Re: Re: Whole and Parts (and boundaries)
Murray,
In any ontological study, purpose is the most important
issue to be considered: Why are you trying to make one
distinction rather than another?
MA> For my needs, I've considered Barry's mereotopological
> writings as quite functional, and in the context of use
> this is (to me) at least as or even more important than
> the mathematical foundation. But I'm still working out
> the details of a mereological/topological ontology
> in my head.
To clarify any such issue, I would ask what is the
purpose of the ontology, and how would an answer affect
what you do. What, exactly, are your needs?
Boundaries in mathematics are of one dimension less
than the area (or volume) they bound. (I was using
the term "infinitely thin" as an informal way of
saying that.)
But classical mathematical forms, such as circles,
spheres, and even toruses (e.g., idealized coffee
cups) do *not* exist in the physical world. However,
for some purposes, such as computing areas and volumes,
an idealized form might be a useful approximation.
But if boundaries are really important, I can't imagine
any application in which their thickness is irrelevant.
In fluid mechanics, physicists and engineers talk about
"boundary layers", where the fluid speed varies from
supersonic to zero in a very thin, but definitely finite
layer, whose thickness is very important and very
definitely measurable.
When you get to the microscopic layer (not even the atomic
or subatomic level), nothing is as clear or simple as it
might appear to the naked eye. Boundaries for so-called
"smooth" objects are very unclear, messy jungles.
And even for the most highly polished diamond surfaces,
the atomic level is definitely *not* sharp. There are lots
of ripples with air molecules adhering to the carbon and
even occasional carbon molecules bouncing off. For
anything less sharp, the boundaries at the atomic level
are much thicker and much more chaotic.
And at the subatomic level, the wave functions for every
particle are continuous functions that are significantly
larger than zero at distances much larger than the quoted
diameter of the particle.
For practical applications (i.e., where even a good
magnifying glass is not being used), boundaries are
"thick", poorly defined, and definitely not even
close approximations to ideal geometrical forms.
Barry Smith made the distinction between "natural" and
"fiat" boundaries. That is a traditional distinction,
but it is always a very practical one in which nobody
depends on any such boundary being infinitely thin.
Typical "natural boundaries" are things like rivers
and oceans as boundaries of countries instead of
geometrical lines on a map. But if you look at actual
rivers and oceans, the boundaries are *all* by fiat:
1. On any detailed map, you'll see that the political
entities on both sides have agreed to "fiat"
boundaries running down the middle or to the side
of the so-called natural boundary.
2. And if you ask what makes a river a "natural" boundary
and a brook or stream "unnatural", the answer is
simple: a river is more difficult for an army to
cross, but a brook or stream can be forded. In other
words, the so-called "natural boundary" is more
precisely an obstacle to a very specific kind of
activity: armies moving to attack or defend. And
purpose is the most characteristic triadic relation:
some agent A has chosen B as a boundary for purpose C.
3. And if you look at real rivers, such as the Mississippi,
you find that they go through "flood plains" because
it is normal for rivers to "meander" (a verb derived
from the eponymous river). And where the river does
become a relatively sharp boundary, it is because some
humans built a concrete channel to force the so-called
natural boundary into a fiat boundary.
4. Similar issues apply to oceans, which rise and fall
with every tide or create new sandbars and wash away
old beaches with every storm. None of these boundaries
are sharp.
I would honestly like to see any practical application of
the distinction between a natural and a fiat boundary that
actually uses any of the formal machinery. I can certainly
see practical uses that depend on purpose. But any real
application that depends on boundaries being sharp (such as
precisely machined parts) always allows a finite tolerance.
John