Phi and Infinity, etc., Part I

My basic premise (assumption) is that infinity is.

Either infinity is, or is not.

Logically, there can be no intermediate possibility. A mix of "infinity" and "finitudes" is oxymoronic. In order to be "finite," an entity must have clearly-defined boundaries. At such a "boundary," "infinity" would of necessity come to a halt. This imposes, in fact, a "boundary" on the "infinite," in which case it is no longer "infinite." It cannot be unending if it has an end anywhere, and that is what such a "boundary" would be.

On the other hand, if existence is therefore "finite," it must exist in some context. If there is a "boundary," what is on the other side? "Nothing?" Problem: "nothing" cannot be "nothing" except in relation to "something." Therefore, "finite" must also exist in relation to "something," even if that "something" is "nothing." Assuming there is a "boundary," that this existence in which we are caught up is a "finite" entity, and that "nothing" is on the the other side of the "boundary," "nothing" must also be a "finitude," as it demonstrably has at least one "boundary." (I.e., the edge of our "finitude," i.e., our "something.")

So what then? A series of "finitudes?" And--where does this series end? Obvously, it cannot, because however far it continues, if an attempt is being made to justify the concept of "finitude," there must ultimately be a "nothing" at the end, in which case one ends up at the same place.

Again, logically, there can be neither end nor beginning to such a series, because "nothing" is at some point a necessary postulate. Therefore, one is left with an "infinite" series of "finitudes,"
thus, "infinity." This is obviously self-contradictory: "infinity" ipso facto precludes "boundaries," a necessary condition of "finitude." "Finitude," therefore, is in appearance only.

It will be found that no entity in our extant universe is in fact "finite." It is really a matter of scale. For example, an apparently solid table on a different scale is a swirling mass of atomic and sub-atomic particles with enormous spaces in between, and merging with the surrounding atmosphere. Neither are words, numbers, ideas, and so forth, sharply defined: their "boundaries," upon closer examination, prove to be fuzzy given the right scale. Music and the other arts are precisely imprecise.

Much of the issue is semantic. Mathematicians (Geometricians, et al.) speak of a variety of "infinitudes," various "infinite" sets, etc., etc. However, these, I am convinced, will prove at some perspective to simply be "segments" of "infinity," with "fuzzy edges," i.e., "boundaries" (delimitations) that are ultimately imprecise and merging with the surrounding "infinitude" of which they are part and a human attempt at expression thereof.

For example, consider the "infinite" ratios, Pi, 3.1416..., and Phi, 1.618... I have come to think that the latter "Golden Ratio" is, in the numeration available to us, the fundamental ratio of "infinity." Further, I now assume that Pi is a "segment" of Phi as it exists--as a descriptor--of our universe. The extrapolation of these ratios, and that of their relationship to one another, into not only three dimensions but into the larger multi-dimensionality projected onto our universe by, e.g., astronomers and other scientists, will provide, I believe, a more accurate descriptor of what I can only speak of as the "convergence" of "infinitudinal segment 'vectors,' "
i.e., "infinite" sets and the like, geometrical projections, parallel straight lines, the observed expansion of the "Big Bang" (hypermolecular hypothesis), the continually "im"-panding [coined antonym of "expanding"] world of sub-atomic particles, and on and on...

This also connects with the ideas I have expressed elsewhere about the general concept of "swarms," and it seems logical to me that Phi would play a role in this matter, especially as a descriptor of the average rate of proliferation and resultant coalescence of various phenomena.

If one considers the vastness of the "universe" outward, one should also consider the vastness of the "universe" inward. The logarithmic spiral deduced from Phi does a nice job of modelling this view of "infinity." One must, I think, also be led to ask if one can really appropriately speak of a larger or smaller "infinity?" Is the sub-atomic world, for example, realy smaller than the galactic world IF both are indeed "infinite?" [N.B.--If my conjecture is accurate, somehow expressible as positions on that "infinite" Phi-based logarithmic spiral.] Which is "up" and which is "down" is really just a matter of perspective and, for want of a better word, "direction," if the concept of "inifinity" has as much a valid basis in reality as I assume it does.

(to be continued)

Swarms (Coalescence) and Mathematics

Note I (on swarms):

I have been thinking recently about the mathematics of swarms, or more accurately, about the fact that there is a branch of mathematics that deals with the formation of swarms.

It occurs to me that the word "coalescence" might also be useful as a quasi-synonym for swarms.

Why? Because I suspect that the mathematics of swarms could explain such diverse phenomena as, e.g., cities on the one hand, or on the other, the formation of black holes.

I think there is also a relationship to what we call "culture."

If there is only a single particulate entity, there is obviously no relationship. If two such come together, some sort of relationship between the two exists. As a third and fourth such come together, the relationship becomes much more (factorially) complex. There will be some basis of attraction implicit in such gatherings, whether chemical, gravitational, or other. As interaction proceeds, "rules" of association will form, consciously or driven by physics (or other). This begins to take on the nature of "culture," and as the numbers grow larger and the "culture" more complex, begins to be a rule not only for inclusion but also exclusion.

