Re: SUO: Re: Proposed SUO Content Outline
Pat,
It is much easier to state axioms that are independent of any
coordinate system than to state axioms for any particular
system.
>>.... We will need to write axioms which say that spatial
>> coordinate systems exist, without saying anything in particular about
>> any spatial coordinate system.
Writing such axioms is a trivial exercise:
A coordinate system is nothing more nor less than a set C
called the coordinates, a designated region R of space-time,
and a function that maps C into points of R. This is true
of coordinate systems tied to the earth, to the sun, to the
center of gravity of the universe, or whatever. It is true
of 4D systems and 3D+time systems. It is true of inertial
coordinate systems and arbitrarily accelerated systems.
For an example, see Section 2 of my paper on processes
and causality:
http://www.bestweb.net/~sowa/ontology/causal.htm#s2
In that section, I assume that C is an open subset of Euclidean
4-space, which is general enough to include nonEuclidean
spaces that are locally Euclidean (which includes spherical
coordinate systems, such as the lattitude, longitude, and
altitude on the earth.)
The only thing that the axioms have to say is that the set C
and the set of points of R exist and there is a mapping from
C to R that is (locally at least) one-to-one, continuous,
and differentiable. A total of less than one page of axioms
will cover any and all coordinate systems that physicists,
engineers, mapmakers, ship captains, and NASA ever use.
Recommendation: State all axioms about space, time, and
space-time in terms of arbitrary sets C, R, and functions
from C to R. Then for any particular coordinate system,
you give a separate set of axioms that define the mapping
from C to R. But all the basic axioms (which include all
versions of Euclidean and various nonEuclidean geometries)
remain unchanged.
Furthermore, the axioms are completely independent of anybody's
notions about continuant, occurrent, endurantist, perdurantist,
obdurantist, or any other philosophy of space and/or time.
And if you want to define time in an arbitrary coordinate
system, I suggest Eddington's "arrow of time", which at any
point is the direction of the maximum gradient of entropy.
This arrow may be undefined in certain regions, such as black
holes -- and that is as it should be. But for ordinary regions
of the universe, it is well defined. For those people who want
a "plug & play" ontology, certain predefined modules can be
specified for the most common systems. (But then, of course,
we have to face the question about specifying modules in KIF,
which I believe is essential for many other reasons as well.)
John