Truth and Justification in light of Godel

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  • #26
Dichter
honestrosewater said:
So I'm really trying to make sense of what you're saying, and if you can explain it, please do.

I appreciate your efforts but I think I'd be wasting your time if I proceeded with this discussion. I'll keep reading your posts and make comments on this topic when I sense I can make a meaningful contribution.

What do you think of the idea that to know what a statement means is to know the conditions under which it is true under a given interpretation? If someone said Bob's shirt is wet when Bob's shirt wasn't wet, wouldn't you think that they don't know what Bob's shirt is wet means?

They could be lying. And this goes straight to my point: ultimately, there's no way to establish if Bob's shirt is wet on a purely linguistic basis. I can't write a computer program to evaluate the truth of the statement "Bob's shirt is wet" by simply evaluating the meaning of each term in the sentence. There is more to language than what you can express with language itself, but of course when I make a statement like that using language, it sounds ridiculous - because it's true!
 
  • #27
honestrosewater
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Dichter said:
I appreciate your efforts but I think I'd be wasting your time if I proceeded with this discussion.
Whatever you think is best, but if I waste my time on here, it's my own fault. AFAIK, no one else knows how meaning works in natural language either. So we're all in the same boat; some are just paddling harder than others. (If they'd only stop paddling in opposite directions, we might actually get somewhere! (Wow, I could take that metaphor, like, a brazillion places, but I'll be nice.))
They could be lying.
Granted. They could also have Tourette's syndrome or be wandering away from the point.
And this goes straight to my point: ultimately, there's no way to establish if Bob's shirt is wet on a purely linguistic basis.
That might be true in a sense merely by definition. In such a sense, there's also no way to establish the cat that I dropped from my roof's velocity when it hit the ground on a purely mathematical basis because that's just not what math and mathematicians do. (Sorry, deathtrap of a sentence there.) As you start asking what is true of the real world things and events to which expressions refer, you start stepping away from what language and, by extension, linguists do. Linguistics is an observational science, but it observes language, not the wetness of people's shirts.

The problem that you're bringing up is that in order for the speakers -- be they apes, programs written by apes, programs written over millions of years by physical forces acting in increasingly complex ways, or whatever else can act as a speaker -- to determine how well their model, i.e., their language, models the real world, they need to actually observe the real world. We can avoid this complication by replacing the real world with a world of pure imagination, say, a mathematical structure, e.g., some version of set theory or Pfearth, which could be a model of the real world if you so desired.

(Actually, you could consider still other structures intervening between language and the real world: your thoughts. But things are complicated enough already.)

In a more optimistic sense, what a language can and cannot determine depends on the language and where and in what form the necessary information is stored.
I can't write a computer program to evaluate the truth of the statement "Bob's shirt is wet" by simply evaluating the meaning of each term in the sentence.
An important idea in the study of meaning is something called compositionality -- the idea that the meaning of a complex expression is determined by (i) its form and (ii) the meaning of its constituent expressions. That is, each meaningful part contributes to the meaning of the whole in a systematic way. If your problem is that you don't know how natural language assigns meaning to the parts or puts meanings together, you're not alone. But if you're trying to give a computer program what is by definition the impossible task of accessing information that it doesn't have access to, I'm throwing you overboard. (Sorry, couldn't resist. Snicker, snicker.) Either way, I think you're underestimating how much information is already built into a language. Compare

(1) Bob's shirt is wet.

with

(2) Bob's shirt is alive.
(3) a. Bob's shirt is wet and not wet.
(3) b. Bob's shirt was wetted and not wetted.
(4) Shirt wet Bob's is.
There is more to language than what you can express with language itself, but of course when I make a statement like that using language, it sounds ridiculous - because it's true!
Considering language as a model, there is of course 'something more to it': whatever it is modeling. Considering language as a physical system, there is of course 'something more to it': the physical systems with which it interacts. And so on. What is the 'something more' that you had in mind?

