Is Language Bound By The Incompleteness Theorem?

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Is Language Bound By The Incompleteness Theorem?

Postby 6453893 » Sun Jun 28, 2009 1:47 pm UTC

I am a firm believer in the practical applications of Gödel's incompleteness theorems. As somebody who likes to make systems and logical categories out of everything, it has served me well to remind myself that no system can be both consistent and complete. You may try to establish a genre system for all of your music, but there will always be one band or album that either breaks the genre classifications, or remains just outside their reach. You will never find a perfect method of sorting my files or classifying personalities or winning at Go, because reality has a habit of breaking formulas. I know this is a complete mockery of Gödel's hard work, but it really does help to remind yourself of the theorem when trying to devise a system.

Is the same true for language? Never mind the grotesque behemoth that is English; if we could build a language from scratch, could we ever design one that is both consistent and complete, so that every thought can find expression? On the other hand, if a language is simply the attribution of meaning to the meaningless, as in that of concepts to words, words to combination of characters, and characters to black squiggles, then the system of Red for Hot and Blue for Cold seems pretty consistent and complete.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby gmalivuk » Sun Jun 28, 2009 2:48 pm UTC

There's no logical reason why no genre system could possibly be complete and consistent. Maybe you just suck at assigning genres? So I don't see how invoking Godel at all is legitimate here...
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Dibley » Sun Jun 28, 2009 5:05 pm UTC

Indeed, there's the trivial case to assigning every last song in existence to its own genre, which would be both complete and consistent, and so it's at least theoretically doable. You can then clump songs together until you feel you've pared it down sufficiently.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby Qaanol » Sun Jun 28, 2009 6:47 pm UTC

6453893 wrote:if we could build a language from scratch, could we ever design one that is both consistent and complete, so that every thought can find expression?

Clearly some thoughts correspond to concepts in axiomatic set theory. If we agree that every concept in axiomatic set theory corresponds to a thought, then our ideal language must be able to describe all of axiomatic set theory. I think you see where this is going.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Zamfir » Tue Jun 30, 2009 11:47 am UTC

Shouldn't the relevant sentence be "This sentence cannot be written in language $X" ?

It's not the completeness that is the issue, but consistency. A consistent language should not be able to grammatically form an equivalent of the sentence above, and I am not sure we would call such a system a "language".

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Re: Is Language Bound By The Incompleteness Theorem?

Postby 6453893 » Tue Jun 30, 2009 1:50 pm UTC

Zamfir wrote:Shouldn't the relevant sentence be "This sentence cannot be written in language $X" ?

It's not the completeness that is the issue, but consistency. A consistent language should not be able to grammatically form an equivalent of the sentence above, and I am not sure we would call such a system a "language".


But a Language does not factor truth value in to its structure. That statement conforms to the proper usage of every individual component, regardless of whether or not it is valid.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby Qaanol » Tue Jun 30, 2009 3:37 pm UTC

You say "complete" means that every thought can be expressed in the language, but we do not have a well-defined "set of all thoughts". Also, you have yet to provide an applicable definition of "consistent". In math and formal logic, a system of statements is consistent if it doesn't contradict itself. However, you claim to accept contradictory—even self-contradictory—statements as valid in language.

Perhaps you intend "consistent" to mean "each statement in the language corresponds to a specifc thought", so the language is incapable of forming ambiguous statements, nor statements without meaning. For example, does the statement, "This statement is false" have any meaning? In any case, you need a well-defined metric for differentiating or establishing equivalence between two thoughts, as well as the aforementioned set of all thoughts.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby gmalivuk » Tue Jun 30, 2009 3:48 pm UTC

Or, we can just accept that language is not the sort of thing the Incompleteness Theorem is applicable to in the first place...
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Birk » Tue Jun 30, 2009 3:56 pm UTC

gmalivuk wrote:Or, we can just accept that language is not the sort of thing the Incompleteness Theorem is applicable to in the first place...



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Re: Is Language Bound By The Incompleteness Theorem?