Now, new particulate entities attempting to join the growing mass of such must more and more conform to the prevalent "culture" in order to get in. Culture per se serves the function of screening just who/which belongs and who/which does not. It strikes me that a major role of all that we call art, music, and literature is the function of defining "swarms," even "swarms" within "swarms." Is not, e.g., concert-going a form of swarming activity?

A two-fold process of selection is involved: the prospective entrant self-selects, and the "swarm," having coalesced to whatever degree, now defines, through its evolved "culture," who/which can or cannot get in. Bear in mind that I am including not only conscious "culture," but also the evolving "culture" of collective physical attributes, which in the case of swarming sub-atomic particles would be of the nature of electromagnetic (or gravitational) phenomenological characteristics, or in the case of insects probably in large part chemical (pheromones).

I believe this phenomenon (the evolution of any given swarm) is likely connected to the development of intelligence and language, and at a more basic level, the evolution of species (as the numbers of the group/swarm increase, the acceptable characteristics for inclusion become more defined and more constant, hence affecting species' reproductive characteristics).

Intelligence, because as the numbers increase the need arises for the retention in some form (even if merely a sort of tropic awareness) of a sense of "place," i.e., where one belongs in the relationship(s), a sense of who/which belongs where, and a sense of who/which may join in the gathering. As the "culture" grows, the need for some sort of comprehension of its increasingly complex "rules" also must grow. At the same time, there is a need to communicate with all the other particulate entities in some form, hence "language" arises. Now, I am using these terms in their very broadest meanings, and this is inclusive of biological, chemical, electromagnetic, gravitational, and other media as the context in which "intelligence" and "language" operate.
Crystalline structures would be a very good example of a non-conscious coalescent process which nevertheless displays these characteristics I have just described.

I like the joint use of the word "coalescence" because I think the word "swarm" implies a certain fluidity which some of the aggregates I have cited might seem to lack. (Although I would argue that, given the appropriate time-frame, that would not actually be the case. Cities, for example, come and go, and if one could film a city over a few decades or centuries, then play that back at high speed, one would see that cities are just as much in a state of flux as a swarm of bees.)

October 31, 2006: [Sequel to the above:]

Note II: on the basis of formation of Swarms (includes flocks, herds, etc.)

[N.B.—I have below used the well-known terms “Byte,” “Paradigm,” “Algorithm,” and “Matrix” in a somewhat specialized way, each of which is defined as I go.]

Byte = basic unit; unit (inherently) of “information:” verbal, chemical, electro-magnetic, gravitational, et al. A “signal” signifying in essence “I am here” by one of these means or another will be emitted one way or another simply as a function of occupying a niche in the space/time continuum. At some point, another “Byte” of some sort (not necessarily the same sort) will react to this, thereby establishing a relationship. The “reaction” might be, e.g., simply one of bumping into the initial “Byte” enroute through the space/time continuum, or it might be a sympathetic resonance to an emitted electro-magnetic wave amplitude, etc., etc. Eventually, such a random occurrence (relationship) will become permanent in many instances, and the relationship thus established becomes a pattern attracting other (quasi-) particulate entities. At that point, it functions as a:

Paradigm = “grammar/syntax” (verbal, chemical, electro-magnetic, gravitational, et al.) by which selection/self-selection and accretion of added “Bytes” takes place, leading to what I will call a “Poly-Paradigmatic” state of accumulation through multiple "Paradigms" so combined that they form a larger and more complex set of patterns which begins to more distinctly define parameters of whatever conglomerate is forming. At a certain point in each such process a critical mass is reached, at which stage the complex interrelationships become an:

Algorithm = here, a collection of Paradigmatic arrangements of “Bytes” so that they form a model arrangement of “Bytes” which becomes the rule for the selection of and futher grouping of “Bytes” into a “Swarm.” The “Paradigms,” having evolved from the initially random coalescence of “bytes,” determine which “bytes” are subsequently admitted to the process. The “Algorithm” is the next step along the process, in which the evolved coalescence of “Algorithms” now determine which, if any, new “Paradigms” are admitted to the fundamental process of structuralizing the growing “Swarm.” “Algorithms” arise in this context as the result of inference (or deduction) extrapolated from the grouping of “Paradigms” and hence becoming the rule-base for the inclusion, and even generation, of new “Bytes,” “Paradigms,” etc. The last step of this building process is the further evolution of multiple such “Algorithms,” and thence the encoding of such a collection of previously individuated “Algorithms,” into a “Matrix,” a filter through which all further “Swarm” growth is processed. Henceforth, all future “applicants” (“bytes,” “Paradigms,” “Algorithms,” even potential supplemental or alternate “Matrices”) must pass in order to gain “admission” to the “Swarm.” I would suggest, for example, that culture and religion are two human examples of just such “Matrices.” Attendance at a symphony concert, for instance, implies acceptance of the “Matrix” provided by the conventions and protocols of the literature, history, and performance of symphonic music. This is obviously quite different from the “Matrix” provided by, e.g., the world of rodeo.

