saviourmachine said:
Concreet example
I find the concepts of the sets still a bit difficult. Can you give some concrete examples?
Sure, anytime you catagoize something, give it a name, you are thinking of a "set". The elements of the set constitutes the entire collection of things which you have decided are included under that name. Lastly, coming up with a name for the set is not required; at least not in the abstract. I like the following comment on the issue of "abstract" from Robert M. Hutchins and Mortimer J. Adler in the "Gateway to the Great Books": "If you have learned to add two and five without asking "Two and five what?" you already have both feet off the ground--higher than you think. You are now air-born. The rest is just a matter of gaining altitude."
saviourmachine said:
Can I take 'warmth' - with all kind of properties - as set A?
Sure; however, if you do then membership in the set would be anything which you would consider satisfying what you mean by 'warmth' – with all kind of properties. I am not at all sure what you would wish to call 'warmth' – with all kind of properties. A coat? A smoldering fire? A friendly girl? Since I defined B to be a finite collection of elements of A, "the discovery of the concept "temperature" would not belong to B unless you would include "the discovery of the concept 'temperature'" as an example of 'warmth' – with all kind of properties. Why would you include this particular discovery as an example of 'warmth' - with all kind of properties?
saviormachine said:
The discovery of the concept "noise temperature" would again change our knowledge, it's Bj.
Not with any meaning of 'warmth' – with all kind of properties I would consider reasonable. Under your example B
j would be a change in the specifc set of things you catagorize as 'warmth' – with all kind of properties. Your mother bought you some warm cloths or some hot (if 'warmth' – with all kind of properties included "hot") girls walked in the room. That is, the collection of things which you regarded to be in that set 'warmth' – with all kind of properties would change. C is the end result of all changes to date and would be everything you currently regard to be 'warmth' – with all kind of properties.
B is not a subset of A, B is a finite collection of elements taken from A. The significant difference is that it is possible for an element which appears only once in A could appear multiple times in B. A subtly which has profound consequences which I won't get into for a while. Likewise, C also being a finite collection of sets B need not be a subset of A.
saviormachine said:
Message vs label
The elements of C are messages. What would be a message in a concreet example? The elements of B are labels. What is a label in a concreet example? Maybe using some standard mathematical notation - like above - can clear things up too.
C is the collection of information available to you (knowledge of the things you consider to be 'warmth' – with all kind of properties as presented above). The elements of C are changes in that information (knowledge of the things you consider to be 'warmth' – with all kind of properties as presented above) which arrive via the addition of new elements to that set C which are labeled B
j. Message just seemed to be a good reference label to put on "change in information". I have no idea what you meant by 'warmth' – with all kind of properties. You could have meant "putting a coat on" as an example of an element of A. B was a collection of elements: perhaps putting a coat on together with three warm girls named susie, joan and kate in front of a low fire in a fireplace. Call it an experience if you wish.
The real problem here is that the moment you begin to put meaningfull labels on these elements, you are proposing a solution (i.e., you know what you are talking about) and not examing the problem (trying to understand what you are talking about). Read my
post to Les on the "Are Qualia Real" thread.
saviormachine said:
Numeric example
...
Did I got that right? Is the focus on the relationship between the elements of the message, or the relationship between a message and previous messages?
I really didn't understand what you were getting at. I don't think you understand why I am talking about sets. Given any problem concievable, C constitutes the information available to you to solve the problem. C is not A because we can not know for sure that all the information is available to us even though it is assumed that A is what we are trying to understand. And the set Bis required to provide for the possibility of change. That's really all there is to it.
saviormachine said:
Layman example
I sort of understand the numeric example, but when considering something like 'gravity' or 'temperature' I don't know what A, B and C mean in that context. Can you help me with that? Or does the nature of the problem forbid you to fill on some details?
"Gravity" and "temperature" are meaningfull labels of concepts essential to the modern solution to all those environmental problems ordinararly referred to as "physics". Their very existence means that one is working with a solution, not with the problem. One could ask, what was the problem which lead to that solution? That would be the history of physics or maybe the history of science or perhaps the history of man. At any rate, the path to that solution is far beyond what I could write. What I am trying to do is to look at the fundamental nature of all problems themselves.
saviormachine said:
The next message can be as predicted or it can be a surprise.