Meh, I don't know what else to say. Here's a quote about meaning that might provide a little clarity or inspire you to search for some. (In case you aren't familiar with him, Frege was not a lazy or sloppy thinker.)

"It is astonishing what language accomplishes. With a few syllables it expresses a countless number of thoughts, and even for a thought grasped for the first time by a human it provides a clothing in which it can be recognized by another to whom it is entirely new. This would not be possible if we could not distinguish parts in the thought that correspond to parts of the sentence, so that the construction of the sentence can be taken to mirror the construction of the thought. ... If we thus view thoughts as composed of simple parts and take these, in turn, to correspond to simple sentence-parts, we can understand how a few sentence-parts can go to make up a great multitude of sentences to which, in turn, there corresponds a great multitude of thoughts. The question now arises how the construction of the thought proceeds, and by what means the parts are put together so that the whole is something more than the isolated parts." (G. Frege: "Logische Untersuchungen. Dritter Teil: Gedankengefüge"; translation taken from Heim and Kratzer 1998:2)​
 
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  • #28
Dichter
honestrosewater said:
AFAIK, no one else knows how meaning works in natural language either. So we're all in the same boat; some are just paddling harder than others.

Well, the hard-disk-space police has been around, sometimes you have to be careful.

That might be true in a sense merely by definition.

It seems to me most philosophical discussions are little more than attempts to refine our definitions. It's really hard to disagree about facts.

In such a sense, there's also no way to establish the cat that I dropped from my roof's velocity when it hit the ground on a purely mathematical basis because that's just not what math and mathematicians do.

I disagree, because "velocity" itself is a mathematical entity, whereas "wetness" is not.

(of course you can start a debate on how much moisture you need before you can call something "wet", but that is precisely because the word "wet" cannot exactly convey what a speaker has in mind when he uses the word)

As you start asking what is true of the real world things and events to which expressions refer, you start stepping away from what language and, by extension, linguists do. Linguistics is an observational science, but it observes language, not the wetness of people's shirts.

I was hoping we wouldn't have to delve deep into linguistics to discuss this, because I'm not familiar with the associated jargon. As I said, what I have in mind is something ultimately simple: the fact that formal languages, like computer languages, lack "something" which prevents them from being able to express certain ideas.

The problem that you're bringing up is that in order for the speakers -- be they apes, programs written by apes, programs written over millions of years by physical forces acting in increasingly complex ways, or whatever else can act as a speaker -- to determine how well their model, i.e., their language, models the real world, they need to actually observe the real world.

Not really. What I was trying to say is that in order for a language to serve as a model, it needs to be "ambiguous". By "ambiguous" I mean, for instance, the fact that the meaning of the word "wet" is not formally established, and you may have a situation when it's not possible to know if Bob's shirt is wet or not, not because you cannot touch the shirt, but because you don't know if the word "wet" applies to the particular level of humidity you are sensing.

So what is seen by formalists as a defficiency in language is actually what gives it power. This was my original argument.

We can avoid this complication by replacing the real world with a world of pure imagination, say, a mathematical structure, e.g., some version of set theory or Pfearth, which could be a model of the real world if you so desired.

I think you can only avoid "ambiguity" if your language is perfectly isomorphic with the world. But if you achieve that, you would have blurred the meaning of "language" and "world". In a language that is perfectly isomorphic with the world we couldn't even be having this debate, since there would be nothing to argue about.

"It is astonishing what language accomplishes."

I don't dispute that. It is indeed quite amazing how a vocabulary of about 1,000 words is enough to express almost every fact about the world.
 
  • #29
matt grime
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Dichter said:
I disagree, because "velocity" itself is a mathematical entity,

no, it is not, it is a physical property of directed relative motion. De Rham Cohomology is a mathematical entity, groups are mathematical entities, gorenstein singularities are mathematical entiteis. velocity is not.

whereas "wetness" is not.

again, no, it is not necessarily quantifiable in the same way as velocity, though, but we can often all agree when something is wet by touching it. There is a whole theory of the supposed quantification of such subjective things, by the way. See eg the University of Bristol's Engineering Mathematics department (AI and reasoning in the absence of full information).



