Quoting ERIC P. CHARLES circa 09-12-29 11:09 AM:
> Well, of course, all of this (Glen and Nick's posts) is ignoring the obvious
> fact that ambiguity is the antithesis of mathematics.
That's just not the case. Mathematics easily captures the concept of
ambiguity. Hence, ambiguity can't be the antithesis of math. Math is a
language. Languages describe. All languages can be reflective
(circular). In English, we can describe circularity with sentences
like: "This sentence is false." In math, we can describe circularity
with sentences like: "Let X = {a,b,X}." Ambiguity is, formally, just a
type of circularity. In generic circularity, the evaluation of some
predicates on X is undefined. In ambiguous cases, the evaluation of
those predicates simply have multiple values. I.e. the mapping is 1 to
many.
This isn't, in any interpretation, the antithesis of math. It is well
described by math, less well described by English.
> So, how do we
> reconcile claims that ambiguity is at the heart of mathematics with the
> obvious
> truth that mathematicians really like producing, teaching, and preaching about
> unambiguous things?
Because math is a _means_ not an _end_. Ambiguity is at the heart of
math because math is our attempt to disambiguate the ambiguous ... to
refine what is coarse ... to peek into the little nooks and crannies
created by our prior theorems.
> Also, re Glen's post specifically, I think there is value in discriminating
> between accidental and intentional ambiguity. Not all claims of ambiguity is
> are claims of ignorance, sometimes situations are actually ambiguous and
> therefore claims of ambiguity are claims of knowledge.
Again, I have to disagree. All claims of ambiguity are statements of
ignorance. Granted, we can whittle away at the ignorance and refine the
ambiguity to a very fine point (which is what Rosen does). But in the
end, ambiguity ... as Byer's and the rest of us use the term [grin] ...
means "We don't know what makes it take value X as opposed to value Y in
this evaluation."
A great example is the square root of 4. There are 2 answers: 2 and -2.
In order to know whether the answer is 2 or -2, we need more
information. I.e. we're too ignorant to answer that question.
> For an example of the latter, we
> might ask whether George W.'s "Free Speech Zones" were protecting people's
> freedom of speech. One possible answer to that question, one that expresses a
> good understanding of the situation, NOT severe ignorance, might be "In some
> ways it technically was, but in other ways it severely undermined freedom
> speech, so the situation is ambiguous." On a lighter note, many jokes an
> innuendo take advantage of ambiguity, and if you don't think the situation is
> ambiguous, you won't get it. For example, I once shot an elephant in my
> pajamas..... what he was doing in my bedroom I'll never know.
It's still a statement of ignorance, though perhaps not of _severe_
ignorance. As I said before, the use of the word "ambiguity" may well
be taken as a methodological point, not an ontological or
epistemological one. Whether "Free Speech Zone" evaluates as "protect"
xor "undermine" is definitely an expression of ignorance. Ultimately,
due to our lack of understanding of the social processes involved, we
can't state for sure whether it is one or the other. Further, we can't
even state that the two are disjoint! Perhaps one MUST undermine
freedom of speech in order to protect freedom of speech, in some
circumstances? The point is that we don't know. We are ignorant.
In the case of shooting an elephant in pajamas, again, it's a statement
of ignorance. We can resolve it by 1 question: "Was the elephant
wearing your pajamas?" Prior to the answer, it's ambiguous. Post
answer, it's not.
--
glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com
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