Nick -
...
It offers a picture of a three dimensional structure as a model for
goings-on in an N dimensional space. Not at all clear to me that the
intuitions drawn from a three dimensional model have any use at all in
n-dimensional space.
Reread Edwin Abbot Abbot's "Flatland: a Romance in Many Dimensions" ?
There are qualitatively new properties that appear in higher dimensional
space which in fact are hard to think about in lower dimensional
spaces. Very specifically, 0-D space has no "room" for distinct
objects... go to 1-D and you can now have objects which are located
uniquely along the "number line"... go to 2-D space and said objects
can now have relations with eachother (connections as in a graph or
network) other than those "adjacent" along the number line.... in 3-D
you find that you can make those same *connections* arbitrarily without
having "edge" crossings (e.g. a road network requires over/underpasses
to avoid crossings while in principle the flight paths of aircars do not).
In the case of an N-Dimensional manifold and 2D surfaces embedded in a
3D space... the idea of a "basin of attraction" is intuitive if we use
the landscape metaphor to think about it. In a hydrological landscape
(watershed) we have the concept of "drainage basins" which are fairly
easy for people to apprehend but invoke all kinds of other thoughts
which are *not* necessarily relevant to the problem at hand. For
example, there is not really a concept of "flow" within the basin, nor
is there one of "erosion" *of* the basin, nor is there an idea of
"filling" (like a lake) which is apt to the problem.
I have always been persuaded that a model that requires a model to
make it intelligible is no model at all. I mean, either a model is
sufficient to bring a phenomenon within the range of some set of
useful intuitions, or it is of no value.
In the above example, the 2D surface in a 3D model with 2D bounded
regions is a valuable *model* of the mathematical abstraction
involved. We *add* the landscape metaphor to it to make it more
usefully familiar. If we see the "surface" as a complex of
"watersheds", it is perhaps a quicker if not more accurate way to
explain the situation.
As usual, our language can help or hinder our understanding. In this
case, what we mean by "model" and how that relates to "metaphor". I
usually think of *mathematical* models, I suspect you think of
*conceptual* models and I'm not sure how you use *metaphor* in this
case, perhaps you don't if you are thinking strictly in the sense of a
literary metaphor. I use metaphor specifically to be a complex analogy
between one domain (target) and another (source). Both domains are
ultimately "models" in the sense that the map is *never* the
territory. Ideally, the target domain is a very simple abstraction of
the territory in question. In our example above... the "territory" is
the socioeconomic status of populations and the "map" is a set of points
embedded in the parameter space (age, race, gender, income, education,
....) along with an Evolution Function, or essentially the "local" rules
(in time) for how an individual "moves" through that space. For
example, individuals educational level is a monitonically increasing
function with time while their income and assets may trend that way but
are NOT strictly monotonic (take a cut in pay, spend savings, etc.).
To *then* translate that geometric description into a more familiar one
(watershed), adds a level of familiarity to anyone with limited
experience with such geometric spaces but at the same time, it adds
potentially unwanted/irrelevant/distracting properties to the
understanding/discussion.
So... said simply, I think we "layer" models (both mathematical and
conceptual) all the time for various reasons, but when we actually shift
to *metaphorical* descriptions to make them more intuitively accessible
(especially to laypersons) we also risk *mis*understandings.
I too, look forward to other folks weighing in from other
perspectives. I believe that our "common understanding" of such
problems as gender/race inequalities tends to be too "simple" which
might explain why progress in the domain is both slow and somewhat
herky-jerky.
- Steve
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