Barbelith Underground > Head Shop > Language: speaking about speaking.

Off the tip, top, the tippity-top, I’d like to say hey stop with the “yeah erh just to qualify i am only a layman and am no philosophy graduate so sorry if my questioning is random and halfbaked...” I tend to feel that every human is a philosopher, artist, and scientist—if we can’t communicate our experiences amongst ourselves, then we’re pretty screwed. What I mean is all these questions are good questions and don’t worry about qualificatin’ yer status!

“what i wondering about is Bertrand Russell and his attempts with Whitehead (?) to create a system that didn't contradict itself. i know this was refuted by Godel but there are lots of puzzles there.”

What you are referring to here is the work that R & W did to produce the Principia Mathematica. This work stems from work that Frege did with predicate logic and mathematics. In the Principia we find the formal beginnings of what is called “logicism.” This is the attempt to frame mathematics in terms of logic. So here we can note a distinction between what gets called a “formal language” (maths and logics) and a “natural language” (English, Greek, Swahili, etc.). Where the Principia runs into problems is that for much of its important work it needs to assume “the axiom of infinity.” This axiom is problematic in some systems, and some people, finitists such as Hilbert, wanted a consistent way to talk about infinity that stemmed from proofs using finitary means.

Now, it’s true that Godel showed that a formal language can’t be both complete and consistent. There is a thread on the proof here. To me, this is again connected with problems with infinity. There is a thread on infinity here. Myself, I feel that I ought to be better able to explain Godel’s proof, but I still currently “get it” at an intuitive level. Rudy Rucker gives a good discussion of it, that is comprehensible without having done any logic courses, in his “Infinity and the Mind.” Look for it at yer library! In a sense, it has to do with figuring out a way to code up an infinite number of infinities in a way that shows that some statements are True, but the language does not have the resources to prove that these statements are true. Here’s Rucker’s take:

1. Someone introduces Godel to UTM, a machine that is supposed to be a Universal Truth Machine [i.e., it operates with a language that is complete—it “knows” everything and it is consistent—everything it “knows” it knows to be true], capable of correctly answering any question at all.
2. Godel asks for the program and circuit details of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
3. Smiling a little, Godel writes out the following sentence: “The machine constructed on the basis of the program P(UTM) will never say that this sentence is true.” Call this sentence G for Godel. Note that G is equivalent to “UTM will never say G is true.”
4. Now Godel laughs his high laugh and asks UTM whether G is true or not.
5. If UTM says G is true, the “UTM will never say G is true” is false. If “UTM will never say G is true” is false, then G is false (since G = “UTM will never say G is true”). So if UTM says that G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM only makes true statements.
6. We have established that UTM will never say G is true. So “UTM will never say G is true” is in fact a true sentence. So G is true (since G = “UTM will never say that G is true.”).
7. “I know a truth that UTM can never utter,” Godel says, “I know that G is true. UTM is not truly universal.”

Like Rucker goes on to say, this is similar to the “Liar’s Paradox,” but Godel’s proof has more depth and subtlety. It is not the mere inability to evaluate the sentence, “This sentence is false,” but rather, the proof shows that we can know a statement in a formal language is true, but the language itself does not have the resources to show that this statement is true.

So, when you ask, “if a system is ‘sufficiently complex’ a strange loop occurs and meaning arises despite itself?” some might ask you to be more precise about some of your terms, but I’m gonna’ stick my neck out (but only a little, really) and say that yes, what you are saying here makes sense to me. It does seem that a complex enough system (even if the axioms—the “program” in Rucker’s example—are finite) will create a sort of infinity that is simply neither assessable or accessible to our best possible formal language. And I do tend to feel that this is due to a strange loop—one created in contradiction, paradox, and absurdity—and that it is this loop which is the generator (of meaning, of things, of relations, etc.). I also feel this is entirely a function of self-referencing. I think that Godel’s formal proof relies on showing how it is possible to code up a sentence in arithmetic in such a way that we get a whole infinite set of sentences that are True, but arithmetic can’t say that they’re true because it would be false for it to say so, and each of these sentences relies on a self-referencing property of the coding protocol.