Postby diotimajsh » Wed Jul 01, 2009 6:35 am UTC

6453893 wrote:You may try to establish a genre system for all of your music, but there will always be one band or album that either breaks the genre classifications, or remains just outside their reach. You will never find a perfect method of sorting my files or classifying personalities or winning at Go, because reality has a habit of breaking formulas.
Erm, what? I'm going to echo others here in that the incompleteness theorem doesn't seem applicable to the music genre question. It is difficult to assign genres, but I believe that's a problem with the loose way we use genres, nothing to do with Gödel.

Let me propose a new genre system, one that shows it is possible to make an "unbreakable" way to classify genres. My new system has but two genres within it: genre A and genre B. Genre A comprises the set of all pieces of recorded music with a length less than or equal to precisely four minutes and thirty three seconds. Genre B comprises the complement of genre A - that is to say, the set of all music with a length that is precisely greater than four minutes and thirty three seconds. What kind of music could break this classification system? I can't imagine any, since as far as I know, we must be able to attribute some length of time to music in order for it to qualify as 'music'. And anyway, how would it be related to the incompleteness theorem?

Similar arguments could be given for sorting files and classifying personalities. The problem you identify is a lack of specificity and precision, not something that arises from the inherent nature of logical relations/systems.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Masily box » Wed Jul 01, 2009 9:34 pm UTC

Along with everyone else, I don't think that the way you're trying to invoke Gödel applies here.

But I recognize that you're just using it as a heuristic to remind yourself of something else. I think that your basic insight about language (which includes the question of music genres... after all those are part of language) is a very important and profound idea; the world would be better off if everybody thought about things along these lines.

The important idea is that language doesn't work in terms of strict meanings and definitions. Every word represents a fuzzy cloud of meanings; when we think of the word, we tend to think of examples of it that fit near the center of that cloud. So when someone says "sport," we think of something like football. But other examples, like basketball and hockey, are pretty similar to football: they involve (differently shaped) goals, two teams on opposite sides of an arena, an object passed back and forth, and so on. So we're pretty comfortable calling those "sports". Golf and bowling are rather different, but the word still applies. Hunting, fishing, poker, competitive eating, jogging, academic bowl.... are they sports? We decide based on how similar they are to the prototypical examples, and in what way they're similar.

The thing to realize is that the question "Are they sports?" really means "Is the word 'sport' a useful name for these activities?" There's no ultimate nature of "sportsness" out there in the universe that an activity either does or doesn't have. This is why language behaves in such a funny way: it doesn't work on strict definitions like math does. (It's also why applying Gödel to language is questionable)

The philosopher Ludwig Wittgenstein wrote a lot about these ideas (in his Philosophical Investigations); it's also an important branch of cognitive science/linguistics referred to as "Prototype Theory". (A place to start might be George Lakoff's books... Metaphors We Live By and Women, Fire, and Dangerous Things. Thinking in terms of categories and prototypes also tends to lead to a philosophical viewpoint called anti-essentialism (which tends to be a pomo/feminist/culture studies/etc buzzword).


The ultimate point is: if you come up with a system that assigns mathematically precise meanings to its symbols (like a red-blue gradient, each of whose shades of color signifies a precise temperature), I claim that your system will be sufficiently unlike English, Chinese, and other natural languages to the extent that I'd be uncomfortable calling it a "language," per se.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby gmalivuk » Wed Jul 01, 2009 9:55 pm UTC

Yeah, if what you're really getting at is the problem of vagueness, then I agree that it's relevant to both language and genre assignments, and believe that reasonable human languages are necessarily prone to this "problem" because that's how our brains work.

It's not related to Godel, though, since Godel applies in situations which are very precisely *not* vague, since mathematics is, well, mathematically rigorous.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby 6453893 » Thu Jul 02, 2009 2:31 am UTC

Masily box wrote:The ultimate point is: if you come up with a system that assigns mathematically precise meanings to its symbols (like a red-blue gradient, each of whose shades of color signifies a precise temperature), I claim that your system will be sufficiently unlike English, Chinese, and other natural languages to the extent that I'd be uncomfortable calling it a "language," per se.


