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I Really Didn't Intend To Upset You.

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Posted by Richard D. Stafford, Ph.D. on January 6, 2005 01:12:27 UTC

Hi Wanda,

Please don't take offense, I have no intention of impugning your intelligence and, if you got that impression I apologize sincerely. You are merely failing to comprehend what I am doing. If you stick with me, I think the light will eventually come on. I also apologize for my failure to make things clear to you; I am trying to do my best.

There are a number of issues evident in this latest note which need to be addressed. When you say "mapping" is another word for meaning, I think you are missing a subtle but important difference. When I say mapping, I mean nothing more than straight away mapping; however, in most cases, a specific mapping, particularly in association with a particular explanation is ripe with meanings (such as the examples you tend to give). These meanings are in reality statements of relationships fundamental to the explanation. It is very difficult for most people to comprehend the great number of relationships that are already presumed when they begin to attach meaning to the elements being mapped. Go look again at the web page:

http://www.omniglot.com/writing/lineara.htm

Under the comment, "There is no [consensus] on how to transliterate the Linear A symbols - the method shown below is one possible transliteration" there is a chart. Note that there is little real "meaning" embedded in that chart. "Parsing" the information and attaching convenient symbols to those parsed elements is the first step in organizing information for analysis (if you don't like numbers, think in terms of breaking a secret code). Commonly, when people are trying to comprehend things, meanings are presumed at the very moment those symbols are attached. What I am doing requires a careful and detailed approach so I regard these issues as very separate and not to be taken lightly.

As PSI is a symbol used to stand for the algorithm to generate your expectations, I would tend to think of "meaning" as the connector between the specific mapping and the specific algorithm. In some ways I see it as more attached to PSI than to the mapping; in many ways it serves as a constraint on PSI. And I would have to agree that, since the mapping is arbitrary the associated PSI would also be in the same sense arbitrary; however, these are certainly not independent entities. They are part and parcel of the same explanation; remember, I have defined an explanation as consisting of those two essential parts, the references and the algorithm.

And, exactly what was it that the specific explanation under discussion (the one attached to that particular specific mapping) was designed to explain? It should be clear that it was designed to explain C! That is, under the specific mapping given, the algorithm is a method of determining the specific expectation for the B's which go to make up C. Everything above is entirely consistent with the idea that PSI is very much dependent upon C. One very serious question can be asked: does an explanation (i.e., a method of determining those specific expectation for the B's) exist for every possible mapping. The answer is yes and and the proof is actually quite straight forward.

Since every B is constructed from a finite collection of elements and C consists of a finite collection of B's, the fact that the expectations of specific B's in C is well determined should be obvious. Just take the mapping you have been given and count the number of occurrences of each B in C. The expectation of a given B is just the number of times it occurs divided by the total number of B's in C. If the proposed B does not occur in C, then the expectation it will occur would be zero. Now that may not be the most useful explanation or the simplest explanation (by simple I mean the least amount of work) but, by my definition of an explanation, it qualifies and it is, by construction, perfectly consistent with C. It merely says "what happened was what happened and your expectations should be more of the same". Not the greatest explanation in the world but an explanation none the less and one frequently used throughout history.

With regard to your complaint that I say C has nothing to do with the mapping while, at the same time saying that PSI depends on the mapping and yet must yield exactly the correct probabilities for the specific B's making up C, I need to point out the following: PSI depends on the mapping in the sense that the arguments of PSI are directly dependent on that exact mapping (in fact, they are the arguments); change the mapping and the set of arguments corresponding to a specific B in C may change violently. However, if the algorithm is a valid explanation of C under that mapping, it must give exactly the same probability for that particular B in C , even if the appearance of B under the new mapping bears no resemblance to B when represented by the previous mapping. As a function of B, PSI is determined, but as a function of the element mapping which are used to define B, the appearance of PSI can be quite different.

Perhaps if we go back to that philosophic problem of defending induction you will gain some insight into what I am doing. Your entire mental image of the universe you find yourself in is the consequence of a vast inductive analysis far beyond detailed examination. One can accept induction (that the mappings we use are the only possible mappings) as a valid approach, and thus accept without question our current mental image of the universe or one can drop induction as a valid method of analysis and perhaps see things not otherwise conceivable but if we are going to do that we must be very careful of our definitions, making sure we are not relying on inductive results.

Thanks for sticking around Dick

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