Definitions, Significance and Goedel

[Matt Gregory]

I've been stuck for a long time on what a definition actually is, but I think I've finally figured it out. I apologize ahead of time if this post is unclear; it's the best I can do at the moment. Most of you are familiar with these kinds of definitions; I'm just trying to put a different spin on them. What I think is that Goedel actually formalized a fundamental property of thought, not just mathematical thought, but of consciousness itself.

Consciousness, as I see it, is focused like a flashlight that can only shine on one thing at a time. That thing can be very complex and made up of many parts, but in a single thought, if that thought is rational, all the parts and complexities are dealt with in a uniform way and captured within that thought. So, in relation to the category of the totality (i.e. everything that exists), a thought is a boundary in the totality, and a definition is the same thing as a thought, but the term "definition" implies that the thought is constant within a certain context (as opposed to its usual definition as the verbal description of a thought). These boundaries come about because of the finite nature of consciousness. Consciousness has to, by necessity, focus on one, finite thing and the boundary determines the focus. In other words, it's significance. There are, of course, different levels of significance that are distinguished by one's overall purpose in life, the purpose of one's stage in life, the purpose of the day, the purpose of the moment, etc. All these things make up the context of the boundary. Context is the boundary as seen from outside. It directs someone to the boundary if they don't know what it is. The person who originally thought of it always knows what it is, because it's his thought, but another person has to figure it out by looking the contexts in which the author is thinking.

This is where Goedel comes in. From what I understand of his work on the incompleteness theorem, he proved that when the context of a boundary is ignored, then the logic that operated within that boundary is no longer applicable. I believe he used the term "system of logic", but that can easily be seen as a description of a boundary. A system may be a complex series of thoughts, but they can all be categorized together, hence encapsulating them with the term "system". It's absolutely true that the context can ignored and we can think outside of the system, but once you do that, the boundary that defined the system is no longer in consciousness and the system is effectively destroyed because you're no longer working within the same context. Absolute truth can only be found within a system (i.e. a consistent context) because systems are necessary for logic to function in the same way that steady ground is necessary to perform the act of walking. To put it as simply as possible, changing the focus of consciousness doesn't have anything to do with the logic that operates within a specific context. Logic has to operate within a specific context.

[Trevor Salyzyn]

Well, you have identified two aspects of definitions, significance and context, but I don't think that's a complete description.

I was trying to solve a similar problem earlier, which I decided would be the only way to guarantee that a definition was complete: if a computer could generate one, from context, using some algorithm or other. An algorithmic description of a definition would make it complete. The extreme example was a computer taking the entire Introduction of the Critique of Pure Reason, providing it had a definition of every other word, and providing a definition of a priori without recourse to a dictionary. What a "definition" is becomes totally obscure with this example. A simpler example is having a computer define any word in the sentence "a cat can walk", which, even given a dozen other sentences to give context, is already beyond the capabilities of any programmer.

Significance cannot be handled in binary.

(Slightly unrelated, but I had some meditations on the topic of A.I., that arose from this problem, which I handled quite farcically. Nat/Uni published it up at His Website)

[Ryan Sepulvado]

A bus isn't always a bus, but that's what we say it is.
Anyway, it seems contradictory of yourself to attempt to define definition.
Defining is never as useful as it seems.

NEVER EVER

[Matt Gregory]

Well, you have identified two aspects of definitions, significance and context, but I don't think that's a complete description.

What made me think of it is the idea of conceptualizing a physical object. Logic operates independently of what the senses give us, or at least it can. So when the senses give us the boundaries of objects, what they are really doing is suggesting what part of reality is significant. Same goes with thoughts. We can have a thought, judge it to be insignificant and discard it, as Ryan did in the previous post.

I was trying to solve a similar problem earlier, which I decided would be the only way to guarantee that a definition was complete: if a computer could generate one, from context, using some algorithm or other. An algorithmic description of a definition would make it complete. The extreme example was a computer taking the entire Introduction of the Critique of Pure Reason, providing it had a definition of every other word, and providing a definition of a priori without recourse to a dictionary. What a "definition" is becomes totally obscure with this example. A simpler example is having a computer define any word in the sentence "a cat can walk", which, even given a dozen other sentences to give context, is already beyond the capabilities of any programmer.

Significance cannot be handled in binary.

You mean in binary arithmetic? Computers are more than just binary calculators. They normally used to model things (see |read|'s post) and there's no reason why the concept of significance can't be modeled. In fact, that's exactly what search engines do, sort web pages by significance.

(Slightly unrelated, but I had some meditations on the topic of A.I., that arose from this problem, which I handled quite farcically. Nat/Uni published it up at His Website)

I'll check it out.

[Matt Gregory]

A bus isn't always a bus, but that's what we say it is.

But when we're saying it, then it is a bus.

Anyway, it seems contradictory of yourself to attempt to define definition.

So does self-consciousness.

Defining is never as useful as it seems.

NEVER EVER

Maybe your conception of it isn't fundamental enough.

[Trevor Salyzyn]

You mean in binary arithmetic? Computers are more than just binary calculators. They normally used to model things (see |read|'s post) and there's no reason why the concept of significance can't be modeled. In fact, that's exactly what search engines do, sort web pages by significance.