OK, so I also feel that mathematics and logics are languages that are most clearly metaphorical in nature—they are, after all, often called “abstract.” Maths talk about numbers as its objects; in other words, a numbers is assigned to something as if it were a number. Logics talk about statements as it objects; in other words, letters and symbols are assigned to something as if it were letters and symbols. In both case it seems to me that we are “applying an name or description to something which it is not [logically] applicable.” In any natural language this also seems to occur. For example, in English we assign the word ‘dog’ to something as if it were a dog. Our meta-phoros brings “dog” to whatever phenomena we experience that seems dog-like. Again, we seem to apply a name or description which is not logically applicable to the thing. In a sense, the self-referencing of the language is ourselves: we cannot, as speakers of English really answer the question, given our experience of a dog-like phenomena, “Was that really a dog?” because in doing so we can only rely on the fact that we have arbitrarily assigned the word ‘dog’ to the dog-like phenomena. In other words, we know that we have experienced a dog, but we cannot rely on our language to consistently prove that we have indeed experienced a ‘dog’. The proof is self-referential with respect to the very meaning of the word ‘dog’ and that meaning is found in 1) the shared universal meaning of ‘dog’ in the English language, and 2) the particular internalized meaning of ‘dog’ with respect to the individual user of English.

This is a little of an aside, but if our best formal language cannot be both consistent and complete, then how in the world do we expect our natural languages to adequately convey the “whole of the truth,” i.e., the entirety of what we intend to relate of our experience? Put differently, if there are an infinite number of knowable true sentences in arithmetic that the language of arithmetic itself cannot say are true, then how many more true sentences are there in, say, English—a natural language less precise and defined than a formal language—that are knowable as true, and yet not provable to be true. An example might be “ESP exists”—who knows?

“also there are questions about etymology in language and meaning and seemingly non metaphorical connections between words dependent upon origin.”

Always questions about etymology—words are power, and part of the power, as in any good witch’s brew, comes from the roots! I would argue (like I have been) that there are only “seemingly non-metaphorical connections between words” and things in any linguistic circumstance. I feel very agreeable with C.E when s/he says “Speech definitely started as metaphor and then became more and more abstract over time.” I would say that, yes, in a sense it became more abstract over time as the same sound (and later picture, symbol, or text) became associated with the same phenomena; thus, we get the universalization of the meta-phoros—it dies and becomes the truth: the sound, image, symbol, or text ‘dog’ truly means dog-like phenomena.

“makes me think of heidegger and his belief that german (?)language was based on ancient greek (logos) where/when the word and its meaning was united...”

My Heidegger is rather sketchy—I’d have to do some research to comment. You could, if you can, say more also…

“also thinking of derrida and play... with the "death of god" and the lack of a transcendental signifier in the language system is there a lack of legitimacy and logic in...”

Hmm, I think I get a sense of what you might be saying about Derrida and “play,” but are you clear on what you’re trying to say? What does Derrida’s notion of “play” mean to you (don’t worry about the depth of your answer, but please, try to answer)? Derrida put forth the idea that there is “nothing outside the text.” While I think that, in a certain sense this is pure rubbish, in another sense it is right on the money. What I feel is right about this is that the words only refer and defer their meanings to and through one and other, or to use Derridese there is “differance” in the “play” of words.

“Death of God,” I think this is a Nietzsche thing and not Derrida (although Derrida might certainly “play” with some of N’s ideas and works—reading N with a “deconstructionist” interpretation). Lack of transcendental signifier might be Saussure, but again, I’m getting into sketchier territory here and would have to do research to comment. And for you, dear unheimlich, it sounds to me that you’d quite enjoy a trip the library—as the ancients said “Lege, lege, et relege” (“read, read, and reread!’).

—| x |—

07:53 / 01.07.03

First, while the “thread-rot” might not seem directly related to the original discussion, it is enlightening perhaps to some who might want to know more about Godel and formal languages. Formal languages do fit into the topic of discussion: they are, after all, languages and we *are* speaking about speaking. So no, please don’t have anything deleted—at worst merely move the posts of mine and Lurid’s to the Godel thread (if that is somehow possible).

Second, I’m not sure if this discussion is “out of juice,” perhaps Lurid’s and my discussion is on its final legs, but not the topic overall, OSISTM—why have to start a new thread?

Third, Lurid, I don’t think “my style of argumentation” has anything to do with the difficulties that you and I are having communicating here. I simply don’t think you have a good grasp on metamathematics and formal languages. I mean, you may be a damn good mathematician, but that doesn’t make you a good logician. I’ve looked at the link you provide—here are some quotes:

“Proof theory, a form of metamathematics studies the ways in which proofs are used in mathematics. However, in contrast to common mathematics, statements and proofs in proof theory are purely formal. This means that they are specified in a formal language that usually employs some symbolic logic [italics added].”

Also from the link to Proof Theory:

“…derivations of logical statements [italics added] are constructed. These derivations constitute formal proofs of the statements.”

