good that does. Absent a proof, you don't really know that it's true, so you're still left with the fact that you don't know if it is true.

Well, yes. If you don't know whether something is true, then you don't know if it is true. That doesn't mean that it is neither true nor false.

The term "true" is a label that was part of natural language before people understood formal logic. Thus, it's not clear to me why one should be able to settle on a single definition of what it means in light of Goedel's theorem.

Godel's theorem doesn't change anything about the notion of truth, other than to say that we cannot discover all truths by logical deduction. But outside of mathematics, nobody ever believed this to be the case. If I'm trying to figure out whether lead weighs more than gold, I'm not going to figure out through logical deduction. Some people had high hopes that mathematical truth was different, but it turns out you can't discover all mathematical truths through logical deduction, either.

You could attach it [truth] to all propositions that lack a counterexample or reserve it for those that actually have a proof. I don't see why either is obviously preferable.

Whether a proposition is provable or not depends on what axioms you start with. So it doesn't make sense to equate provability with truth. You choose axioms because you believe those axioms to be true (of whatever subject you are interested in); they don't become true because you declare them to be axioms.

The modern way of looking at this is model theory. Whether a sentence is true or not depends on the interpretation of the symbols (the meaning of the constant symbols, function symbols, relation symbols, the domain over which quantifiers range). Given the interpretation, sentences are either true or false for that interpretation. Whether they are provable or not depends on what axioms you are assuming.

Likewise, given that both theories define all the useful properties of the natural numbers, it's not obvious to me why one is "not a theory of the natural numbers."

It's just a matter of definition. pi is not a natural number. square-root(2) is not a natural number. You can very well have a theory about pi and square-root(2), but such a theory will not be about the natural numbers (at least not exclusively). Similarly, if you take the theory of Peano Arithmetic, construct the Godel sentence G, and add as an axiom not-G, the resulting theory will be a theory about objects that are not natural numbers.

The technical definition of what the natural numbers are is defined as follows: The natural numbers is the smallest set containing 0 such that if x is in the set, then x+1 is in the set. If you add extra elements to your model, then you no longer have the smallest model, and so your model is no longer just the natural numbers.

Actually, my view of Goedel's theorem is largely pragmatic: In mathematics, we often make statements about all members of some infinite set. It should come as little surprise that not all of these statements can be settled one way or the other by an argument composed of finitely many inference steps. There's nothing spooky going on. Actually, it would be a lot spookier if every statement about infinities could be reduced to a finite argument.

I think that's a very good insight. But it doesn't imply that there is no truth beyond what is provable.

Mark,
Thank you for the response, it was helful!

I thought SEED sounded familiar, but I didn't google. (See, I'm half offloading on your missing links. ;-) I meant the S.E.E.D Journal ( http://www.library.utoronto.ca/see/pages/SEED_Journal.html ).

Daryl McCullough: thanks for the comments. My only formal exposure to Goedel's theorem was a math logic course when I was an undergrad, so I'll defer to your explanation.

My main take on the various abuses of Goedel's theorem in pop philosophy is that people who want a make a big deal about it (e.g. Penrose) usually start out with a really unrealistic assumption about how smart humans are and how rigorous is the basis of our belief. While it's a very exciting thing (to me anyhow) when you can nail down a belief with a rigorous argument, I also think that it's more realistic to view all our beliefs as wagers. They will pay off or not independent of the epistomelogical basis, which our available data and mental capacity are inadequate to make very rigorous anyway.

ArtK:
If he were a Wiccan, I'd think that he was trying to provide a mathematical basis for symbolic magic: Manipulate the symbol and you manipulate the thing.
/ArtK

I have too use that in a roll-playing game some time. sounds perfect for Unknown Armies or Mage: the Ascension (MtA basic primise is what if the extreme technophiles, the wiccans, the Victorian spiritureists, the conspiracie therists, the young erthers and the new agers were ALL right. ) (the seting is a bit innternaly contridictory but thats the way its ment to be)_

Hi! It turns out this paper is not the worst to be published in Chaos, Solitons & Fractals. It's quite a curious journal!