Distinction in particular or how I'm using it? Not sure I understand your question. If you mean how I am using 'distinction' between the numbers, I simply mean what mutual observable quality each one has conceptually that makes them soley unique to themselves and to each other. Each one holds a unique operation in it's value, and I probably should'nt use that word 'operation' because that probably means something specifically in mathematics. I hope you can follow me here.
Okay, I wasn't sure if you meant something cognitive or something mathematical. That clears things up.
Okay, that helps. Question, on the infinite number line of natural numbers, you're saying you would never refer to 0 as the null set on that line? How is zero defined on that line in mathematics? non additive integer? this is where I am fuzzy.
Null set and zero are never the same, that's correct.
Properly speaking, 0 is simply a number that is a member of certain sets. It is a member of the set of "all real numbers". It is also a member of the set "all integers" which is, technically, a subset of the "all real numbers" set which is a subset of the set "all numbers".
In this case, it is defined as a member of the set "all natural numbers" which you then plot on a number line. It's technically correct to call 0 an integer, but that's not the only thing it is.
I'm not sure what you mean by "non additive integer", and the only result on Google is this thread.
Yes, that is what I mean. I'm fuzzy on the mathematic terminolgy here, sometimes I say cardinal when I mean natural! gaffes galore, thank you for your assistance.
It's fine, the distinctions are often mixed up.
It's logical expression is true, false, unknown, and I certainly relate it to numbers, the basic ordering and counting that we are born with, hardwired with. But that's just it's expression. It's also, each one of them, a distinction in subjective experience. I assume for the sake of argument it's all the same 'hard wiring', but that's because I can't imagine it would be anything else than hard wiring.
So this subjective experience tags the 'meaning' of the event as 'true/false/mystery', and then what ever process occurs from there, concept, idea, language, behavior follows in accordance.
All ideas can either be true, or false, or mystery because all we can experience can either be true, false, or mystery.
I see, so this is getting back to the philosophical realm. Well, mathematically speaking, there is also probability, which is a way of expressing and evaluating different
degrees of certainty. I suppose that wouldn't really affect what you're talking about.
I do, however, agree with an earlier poster who mentioned that in the strictest, "all knowledge is available" sense, every statement is either true or false. The concept of unknowability is a function of limited knowledge or limited understanding.
Also, I'm wondering if probability would have a role in human thought processes as well. We may not innately calculate it, but we do experience different degrees of certainty, that is, different degrees of "mystery".
That is exactly what I mean. That natural sense of ordering and counting. I think that is common knowledge that is hard-wired. I'm saying that foundation for number counting, and I am saying this philosophically here, I'm saying that those basic distinctions we use there, 0, 1, 2, recapitulate further than just number, i.e. how we use those numbers, we are also using our ideas.
I think I'd need to see more research on the link between numerical thinking and ideas, unless you mean the idea of breaking things down into categories like "true" and "false" (and yes, "unknown"), which I think is a function of the fact that real objects either are, aren't, or we don't know if the are or aren't. If that's what you mean, then I suppose it follows from our grouping and separation of objects.
yes, that makes sense.
But here I can suggest, philosophically, that 'honesty' and 'rationality' are also 'hard-wires', or exansions on some pretty basic wiring, although research shows not in everyone, I believe. I forget the percentage, but we have to accept a small number of the population are socio-paths, and do not have this hard-wiring past a certain point, they can't be honest and self reflect, they have no nuerological framework for it.
Yes, I think those concepts are in some ways "hard wired" into use.
I think that concepts like "honesty" are things developed over millions of years of human biological and social evolution. Those concepts don't exist innately in all animals, but they have a distinct advantage in human society. I think it can be argued that those traits provided a survival advantage in the form of an improved society.
Sociopaths, as far as I know, display different brain chemistry, resulting in their dramatically different behavior.
So episteminologically, it makes sense that when you combine the natural ordering and counting, which really is both a logical and artistic synergy, with the hard wires of honesty and rationality, you will eventually come to the higher maths. You can't be dishonest when your doing math.
Well, you can be dishonest doing math, but you won't arrive at the correct answer.
I agree that rationality is key to developing higher math. You have to observe, model, refine the model, etc. a lot to develop concepts. Once you have a decent set of mathematics, you start to refine within the context of math itself with concepts like number theory.
