Hi Anssi,
Your posts are delightful. You make it quite clear that you think deeply about what I say; something I wish some of the other people reading this forum would do. (Who knows, maybe there are others who have a grasp of what I am saying, If they are out there, I wish they would comment.)
AnssiH said:
Yeah, so this step of the indexing process doesn't imply any specific ontology either?
Of course not. You cannot have an epistemological solution to any problem without an ontology to build that solution on. And, you certainly cannot explain that solution to anyone without communicating the required ontology; so, if we can find a valid epistemological solution, we can certainly refer to the required ontological elements. That is what language is all about: mere symbolic representation of concepts we feel are important so we can communicate those thoughts with others.
An Aside: (you can skip this if you want!) There is a short column in the April 14, 2007 issue of "Science News" ("Rats take fast route to remembering") where the authors say,
Prior studies, which have focused on task learning unrelated to preexisting knowledge, indicate that a brain region called the hippocampus incorporates new facts and events into memory. The hippocampus gradually yields to another structure, the neocortex, as new memories become stronger. [And correlated into preexisting knowledge.][/color] This process typically takes at least 1 month in rodents and a few years in people.
The blue comment is mine. As I told my wife, that sort of means rats are pretty smart. I guess we should be thankful their life span is short and they haven't come up with language yet or they would take the world over!
But really, I think the difference might very well be that the rats are hard wired for specific types of memories and don't waste any time trying to think of alternate explanations whereas the essence of human success is that they spend a lot of time (as a species, not as individuals) considering alternate possibilities before new information is correlated into preexisting knowledge. Of course I could be wrong.
Just a comment on the importance of learning a language.
AnssiH said:
Yeah, and so one example of assigning the same label to a number of fundamental elements would be when one ends up defining that many electrons exist at a specific "present"?
Yes, exactly. Another good example would be that family tree of the primates I brought up. How would you show multiple entries for the same species? You already use horizontal displacement to indicate different species and vertical displacement to indicate time and you would have to include another axis if you wanted to show the time change in populations.
AnssiH said:
Hmmm, I think at this point it would be helpful for me if you could explain how you end up manipulating this representation for some useful end. I think it might clear up some things that might be little bit blurry to me right now.
The useful end is the organization of your thoughts and that organization yields results almost beyond belief. That is exactly where I want to lead you.
AnssiH said:
i.e. if it cannot explain your past?
If any explanation turns out to be counter to my past (i.e., inconsistent with what I know to have happened beyond doubt) I certainly wouldn't accept it as valid. Would you?
As far as "a useful end" is concerned, we need an exact definition of "an explanation" (otherwise, we don't know how to go about explaining things). That is why I defined an explanation to be a method of obtaining one's expectations from known information.
Under that definition, the structure of the "what is", is "what is"[/color] explanation is quite simple in that it is no more than a table of "undefined ontological elements" going to make up every discrete present going to make up that "past" which constitutes "what one thinks one knows". Since "what one thinks one knows" is undefined we can represent each element with a number. One's expectation are no more than a "true/false" decision on any given present. In the "what is", is "what is"[/color] explanation, the method is no more than "look in the table". If a particular list is in the table the answer to your expectations is, "true". If it is not there, the answer is false.
If we could really contain, in our minds, a complete collection of all "presents" going to make up our past, then that might be a useful view but that feat is somewhat beyond our mental capabilities. What we would really like is a procedure (think of it as a fundamental rule) which would accomplish that result for a any single ontological element. In such a case, we need comprehend only that element in our logic, taking the rest as "understood": i.e., as established by that rule. So I will show you a way of accomplishing such a result by including intentionally invalid ontological elements, an extremely powerful procedure. After all, if you can't prove that your explanations of reality include no "invalid ontological elements" how can you constrain me to a presentation which excludes such things? Particularly if I explicitly declare these additions to be "invalid".
The first "invalid ontological elements" I would like to add, is a very simple set. As defined, all real presents consist of specific changes in my knowledge of valid ontological elements. I have already eluded to the fact that I am using numerical labels because I can then talk about that "method of obtaining one's expectations" as a mathematical function. The "true/false" can be seen as a "one/zero" dichotomy and I am using numerical labels for that "known knowledge" (those specific "valid ontological elements" which constitute the "reality" of any given "present") so the method is a mathematical function: i.e., it transforms one set of numbers into a second set (you give me a set of numbers which could possibly be a real "present" and that "mathematical function" returns either a one or a zero (depending upon whether or not that collection of numbers is in that table of my "what is", is "what is"[/color] explanation.
But this is a very strange "mathematical function". The number of arguments for any particular "real" present is neither fixed or known.
In order to simplify the situation (given that I have a specific epistemological solution to represent), I will simply add a sufficient number of "invalid ontological elements" (additional numbers) to each "valid present" until all cases have exactly the same number of arguments. Now you have to understand that, after I add these "invalid ontological elements" there need be no method within my finished explanation (where I am going) to tell the difference between the valid and invalid elements; in fact there cannot be such a method for if there were, it would constitute a flaw in the epistemological solution (invalidating that ontological element). Notice that the numbers I have added to the collection are totally arbitrary; counter to the valid ontological elements which are immutable. (This fact will become extremely important down the line a ways.)
So, after that agumentation, it is not a very strange function at all, it has a clear set of arguments (that total number of ontological elements that flaw free epistemological solution presumes makes up all presents, some of which are valid and some invalid). My flaw free epistemological construct must yield a one or zero for each and every such possible collection.