For insects, the matter could well be defined by pheromones. Occasionally one might see a random “Swarm” of mixed insects ( I do not think hetero- or homogeneity ipso facto is a necessary precondition for the formation of a “Swarm.” However, part of my point is that a “Swarm” becomes, de facto, a self-defining, self-contained homogeneity for the duration of its existence) which might gradually metamorphose into a “Swarm” of all the same kind of insect, BECAUSE of the “Matrix” provided by pheromonal distinctions. The initial distinctions of the proto-“Swarm” of mixed insects would have included a “Matrix” that would limit membership to very small flying creatures who don’t smell at all like Birds, Dogs, Lizards, etc. The further evolution of my putative “Matrix” would then be toward one in which only pheromones broadcasting “Wasps!” or “Bees!” or “Flies,” etc. would be included in the “Matrix.” Thus, it would tend to become more specific. My speculation is that all “Swarms” tend to become more specific, then to reach a stage of stasis, and from that point, one of instability, and finally disintegration. (One might well graph these developments as complex wave-forms involving additive, diminutive, and interference patterns.)

[N.B.--Here, I suspect, is a fundamental pattern of nature which involves the range from Pi to Phi, and the relationship of each of these “infinitudes” to the other. (As a small side topic, I note that it seems to me that to speak of a definition of Pi or Phi, mathematically or otherwise, given their infinite nature, is inherently oxymoronic.)]

A long-term example of the process described above is the evolution of European-based music from, e.g., 6th- and 7th-century Plainsong through Organum, subsequent Medieval Polyphony, through Renaissance Polyphony, Baroque Counterpoint and early Functional Harmony through Beethoven and the ever-increasing chromatic accretions of the 19th-century, especially Wagner, et al., and into post-Romanticism and finally, Schoenberg’s Pantonality in the form of his 12-tone method, which is, after all, the evolution of all the foregoing into a Matrix encompassing all of the previous developments and clearly pointing the way for future inclusions, as has in fact happened. I believe this can be reasonably described as a centuries-long “Swarm,” or “Swarming” process.

I would also contend that by the time the “Swarm” has reached the level of “Algorithms,” said “Swarm” has evolved what may be said to be a “proto-Culture,” and by the level I have described as filtered through the “Matrix,” it constitutes a distinct “Culture” in the broadest sense of that term, whether bacterial or symphonic. Initial all-inclusivity has metamorphosed into partial-inclusivity commingled with a larger exclusivity. This is the self-defining “Swarm” as a distinct “Cultural” entity.

The role of “Culture,” then, is as the “Matrix” defining membership in the “Swarm,” whether linguistic, chemical, bacterial, viral, electro-magnetic, or other is secondary to the more general characteristics I have described above. The general principles, it seems to me, are essentially the same.

Aphoristic musings:

Cities are self-sorting information distribution systems.

One of the main functions of humans is the redistribution of trees and other plants.

Another important function is the preservation and nurturing of other animal species.

We are endoskeletal "brains" for the exoskeletal machines we have concocted.

Like Hermit Crabs, we skitter from one housing to another, as it suits us.

We are weird ant colonies, in which we can be workers, soldiers, and so forth, depending upon which exoskeletal housing we use.

Even musical instruments may be seen as exoskeletal extensions of our bodies.

Our highways are surrogate rivers.

Music provides a highway for neurons.

Thus, music is like an electroencephalogram in reverse.

We look for parallel universes: they are right under our noses! How many living creatures are there on Earth? There are at least that many parallel universes.

Our new age reconsiders the ancient idea of reincarnation: but what is DNA?

We exist as highly complex composite waves in time. Does the tiny virus achieve its impact through temporal displacement of an equally tiny segment of our wave-composite, thus altering our periodicity?

Perhaps the virus is the primary agent of evolution.

Evolution has not stopped; are we perhaps, even now, evolving into two or more distinct species? However long it takes, humans will not always be humans.

Technology in its broadest sense is what most clearly distinguishes us as humans. Other species are tool-makers; we make technologies.

(Tools per se are singularities; technologies are systemic.)

Whether or not "God" exists, or there are multiple gods, or the world is animistic, or there is an oversoul as Emerson thought, or a Jungian collective consciousness, or the Tao, or Nirvana, and so forth is really a set of moot points, because we are in the process of inventing IT. Our information age has created global TV, the web, and an infinitude of other such humanoid interconnective phenomena. As a result, we now live in the midst of a transcendental virtual reality mega-consciousness which envelops our planet.

(The Golem has come to life: huge, powerful, neither benevolent nor malevolent.)

Is this OUR mind? Or, are we in the infinite process of discovering deity or a reasonable facsimile thereof? Is this an "I-it" or an "I-Thou" relationship a la Buber? ...All of the above (and probably--no, certainly!--more), I have to think!

All of this has a symbiotic relationship with music--and the other arts.

Music and the other arts are perhaps the most essential component of our planetary mind (or collectively improvised deity). These creative modes of expression are possibly our most important technologies. They provide the most universal and fundamental--and at the same time, intimate--framework for communication.

One last "perhaps" for today:
Perhaps we humans are the "neural synapses" of "Old Mother Gea?"

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