User blog:Edwin Shade/The Timeline Function

If geometry is the study of space and the objects that may exist in that space, then what what mathematical discipline would you say has to do the most with time ?

This may be an easy question for some, as you might say: "general and special relativity". While true, you would be hard pressed to answer with a discipline that treats time in an abstract manner. What do I mean by this ? Well, the real world can be well described by mathematics, but mathematics itself can be consistent without needing to align with our everyday experience. We can mathematically answer such problems as "a point is moving at 5 quadrillion meters per second, how far will the point travel after three seconds ?", even though this question breaks the laws of physics, math doesn't care about the laws of physics, unless we want it to.

It is perfectly valid then to speak of 100-dimensional spaces that can not exist in the real world but do mathematically. Geometry deals with the abstract concept of space, physics deals with the real concept of space and time. So, I'm asking, what is the math that deals with time in the pure, abstract sense ?

The answer is logic.

To see why, we should first get at the root of what time actually is. Or should I say, how it is. To try to define what time is itself is silly, and is kind of like trying to define what empty space is. You must always define it in reference to something, so the concept of space and time should be viewed as basic concepts, which are assumed to just be. Anyways, how is time ? Well, time has cause and effect, so we may think of any timeline as being composed of a set of causes and effects, or states and subsequent states. Sounds familiar ? This is exactly the sort of stuff that logic is made of, that of implying a statement to be true or false based on a prior statement. Therefore, we may say the study of time in the abstract sense is logic.

If we say that an event A causes an event B we may represent it as $$A\implies B$$. If events A and B are dependent upon each other, then we may represent their temporal relationship as $$A\Leftrightarrow B$$. It may seem impossible for two objects to be temporally linked like this, because usually an event only precedes another event, and the effect doesn't cause the cause. But this confusion is only due to our everyday experiences of time. Abstract time need not be viewed as consisting of a strict progression of events, but can include the intuitively paradoxical but logically sound.

There are a few axioms of time to keep in mind, and they are the following:

1.) Events cannot be linked by timelines with discontinuous gaps, (so for instance, an object cannot instantly teleport back in time, but must travel back in time.)

2.) A timeline is a line connecting two points representative of two or more mutually related events on a plane.

3.) There are two types of time, singular time and universal time. Local time is the progression of events in a single timeline, and universal time is the progression of timelines themselves.

There is the old paradox in time-travel stories of meeting your future self, who tells you that you are in grave danger, and then vanishes. When you pursue them you have an adventure, and then time-travel back to remind your past self that you are in grave danger. This may seem to create a paradox however, because if we follow cause and effect we fall into a recursive loop in which the same bit of history seems to play out over and over and over again, and thus you're 'stuck' in a temporal loop forever.

This paradox though isn't a paradox, because your future self who time-travels back would just live through time normally beginning from that point, and so the timeline isn't truly a loop, but rather the causality goes both ways, which breaks common intuition and so we call it a paradox. We've invented the idea of a "temporal loop" because it seems like the only explanation, rather than accept the much simpler conclusion that causality can work both ways. This is an example of a set of events which are dependent upon each other, and when charted on a diagram showing cause and effect would resemble a loop-da-loop shape.

Now, to get to the number-y portion of this. Let us define the timeline function as $$T(n)$$, where n is any natural number including 0. Using all we have learned thus far, let $$T(n)$$ be equal to the number of cause-and effect diagrams that can be made with a set of n-events

Note: I feel this definition is a bit vague, so I will be making it more precise in the future. In the meantime, let me know what you think of this in the comments below.

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