Fwd: Re: Boundaries and parts in the FMA
Barry Smith
Smith concerning the question: Are boundaries parts of the things
they bound (with specific reference to the Foundational Model of
Anatomy ontology)?
CR: Currently, the FMA asserts part-whole relations between A and B
only if A and B have the same dimension. A line has lines as its
parts, a surface has surfaces as its parts, and a 3D entity has 3D
entities as its parts. In other words, a boundary cannot be a part of
an entity that has a higher dimension than the boundary, and a part
cannot be a boundary of an entity that is of a lesser dimension than
the part. In addition the FMA asserts the following remainder
principle: If A has_part B, then there is a complement C which
together with B accounts for the whole (100%) of A.
BS: Boundaries are certainly not parts in the sense required by this
remainder axiom. But what you are describing here are special kinds
of parts -- material parts -- so that is not a problem. For parthood
relations between material entities the remainder principle does
indeed hold. Boundaries, however, are immaterial parts.
CR: Can you enlighten me about the different senses in which we can
consider parts? Do you imply that when viewing boundary in one of
these contexts, then boundary qualifies as part, but it does not
qualify when viewed in another context? What are the two or more
contexts and what distinguishes them?
BS: I think that boundaries are parts of the material things they
bound independently of context. I have three reasons for this.
1. to capture the asymmetry between
x boundary_of stomach wall
x boundary_of stomach cavity
Only in the former sense do we say that x is also part of the stomach wall.
CR: I agree that x as a boundary is a property inherent in the
stomach wall and not inherent in the cavity. But I do not follow why
this property must be thought of as having a part, and why it cannot
be considered as distinct from parthood. The asymmetry is entailed in
spaces not possessing the property of inherent boundary whereas
material objects do. If there weren't a stomach wall, the cavity
would be infinite.
BS: The second reason is:
2. To do justice to the fact that the relation between point and line
is the same as the relation between line and surface which is the
same as the relation between surface and 3D material object. For the
first two 'part_of' is the appropriate relation, therefore this is
appropriate for the last also
CR: I can appreciate that the relation between a surface and a 3D
material object is the same sort as those between points and a line,
and between lines and a surface, but I do not see why this relation
is the same sort as the one between a subsegment of a line and the
whole line, or a part of a surface and the whole surface, or one of
the 3D constituents of a material object and the whole object.
BS: 'Part' is a very general relation. There are consistent formal
theories of part which (provably) allow transitive inference here:
if the North Pole is part of the Northern Hemisphere and the Northern
Hemisphere is part of the Earth then the North Pole is part of the Earth
CR: You imply that the relation between point and line and line and
surface is part_of. But how come? A point has zero dimensions and a
line has one dimension. How can two or a hundred zero-dimensional
entities add up to [constitute] an entity of one dimension? They
could only do that if each point had at least some one dimensionality
to it. If they don't they cannot stack up to form a line.
BS: In fact the standard mathematical theory of lines and surfaces
precisely supposes that you CAN build n+1-dimensional things by
stacking up n-dimensional things. But fortunately my argument does
not depend on this weird idea. Rather I prefer to start the other way
round. In anatomy, at least, the entities with which we start are
extended in three dimensions, e.g. the human body. I believe that
there are boundary entities (of 0, 1 or 2 dimensions) only if there
are entities of 3 dimensions which they are the boundaries of. Hence
it is not a question of stacking up these lower-dimensional entities
to make higher dimensional ones as if by magic. Rather, when we
examine the higher-dimensional entities we can see the
lower-dimensional entities already within them: we can see, for
instance, the mid-point of a line.
CR: The way I see it is that in reality points exist only as
boundaries of lines, lines exist only as boundaries of surfaces and
surfaces existing only as boundaries of 3D entities;
BS: Absolutely; here we agree. This is what (in the paper referred to
below) I call 'the Brentanian axiom' (here agreeing with Brentano,
and disagree with the standard mathematical theories of boundaries,
points, etc., which seems not easily applicable to boundaries in the
sense we encounter them e.g. in biology):
http://ontology.buffalo.edu/smith/articles/mereotopology.htm
3. The third reason rests on the following thought experiment:
imagine that you have a 3D object, e.g. an apple, and that there is
some creature, say a maggot with quasi-divine powers, eating away
from the inside at the substance of the apple. At every stage of the
eating process if we have something left then what we have left is
still a part of the apple with which we began; suppose, now that the
maggot is approaching the surface of the apple, eating his way very,
very (mathematically) carefully towards the limit; then still, at
every stage, what we will have left is a part of the apple; as the
maggot advances closer and closer to the limit then what we would
have left would still, at every stage, be a part of the object with
which we began
Of course the maggot cannot advance all the way to the limit, so that
we just have the boundary remaining behind; but still there is a
homogeneity between the smaller and smaller parts and the smallest
(thinnest) possible part which is the boundary (much thinner than any
thin layer, of course).
CR: We have done this experiment many times and I always perceived a
fallacy in it. In my view you make the same mistake each time. The
maggot CAN eat his way through to the surface and he CAN eat the
whole apple [there is no 'of course' about it], leaving behind a fat
maggot sitting on the table when the whole apple has vanished and the
surrounding space expanded to fill the region previously occupied by
the apple. The boundary of the apple will have disappeared with the
apple. As long as there is a thin shell of apple that the maggot has
not yet eaten, there is still some apple left, because there is depth
or thickness, a third dimension to the shell, however thin
mathematically it is. And that is not boundary; that is still apple.
Once the maggot has eaten all of the 3D entities that constituted the
apple [including all the depth of the shell], the boundary will have
also disappeared.
Yet the maggot could not have eaten the boundary, because the
boundary is 2D and neither you nor the maggot can eat immaterial things.
BS: If I am right then we are constantly eating boundaries -- since
when we eat (up) an apple, then we eat (up) all its parts, including
its boundary parts (which do not take up any extra eating effort).
CR: The boundary has disappeared when all material parts of the apple
have disappeared because boundary is an obligatory property of
material objects the same way as qualities such as mass or
temperature are obligatory properties. Do you agree that mass and
temperature are distinct from parts? Why do you not accept that
boundary is likewise distinct from parts?
BS: This comparison between boundaries and qualities is precisely
what I am objecting to -- along with Aristotle, incidentally. The
relation between a substance and its qualities (such as mass or
temperature) is quite different from the relation between a substance
and its (internal and external) boundaries, and this (and only this)
is what the maggot thought experiment is trying to prove. For the
maggot shows that we can start with a 3D thing and imagine a gradual
step by step process of taking away parts which tends, in the limit,
to give us a 2D thing; of course as you point out, this process could
never end in the precise way I require (this is true of mathematical
limit processes in general). But for e.g. temperature we have NO
ANALOGOUS PROCESS AT ALL. There is no way, even in the limit of a
never-ending process, to turn an object into its own temperature.
We need to check, here, that we agree on intuitions.
Do you think that the center point of a solid sphere is part of the sphere?
Do you think that a circular plane within the sphere running through
this center point is part of the sphere?
Do you think that a circular plane running through this center point
that is twice the diameter of the sphere has parts that are part of
the sphere and parts that are not part of the sphere?
Do you think the North Pole IS part of the Earth? Do you think that
it is NOT part of the atmospheric region which surrounds it?
Note, again, that I do not hold that every boundary is a part of the
entity that it bounds; the boundary of the earth is part of the
earth, but it is not a part of the surrounding atomsphere, though it
is a boundary thereof.
CR: Many thanks for your last response. I now understand where you
are coming from; and I am convinced. I will make a modification in
the FMA and will check it out with you.