Let us look carefully at this statement:
"In math, we don't care what the 'set of elements' is, it is just treated as an abstract object and the elements could be anything you wish to consider as a 'set'. However, in the application of mathematics (e.g., to philosophical issues), you have to tell us what you want the "elements of set A" to apply some kind of meaning. So, for example, if you want the "elements of set A" to refer to all the different animals at Seattle's zoo, then I know that you where you want to apply your math."
Harv, right here you have completely omitted a very important issue. Exactly how am I supposed to tell you what I want the "elements of set A" to mean other by communicating with you? You are, at this point, presuming a great portion of the problem before you is already solved: i.e., you are presuming that you and I have exactly the same meanings associated with the vocabulary to be used in that communications from whence we affirm our understanding of the problem under discussion.
Now it may very well be true that we do have identical meanings associated with the symbols we are using: however, unless you can prove no other interpretation of any symbol used is possible, you must maintain a representation capable of yielding to corrections of all possible misunderstandings. I am constructing a model of an explanation here, not explaining anything (other than the model itself that is). The model must be capable of modeling any explanation including any and all information required to make our communications clear,
It should be eminently clear to you that all vocabulary necessary to present the problem is part of the statement of the problem. That is why an education in French is required to understand a statement of a problem in French. Or, an education in Mathematics is required to understand the statement of a problem in mathematics.
So, instead of presuming the meanings of all these peripheral symbols are clearly understood, I merely include the messages used to clarify those meanings as more elements in A. That is exactly why I state that any explanation must include [b]"A set of reference labels for the elements of A (so that we may know and discuss what we are dealing with)."[/b]
With regard to your second statement, "My understanding is that the elements of A could be anything (e.g., zoo animals) and that has nothing to do with an explanation. Why do you want your math applied to connect these apparently 2 totally different concepts?"
I thought I had made that clear; however, you apparently missed the connection. The concept of an explanation certainly seems to me to be lacking something if there is nothing to be explained! A is what is to be explained!
If I may quote another on the subject of modeling explanation expressing their position that a model of "explanation" already exists:
"Initial data, information, pattern, initial Model (or paradigm), more data, information, pattern, change of Model (or paradigm), back to more data."
And my answer to him:
Yes; but your model is, and has been, the standard model of explanation for eons and has, after thousands of years of analysis, yielded no predictions in and of itself at all. Of exactly what value is a model if it yields no testable deductions?
My model is certainly superior to yours as mine at least puts forth some testable deductions. Now, if you can show that there exists a set which cannot be modeled by my procedure; or if you can show a specific error in my deduction (that would be my deduction of that fundamental formula) then you would have reason to prefer your ancient perspective to mine. Why don't you give it a little serious thought?
Have fun -- Dick