I think you misunderstood what I was asking. First, I was talking about a representation of C (that upon which your solution is to be based). What I had pointed out to you was that, since C was a finite collection of sets B (which I had labeled as B
j) and every B constituted a finite collection of elements of A (by definition) I could put arbitary numerical labels on each element of every B and then have a specific list of all B in C. Possessing this list I could answer the question, "is this specific B (specific in the sense that I have a set of numerical labels which specify the B I am talking about) in the complete set C?"
I should comment about this circumstance. Essentially I have not given those numeric labels any meaning at all and yet I can answer a very specific question; however, my ability to answer that question depends on those numerical labels I have assigned. Again, you need to recognize that meaningfull labels are part of a solution and not a mere formality. As soon as I give two elements somewhere in this set the same label (numerical or otherwise), I am fundamentally asserting they are the same element. How in the world did I know they were the same? For the moment, let us just say I guessed and maybe I guessed right and maybe I guessed wrong. Later I will point out some interesting aspects of having the freedom to label things anyway I wish. At the moment, in order to avoid making presumptions, I must assure that, whenever I make an assertion about C, that the assertion must be true no matter what labeling proceedure was used.
But meanwhile, you have a list of every B you have ever seen (in the problem so far) and, having labeled them, can refer to any specific one via the list of labels assigned to it. Against that backdrop I asked, "suppose, for the fun of it, someone gave you the labels of all the elements but one from some B and asked you what element could be added to make that list a valid example of one of the sets you were given." Your answer, "the next message can be as predicted or it can be a surprise", presumes we are talking about a new message. We are not, we are talking about answering questions about the C which is available to you.
There is a very simple proceedure which will create a set of lists from the lists you have prepared where this second list will asnswer the above question. First, take each list you had to start with and, for any two lists which are identical, add a label which does not exist on any list (since we are using numerical labels and the number of elements already labeled is finite, there exist a plentyful quantity of unused labels). Now, for each of these altered lists, make every list possible which is identical to that list except for the absense of one element. Now repeat the first step above, eliminating any duplicate lists created in step two, Finally, replace the labels which were removed in step two.
The result has a very curious property. That is the fact that no matter what list you choose of the finished collection, you can remove any element and there will be no list in the remainder of the lists which have the same set of numbers you are looking at. This property is achieved no matter what the original label assignment was. Given this new set of lists, if someone gives you the labels of all the elements but one, there is only one possible missing element so the question asked above may be answered.
I suspect you are questioning what value lies in the above performance. The value arrises when you realize that the collection of lists you have constructed can be seen as a mathematical function. The numerical result (the missing label) is given by the table of all lists missing one element (it is not a continuous function as it is only defined for the arguments in the table). It is a function of many variables (the list of B labels missing one element) which results in the label which is missing. Mathematically this can be written:
[tex] x_n=f(x_1,x_2, \cdots, x_{n-1}) [/tex]
If that is true, then it implies there exists a function F, defined by
[tex]F(x_1, x_2 , \cdots, x_n)=f(x_1,x_2, \cdots, x_{n-1})-x_n [/tex]
where the rule which tells you whether a particular list is valid is given by F=0, a very simple rule. This result was achieved by imagining the existence of some elements which were not part of A: i.e., they were totally made up in order to achieve that simple rule. What I have shown is that the existence or non-existence of any finite collection of elements can be constrained to exactly what has been seen (no matter what that collection is) by the simple rule F=0 and the introduction of imaginary or make believe elements.
Now what does this have to do with reality? Throughout history, scientific hypothicies have consisted of invented entities and invented rules. The rules are supposedly invented to explain what we see and the invented entities are then invented to correct for the problems in what we see. Every time we are able to prove that what we see is consistent with the entities and the rules (we think are true) we consider it proof that that they both exist. If you are going to give me the freedom to invent any entities I want and the freedom to propose any rules I want, you have given me more freedom than I need to explain everything which is known (since "everything which is known" is finite). I can construct a very simple universe based on that simple rule F=0 which is quite interesting. That is, if one is interested in thinking.
Have fun -- Dick