Thank you. I did not mean for this to become a discussion of the incompleteness theorem and its relevence. I was really just trying to illustrate what Box put so eloquently above. To carry your example, one might decide to call Mathlympics a 'sport', but this context lies on the periphery of that fuzzy cloud of appropriate usages of sport. It can be corralled either in to or out of the 'sport' definition depending mostly on line-drawing and subjective definitions (Tests the ability of a muscle so it is, no interaction between sides so it isn't, &c). But what would English look like if we gave those nebulae of meaning concrete definition?

One solution is to remove higher-tier terms like 'sport' altogether, which correlates to the genre solution. Mathlympics would be nothing more or less than Mathlympics. I know a language structured this way would never be functional, that you can't define words for people who do not know them. Every term would become inextricable from what it describes, like a letter to its character. There would be no way to explain Mathlympics except to demonstrate it. And even then, you cannot use one word to describe every game of Mathlympics, because each one is unique. At its logical conclusion, a language of this structure would boil down to a serial system, where one specific coat of plastic on one specific aglet of one specific shoe would be 46abb8c7883db89a9fb95cdd2116c033, and the other aglet of the selfsame shoe would be c961662228260e9b04e3da980e346414.

I do not, however, think a system would be infeasible. Not necessarily as perfect as what was stated above, but closer to it than English is. Of course, I have not seriously considered any other language prior to now. Japapnese is pretty good as far as definitions go, but the language is perverted in practice by excessive implication and contextualization. And of course the fact that half of it is katakana English.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby n7a7v7i » Fri Jul 03, 2009 7:48 pm UTC

diotimajsh wrote:
6453893 wrote:You may try to establish a genre system for all of your music, but there will always be one band or album that either breaks the genre classifications, or remains just outside their reach. You will never find a perfect method of sorting my files or classifying personalities or winning at Go, because reality has a habit of breaking formulas.
Erm, what? I'm going to echo others here in that the incompleteness theorem doesn't seem applicable to the music genre question. It is difficult to assign genres, but I believe that's a problem with the loose way we use genres, nothing to do with Gödel.

Let me propose a new genre system, one that shows it is possible to make an "unbreakable" way to classify genres. My new system has but two genres within it: genre A and genre B. Genre A comprises the set of all pieces of recorded music with a length less than or equal to precisely four minutes and thirty three seconds. Genre B comprises the complement of genre A - that is to say, the set of all music with a length that is precisely greater than four minutes and thirty three seconds. What kind of music could break this classification system? I can't imagine any, since as far as I know, we must be able to attribute some length of time to music in order for it to qualify as 'music'. And anyway, how would it be related to the incompleteness theorem?

Similar arguments could be given for sorting files and classifying personalities. The problem you identify is a lack of specificity and precision, not something that arises from the inherent nature of logical relations/systems.


This is a really good try, but...

In order to claim that all music could be classified as either A or B, you would have to define what exactly music is. What counts as music? Only digital recordings? What about analog? How about pieces of music that have never been recorded, but simply passed on through generations?

A solid this-is-red-'n-this-is-blue system is useless if you cannot agree on color.
And I feel that even if we could agree that a word like sport is either too vague to be useful as a word in a precise language, or if we could actually make a better definition for it, a whole new problem would appear. How the hell do we define those individual sports?

I think language is fuzzy for a reason, and thus is not mathematical in nature and likely shall never be. Attempts to make a perfectly logical language like Esperanto or Loglan are certainly not a threat to anyone... but at some point, a language becomes too precise to be useful to its users, and at that point, what's the point?

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Re: Is Language Bound By The Incompleteness Theorem?

Postby 6453893 » Thu Jul 16, 2009 11:32 am UTC

Of course, I forgot about programming languages. I am not very familiar with programming, but from what I've picked up of LISP and C, those languages seem pretty close to my vision of this hypothetical precise language. Of course, at the same time I question whether the use of the word "language" in that context isn't something of a misnomer. You aren't using this language to communicate in any meaningful way, only to design a function. It seems to me more like a mental toolbox, used in the construction of a kind of machine. Please, any coders here, feel free to detail your understanding of programming.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby tamerlane » Sat Aug 01, 2009 9:50 pm UTC

gmalivuk wrote:Or, we can just accept that language is not the sort of thing the Incompleteness Theorem is applicable to in the first place...


