Ok.. lets forget all of that law of the excluded middle stuff. It's just a red herring for where your real confusion lies.
Treatid wrote:Twistar, I think you have a good grasp of what I'm getting at.
You are right that the bar for 'description' is too high for axiomatic mathematics to reach (I think arbiteroftruth also agrees with this). However, that bar is set by Axiomatic Mathematics itself. Axiomatic Mathematics specifies exactly what is required to rigorously describe a system. I think the problem you have with my high bar for description is the problem I have with axiomatic mathematics's use of description.
"However that bar is set by Axiomatic Mathematics itself. Axiomatic Mathematics specifies exactly what is required to rigorously describe a system."
Where do you get that idea? You need to explain this better. Can you describe this bar to me? Can you also point me to a resource which shows text where axiomatic mathematics is supposedly specifying this bar?
Here is my understanding of "the bar" we are talking about. The bar specifies a level of description whereby I or anyone could describe a system to that level and anyone I am talking to would unambiguously understand what I am talking about with no chance of miscommunication. I think this is what you mean by "rigorously describe".
However, and here is my main point to you:
Logicians and mathematicians DO NOT CLAIM that axiomatic mathematics attains that level of description. And I'm not just making a point to you, I'm trying to point out something about which you are wrong. You have this misconception (misconception #1) that logicians purport axiomatic mathematics to describe things to this level, so you see a contradiction. HOWEVER, logicians DO NOT purport that axiomatic mathematics describes things to this level.
Now, this isn't an unreasonable misconception. You could probably easily find example of people glibly talking about how math is absolutely right and unambiguous. People making those statements probably haven't thought hard about formal logic or formal languages. I would say that they are wrong. But if you talk to people who really have thought hard about these things my bet is they would agree with what I'm saying here. But the point is, now that you're thinking about it a lot, you need to realize that this is, in fact, a misconception.
So then your next worry is that:
You are also right in your characterisation of what would happen next; that partially-describable is no different than indescribable. We are left in a position where we either know everything or nothing.
Your next misconception (misconception #2) is that if we can't describe something to the level of this "bar" that we are speaking about then we can't describe ANYTHING and we then know nothing. This is a misconception. Let me first tell you what is correct and what I agree with:
-It is correct that for any given thing we can either describe it above the level of the bar or we cannot.
-I agree with the statement that NOTHING (including axiomatic mathematics) can describe ANYTHING to the level of this bar.
But here is where I differ with you:
-However, just because we can't describe anything to the level of this bar, that DOESN'T mean we can't describe anything at all. We can describe things to lesser degrees than the bar. Yes, there will be some ambiguity, but that doesn't mean there is no description. You're thinking of it as binary. That is the wrong way to think about it. it is actually a spectrum. It's like a dial. Say we can rate a level of description from 1-100. Something that is a 1 is not describe at all. Something that is 100 would be AT the bar we are talking about. The scale doesn't go above 100 because the way we have defined the bar means there is no higher level of description. Let's talk about describing the weather tomorrow.
-Say you ask me what the weather is going to be tomorrow and I say "I dunno". That would be a 1 or 2 on the level of description. I haven't really told you anything about the weather except maybe the fact that it is unpredictable at the moment by me.
-But say instead I say "It's supposed to be nice". Now maybe I've given you a description which is at a level of 10 or 20 or something. Maybe I've given you all of the information you need and you can go on with your day. But there's still really a lot of ambiguity there.
-Now maybe I can get more descriptive. I can say "it's supposed 50 F at 06:00 rising up to 70-80 F from 11:00 - 18:00 and settling back down to 60F from 19:00 on." This is now starting to rank at maybe a 50 on the descriptive level. However there is still a lot I'm not describing.
-Now, maybe you could talk to a weather scientist or meteorologist who could tell you the absolute state of the art prediction for the weather tomorrow. This would include by the minute temperature predictions, pressure reading, wind patterns, local and global weather patterns etc. This is now getting up to maybe an 80 on the descriptive scale. There is still a lot of uncertainty but the level of description is getting higher.
-However, we can imagine
a description which is even better than that. Maybe I could tell you what all of the individual atoms making up weather patterns are going to be doing tomorrow. And from that we can extract temperature information on a microscale as well as predict the answer to any question about the macroscopic weather that you might be interested in knowing about. Maybe this is a 97 or a 98 on the scale.
The key is that for the final description it is STILL not 100. Now the level of descriptiveness is not limited by our current meteorological models, it is limited by this fundamental lack of an ability to describe anything 100%. But the point with all of this is that even if we can't describe something 100% we can STILL describe it TO SOME DEGREE. It is not black or white, it is a spectrum.
So it is with axiomatic mathematics. I think on this scale most logicians would agree that the goal of axiomatic mathematics is to describe something to 99 or 99.9% or whatever. But they're not claiming it's 100%. That is the distinction.
Let's get a little epistemological for a second. Your argument is that if we don't know something 100% then we don't know it at all. If we can't know anything 100% then the composition of these two statements implies we don't ANYTHING. I disagree with the first statement. The only claim that be made is that "we don't know anything 100%". But what can still be said is: "we know a lot of things 99%". or "we know a TON of things 60%".
So to repeat your two misconceptions:
1) Logicians purport that axiomatic mathematics describes things 100%
2) If we don't know something 100% then we don't know it at all.
These are the two things which I and everyone else in this thread disagrees with. And I really think it ALL boils down to these two misconceptions that need to be worked on.
The first one is a simple matter of fact. You can look up what logicians actually say or just talk to them. It's not a statement about knowledge or mathematics or anything. It's literally a statement about what people say. The subject of the sentence is logicians, not axiomatic mathematics.
The second one is a bit philosophical and one that I have grappled with in the past. You know, maybe the notion that we can't know anything 100% throws us into a nihilistic nightmare but I don't actually think it's that bad. You just have to accept that our world just isn't like that. In the end, this idea that we can't know anything 100% doesn't change much about what actually happens in our lives. Furthermore, and this is your main concern, it doesn't change much about how math actually works either, vis-a-vis misconception #1.