I think the hard nuts of what I suggest is common knowledge. I wish I had the exact study on hand right now. I am still exploring this issue further with a friend of mine experienced in such matters. He showed me brain scans that he said supported what I suggest here, with 0, 1, and 2. That's what sort of got him interested in what i was saying. I can't for certain say I can repeat accuratly what he was saying, he was just pointing to three sections of the brain that light up when there is emotional/intutive activity, rational activity, and then where there is the negotiation between the two. I'll be getting more information on this at some point.
That would be interesting.
As a die hard skeptic, I'll have to suspend belief until I see something definite. Keep us posted.
hehe, yes, and that is how I was using 'verb' as a metaphor earlier in to describe infinity to the 'noun' of finite. Sorry, I know that was creative, it wasnt technical, I was appealing to your intutition.
It's okay. As long as you're not going around trying to solve 2x = y where x = infinity, that's fine.
Does not 'one million' mean one million 1's? 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,....... takes a long time to write out that way. We see each other's meaning here, yes?
Yes, but my argument is that no one conceptualizes it that way. They conceptualize the abstract concept of the million rather than the idea of "one million repetitions of the unit one". Especially with large numbers, the conceptual framework has to extend beyond imagining a group of
n objects.
We talked about recapitulation earlier, I said these three distinctions that we make about 'reality' at a very primal level. My exact phrase was I am addressing the conceptual in natural numbers that are found repeating and recapitulaing elsewhere, i.e. outside of number.
You know, I'm going to turn it around here: I think that numbers are a representation of the external reality. Sure, there's a distinction... because we created it that way. Numbers exist as a way of understanding and working with concepts like "I see a bunch of apples here". If that's what you mean to say, then I agree.
I just wouldn't say it as if the concepts are found outside of numbers. That seems to imply that reality is comforming to number theory, rather than the other way around.
Let me try putting this way, which is partially intuitive and partially rational.
We can only understand reality through our concepts about reality. I'm not trying to get deep here, this is actually a very superficial distinction. We have ideas about things and for things, and without these ideas, we dont really have any things.
Some of our primal ideas are ordering and counting. All of our concepts of reality are structured into these three 'limits' of perception, 0, 1, 2. We can decontruct our ideas back into these three mutually held distinctions, our basic operations, and see them 'reflecting' (recapitulating each other). I do that in dialouge, that's what i do here, I deconstruct our dialogue back to these simple and shared places. That's what many people do naturally too. The only problem is some frameworks, like bivalency, over ride it, confuse it as something it's not.
I could go on more here, but I fear I already opened up a can of worms with that one. Don't hate me. I'm trying to explain something very tricky.
Well, just as true/false has limits, so does true/false/unknown. For example, T/F/U fails to address degrees of certainty or probability. Sure, you could address that casually, but in casual discussion you can also include "unknown" into a generally true/false discussion. You can also divide things up into multiple frameworks, if it would be useful.
I think it's extremely difficult to describe human experience in precise, non-amibiguous terms, so we invent all kinds of systems to describe different aspects. We invented mathematics to define our perception of objects and quantify our experience with the real world. We invented logic frameworks to help describe concepts like "truth".
As far as I know, we don't get to step outside ourselves. That sucks, but we just have to deal with it, so we invent systems to help convey things to other people.
yes, I am saying that, and no I'm not. I'm not saying these three values exist 'out there'. I'm saying they exist 'in here', and we can't percieve the world 'out there' with out them.
Think about mystery. If you dont think mystery is a truth value we contemplate very deeply, watch your reaction next time someone unintentionaly sneaks up behind you and scares the living daylights out of you. You're reaction is most likely akin to your reaction to death. Dying is a mystery that humanity has been dealing with for quite some time, and it's other side, nothingness or not, is completly shut off from objective verification (t/f). A long time ago, the mystery could kill you, you didnt know when your death was coming when you are being hunted or attacked. Martial arts teach us that, they teach us to be 'calm' in the face of that reaction in the body. In Aikido, this becomes really refined. You face your fear of 'mystery' in aikido, and it is in that space that you supply your counter move. You then use your opponents fear of death to react irrationally, giving you the ease of moving them.
So we also have a basic fear of mystery at a deep level, if my argument here is true.
I don't think anyone here has been arguing that the unknown is not a component of human experience. I agree that humans often have a deep fear of the unknown.
Put from the position of probability, you could say that humans have a fear of situations in which they have a low percentage of certainty regarding the outcome. The degree of uncertainty and specific situation required to cause fear would vary from person to person, I suppose.
you already understand what I was meaning to say, I was just using inappropiate mathematical language so I confused you, sorry.
