[om-list] Functional dependency of binary relations

[Mark Butler]

Tom wrote:

>     Do you mean that a single valued attribute can vary in time but not in
> space?  

Single valued has two different meanings - one that there is no functional
dependency at all, an idealization that does not exist for relationships
between physical objects, or two that there is only one value for any given
set of arguments.  What I meant to say is that the simplest possible real
world relations - relations that are single valued in the first sense in the
snapshot picture of things automatically become unavoidably functionally
dependent on time.  

Adding spatial dependency makes a general binary relation into a eight
dimensional function of 2 sets of space / time coordinates, which is much more
difficult to model, and a little overkill for our application.

The most fundamental of physical objects is a particle that describes a world
line through space time.  It is no accident of human perception that none of
the known laws of physics relate the world lines of two different particles in
any temporal manner other than through interactions that occur at the same
time.  

So other than relationships between idealizations like events, the first
simplification that we can make is to store most relationships as a function
of a common time for both objects.  This merges the two time dimensions,
reducing the number of independent dimensions to seven.

If your objects (or their centroids) can be treated as points, that eliminates
the spatial dependencies, leaving only one independent dimension, time.  This
is the case that is relatively easy to model.

You can model higher order relationships using differential equations, but it
is technologically impossible to store any significant amount of data in a
large number of dimensions.  For example, if you are modelling light
propagation in a room and use a relatively coarse 1000 x 1000 x 1000 x 1000
hypercube, you need 1,000,000,000 records just to store the initial
conditions.  Do you see why the weather is so hard to simulate?

Until we have a multi-dimensional differential equation representation I do
not see how we can practically store seven or eight dimensional relationships
between four dimensional objects, let alone store four dimensional objects
using four independent dimensions.  Right now we are pretty limited to samples
of functions in one or maybe two dimensions.

> This may often be the case, simply out of the fact that we humans
> look for such asymmetrical attributes because we have "issues" with the time
> dimension: we look at it differently than space.  

We just don't look at it differently - every known physical and natural law
treats time fundamentally differently than space.  The closest scientists have
ever come to space/time unification is in a four dimensional version of
electromagnetism, and there time has to be treated as an imaginary dimension
so that the speed of light remains constant.  

If time and space were remotely symmetric, light would be observed to travel
at all speeds.   Instead, no matter how fast or in what direction you travel,
light from any source is always observed to travel at specific angles
(velocities) in space time relative to you.  This leads to all sorts of
unusual phenomena, but suffice it to say you just aren't going to find any
local physical object that is experiencing time in one of our space
dimensions. 

In my opinion, a more useful theory (although not particularly practical for
what we are doing) is to consider is that time may not be a fundamental
dimension at all, but rather some sort of path length in a world composed of
four space dimensions where the whole local part of the universe is falling in
the same direction, making that "dimension" both irreversible and invisible.

> But there are certainly
> attributes which vary in time and space, such as measuring the temperature
> at points within an object which has extension in space and time.  Another
> one is colour.

Definitely.  We are not going to get very far storing them as four dimensional
data sets, however.

- Mark