As I said, what I have in mind is something ultimately simple: the fact that formal languages, like computer languages, lack "something" which prevents them from being able to express certain ideas.

You are demonstrating perfectly that the same thing applies to human languages.
 
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  • #30
Les Sleeth
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matt grime said:
You are demonstrating perfectly that the same thing applies to human languages.

This doesn't have anything to do with Godel's theorum, but since you mentioned it . . .

I think the "same thing" applies to written language, but not to spoken language. I usually can spot sincerity, for example, in a speaker. One might mathematically and linguistically represent a James Taylor ballad, but there would be a lot not communicated symbolically that James manages to communicate through his being.
 
  • #31
matt grime
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Les Sleeth said:
This doesn't have anything to do with Godel's theorum

None of this thread has anything to do with Goedel (apart from those bits explaining why it has nothing to do with his theorem).
 
  • #32
Dichter
matt grime said:
None of this thread has anything to do with Goedel (apart from those bits explaining why it has nothing to do with his theorem).

Sure but the critics are not helping much. There is a struggle from non-mathematicians, like myself, to understand what all the hoopla about Gödel is about, and all we get in response is pedantry. Come on, enlighten us instead of simply keep saying we're in the dark!

For anyone interested in a better understanding of "Truth and Justification in light of Godel" and skip the pendatry, I've found a very interesting link:

http://www.edge.org/3rd_culture/goldstein05/goldstein05_index.html
 
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  • #33
matt grime
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And most mathematicians are completely unmoved by Goedel's theorem, most of them won't even know what it is; heck I don't even know the precise statement of the theorem and end up guessing most of the time. The fascination amongst lay audiences the amazing thing.

Maths in some sense 'is' pedantry, or at least the application of rules and formal reasoning. Goedel's theorem is explained all over the web, and in the forums. I think it states that in a finite recursively axiomatized system that is strong enough to define the natural numbers (ie in which we can do induction) then there is a statement such that neither it nor its negation is derivable from the axioms.

Examples: System is the standard Zermelo Frankel set theory axioms, then the continuum hypothesis, the generalized continuum hypothesis and (I believe) the axiom of choice are all consistent with the axioms (ie there is a model of ZF where they are true) as are their negations.


It is a statement about mathematics, finite recursive axioms, and induction. Nothing to do with language, or real life, or anything. It also has practically no bearing on 'doing' almost all mathematics. In fact outside of this forum I have never had any reason to talk about it, but then I'm not a set theorist, or logician. If I want the axiom of choice I use it, I don't care whether it is or isn't independent of ZF.

Everything from post 7 pretty onwards has just not been about mathematics, if that observation is pedantry that annoys you then so be it.
 
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  • #34
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matt grime said:
And most mathematicians are completely unmoved by Goedel's theorem

So much so they don't even know how to spell his name...

(even if you can't type Gödel, with the umlaut, you still don't have to add an extra 'e'...)
 
  • #35
matt grime
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I do know how to spell his name, I do not know how to typeset umlauts in html, and nor do I care to learn since adding an extra e is perfectly acceptable (in my opinion) as a workaround (nb, see your own post 10, if we're being picky) just as it is acceptable (even preferred, these days I believe) to use ss and not the 'beta' like symbol in German (it annoys me I can't remember its name) as well as ascii-tex for maths. I would much rather offend your sensibilities on that, than misapply his theorem. Now, do you have a valid mathematical point, or would you like to spell Tchebyshev (aka Chebyshev, Chebytchev or God knows how many variations; there is supposedly some book with 22 different spellings of the one name (all for one person) in the index).
 
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  • #36
Hurkyl
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Since we've strayed from talking about Gödel's theorem, and are beginning to degenerate into personal attacks (:grumpy:) I think it's time that this was closed.
 
  • #37
HallsofIvy
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Actually "oe" in place of "ö" is fairly common even in Germany.
 

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