I don't see why that's relevant to clarify, but I'll give it my best shot.
-There is no possible ambiguity about a "standard number"- 1, 2, 3, etc. If you specify that decimal numbers, 0, and negative numbers don't count, no more definition would be needed. A system which includes decimal numbers and negative numbers need only clear up ambiguities, say about irrational numbers, and it will work as a working "definition". -Take a hypothetical world where the word "sport" is used to refer only to football, soccer, and basketball, and no other sports exist. Also assume that any new "sport" will automatically be dismissed as not one because it is none of these, including any deviation from the standard rules of the three sports. This may not be a useful concept, but it is a completely coherent one (as long as the issues is dealt with of differing interpretations of the rules)- further definition is not required. -There is no inherent contradiction in free will existing. It does not actually exist, but in a theoretical world with limited or no actual casuality it could exist. Such free will would consist of a completely uncaused force making decisions for no reason whatsoever. This wouldn't be much like the stereotype idea of free will, but I think it would deserve the name.
If you specify that decimal numbers, 0, and negative numbers don't count, no more definition would be needed.
My chair is not a decimal number, zero, or a negative number, but I suspect it's not what you meant by "standard number".
Hence my referring to "1, 2, 3 etc". Yes this appeals to a little bit of human intuition, but so do all definitions- even a theoretical world in which absolutely everything was defined in terms of other definitions would be useless if a human did not have conceptions of what some of the words meant using intuitive associations.
This gets rid of all ambiguity as to standard numbers.
Human intuition is not always unambiguous.
It's been proven that there is no way to unambiguously refer to natural numbers using only first-order logic. Human intuition is part of a human brain, which can be fully described through first-order logic. Thus, human intuition must still be unambiguous.
Doesn't such research presume a formal definition? What possible ambiguity could exist in my quasi-definition?
You appear to be contradicing yourself regarding human intuition.
Doesn't such research presume a formal definition?
No. Not an unambiguous one, at any rate.
What possible ambiguity could exist in my quasi-definition?
The usual definition of natural numbers is that every natural number has a successor, only 0 has no predecessor, any two numbers with the same successor are equal, and induction works. This is not sufficient to prevent you from having an infinite chain of "natural numbers" extending in both directions.
If I remember correctly, the idea is that you can construct a sentence that can neither be proven nor disproven. There is another theorem that I'm less familiar with that says that there's a model of the natural numbers where it's true, and another model where it's false. You can add this sentence as an axiom, but you can still construct another such sentence. Any computable, consistent axiom schema will have such a sentence. Even if it's as complicated as "simulate 3^^^3 mathematicians for 3^^^3 years and any axioms and computable axiom schemas that they think up".
I think it's Gödel's sentence. It's basically that you find a way to encode a sentence into a number, then you find some crazy mathematical sentence that basically claims a given number corresponds to an unprovable sentence. You make another sentence that results in the truth value of the given sentence being plugged into itself. By plugging the number for the first sentence into the second, you have a sentence that states that it's false.
You can say that Gödel's sentence is false, and the one that's made with that, etc. are all false, but that axiom schema introduces a new sentence. You can do that again, but then there's another level. You can do it for an infinite number of levels, but there's another level after that. You can only do so much recursion.
Take a hypothetical world where the word "sport" is used to refer only to football, soccer, and basketball, and no other sports exist. Also assume that any new "sport" will automatically be dismissed as not one because it is none of these, including any deviation from the standard rules of the three sports.
Your idea of sport implies that concepts can be non-sports. Red is a concept that's neither a sport nor a non-sport. Red can't be a non-sport because it's no activity. If you don't give me enough knowledge to separate non-sport from concepts like red than your sport concept is useless. If you do give me that knowledge than your sport concept is well defined.
If you talk about free will on the other hand it's more difficult what you mean with non-free will. Here I think you actually define the concept of free will as something that happens in absence of causality.
A non-sport would simply be anything that is not a sport. I don't see why saying that anything that isn't football, soccer, or basketball is not a sport automatically creates a new category of non-sports other than not being a sport.
On free will, it could actually be a relatively coherent concept close to free will as we understand it (if, as with any creation of a coherent definition where there was none before, very minor alteration of the concept). A conscious being which is simultaneously an acausal force which does things for no reason whatsoever. You could even add that said being chooses for no reason whatsoever and yet has limited options to choose between.
It may not actually exist, but there is nothing logically impossible about it.
A non-sport would simply be anything that is not a sport. I don't see why saying that anything that isn't football, soccer, or basketball is not a sport automatically creates a new category of non-sports other than not being a sport.
Normally that's included in the idea of sport. Walking might not be a sport but it's different than red in relation to being a sport. Useful concepts have that property.
It may not actually exist, but there is nothing logically impossible about it.
How do you know that "conscious" and "acausal" are concepts that are logically coherent?
Why need this one? I'm merely saying it's coherent, not that it's true or useful. Ideas of non-sports are a prime source of ambiguity anyway.
"Conscious" is something which exists in the actual world. I don't quite understand how it works, but I can rip off that.
Claiming "acasual" to be incoherent assumes casuality in the universe. Casuality may exist in the actual universe, but there is no logical necessity that it must.
To be clear, "conscious" is a label which we slap on certain types of behaviour for certain types of very large, complex machine. Similar thing with "red" - both "consciousness" and "red" are labels we slap on certain features of the universe. Neither of those things are fundamental to the universe.
Are you confusing map and territory?
Yes, but the behaviours described as "conscious" do exist in the actual territory. Same with red. A conception of what it means to be "conscious" has been constructed, which as I said I can rip off for my purposes. My definition of free will is still coherent.
How can you tell that something "can exist without any ambiguity within it", if it "is undefined, or even cannot be defined"?
And what constitutes being "still a tenable concept"? What exactly are you trying to defend these undefined terms from?
Your examples seem less than convincing.
I have the impression -- perhaps quite wrongly -- that there's some particular example or class of examples that you have in mind but haven't listed here, which provides the actual motivation for this. Certainly the examples you've given don't seem to me like cases where (1) a concept is undefined or even indefinable, (2) it's clear that it "can exist without any ambiguity within it", and (3) there's any call to defend it from charges of untenability (whatever that actually means). Am I wrong?
(It occurs to me that what you've written would be more credible to me -- though I think I'd still disagree -- if by "ambiguity" you meant not "multiple or indefinite meanings" but something more like "internal contradictions". That seems like an odd misconception to have, but such things can happen to the best of people. The only purpose of this paragraph is to give you a chance to notice and clarify in the unlikely even that that's what's going on.)
Sir Karl Popper was also not keen on definitions. From memory (so please check my claims) he said all definitions lead to infinite regressions of 'yes, but what does that mean?'
I'm not sure about this, but presenting it anyway for scrutiny.
I was thinking that it doesn't matter if a concept is undefined, or even cannot be defined, if hypothetically speaking said concept can exist without any ambiguity within it then it is still a tenable concept. The implications, if this is true, would be that it would knock down Quine's argument against the analytic-synthetic distinction.
Your thoughts, Lesswrong?