10. Serial and Parallel Sympositions and Non-Hierarchical Partonomies

So far, I have not explicitly juxtaposed time with space in the context of symposition, and consequently the concept of sem-linking has by default referred to the collecting of symponents in a region of space at some specific moment. Sem-linking does not have to be restricted in such a way and can collect symponents that have spatiotemporal instead of just spatial pose. {volcanic eruption} is a symposition that is not localized at one specific moment, as is {butterfly} with its splendid metamorphosis. There are two things fundamentally different between time and space that complicate the generalization of symposition to the temporal dimension, however. First, note that if space was 2D instead of 3D—a plane instead of a volume—and was similarly occupied by point-like physical primitives, then the symposition of these primitives into a Logos would be basically the same process as if it was 3D. This holds also if space was 4D and again occupied by point-like physical primitives. The problem is that physical primitives are not point-like in our 4D spacetime; they are line-like, tracing extended paths through the dimension of time.

The other difference between time and space is that, given a bunch of points in space, there is no natural ordering to them, they’re just all there at once. On the other hand, given a bunch of points in spacetime, there is a natural ordering to them: the chronological order of first to last point.1 A consequence of this is that when a partonomy is serialized in order to put it into writing, for example with a hydrogen atom: {e, {qu, qu, qd}}, arbitrary choices must be made to put the electron before the proton and then the up-quarks before the down-quark. These choices are not arbitrary with the temporal symponents of a symposition; just put the first one first. Consequently, a clear distinction can be made between serial sympositions, whose symponents can be organized temporally in a series, and parallel sympositions, which are sympositions as originally conceived at a given moment of time whose symponents appear together in parallel.

Now, what is the partonomy of a serial symposition and how might it be different from the partonomy of a parallel symposition, and what happens when these partonomies are put together? First, let’s consider a symposition that for at least some stretch of time has parallel symponents that do not move—our atom of hydrogen, perhaps, at a temperature of absolute zero—and we can say that such a symposition is “frozen.” The self-same parallel configuration of the hydrogen atom is present at each snapshot of time, such that there is a series of {e, {qu, qu, qd}}, {e, {qu, qu, qd}}, {e, {qu, qu, qd}}, etc., that is ordered by time non-arbitrarily. Thus it is also a serial symposition. All frozen parallel sympositions are also serial. How can all of these identical only-parallel partonomies be fit into a single serial+parallel partonomy? There is a range of possibilities with two extremes; we can serially sem-link at the Dust or at the top—either as {{e, e, e, …}, {{qu, qu, qu, …}, {qu, qu, qu, …}, {qd, qd, qd, …}}} or as {{e, {qu, qu, qd}}, {e, {qu, qu, qd}}, {e, {qu, qu, qd}}, …} or in between as {{e, e, e, …}, {{qu, qu, qd}, {qu, qu, qd}, {qu, qu, qd}, …}}. They have, respectively, four, one, and two ellipses (…), two extremes and one intermediate, corresponding to the number of nodes at each level.2

Recall that a parallel partonomic gap, if sufficiently large, results in the level below the gap relating quasi-continuously to the level above the gap. The ellipses in the previous paragraph represent serial partonomic gaps, but how large are they? The fact that primitives are lines in spacetime means that they are continuous and that there are infinitely many points between any two moments of time.3 The serial partonomic gap is so large it’s infinite. Whereas parallel partonomic gaps only approach continuity, serial partonomic gaps can actually get there, and this holds regardless of at which parallel partonomic extreme of bottom or top or where in-between the snapshots are serially sem-linked. So what is the spacetime partonomy of a symposition? A final answer requires choosing a level where the serial sem-linking occurs. Such a choice seems arbitrary, however, at least for frozen sympositions, so I won’t make it, instead preserving the tension.4 Regardless, we can see that serial sem-linking in frozen sympositions is always continuous.

Two questions related to each other arise. Can there be continuously serial sympositions that are not frozen? And what would it take to have a discretely instead of continuously serial symposition, which of course could not be frozen? For a symposition to not be frozen, poses must be changing somewhere in it over time. Recall that a symposition is more than its partonomy. Though frozen sympositions clearly also have frozen partonomies, sympositions with frozen partonomies do not necessarily need to be frozen themselves. Take the Solar System. The partonomy of the Solar System, at least considering only the planets and the major moons, has not changed in millions years. Still, the earth and the sun have completed millions of loops of a continuous sequence of poses since then, and similarly all the other planets and moons. Importantly, if we tried to discretize the sequence of poses, we would have to make arbitrary choices.

