Mathematical abstractions of space, time, and kinematics

There is a question of what mathematical structures to use in order to represent space, time, and related concepts. It is well known that time can be represented as a partial order relating events that are caused by one another. Space on the other hand, can be represented in multiple different ways, but generally we need some concept of a metric to determine distances between points. Entities in a space can only most readily effect other entities that are most near to them, as determined by the distance relation.

One property of these abstractions of space and time is that they both have a topology associated with them, the metric topology and the order topology. There is a question as to rather or not space and time are discrete or continuous or not, this can be determined by the topology of these structures. In particular, a discrete space will have a discrete metric and discrete time will have a discrete order. Another issue is rather time is absolute or not. An absolute time will use a total order and a relativistic time will use a partial order. The overwhelming evidence for relativity favors a partial order for time. The theory of causal sets uses a discrete partial order to define spacetime, for example, but sometimes it might be useful to have a different abstraction of time, or to have a way of representing space, since geometry is clearly so fundamental.

• Space : metric
• Time : order

These are mathematical abstractions of space and time, and therefore they don't have to be used to model actual physical space, but rather, they can be used to model anything analogous to them. If space is discrete for example, it may be useful to define it simply by a connected graph, then the metric will be simply be the path length. For example, for a cellular automata like Conway's game of life, the space is the connected graph of the max degree eight neighborhood relation between squares and the time will be an absolute, or total, discrete ordering. For a simple game like checkers, the space could be the squares pieces move on, and the time could be a discrete total ordering, and so on. The space could also be a manifold of any sort, like a two dimensional manifold, assuming there is a two dimensional space, like flat world. Or one could imagine a manifold with many dimensions greater then are own, or with some extra structure to it. This gives us an idea as to how to model all the different possible universes and environments an agent can be embedded in or involved with.

Of course, space and time, cannot be considered completely separate concepts even though concepts like order and metric can be considered separately from one another. One reason that they are related to one another is kinematics, the study of motion, which connects the two together to form spacetime. If one accepts a process ontology, which says that everything is an event, then one could suppose that the meaning of space in a kinematic universe is to define the nature of motion events. It is helpful to examine the relation between them by determining rather the amount of spatial distance between events is greater then the temporal distance, to determine if they can be causally related. In particular, one can define the average speed by the spatial distance divided by the temporal distance as it is typically defined in kinematics. In a discrete universe, the average speed is simply the average amount of time units spent on discrete spatial jumps. With these concepts we can define the equivalent of light cones used in relativity to determine causality.