At this point, I would like to add a second set of invalid ontological elements. Again, I add these elements for my own convenience as they will make that explanation I am looking for (that mathematical function which constitutes the "fundamental rule") simpler. As that mathematical function (which is a direct explicit expression of our explanation) now stands (per what I have laid out above) there could exist identical "presents". That issue is the source of some conceptual difficulties. All of my presents are supposed to have a unique index on them and that unique index can not be established by my proposed epistemological solution unless the value of that index is embedded in the collection of presents themselves. If two presents are identical, the index can not be embedded in the collection: i.e., no epistemological solution based on that collection of ontological elements can yield a different index for those two "presents".
The solution to this difficulty is very simple. All one need do is find all identical entries in that table of our "what is", is "what is"[/color] explanation (where we have already added the entries which made all presents have the same number of arguments). We can now add another entry (just another invalid ontological element) to every present, making sure that the entry is different in every case where the earlier table had identical entries. Now every "present" going to make up our "what is", is "what is"[/color] explanation is an identifiably different case. This provides us a direct procedure for obtaining that embedded index. You give me any hypothetical entry for that table and I can examine the table and tell you not only if it is a member (give you the true/false answer) but I can also give you the "t" index for every true case. By the way, I am not suggesting this as a reasonable way of explaining reality, I am simply saying that it must work as the collection of table entries is finite so the job can be completed.
So, let's extend this idea of adding invalid ontological elements to simplify the problem one more very subtle step. Let us make a new table consisting of a list of all entries in the table we now have but omitting one number (that's one of those reference labels) from each specific present. To make what I am proposing very clear: if every present in the current list (that is both the additions above have been done) consists of n numbers, this new table will have n entries for every specific "t" index: each one being the entry for the "t" present with a different specific numerical reference removed. We can call this subsidiary table, "table number two".
Again, after removing one number, we introduce the possibility that this second table will have some identical entries. We can once again get rid of identical entries by adding more "invalid ontological elements" (using the same method described above) until table number two consists of totally different entries (please note that, since nothing has been said about order in those arguments, the same set of numbers listed in a different order will be considered to be identical lists). This step may be quite extensive but it is nonetheless finite and can thus be accomplished.
Now this augmented table number two can also be seen as a tabular representation of a function (which I will call function number two). A function which yields a one/zero result for each of all possible collections of arguments (including that "t" index): one for "true" (that set of numbers is in the table or) zero for false (that set of numbers is not in the table). These two tables (the table yielding probabilities and table number two), taken together provide the definition of a new function with a very interesting property. Given that the original table upon which table number two is based (that primary table being augmented with those new "invalid ontological elements") has n entries; given any possible set of (n-1) arguments, one can find first if that set is an entry in table number two (in which case there is either one or zero entries). Since that entry includes the "t" index, the associated entry in the primary table can be examined. That entry will have exactly the same arguments as the set which was given plus one more additional argument: the entry which was removed to create table number two.
What I have just described is a method of finding the missing number given all the labels except the missing label. That means that, if I have a flaw free epistemological solution to this uniquely augmented "what is", is "what is"[/color] explanation, there must exist a mathematical function of all but one argument which will yield the missing argument (I have just explained how to construct such a table). Now, it may be true that I only have given the mechanism for constructing a table of the results which corresponds to my presumed past (what I think I know, including those invalid ontological element) but it should be clear to you that the procedure must also yield all of the known "valid ontological elements". What I have just proved is that, if I have a flaw free epistemological solution, I can use that solution to build a tabular function which will yield the missing argument for every valid set of arguments where one argument is missing. That function can be written as
x_n(t) = f(x_1, x_2, x_3, \cdots, x_{n-1}, t)
or,
F(x_1, x_2, x_3, \cdots, x_n, t) = x_n(t) - f(x_1, x_2, x_3, \cdots, x_{n-1}, t) = 0.
Note that, since order of arguments is of no significance, x sub n can be any element in the set. To clarify what I have just proved: Given a flaw free epistemological construct based on the collection of valid ontological elements plus a designed set of invalid ontological elements, there always exists a function (which I will refer to as the function F(
B(t) ) of those numerical labels which will yield exactly that "what is", is "what is"[/color] table under the very simple rule, F=0. Likewise, given that table, there exists a function (which I will call P(
B, t) ) which yields the probability the collection of arguments
B exist in the particular present indexed by t: i.e., that function will yield either one or zero to indicate that
B(t) is or is not an entry on the table.
Now, not only must such a functions exist, but anyone with a little mathematics training must realize that an infinite set of functions satisfying that constraint exists for every possible set of valid ontological elements. These numbers constitute a finite set of points in that (x, tau, t) space and there are an infinite number of functions which will fit that set of points exactly so no constraint whatsoever has actually been placed on the future (which is, by definition, what I do not know). In other words, there exists no epistemological solution based upon any set of valid ontological elements which can not be expressed by a specific P(
B[/b), t) under the simple rule that the only constraint on the numerical references is that they satisfy a relationship which can be written: F( B(t) ) = zero.
The only difference between this mathematical representation and the specific explanation it represents is the fact that I have added one hell of a lot of "invalid ontological elements": i.e., an epistemological construct invented by a theorist could possibly contain fewer "invalid ontological elements" but it certainly could not depend on a simpler rule ( F=0 is a pretty simple rule).
I think I have given you enough to think about for the moment. Check out what I have said carefully and if you find any part of it confusing, I will do my best to clear things up. When this all makes sense to you, I will take you to the next step. Let's see if you can get your head around the above exposition.
Have fun -- Dick