Yea verily. The incompleteness theorem was only intended, and formally only does, apply to axiomatic systems, of which natural language is notably not one. In order for the proof to work, each term needs to be represented by some other symbol in a meta-language which directly corresponds to the object language; Goedel invents "Goedel numbering" for this purpose. The proof works by creating a liar's paradox between object and meta-language. You can't mirror natural language in formal language without tremendous loss of meaning (it's the same problem with hangs Tarski's attempts to give a theory of Truth). Natural language(s) may be "logical" in the colloquial sense, but none are by any means formally so.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby u38cg » Mon Aug 03, 2009 4:19 pm UTC

This sounds an awful lot like equivelant discussions of pidgins, creoles, and dialects. I'm also surprised no-one has mentioned 1984 yet.

I think the generally accepted answer to your question is, yes, you could; however as soon as that language had been learned by a generation of kids, they would naturally extend it to cover the full range of expression.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby PM 2Ring » Sat Aug 08, 2009 6:20 pm UTC

As other posters have said, Gödel's theorems apply to formal systems, not languages, per se.

6453893 wrote:at the same time I question whether the use of the word "language" in that context isn't something of a misnomer. You aren't using this language to communicate in any meaningful way, only to design a function. It seems to me more like a mental toolbox, used in the construction of a kind of machine. Please, any coders here, feel free to detail your understanding of programming.

Programs aren't just read by machines, they are also read by other programmers, so they are a form of human communication, albeit limited.

FWIW, virtually any kind of methodical process can be described in a programming language, not just stuff related to operating the computer's hardware.

As for Loglan, it's a language that was designed for human communication, but it's usefulness in man-machine dialogue was recognised early in its development. It's extremely logical and very regular, as human languages go, but it's main strength (IMHO) is that it's syntactically unambiguous. (Semantic ambiguity in language can be reduced, but never eliminated.)

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Re: Is Language Bound By The Incompleteness Theorem?

Postby 6453893 » Sat Aug 08, 2009 11:15 pm UTC

tamerlane wrote:Yea verily. The incompleteness theorem was only intended, and formally only does, apply to axiomatic systems, of which bla bla bla bla bla


PM 2Ring wrote:As other posters have said, Gödel's theorems apply to formal systems, not languages, per se.


Why do you people still feel compelled to point this out almost a month after I agreed that the discussion is not and never really is relevant to Gödel's actual theorem? I'm pretty sure tamerlane is some kind of math major who only read the title and decided to storm in on his high horse.

PM 2Ring: Is that really communication between developers though? Does a car act as a means of communication between two mechanics working on it?

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Re: Is Language Bound By The Incompleteness Theorem?

Postby PM 2Ring » Mon Aug 10, 2009 2:04 pm UTC

6453893 wrote:
PM 2Ring wrote:As other posters have said, Gödel's theorems apply to formal systems, not languages, per se.


Why do you people still feel compelled to point this out almost a month after I agreed that the discussion is not and never really is relevant to Gödel's actual theorem?

Sorry, my remark wasn't directed at you; I was just giving my general agreement.

6453893 wrote:PM 2Ring: Is that really communication between developers though? Does a car act as a means of communication between two mechanics working on it?
I don't think the car analogy is very close - program design is far more flexible than car hardware design.

Almost all programming languages have some way that programmers can include natural language comments in their source code to help others who are reading their program. But that's not what I was talking about.

Over on the coding forum, we often get people asking how to perform some task or other. The answer is usually given in the form of source code (often in Python or C, sometimes in Lisp), with some explanation written in English. Using a programming language like this is succinct & unambiguous. Sometimes, such example code is a fully-fledged program ready to run, but quite often it's just a program fragment that needs to be incorporated into a larger structure, possibly with modifications. If the code is runnable, that's a nice bonus, but such code is primarily posted to impart knowledge to another human, not as a set of instructions to be fed to a machine.

If I were trying to describe the quicksort algorithm to someone, and to compare it with other sorting algorithms, the simplest way would be for me to show them some source code. Trying to do the same in English would take at least three times as long, and there'd be the possibility that my explanation would be misunderstood or misinterpretated, no matter how careful & precise I was in my use of language.

