Re: SUO: Re: Proposed SUO Content Outline
Pat,
At the end of this note is the beginning of Section 2 of
my paper on causality, which talks about differentiable
manifolds. I would propose that such a treatment be included
in the SUO. It will be necessary for almost anything that
involes space, time, causality, and practically anything that
has to deal with physics in one way or another. In this note,
I'll refer to those definitions from time to time.
>Yes, but there's a lot it doesnt say, and in some ways it assumes too
>much. How will you characterise the general notion of dimension? This
>is a famously tricky thing to do in any general theoretical
>framework: for example, in topology, a space may not have a single
>global dimension, and there can be infinite-dimensional spaces,
>etc.,. One can fudge the issue by just imposing it by fiat, of
>course, and this may be OK for ontological purposes. (Similarly for
>orthogonality, another tricky one.)
For complete generality, I left C undefined. But in definition
2.1 below, I assume a local homeomorphism to and from Euclidean
4-space (except at singularities, such as black holes). I think
that we can safely assume 4 dimensions. (If the superstring
people come up with more, they still provide a projection to
4 dimensions, which is adequate for most purposes.)
>What about spatial models in which there are no points (like many
>mereotopological theories)?
That is no problem, because you can use Tarski's construction
to define points. I summarize his axioms in Ch. 2 of my KR
book, but the basic approach is simple:
1. The only primitive spatial things in T's theory are
spheres (of any arbitrary size).
2. All the fundamental axioms are stated in terms of
spheres.
3. But a point can be defined as a converging sequence of
nested spheres. In effect, spheres are the "real" things
in the space, and points are metalevel constructions used
for talking about limits.
4. Then T. proves that any of Euclid's (or nonEuclid's) axioms
concerning points can be mapped to the so-called "points"
that consist of sequences of spheres. Therefore, you have
a mapping between the two. That can be used to support a
coordinate system.
This is just one kind of construction that can be assumed (you
don't have to worry about it in your axioms, since after the
mapping has been proved to exist, you can safely ignore it,
except when it turns out to be useful for some purpose).
>Yes, OK, I agree that is likely to be a pragmatically useful way to
>do this...
>Perhaps one could assert that any such spatial model be mappable into
>a coordinated space, thus allowing the mereology to ignore the finer
>resolution wihtout actually denying it. But those mereo-to-locally
>Euclidean mappings will need to be described somehow.
Yes, Tarski defined them in his 1926 paper. In his later
papers, he gave various axiomatizations in terms of points,
any or all of which could be mapped into the mereological
version -- or they could be ignored, if you aren't interested.
That's a good example where you can have separable modules that
don't get "in the way" unless you need them for some purpose.
>I'm not so sure. You refer to time as a dimension, and even give a
>Minkowski diagram. I think you are a closet perdurantist :-)
Every theory of physics makes a distinction between the
3 space-like coordinates and the fourth time-like coordinate.
I agree with Doug McDavid that it is silly to make a sharp
distinction between two different kinds of people or ontologies.
You should be able to mix and match them at any time and use
whichever one is appropriate to the problem at hand.
Tarski, for example, showed how to mix and match a mereological
version of space and a point-set version. The algebraic
topologists don't even care what space is made of -- it is
irrelevant for many purposes, and when you want to map to
points or to pointless versions, you just load the module that
does the mapping.
The same kinds of transformations should be possible with the
various theories of space and time. I just think that the
arguments about whether a 3D approach is better or worse than
a 4D approaches are silly. You use whichever approach is best
suited to whatever problem you happen to be dealing with.
>Nah, to hell with physicists notions of time. Lets just treat time as
>a dimension; nobody but a few theoretical phsycisists is going to
>seriously worry about which way time is really going.
I don't think that anybody seriously considers it to be a
problem. Physicists have a fine way to deal with it that
has a mapping to everybody else's "common sense". So I
wouldn't worry about it.
>For human
>beings a better way of specifying it is in psychological terms, in
>any case: the past is the stuff you can remember, and the future is
>the opposite direction (you know, where y canna' see, and all the
>doot and fear is.)
That's fine. And that view can be mapped to a definition
in terms of entropy. That should make everybody happy
(i.e., it should accommodate "commonsense" problems as well
as highly sophisticated physics problems).
