Hi Tim,
My wife and I are headed off for a week and there is no Internet access available where we are going so I hope my comments will be sufficient for the time being. I think you may be trying to put more in the section you refer to than is there.
Actually, what I am saying is quite simple. First, I am working with some information. Since I have no idea what the symbols, concepts, words, messages or whatever actually mean, I am simply going to label the underlying elements with numbers. Thus it is that when I examine the information which comes to my conscious attention, I am examining a set of numbers (this is no more than an abstract representation of any message). In my paper, I have defined an observation to be a particular set.
So essentially, what I have to analyze is a sequence of sets of numbers. I label each observation with the parameter t. Once I have these observations, they do not change; they "are" what I know: i.e., the information I have to work with. You can picture this as a humongous collection of lists of numbers.
What I do, is look at all this information and see if I can deduce some rules which will tell me something about how these numbers are related. My first step is to look at the collection of lists of numbers (tack them up on the wall of my infinite office in order of that parameter t (which I will just call time). Now, if you give me a particular number for t, I either have a list with that parameter attached to it or I do not. If I don't, I will just say "well I don't have that observation" but I may have one on each side of it (if I have any observations at all, I have one on one side of it at least).
At any rate, for lots of values of t, I have a list of numbers. That list is called a "function" of the parameter t. People often think of a function as something like sin of x or x squared, but a tabular function is also a function. You tell me t (one I have) and I will give you the list; "the list is a function of t.
Now, if all these lists are different, and I were given a particular list (one which was in my collection of observations) I could go look at that collection of lists and find the one which was the same as the one you gave me (that means the parameter t is a function of list numbers). For that reason, I add "unknowable data" to my lists so as to make all the lists on the wall of my infinite office different. One could then say that the time is an implicit function of the list (of course, when you give me a particular list, what you give me must include the values for the "unknowable data")
Now, if those lists were produced by a rule! And I were God and knew the rule, I could just give you the list for any particular time you wanted. But of course, I don't know what that rule might be; however, since the number of lists (and the number of numbers on a list) are finite, it turns out that there are an infinite number of possible "rules" which will reproduce the lists we have already obtained exactly. (Try a power series fit to the data.) The problem is, there is no way to know which one is right (no matter which one we pick, the next new observation will probably invalidate it). What is important here is that there always exists "a rule" which will exactly produce everything we know (it may not be simple but it always exists).
So, I then say, "Let me add additional unknowable data such that every possible observation consists of a unique pattern even after any arbitrary element is removed from that observation." If I knew the "correct" rule for that set of lists, I could produce the entire set of lists. Since every possible observation consists of a unique pattern even after any arbitrary element is removed, there would only be one list which would match the one you gave me (even though you had removed a number). All I would have to do is find that list and match the numbers you gave me and the remaining number in the list would have to be the one you didn't give me.
This means that the value of the removed element may be determined via the rule and the remainder of the pattern: the removed element is a "function" of the given list of n-1 numbers. This implies that knowing the correct "rule" would allow us to write down the function defined in 1.1 as a tabular function. Equation 1.2 is nothing more than a rearrangement of terms. Thus it is that one can conclude that the "rule", no matter what the rule may be, can be written in the form shown as equation 1.2.
I hope that makes my reasoning clear.
Have fun - Dick |