Good programmers have a good writing style, and much energy has been expended over the last several decades discovering and debating what constitutes good programming style. Style only matters for humans reading one's code (including oneself); the machines happily execute code whether it's nicely structured or a veritable rat's nest. Well structured program modules are easier to maintain, enhance & reuse. Large modern applications & operating systems could not exist without code that was well written & well organized. (I'm not saying all code in these big programs & systems is well-written, but a large majority of it has to be, or it won't survive.)

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Re: Is Language Bound By The Incompleteness Theorem?

Postby userxp » Fri Sep 04, 2009 2:22 pm UTC

Language is used to communicate a concept from a human to another human.
Imagine this: while you think about what you want to say, a machine makes a full molecular scan of your brain in a digital form. That scan surely contains the precise concept you wanted to communicate. Could this be considered a language? Why not?

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Re: Is Language Bound By The Incompleteness Theorem?

Postby FrankManic » Mon Sep 07, 2009 1:30 am UTC

I thought English could express every possible communicable thought. It just makes up new words to convey the thought, uses a lot of other words to convey the thought, or steals an appropriate word from another language.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby 6453893 » Mon Sep 07, 2009 1:55 am UTC

FrankManic wrote:I thought English could express every possible communicable thought. It just makes up new words to convey the thought, uses a lot of other words to convey the thought, or steals an appropriate word from another language.
Huh? I'm not sure what that has to do with anything.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby Josephine » Wed Sep 09, 2009 12:42 am UTC

Keeping up with the computer science metaphor, I'm convinced that, contrary to the Sapir-Whorf Hypothesis, nearly all languages are 'Turing Complete'.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby The Mighty Thesaurus » Wed Sep 09, 2009 12:58 am UTC

userxp wrote:Language is used to communicate a concept from a human to another human.
Imagine this: while you think about what you want to say, a machine makes a full molecular scan of your brain in a digital form. That scan surely contains the precise concept you wanted to communicate. Could this be considered a language? Why not?

Noam Chomsky wrote:From now on I will consider a language to be a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Masily box » Sat Sep 12, 2009 2:11 am UTC

Dude, that Chomsky quote must be from super early on. I'm not sure how well it accords with traditional views of Chomsky's take on language, anyhow, seeing as how his syntactical rules allow for the construction of sentences of unbounded length.

To respond to userxp's post (though I'm not quite sure how it's relevent), sure, I'd consider it a language, if we understood the brain well enough to translate back & forth between the scan image and a written/spoken sentence. In fact, then the scans would be nothing more than another modality to represent the same language. (Although, I actually challenge the notion that "precise concept" is itself a very meaningful term... in fact, that's what this thread is all about.)

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Re: Is Language Bound By The Incompleteness Theorem?

Postby davidbackslashse » Wed Oct 28, 2009 11:47 pm UTC

Masily box wrote:Dude, that Chomsky quote must be from super early on. I'm not sure how well it accords with traditional views of Chomsky's take on language, anyhow, seeing as how his syntactical rules allow for the construction of sentences of unbounded length.


(I'm a little late to the party here but) this quote

Noam Chomsky wrote:From now on I will consider a language to be a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements.


isn't about natural languages; it's about formal languages. Natural languages (the kind we speak) are recursive and are capable of generating sentences of infinite length (in practical terms, though, we're really only capable of processing up to about three embedded clauses). Formal languages, as in the quote, are defined over the set of strings in full that make them up.

http://en.wikipedia.org/wiki/Chomsky_hierarchy

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Re: Is Language Bound By The Incompleteness Theorem?

Postby Eebster the Great » Sat Oct 31, 2009 11:18 pm UTC

Qaanol wrote:Clearly some thoughts correspond to concepts in axiomatic set theory. If we agree that every concept in axiomatic set theory corresponds to a thought, then our ideal language must be able to describe all of axiomatic set theory. I think you see where this is going.


That sounds rather . . . naive

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Re: Is Language Bound By The Incompleteness Theorem?