John
_______________________________________________________________
For a more nicely formatted version of this excerpt, see
http://www.bestweb.net/~sowa/ontology/causal.htm#s2
2. Continuous Processes
The known laws of physics, which are the best available approximations
to the ultimate principles of causation, are expressed in differential
equations defined over a four-dimensional space-time continuum. Those
equations relate functions, such the electric field E, the magnetic
field H, and the quantum-mechanical field Psy. A continuous process
can be defined as a collection of such functions on a differentiable
manifold M, which represents some region of space-time. Such manifolds
can represent the warped universes of general relativity, but they
can also be treated as locally flat: around every point, there is a
neighborhood that can be mapped to a Euclidean space by a continuous,
one-to-one mapping called a homeomorphism. Since a one-to-one mapping
has a unique inverse, the four coordinates that identify points in the
Euclidean space can also be used to identify points in the neighborhood.
2.1 Definition of continuous process: A continuous process P is a pair
(F,M) consisting of a collection F of differentiable functions defined
on a four-dimensional manifold M.
* Every point p of M has an open neighborhood U that is homeomorphic
to some subset of four-dimensional Euclidean space, E4. The
homeomorphism at p determines a coordinate system x1, x2, x3, x4
over the neighborhood U.
* A path through M is the image of a continuous map m from a real
interval [a,b] into M. The point m(a) is called the beginning,
and m(b) is called the ending of the path.
* The coordinate x4 of a point p, which may also be represented as
t(p), is called time.
The mapping from each neighborhood into Euclidean 4-space determines a
local coordinate system around every point of the manifold M. Axiom 2.2
introduces the option of joining overlapping neighborhoods and extending
the coordinate system to the larger neighborhood. Any finite number of
unions of neighborhoods is also a neighborhood, but a covering of the
entire manifold may require a countably infinite number of neighborhoods.
In the general case, the entire manifold might not have a globally unique
coordinate system. A spherical universe, for example, has no one-to-one
mapping into a flat Euclidean space.
2.2 Axiom of extensibility: For any continuous process P=(F,M), the
manifold M can be covered by a countable union of neighborhoods. If U
and V are two overlapping neighborhoods, then their union UÈV and their
intersection UÇV are also neighborhoods with homeomorphisms into E4.
The local neighborhoods of the manifold M may be extended around, but
not across singularities, such as black holes. If the manifold happens
to represent Newton's conception of absolute space and time, the local
mappings could be extended to a global isomorphism of M to Euclidean
4-space. Definition 2.1 and Axiom 2.2, however, are general enough to
accommodate the geometries Einstein used for the special and general
theories of relativity. They could even be generalized to more exotic
conceptions, such as a multidimensional superstring theory.
Calling coordinate x4 the time coordinate does nothing to distinguish it
from the three space-like coordinates. To distinguish time from space,
Axiom 2.3 adds a constraint, which implies that time travel to the past
is impossible. It is a safe assumption for any processes that have been
observed within the solar system, but there is still some controversy
about its universal applicability to remote regions of the universe
(Yourgrau 1999).
2.3 Axiom against time loops: No path in which the time coordinate
increases monotonically along the path can form a loop. Formally,
if m: (a,b)->M is a continuous map, and for any two points x,y on the
interval (a,b), x<y implies t(m(x))<t(m(y)), then m must be a one-to-one
mapping.
Although Axiom 2.3 prohibits time loops,there is no prohibition against
loops in the space-like coordinates. On the earth, for example, the
coordinates of latitude and longitude are locally flat, but it is
possible to have a loop in which the longitude increases monotonically:
on a trip around the world, a point at longitude 360°+x can coincide
with a point at x.
The fundamental laws of physics govern all processes from intergalactic
motions to subatomic interactions. Derived laws, such as the equations
of fluid mechanics, involve functions, such as temperature or fluid
density, which are averaged over regions that may be small compared
to M, but large compared to individual atoms and molecules. Other
functions could be defined in terms of arbitrarily large configurations
of space-time. The predicate isInObject(p,e), for example, might mean
that the point p is inside object e. That predicate could be specialized
to more detailed predicates like isInHuman(p,e) or isInChair(p,e), which
would mean that p is inside a human being or a chair.