I'm not altogether sure that you are going by the same definition of significance here as you were in your opening post. From the OP, I got the impression you were talking about meaning, not probability. That's why I brought up artificial intelligence. A search engine is a different beast.

The binary comment was a bit more basic even than arithmetic. As I had assumed you were talking about meaning, I was pointing toward the idea that the meaning of a statement involves much more than its truth value, which by your comments regarding context, should have been a given. In cases where the truth value cannot be determined, or is completely irrelevant, it'd be one hell of a trick to get a computer to recognize absurdity or context-without-truth.

I mean, there's already a lot of philosophers (Quine, for one) that refuse to recognize absurdity, and consider the description "meaningless proposition" a cop-out.

[Matt Gregory]

You mean in binary arithmetic? Computers are more than just binary calculators. They normally used to model things (see |read|'s post) and there's no reason why the concept of significance can't be modeled. In fact, that's exactly what search engines do, sort web pages by significance.

I'm not altogether sure that you are going by the same definition of significance here as you were in your opening post. From the OP, I got the impression you were talking about meaning, not probability. That's why I brought up artificial intelligence. A search engine is a different beast.

I'm talking about it in general. Significance is contextual, so it is decided by different things in different situations. Probability is a valid method of determining significance in a lot of situations, like search engines, polling, quantum physics, etc.

The binary comment was a bit more basic even than arithmetic. As I had assumed you were talking about meaning, I was pointing toward the idea that the meaning of a statement involves much more than its truth value, which by your comments regarding context, should have been a given. In cases where the truth value cannot be determined, or is completely irrelevant, it'd be one hell of a trick to get a computer to recognize absurdity or context-without-truth.

The more you limit the situation, though, the easier it becomes to create a model for it. But yeah, if you wanted to do it for all situations at the same time, that would be one hell of a trick.

I mean, there's already a lot of philosophers (Quine, for one) that refuse to recognize absurdity, and consider the description "meaningless proposition" a cop-out.

How do they feel about "contextually irrelevant propositions"? :-)

[Trevor Salyzyn]

I'm talking about it in general. Significance is contextual, so it is decided by different things in different situations. Probability is a valid method of determining significance in a lot of situations, like search engines, polling, quantum physics, etc.

Okay, there's a possible direction you can go with this idea to make it more clear. Since I got caught up on significance, just break it up into its different modes and test the scope of the word. If it contains too many modes, then it might be better as two distinct parts of a definition.

But yeah, if you wanted to do it for all situations at the same time, that would be one hell of a trick.

Just so you don't poo-poo this idea... I got back an essay and discovered I used the word "counterexample" where I should have used "countervailing position". For some reason, my WrongWordCheck didn't pick it up. We're living in the Dark Ages.

How do they feel about "contextually irrelevant propositions"? :-)

I'd like to think someone saying something like that is what ultimately killed Quine. He couldn't give logical positivism an inch, even if it meant for him to be somewhat reasonable. Either his brain melted, or exploded. I haven't yet worked out the details of this fantasy.

[Matt Gregory]

I'm talking about it in general. Significance is contextual, so it is decided by different things in different situations. Probability is a valid method of determining significance in a lot of situations, like search engines, polling, quantum physics, etc.

Okay, there's a possible direction you can go with this idea to make it more clear. Since I got caught up on significance, just break it up into its different modes and test the scope of the word. If it contains too many modes, then it might be better as two distinct parts of a definition.

Well, we only need to look at significance dualistically to interpret it as a boundary. There's the significant and the insignificant. Every method of determining criteria is going to have this distinction.

But yeah, if you wanted to do it for all situations at the same time, that would be one hell of a trick.

Just so you don't poo-poo this idea... I got back an essay and discovered I used the word "counterexample" where I should have used "countervailing position". For some reason, my WrongWordCheck didn't pick it up. We're living in the Dark Ages.

We're working on it, though. They'll come up with some breakthrough that allows computers to process language better.

How do they feel about "contextually irrelevant propositions"? :-)

I'd like to think someone saying something like that is what ultimately killed Quine. He couldn't give logical positivism an inch, even if it meant for him to be somewhat reasonable. Either his brain melted, or exploded. I haven't yet worked out the details of this fantasy.

I don't think Quine has gone anywhere. He's still alive and kicking.

[Dan Rowden]

I'm going to bump the tread up.

Damn, nothing worse than illiterate spammers!

[Matt Gregory]

I just had a fascinating insight into this as I was trying to get to sleep... (well, maybe not to you guys, but it was to me :)

Another reason for thinking that boundaries = significance is that the more you think about something, the deeper its impression becomes in your mind. It's like its boundaries get more pronounced. Also, the deeper your mental impression of something is, the more likely you are to fall into thinking of it again. Consequently, you'll think about it more often. Hence, it becomes more significant to you. It's so simple.

I was thinking about breath counting meditation and how it's supposed to relieve stress, and I think it does that because, seeing how it makes you think about one thing all the time, it creates a deep impression of this thing so it causes you to place a lot of significance on it, thereby taking significance away from other things in your life. And if you're placing significance on something that can't really be taken away from you, then you can't really get stressed out about it. But if you're meditating on your breath, I wonder if that increases your fear of drowning and suffocation. That would follow from my theory.