Example

    Composer, teacher, and author
    Born November 4, 1941 in Oklahoma City
    B. Mus. (Oklahoma City) 1964
    M. Mus. composition (Oklahoma) 1965
    Ph.D. composition (ESM, Rochester) 1972



Wednesday, September 20, 2006

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.

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