Monday, February 28, 2005

Flea fractality

Captain Ed parodies Jonathan Swift in his header:

"So each blogger in his kind is bit by him who comes behind."

Wrote Swift:

"So, naturalists observe, a flea
Has smaller fleas that on him prey;
And these have smaller still to bite ’em;
And so proceed ad infinitum." (

Swift's little gem describes a fractal - progressively smaller and smaller chomping insects.

I suggest that Captain Ed's blog manifests fractal structure, as well, particularly when contrasted with a recent Jonathan Chait column he critiqued (

The Cap'n quotes Chait: "What's uncanny about the Bush administration is that its dissidents invariably recant, usually in zombie-like fashion." He then notes, "Wead has never been a 'dissident' in the Bush administration -- he's never even held a job with Bush, either in a campaign or in the administration."

So what does this have to do with fractals? Chait tried to interpret the Wead imbroglio as the remarkable ability of Bush & Co. to deal with its problem people. Chait created a template based on one supposed example and then applied it to a second. He thought he saw the simplest of fractals, but, the Cap'n called him on it. The structures in either case Chait cited didn't fit the defined template, but Chait applied it anyway.

Consider the recent Eason Jordan saga in which Captain Ed (with a number of other bloggers) played a prominent role (
link). At first, bloggers were intrigued by the intitial account of the Davos meeting and began chasing confirmations from others present (and got them). Of particular importance were confirmations from "non-conservatives", even given that the post that started the swarm was by an essentially apolitical person. But, more importantly, was recognition of the importance of previous comments by Mr. Eason, especially in Lisbon. So self-similarity was found in multiple attestations of the Davos comments and in previous comments in foreign venues. The bloggers developed a fractal that ensnared a prominent newsman.

Symbolically Challenged

Do mathematical equations, however simple, belong in any "intelligent" conversation. It takes a special kind of hubris to imagine that anyone but an engineer, scientist, or mathematician would try to read on when that space and indentation appear, followed by Greek and English (occasionally even Hebrew) letters and numbers -- as subscripts and superscripts and brackets, not to mention strange symbols like ∂, ∫, or √.

When it comes to concepts like fractals, a coffee table book might be acceptable -- pictures with an apparent, if questionable, aesthetic appeal based on some mumbo-jumbo created by somebody named Mandelbrodt. Didn't he have something to do with the IBM PC?

Finally, there's entropy. A post-impressionist or even modernist word, isn't it -- sort of a rationalization for Roaring Twenties degeneracy or Seventies consciousness-raising: "not with a bang, but a whimper."

There's been a kind of unconscious feeling among many that there's a connection between entropy and fractals. Google "entropy fractal". At the time of this post, I get 117,000 hits. Delve more deeply however into these links, and the connection (with few exceptions) is esoteric at best.

Fractals are peculiar components of natural or human-created structures, phenomena, or processes that show
self-similarity at a variety of scales.

Entropy can be thought of as a measure of ignorance or, conversely, information.

In a fractal, repetition of a structural leit motif over multiple scales implies that the particular information nugget that describes the leit motif exists at each scale. In a human creation the leit motif might be constructed by algorithm or formula.

The technique of maximum entropy, popularized by physicist
Edwin Jaynes (1922-1998), is a method of interpreting a phenomenon (or process) for which some mathematical structure is inferred and some "inadequate" constraints are known. By application of the technique (or principle), the most probable state of the phenomenon given the constraints may be derived.

In the case of fractals, their occurrence suggests a possible information content constraint at each scale. In a particular geographic area (e.g., Southern California) earthquake magnitudes and frequencies show fractal behavior over a broad range of earthquake energies. This implies that the processes that generate earthquakes (sudden fracturing of rock under dynamic stress) operate in a consistent manner over a wide scale. Small areas have the same kind of distribution of earthquake magnitudes and frequencies as large areas. Small earthquakes occur more frequently than large earthquakes.