Postby bebboe » Sun Nov 01, 2009 5:44 pm UTC

davidbackslashse wrote: Natural languages (the kind we speak) are recursive and are capable of generating sentences of infinite length (in practical terms, though, we're really only capable of processing up to about three embedded clauses). Formal languages, as in the quote, are defined over the set of strings in full that make them up.

This is a common misunderstanding. There is no such thing as a sentence of infinite length, at least not in natural languages. The length of sentences in natural languages is unbounded, but not infinite. There is no upper bound to the length of sentences, that is, given any sentence, you can always construct a sentence that is longer than it, but this is NOT the same as saying that there are sentences of infinite length (though it follows from this that the cardinality of the set of English sentences is infinite). Not that this really matters in this discussion, but since even some of the most educated people I know tend to claim that Chomsky claimed that there were sentences of infinite length, I felt like clearing this up.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby Makri » Sun Nov 01, 2009 8:42 pm UTC

Yea verily. The incompleteness theorem was only intended, and formally only does, apply to axiomatic systems, of which natural language is notably not one. In order for the proof to work, each term needs to be represented by some other symbol in a meta-language which directly corresponds to the object language; Goedel invents "Goedel numbering" for this purpose. The proof works by creating a liar's paradox between object and meta-language. You can't mirror natural language in formal language without tremendous loss of meaning (it's the same problem with hangs Tarski's attempts to give a theory of Truth). Natural language(s) may be "logical" in the colloquial sense, but none are by any means formally so.


Well, if I understand Gödel correctly, his point was that you could make a language that was able to express arithmetic self-referent, and that that possibility self-reference allows you to formulate a liar's paradox.
But natural languages has this capability for self-reference, as the liar's paradox was obviously conceived within natural language; so natural language is obviously bound by Gödel's results. It's basically the prototypical case! Besides, I don't see the principled difference between formal languages and natural languages.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby gmalivuk » Sun Nov 01, 2009 10:29 pm UTC

Makri wrote:Besides, I don't see the principled difference between formal languages and natural languages.

Formal languages tend to lack the ambiguity and vagueness of all natural languages, which seems pretty important to me.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Makri » Sun Nov 01, 2009 11:52 pm UTC

Formal languages tend to lack the ambiguity and vagueness of all natural languages, which seems pretty important to me.


Yes, but what's principled about this? Nothing prevents me from having my predicates in first order logic interpreted as vague. It's just accidental because we use natural language to talk about the world around us - in which case vague predicates are useful -, which we rarely do with a formal language. Except if we are formal semanticists; and then of course we have vaguely interpreted predicate constants. (But these are basically the least interesting part for formal semanticists.)
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Nicholas T » Mon Nov 02, 2009 11:59 am UTC

The problem I see with applying Gödel's theorem here is that natural language (along with subproblems like genre classifications) is constantly in flux, not as a consequence of its own internal evolutionary rules but in as a result of its interactions with other languages and cultural systems external to itself. The analogy of having to step "outside" an axiomatic system to perceive its strange loops applies here, and is indeed at the heart of Douglas Hofstadter's conception of intelligence in GEB and other works.

The gulf between natural and formal languages is pretty vast. Try writing a parser or compiler for a natural language and you'll see what I mean. The principle here, once again, lies in language change. You can certainly try to take the whole of language at one point in time and appraise it as a single system (synchronic linguistics, as Ferdinand de Saussure called it), but even then it would be hard to think of language in terms of theorems derived from generative rules when that's not how it's constructed at all. To me, natural language seems not only incomplete but trivially incomplete.

Now, when it comes to formal languages, the logical restrictions on all Turing-complete languages most certainly apply. In the 1950 article in Mind where Turing presents the Turing Test (which in his formulation, was entirely a matter of natural language), he briefly considers a "mathematical objection" to AI from exactly that - Gödel's theorem. I don't quite remember what he said, but I as far as I know it's an accepted consensus these days that strong AI capable of humanlike linguistic abilities would very likely require something more powerful than the Turing machine model of computation.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby Nicholas T » Mon Nov 02, 2009 12:01 pm UTC

I should also say that the objections to applying Gödel's theorem formally don't make incompleteness any less valuable as an analogy. It most certainly is valuable as a critique of any essentialist taxonomy where we classify things into sets according to shared attributes.

