Hi Dr. Dick:
"On that first occasion, I add "unknowable data" sufficient to make all observations unique. I do this purely for mathematical simplification. As I said, everything I know I obtain from a finite number of observations. Since every observation consists of a finite set of numbers and everything I know consists of a finite set of observations, those observations can be ordered (that is basic mathematics: numbers can be attached to each one). Since, in my analysis, I want to order them, I must include that information in my analysis. Is there any easier way then to attach that additional data to the sets? If I don't make them unique, how do I know where to put them in the series? On what is the order based?"
You effectively are saying that you are going to want to put your sets in a specific order.
So you want to identify which one goes where. So you name them.
The "name" is a figment of your imagination. You could name them "John", "Paul", "George" and "Ringo" ?
You attach your names and call it "attaching unknown data". As your model is general in intent, you leave the actual "names" used as "unknown". What counts is that your sets have unknown names attached to them.
You consider the names "bogus", because "there is absolutely no evidence" that these names are part of reality!
Quote: "To add to my model the possibility of significant information which is not knowable via any direct means does not constrain the ability of that model to represent reality. "
At first I thought: "but how can arbitrary nam-calling your sets be "significant information"?"
Then I realised: the names themselves are not significant; but the "fact of attaching unique identifying tags to each set" IS significant.
Am I correct in understanding that here "the significant information" is the act of tagging the sets, not the tag used?
Quote: "If I go a little further and add sufficient "unknowable data" to make every observation unique even when one piece of data is removed, I have created a very strange situation."
So like adding a second name-tag, a second unique-identifier, to each set?
Quote: "The situation is strange because, if I knew the rule which constrains the observations (controls what data may be seen)..."
A rule that controls what data may be seen....
and if you knew this rule....
" and were given all the numbers but one.."
so you know the rule to control the numbers, you know all but one of the numbers...hmmmm...that will not tell you what number is missing but...
"I would know exactly what observation I was looking at"
I wonder why? Since you gave each set two independently unique name-tags; your rule has given you either one name-tag plus all the non-bogus data in the set; but left you minus the missing name-tag (the missing unknown-data item)
the rule has given you two-unique-name-tags plus not quite all the non-bogus data; and left you minus one piece of non-bogus data.
So: the rule controls the real data; you know the rule; so any bogus-data in your rule-given set will stand out as bogus (out-of-control? Free of constraint?).
Case (a):If there is one piece of bogus data in your set after subtract an item; your rule showed it up; so you know which set it is as the bogus item is one of the sets' two names you gave it.
Case (b): If there are two pieces of bogus data in your set; your rule shows them up as a pair. Since your second exercise in naming your sets still made them all unique; any pair of names is a unique pair. So you still know what set you are looking at.
Quote: ..."and only one possibility would exist for that missing number. That means that for every set of observations (missing one number) there would exist only one possibility for the missing number."
Why only one possibility for the missing number?
In case (a): you know the missing number is the other name you gave the set. But do you know which of the two names the missing number is? Which of the two unknown-data attachments?
In case (b): You know the missing number is the one that wasn't delivered by the rule. If the rule really delivers all of them, I guess you know the missing number by default.
Problem?: If your "unknown data" name-tag wasn't unique, it might have got muddled with an identical real-data in the set. But you specified it was unique.
Quote: "That means that for every set of observations (missing one number) there would exist only one possibility for the missing number."
Namely: the missing number is EITHER:
the other of a pair of "unknown-data" names you attached to the set;
OR; it is the number missing from all the numbers the rule gave.
(On re-reading, something bugs me. Aurino, Paul: is the logic sound? Seems fishy somehow. I know for instance, that in reality: a plural means contents are "already unique". If things were not different in at least one way, they would be "it", you couldn't use a plural word. And "order" seems to be ingrained: they must have some starting order.)
Equation 1.3 means.........?
And I'm thinking:
My explanation looked so different, but it mapped equations 1.1 and 1.2. So it must be possible to show how it transforms into your method.
You are dealing with categories here: because a category has a "naming function" when it intersects another category! In fact, it may be regarded as "naming it" with unknown data? (If all you know is that two categories, two words, contain each other; each has been named by the other.)
"Telephone" names "communication". But exactly where do these intersect? The naming is partial.
The choice of categories can remain "unknown:; what counts is that they named each other; they identified each other at the point of crossing.
We make the crossing unique by specifying that it is unique?
I think it will turn out that your little inter-change game with "rule-constrained data" and two unique set names (two unknown data) maps what I call "musical chairs".
The two names can be swapped: musical chairs game 1.
The rule-constraint can give all the real data, or all real data minus one item of real data:
musical chairs game 2.
The missing item could be one of the two unique set-names (one of the attached unknown data), or it could be a piece of real data: musical chairs game 3!
Is not one of these games the difference between the other two?
And of the other two, does not one of them make a connection between the "difference game" and the other game (so it joins the dots)?
Is there not these three games encoded here?
I follow your method: can you see mine?
I've only responded to the last technical part of your post here; but O.K. technically to tackle eqation 1.3? What does it mean?
Thank you for clarifying muddied water.