ONT Re: Extension x Comprehension = Information
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I see that I am using a few bits of language about factors and fibers
that I have not mentioned in quite a while, and not only that but I am
using it in ways that I learned at different times from different folks,
and so I am using it slightly equivocally. So let me try to straighten
that out before going on with the story of Bartleby, his preferentials,
and the apothematic dimensionals of his human, all too human humanity.
Let's say that we have an ordinary function f : X -> Y.
If we pick out one y from the target or the codomain Y,
then it either has x's from the source or the domain X
that are assigned, or mapped, or sent to it by f or it
doesn't. Let's say it does. Then here is the picture
of what I will frequently call the "fiber (of f) at y":
o-----------------------------o-----------------------------o
| Source Domain | Target Codomain |
o-----------------------------o-----------------------------o
| |
| x |
| x · |
| x · · |
| x · · · |
| x · · · · |
| x · · · · y |
| x · · · · |
| x · · · |
| x · · |
| x · |
| x |
| |
| |
| Functional Fiber |
o-----------------------------------------------------------o
Very often the reason that one is interested in these varieties
of fibers under a given function is so that one can follow them
"upstream" or "backward", functionally speaking, in other words,
toward the "source" of the functional value under investigation.
That leads rather naturally to the other mathematical usage for
the word "fiber" that I have in mind. Here are the definitions
as I formulated them in my dissertation:
| The "fiber" of a codomain element y in Y
| under a function f : X -> Y is the subset
| of the domain X that is mapped onto y,
| in short, it is f^(-1)(y) c X.
|
| In other language that is often used, the fiber of y under f is
| called the "antecedent set", the "inverse image", the "level set",
| or the "pre-image" of y under f. All of these equivalent concepts
| are defined as follows:
|
| Fiber of y under f = f^(-1)(y) = {x in X : f(x) = y}.
|
| In the special case where f is the indicator function f_Q of the set Q c X,
| the fiber of %1% under indicator function f_Q is just the set Q back again:
|
| Fiber of %1% under f_Q
|
| = (f_Q)^(-1)(%1%)
|
| = {x in X : f_Q (x) = %1%}
|
| = Q.
|
| In this specifically boolean setting, as in the more generally logical
| context, where "truth" under any name is especially valued, it is worth
| devoting a specialized notation to the "fiber of truth" in a proposition,
| to mark the set that it indicates with a particular ease and explicitness.
| For this purpose, I introduce the use of "fiber bars" or "ground signs",
| written as "[| ... |]" around a sentence, or the sign of a proposition,
| and whose application is defined as follows:
|
| If f : X -> %B%,
|
| then [| f |] = f^(-1)(%1%) = {x in X : f(x) = %1%}.
|
| The definition of a fiber, in either the general or the boolean case, is
| a purely nominal convenience for referring to the antecedent subset, the
| inverse image under a function, or the pre-image of a functional value.
| The definition of an operator on propositions, signified by framing the
| signs of propositions with fiber bars or ground signs, remains a purely
| notational device, and yet the notion of a fiber in a logical context
| serves to raise an interesting point. By way of illustration, it is
| legitimate to rewrite the above definition in the following form:
|
| If f : X -> %B%,
|
| then [| f |] = f^(-1)(%1%) = {x in X : f(x)}.
|
| The set-builder frame "{x in X : ... }" requires a sentence to
| fill in the blank, as with the sentence "f(x) = %1%" that serves
| to fill the frame in the initial definition of a logical fiber.
| And what is a sentence but the expression of a proposition, in
| other words, the name of an indicator function? As it happens,
| the sign "f(x)" and the sentence "f(x) = %1%" represent the very
| same value to this context, for all x in X, that is, they are equal
| in their truth or falsity to any reasonable interpreter of signs or
| sentences in this context, and so either one of them can be tendered
| for the other, in effect, exchanged for the other, within this frame.
|
| http://suo.ieee.org/email/msg07409.html
| http://suo.ieee.org/email/msg07416.html
Jon Awbrey
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