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Re: Is Language Bound By The Incompleteness Theorem?

Postby Pez Dispens3r » Mon Nov 02, 2009 2:42 pm UTC

Foucault (the flaming prat) once repeated what Borges found:

This passage quotes a ‘certain Chinese encyclopedia’ in which it is written that ‘animals are divided into:
(a) belonging to the Emperor,
(b) embalmed,
(c) tame,
(d) sucking pigs,
(e) sirens,
(f) fabulous,
(g) stray dogs,
(h) included in the present classification,
(i) frenzied,
(j) innumerable,
(k) drawn with a very fine camelhair brush,
(l) et cetera,
(m) having just broken the water pitcher,
(n) that from a long way off look like flies’.

What's my point (or rather, Foucaults)? Thing is, your definitions are always arbitrary. Sure, you can make them, but they're meaningless. If this serves, essentially: language can't be such a system, because it's too dynamic, fluid, and real. It will always have a word that contradicts your categories, or else your categories are arbitrary and therefore meaningless (or, they have very limited meaning).
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Makri » Mon Nov 02, 2009 3:31 pm UTC

Pez Dispens3r wrote:What's my point (or rather, Foucaults)? Thing is, your definitions are always arbitrary. Sure, you can make them, but they're meaningless. If this serves, essentially: language can't be such a system, because it's too dynamic, fluid, and real. It will always have a word that contradicts your categories, or else your categories are arbitrary and therefore meaningless (or, they have very limited meaning).


Like this paragraph... ? :wink: Seriously, I can't really understand what you're saying. What does it mean for language to be dynamic and fluid; how is realness gradable so that something can be too real? What does it mean for a word to contradict a categorization? Why does arbitrariness entail meaninglessness?

Nicholas T wrote:Try writing a parser or compiler for a natural language and you'll see what I mean.


But isn't this merely a complexity issue? Natural language syntax is just so complex that we haven't yet figured it out completely.

What is a curious feature about natural language is that the language itself appears to be vague: It's not always clear whether we find a sentence grammatical or not... This may be a significant difference, since, as far as I'm informed, formal languages, as we treat them, are never defined as fuzzy sets.

You can certainly try to take the whole of language at one point in time and appraise it as a single system


You talk like this were somewhat outlandish and absurd...

but even then it would be hard to think of language in terms of theorems derived from generative rules when that's not how it's constructed at all.


Nobody thinks of languages "in terms of"... A language has a syntax and a semantics, and any calculus you might invent is something secondary.

To me, natural language seems not only incomplete but trivially incomplete.


Are you saying it's impossible to find any calculus for natural language, or what?

I should also say that the objections to applying Gödel's theorem formally don't make incompleteness any less valuable as an analogy. It most certainly is valuable as a critique of any essentialist taxonomy where we classify things into sets according to shared attributes.


What do you mean, as an anlogy? I can't see how you would apply the very formal notion of incompleteness analogously. One can - and, I think, should - talk about vagueness completely without mentioning Gödel...
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Re: Is Language Bound By The Incompleteness Theorem?

Postby gmalivuk » Mon Nov 02, 2009 3:54 pm UTC

Makri wrote:What is a curious feature about natural language is that the language itself appears to be vague: It's not always clear whether we find a sentence grammatical or not... This may be a significant difference, since, as far as I'm informed, formal languages, as we treat them, are never defined as fuzzy sets.

Yes, this is part of the vagueness I was talking about before, and I think it is an intrinsic difference and a significant one.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Makri » Mon Nov 02, 2009 3:58 pm UTC

Oh, okay. Sorry. I thought you were talking about the vagueness of concepts.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby gmalivuk » Mon Nov 02, 2009 4:50 pm UTC

Well both, really. But it's the vagueness of grammar that probably creates the bigger and more unbridgeable gap.
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Re: Is Language Bound By The Incompleteness Theorem?

Postby Nicholas T » Tue Nov 03, 2009 1:02 pm UTC

Makri wrote:
Nicholas T wrote:Try writing a parser or compiler for a natural language and you'll see what I mean.


But isn't this merely a complexity issue? Natural language syntax is just so complex that we haven't yet figured it out completely.


