Quoting Nicholas Thompson circa 09-12-29 05:02 PM:
> In any discussion such as this one, lest the discussion just spin out of
> control (which gives everybody a giddy sense of whizzing around but
> eventually gets nowhere) we have to understand which definition of
> ambiguity we are working with.
>
> I suggested that we work with Byers's. The is nothing coarse about Byers
> ambiguity. To be ambiguous in Byers sense, a situation must include
> two well articulated ideas that are mutually antogonistic but bound
> together in the same well articulated system of thought.
>
> To have achieved Byers-ambiguity is to have clarified a lot.
Under Byers' use of the term ("a single situation that is perceived in
two mutually incompatible frames of reference"), ambiguity can be
thought of as a METHOD. Byers is talking about ambiguity as an
attribute of a function/process that takes a single input and produces
multiple outputs. The perception/interpretation according to distinct
and mutually incompatible frames of reference is just an elaborate way
to say that there are details about the evaluation that are obscure, as
in the case of the square root function and the sign of the result.
When the same situation can be evaluated to two distinct results, it
helps to do as Lee suggests and formulate the ambiguity as an explicit
part of a larger evaluation. For example, if the square root of 4 can
be 2 or -2, we need some larger unifying context within which to
reconcile the two answers. I.e. There is some data missing. If we had
the extra data, we would know whether the √4 evaluates to +2 or -2. For
example, perhaps the "4" represents the height reached when you throw a
baseball up in the air. The answers +2 vs. -2 then mean either the
place you stood when you threw it or the place it landed. Obviously,
where you're standing and where the ball lands are mutually incompatible
answers to the same (ambiguous) question. But they fit into a larger,
unifying whole, which is what Byers' talks about on those pages
surrounding his definition. Add more attributes to the evaluation and
the ambiguity disappears.
Hence, as I said, ambiguity is used as a method for refining (from
coarse to fine) a question. If you prefer Lee's "an ambiguity", then
you can say it as: The explicit expression of an ambiguity is a method
for refining a question. It shows us exactly why, where, and when we
need new data.
For reference, here's what I said before... to help those who don't read
carefully. [grin]
>> [Original Message]
>> From: glen e. p. ropella <[email protected]>
>> [...]
>> 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.
--
glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com
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