When would we be able to make a non-arbitrary choice? Consider a planet that is happily orbiting the Sun until an unfortunate confrontation with another object throws it off course indefinitely into a highly eccentric orbit. This is hypothesized to have actually happened in the history of our Solar System to “Planet Nine,” which started out between Saturn and Uranus but was at some juncture deflected by Saturn towards Jupiter and then by Jupiter into the far reaches of the Solar System to live out its life with an orbital period of 10,000 to 20,000 Earth-years.5 A non-arbitrary division can be made in the temporal series before and after the planet’s deflection. The exact moment of the division may not be easy to determine, but the more important and fortunately easier question is whether there should be a division, and the answer is yes. Regardless, note that the Jumping-Jupiter scenario, as it’s referred to in astrophysics, did not change the parallel partonomy of the Solar System at any moment;6 its partonomic relevance is purely serial.

Imagine instead a spaceship that orbits Earth many times and then changes course and orbits Mars many times. The change in course affects the serial partonomy just like above with a clear before and after, but if we construct the parallel partonomy, we see that it is also affected, unlike above with Planet Nine. It goes from {{Earth, spaceship}, Mars} to {Earth, {spaceship, Mars}}.7 Let’s continue with just the initial letter of each and try to construct a full spacetime partonomy. The parallel partonomies at first are the repeated {{E, s}, M}, {{E, s}, M}, {{E, s}, M}, etc., but in the end are {E, {s, M}}, {E, {s, M}}, {E, {s, M}}. If we serially sem-link at the gravitational Dust, then we get {{{E, E, E, …}, {s, s, s, …}}, {M, M, M, …}} followed by {{E, E, E, …}, {{s, s, s, …}, {M, M, M, …}}}. Recall that choosing the Dusty bottom instead of some other level for continuously serial sem-linking is arbitrary in frozen sympositions and in sympositions with frozen partonomies, such as the Solar System in the Jumping-Jupiter scenario. When partonomies are not frozen, however, the choice is no longer arbitrary. If we try to sem-link our spaceship scenario at the bottom, we would need to be able to put {E, E, E, E, E, E, …}, {s, s, s, s, s, s, …}, and {M, M, M, M, M, M…} into a tree. This cannot be done in such a way that reconciles both the initial and final parallel partonomies. We can sem-link at the top, getting {{{E, E, E, …}, {s, s, s, …}}, M, M, M…}, {E, E, E, …, {s, s, s, …, M, M, M…}}}, but we procure a very odd result if we do so: the intrinsic serial partonomy of Earth is sliced in two by an extrinsic spaceship flying to Mars!

We can solve the problem by an analogy with taxonomies. There were two ways in which a taxonomy could fail to be a tree: by the presence of either continuous or cartesian variation. There are also two ways in which a partonomy can fail to be a tree. The first has already been covered above: continuous sem-linking does not provide a tree, nor even a graph, because both of these are discrete mathematical objects. The second way occurs when symponents belong simultaneously8 to more than one symposition, a situation that can still be depicted by a graph if no longer by a tree. Non-hierarchical partonomies are ubiquitous. They are necessary any time a symponent moves from belonging to one symposition to belonging to another. Though non-hierarchical partonomies are first defined here for temporal reasons, we will see later that they are also relevant spatially.


Figure 1. A non-hierarchical serial partonomy.

Though we need non-hierarchical partonomies to fully account for it, spatiotemporal symposition is as full of hierarchical possibility as just spatial symposition. Consider the spatiotemporal partonomy of an individual {football game} or individual {conference}; they both have many levels above the lowest discrete serial symposition. I’ve mentioned several times that the Logos has developed, grown, or changed after the Big Bang. Given serial sem-linking, however, those expressions are faulty. The Logos spans all of spacetime, planted firmly on both the primordial homogeneity at the Big Bang and the ever-present Dust. The Logos hasn’t developed after the Big Bang, but rather the horizon of its revelation has been expanding.

Footnotes

1. Ignoring relativistic concerns regarding frame of reference. Again, this book is focused on the classical realm.

2. In {e, {qu, qu, qd}}, there is one level between the quarks and the atom, but no levels between the electron and the atom. The in-between serial partonomy could alternatively be depicted as {{{e}, {e}, {e}, …}, {{qu, qu, qd}, {qu, qu, qd}, {qu, qu, qd}, …}}, for clarity. Also, there’s no need distinguish the serial symponents, with subscripts for instance, because of the direct mapping between the seriality of writing and the seriality of time.

3. You may object because of the Planck time. Note that the Planck time does not set a discretization of time, but rather a bound for the measurability of time. Time could go unmeasurably crazy at such tiny scales even while remaining continuous, if it behaved something like the Weierstrass function or the Koch snowflake, perhaps.

4. The tension can be explored further in the endurantism vs. perdurantism literature, although it doesn’t explore the in-between possibility.

5. http://www.nature.com/news/evidence-grows-for-giant-planet-on-fringes-of-solar-system-1.19182

6. In all likelihood there was a reorganization of moons, but we are entirely ignorant of the details, so I’ll ignore this possibility.

7. Continuing with the theme of a very pruned Solar System.

8. “Simultaneously” in a timeless, 4D sense, perhaps oxymoronically.

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