The specific aspect of complexity that causes difficulties for systematizing a way to parse natural language is syntactic ambiguity. When parsing formal languages, one of the defining features of complexity is how many symbols you have to look ahead to disambiguate syntax (independent of semantics) - a property of the class of formal grammar that you're working with. (I've been away from doing anything with compilers for a few years, so someone please correct me if my understanding is off.) With natural languages, where speakers are capable of deliberately producing intelligible statements that defy syntactic rules, or statements that depend highly on the semantics to be coherent - we can probably draw all manner of examples from poetry - it's unclear to me that we can even formalize the lookahead class of the grammar in something like, as you say, a calculus.

Makri wrote:
Nicholas T wrote:You can certainly try to take the whole of language at one point in time and appraise it as a single system


You talk like this were somewhat outlandish and absurd...


Not absurd at all. Without looking at languages synchronically we wouldn't get very far in studying the syntax of natural languages to any degree, and linguistics as a discipline would still be trapped in nineteenth-century philology. What I'm saying, though, is that it doesn't prove to be very useful when we're talking about formal limitations on language, which necessarily have to account for diachronic language change.

I would offer the same answer to your question about the possibility of finding a calculus for natural language. A natural language in stasis? Probably; the abstractions we use to describe language would involve lots of special cases and exceptions and not reducible to a small set of axioms, but they would be abstractions nonetheless in a descriptive theoretical system. But if we are talking about the limitations on the power of natural languages, considering them strictly in stasis seems to me to be an incomplete account.

And relatedly (or not), one of the first premises Ferdinand de Saussure works from in A Course in General Linguistics is that when you look at a language as a system in stasis, you can't conceive of any individual word as having a positive semantic value on its own, but as meaningful only in contradistinction to the rest of the system. (I'm not saying this is an uncontroversial claim; indeed, it's curiously emblematic of early-20th-century French thought, if you compare it to, say, the kind of stuff Pierre Duhem wrote about theoretical models in physics.)

Makri wrote:
Nicholas T wrote:I should also say that the objections to applying Gödel's theorem formally don't make incompleteness any less valuable as an analogy. It most certainly is valuable as a critique of any essentialist taxonomy where we classify things into sets according to shared attributes.


What do you mean, as an anlogy? I can't see how you would apply the very formal notion of incompleteness analogously. One can - and, I think, should - talk about vagueness completely without mentioning Gödel...


As with many things analogical (and many things regarding Gödel's theorem), the examples I like to appeal to come from Hofstadter. There's an excellent piece in Metamagical Themas where he writes about Donald Knuth's Metafont and objects to the (somewhat innocent) claim that all typefaces can be described as a function of various parameters (kerns, serifs, and so on). Hofstadter explicitly deploys the notion of formal incompleteness as a metaphor for his objection, that there are clues we use to recognize alphabetic characters that come not from those generative parameters, but from their place in the context of a higher-level system. (The mapping is between the unprovable true statements of a consistent formal system and the visual clues we use to identify written characters that do not result from generative rules.)

Incompleteness is a formal notion, but to use it as an analogy is to map a concept from formally defined systems to informally defined ones. In the informal case, it wouldn't refer to vagueness in general, but to the impossibility of generating all members of a set from a priori parameters (analogous to axioms). You can probably see why this lends itself well to a critique of essentialism even in an informally defined sense - the idea that members of a set can be generated from certain properties and non-members of the set cannot. Nobody would claim that things like nationality or race (or, some would say controversially, gender) are formal axiomatic systems, but an intuitive grasp of formal concepts like incompleteness or undecidability is a surprisingly powerful way of grasping the problems with how we structure or classify ideas.

I also think this is far from the only way in which we map formal (or at least rigorously specified) scientific/mathematical concepts onto fuzzy and informal things. Look at how the language of evolution and competitive natural selection has percolated into so many other disciplines, both rigorously (when it comes to studies of language change) and not so rigorously (like "social Darwinism"); or how Dawkins mapped information-theoretic claims about genes onto memes; or (to use an example that is actually formal and mathematical, and isn't as spurious a comparison) ideas of exponentiation, dimensionality, and so on.


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