From the link to Model theory:

From Model theory:

“A model is formally defined in context of some language L [italics added]. The model consists of two things:
1. A universe set U which contains all the objects of interest, and
2. a mapping from L to U (called the evaluation mapping or interpretation function) which has as its domain all constant, predicate and function symbols in the language.

A theory is defined as a set of sentences which is consistent; often it is also defined to be closed under logical consequence [italics added]. Under this definition a theory is thus a maximally consistent set of sentences.”

And finally, from the link to mathematical logic which simply redirects to Logic:

“…nowadays logic is accepted as an accurate way to describe mathematical reasoning.”

So you might see my confusion when you accuse me of not accepting what you’ve said, not reading or understanding the reference you’ve provided, and etc.. Again, I’ve read the resource: have you?

“Metamath isn't framed formally in any standard treatment. The infinite regress you refer to is the reason why.”

From the links it seems as if metamaths are most certainly tied to logic, reasoning about our reasoning, and formal language. There must be a “standard treatment” of some sort, if by this we mean “formalization.” I do not see how a formal language is expressed in anything other than some more or less standardized notation and according to some more or less standardized logical practices. The regress is largely either ignored or defined away—to a point. The point is to become more clear about the language we use, the standards of reasoning we use, and etc.; thus, we accept that we are not getting absolutely clear about what is going on, but we do gain more insight as we move to examine first order languages in a second order language, OSISTM.

“Although you will notice that absolutely no mention is made of higher order logics. There is a good reason for this, you know. It just isn't relevant.”

Hmm, it seemed to me more because it is not an overly complex presentation of the ideas: why crowd the explanation with something that most people aren’t going to understand anyway? It clearly talks about the important role that logic plays in metamaths as the formal language in which these things are discussed.

See it’s like this:

A first order formal language is the base object language. Wrt many maths the quantifiers range over the objects: numbers. Predicates are extended or given via definition. Any statement in such maths, such as ‘4 + 4 = 8’ can be modeled in a first order language. By analogy, this is like me (or someone like me) saying “That house is old,” in a natural language—in this case English. They are similar because they are both about the objects in some world—in the former case “the world” of numbers (whatever these turn out to be), and in the later case “the world” around us. Both are statements that we can see (by analogy) as “first order” because there is taken to be some kind of connection between the “object” language and its “world.”

Now a second order language is about the statements of the first order object language: it is reasoning about our reasoning. Put differently, the connection between a first order language and a second order language is that a second order language can be seen as an “object language,” however, the objects of a second order language are the statements of the first order language: a step away from the first order language’s “world.”

As a little bit of an aside (but perhaps not so much), the reason I haven’t said much about topology is because I am not very familiar with topology—although by synchronicity my sis gave me a book on topology recently and I’ve been reading it (only my second book on the subject so far). Now this Stephen Barr author guy says that topology is about the invariant properties of certain structures—particularly mathematical structures (he discusses the mobius strip in various ways, the klein bottle, the torus (topologically equivalent to a coffee mug, btw folks), map colouring, the projective plane, and etc.). These mathematical structures are objects in so far as their existence is dependent upon the relations between the objects of the first order language: numbers. The relationships (which are the predicates and functions of the first order language) mark out the various statements we make about the objects in groups (or in some interpretations “sets”): that x is a circle, here is how to determine properties of the circle—we now give a formula with fixed relations, but variables where the numbers would be.

OK, so topology according to Barr says, forget these sorts of variables (size and such isn’t the issue) let’s look a t the invariant properties under various sorts of defined transformations. Thus, we are no longer taking numbers as objects in the language of topology, we are taking the statements made about classes of objects in a first order language and examining them from a “higher” perspective: we now have a second order language (of some sort) simply because we a now quantifying over relations—certain relations of the mathematical structure become variables in the language of topology, OSISTM.

Wrt our analogy between formal languages and natural languages, this is now like allowing variables into the sentence in a different way. Whereas before the object was “house” and the first order language can substitute a plethora of objects in its place. When we move to “quantify” over the predicate ‘is old’, we move into a sort of second order language: we could replace this predicate with a variety of others: ‘is new’, ‘needs paint’, etc.. The difference between a formal language and a natural language here seems to be that a formal language marks a sharp break between first and second order, but the distinction in a natural language is so fluid and common place it goes unnoticed.