It's fine.
I see that, I am refrencing that 1 is the only 'real' number in the conceptual sense, as in how we must conceptualize it to do anything with it. We may need more time on this one. It gets fuzzy for me around here, because here we blend the natural ordering and counting, which is surely mathematical, to something conceptual and subjective, experience, awareness, that sort of thing.
I think I understand where you're coming from with the 1 thing. I've already outlined my objections to the conceptual concept, but that's only my opinion. I would think more research would be needed to know if we really see everything as some conceptual combination of the number 1.
I have absolutly no idea what you mean here, but I trust you do! Imaginary numbers??? That sounds absolutly delightful. They are all imaginary from how I describe it. I would love to know how mathematicians define 'imaginary'
The square root of -1 is what I mean. Wikipedia
has an article on the concept. They're a very abstract concept, but mathematically useful.
that's what I'm saying, but I'm not saying it is the only real distinction, i am saying it is the only real unit. It is the thing unto itself. 1=1. 2=1,1. 3=1,1,1.
1 is real, 2 + are all symbolic. There is no 'two' that exists, only a series of distinctions. 1 is the number of distinction, a conceptual distinction is always 1 whole thing, which can be broken down into perhaps more things.
Make sense?
Kind of, yes. It only works for non-abstract concepts, but I see where you're coming from.
However, what's the deal with fractions, then? If you break an apple into two, are you looking at 1/2's of an apple, or are you looking at two units of 1 apple half? When you look at the apple, are you really looking at several million units of 1 molecule?
Do you see where I'm going with this? You can say that 1 unit is the smallest unit you can divide any real object into, but 1 of what? If you're going to abstract the concept, you might as well go all the way and acknowledge that the concept of "1" is in and of itself an abstraction of what we observe. It's not the
real object, it's an abstraction we use to meaningfully distinguish the apple from the ground.
that's just it, we dont! we dont need to, it's built into our hardwear to make that distinction so we dont have to deal with a conceptual group of 1's, we just deal with 1 and symbols for more than 1. when I say three, you dont intepret 'he means 1 thing plus one thing plus one thing', you get '3'
As I said above, I think that the unit of 1 is just as much a conceptual abstraction as 3, 10, 50, or 5 trillion. When dealing with abstract numbers, I think the abstraction of 1 is lost in the general abstraction that is mathematics.
well we have to be doing it subconsciously, right? we have it built in, we know, by default, that 5 means 1,1,1,1,1. it's an instant computation, natural.
make sense?
Sure, but I don't agree with you. We may know that 5 means five groups of 1 (or 10 of 1/2 or whatever), but we don't conceptualize it like that. 5 exists as an abstract entity just like 1 does.
Even when dealing with physical objects, these are abstract concepts. Do you think of your computer as a group of 8 or so components or 1 computer? Do you think of yourself as billions of atoms?
It's especially difficult when dealing with very large numbers. If you want to see abstraction, try thinking about a trillion. Try to conceptualize just how many "things" that would represent. I would wager that you do not have a distinct picture in your mind of, say, one trillion apples... or anything else, for that matter. It's become a distinct abstract concept from the concept of a singular object in order for you to work with such a large number.
that's usin' the ol 1 and 2! Do you get my meaning here?
Unfortunately, I don't. As these are all abstract models for underlying reality, I'm not sure 1 and 2 have all that much inherent meaning. You might use them to define concepts with inherent meaning, but I conceptualize in terms of different units, or even different objects.
I am really interested in what problems it is useful in, other than computational/computers.
If you wanted to quanitify a problem using "unknown" as a value, it would be useful. Say you wanted to formally propose something like "God exists" (I know, I know... can of worms) but wanted to include the possibility that, say, the concept of God was unknowable at this time.
I think we are in agreement on the 0, 1, 2, it's role on the natural number line, and how that symbolizes a set of distinctions that i am using.
Yes, in that limited context I can see what you mean. In the broader sense, though, I'm still in disagreement.
I think I may have opened up a can of worms on the hard-wiring stuff, or at least I hope I have!
The can of worms is only an abstraction.
An note of irony, perhaps. I have noted from experience that the easiest and most rational discussions I have had are with mathematicians. I wonder if that is due to the neccesity of honesty in mathematics. I enjoy the experience here.
It's because mathematics has defined rules. There is a right and a wrong answer when discussing mathematics, and if something is unknown, it's plainly stated as such (for example, there are plenty of unsolved problems in mathematics).
