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.
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.)
In your theory, are there any restrictions on coordinates? Can a
space be its own coordinate system, with the identity function as its
projection mapping?
What about spatial models in which there are no points (like many
mereotopological theories)?
>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.
Yes, OK, I agree that is likely to be a pragmatically useful way to
do this, though I still think that the problem of dealing with
'pointless' mereologies of spaces is one that needs to be faced.
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.
>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.
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 :-)
>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.
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. 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.)
>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.)
Yes, we have to be able to talk about modules in something, for sure.
Pat
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