What is even more interesting, perhaps, with respect to natural languages is that we can not only see an implicit or fluid “higher order” in our easy use of variables wrt predicates, but we can also see that we move to an entirely different order (which is perhaps in ways closer to the technical notion of a second order language) when we simply make a new statement not about the house or it being old (or by recognizing the possibility that not only could we say “dog” instead of “house” for example, but we can also say “has fleas” in place of “is old), but by attempting to become clearer about what we had said: “By ‘that house’ I meant the brick one and by ‘old’ I meant built more than thirty years ago.” Here we see immediately that infinite regress that lurks when we try to talk about what we mean (or reason about our reasoning): we can further differ and deflect our meaning (or the reasonings about our reasoning’s reasoning) to still further clarification—we can still say more. This is back to Derrida’s differance.

To bring this back to the metamaths: model theory and proof theory rely on logical reasoning which is commonly held to be expressible in symbolic logics (typically some sort of predicate logic: sentential logic is too simple). Proof theory seem to me to have to be second order because it is talking about the proofs about, say, mathematical objects: it takes the statements of mathematics as its objects. Model theory, it seems to me, discovers/creates/examines mappings from one first order language to another (or one domain or “world” described in a first order language into another). Thus, it too takes the statements of a domain and maps them to the statements of another domain; i.e., it uses a language that takes statements for objects: the very definition of a second order language. Of course, there are first-order model theories that examine many models in the same language, and in these cases, we need not necessarily move to higher order langauge.

“Also, since we are both familiar with Hilbert's programme - which was about reducing all maths to logic…”

This isn’t quite right: Hilbert’s program was to get at consistent proofs about infinities that were obtained by finite means—in this way he is more of what gets called an “intuitionist” than he is a supporter of what gets called “logicism” (which is people like Russell and the early Wittgenstein). However, Hilbert is also more of a so-called “formalist” since he wanted to have infinities in math as “ideal” elements so long as the system remains consistent. He figured that mathematics did have an extralogical subject matter (and this too is counter to the position of the logicists who felt that mathematics has no subject matter but was merely pure logical relations), and this is captured in his famous remark: “Points, lines, and planes could be tables, chairs, and beer mugs.” The point being that the formalists don’t care so much what the objects of mathematics are: they are more concerned with maintaining consistent ways to prove things about these objects.

Godel directly undermines Hilbert’s program: we both know this. Logicism isn’t so much affected by Godel’s proof, but has other unrelated shortcomings.

“Yes (although Godel is not restricted to first order, I believe). But incompletely. There are theorems of arithmetic that are not provable in the first order formulation. Godel proved this in the abstract, but there are concrete examples of this, IIRC.”

Godel simply doesn’t matter to much but first order languages (or perhaps some “weak” second order languages?): like I have already said, “…it is an established fact that there are no positive nor negative tests for 2nd order validity in ‘full’ 2nd order logics; thus, they are incomplete.” The theorems of arithmetic are in a first order language: Godel’s proof depends on this fact in order to show that first order languages are incomplete by showing that there are statements in the first order language which we know are true (from a second order perspective?), but which the language itself cannot prove: see Rucker’s take on it above for a more or less “concrete” example.

“One is allowed any inspiration whatsoever, and one can then use that to find a formal legal proof. Undecidability says that the method of finding the proof is not able to be automated. It does not say that there exists no formal proof. Godel says that about some, perhaps most, systems. But that isn't undecidability.”

Inspiration is not proof. What if there is no “formal legal proof” that exists at all regarding our inspiration. Validity is about the form of the argument: if there is no way to know if the form of an argument is indeed valid or not (no positive or negative test for validity—i.e., incomplete) then how can conclusions be assumed to be proven based on the form of argument? Undecidability says that there are some statements that may or may not be of valid form; that is, there might be no end to the computation involved in deciding; thus, we simply can’t tell; however, in first order languages, we can definitely decide if an argument isvalid (i.e., true premises can’t possibly yield a false conclusion—Cambridge Dictionary of Philosophy), but not definitely if it isn’t.

“ Given that we are quoting Godel, I assumed that we were sticking to standard binary logic systems. But you were arguing that natural language is binary valued and I was arguing that it is not…”

Godel was a self-professed Platonist, ya’ know. He figured we had direct intuition (not to be confused with the intuitionsits) of a mathematical realm of ideal objects—and he was a logician (not to be confused with logicism). He certainly believed in only bivalent truth conditions. Perhaps like old Plato himself, either a thing partakes of the form of truth or it does not. I was not arguing that all natural language was binary valued, I was arguing against your claim that all natural language was *obviously* not understood in terms of bivalent truth conditions. I was arguing that at least some (but an apparently large and significant portion) of our language appears to be interpreted as if it was modeled in only bivalent truth conditions under normal circumstances regarding normal interactions in everyday life.

You see what I’m saying about languages here?