It took Pastor-Satorras and Wagensburg (
link) to demonstrate the mathematic connection between entropy and fractals. However, intuition can be used to understand the rationality of the connection (without equations).

Knowing something concrete about a process, having some knowledge of the likely structure of the process, the "most probable" state of the process might be inferred by applying the principle of maximum entropy. In the case of a chaotic process which demonstrates scale-invariant self-similarity we may well recognize entropy (the most probable) at work.

Saturday, February 26, 2005

Intuitive Leap

Even now, 56+ years since Shannon introduced information entropy and 48+ since Jaynes proposed that Shannon's entropy and Gibb's thermodynamic entropy are conceptually equivalent, debate concerning their equivalence rages on. Search Google and see.

With the more recent connection of fractals and maximum entropy (
link), and the existence of fractals in both the natural and human-created world, it is becoming difficult to separate thermodynamic and information entropy. But, in the world of human creativity, to speak of either fractals or entropy is to introduce an alien conceptualization.

Fractals are structures, real or imagined, in which there is scale-independent self-similarity. Look at a coastline from a distance, then up close: the structure -- the ins and outs -- may well appear to be very similar at the two scales. It's the same thing with a stock exchange index. Variability in stock price from minute to minute may appear no different day to day. The latter is very much a human-created process, even if it seems to lack any rational underpinning. The former is natural, although observation and characterization of a coastline, by definition, requires an observer and characterizer.

Fractals appear in music -- harmony itself and repetition of the musical scale are intrinsically fractal. Add in additional accompaniment to the melody and harmony, notes of varied duration, and phrasing: whether classical or popular, fractal structures emerge (

Fractals appear in Tolkien, both in the grand myth of creation (
link) and in the saga of the Hobbits and the Ring. Spiders, big and small, appear and reappear. Villains great and petty confront heroes grand and humble.

Shakespeare's comedies incorporate rhyme, verse by verse, and revisited themes of misunderstood and misplaced love. Puns abound and are, in fact, are the most surprising of fractals, as they surface multiple meanings of contrary connotation and context. More generally, comedy can be seen as a larger scale fractal of perversely associated ideas and events.

Fractals Everywhere -- a coffee table book -- doesn't know the half of it, especially in light of the fratal-entropy insight. Entropy is a human way of understanding the deeper manner in which the complexity of the universe is organized. Given that which is known, measured, or understood, that which is not known, measurable or understood can still be assumed to behave reasonably. In a sense, maximum entropy maximizes our ignorance while preserving that which we do in fact know.

Maximum entropy constrained by our knowledge is most profoundly reasonable.

Friday, February 25, 2005

Conservative >> Fractal; Liberal >> Nonfractal


Here is the hypothesis: The discrepancy between conservative and liberal might be the difference between fractal and non-fractal (or the degree of fractality). The title is to be interpreted as Conservative implies fractality; Liberalism (contemporary) implies nonfractality.

Example: Consider a recent post by relatively conservative Ed Morrissey (Captain Ed;
link), a discussion of a column by relatively liberal Jonathan Chait (link). (Important: This is an analysis of Morrissey's interpretation of Chait not of Chait's column independent of Morrissey.)

The Captain quotes Chait:

"'What's uncanny about the Bush administration is that its dissidents invariably recant, usually in zombie-like fashion.'"

And then, Cap'n notes,

"Wead has never been a 'dissident' in the Bush administration".

Wead "never even held a job with Bush", though "he certainly tried his best to get a position with him."

So where is the fractality and what is its significance?

The point is, in a fractal, there is self similarity at multiple scales within a phenomenon, process, creative product, or natural observation. In the Cap'n's (note the two apostrophes!) analysis, the "self-similarity" claimed by Chait occurs at only two levels, not the multiple levels between.

The same generalization applies to the Eason Jordan flap. First, several of those present at the meeting "heard" the same thing. Then, as the Captain and others discovered, Mr. Jordan had made similar accusations in other foreign venues. There was a consistency of inconsistency. But, the MSM refused to see the fractality. To their credit some liberal blogs acknowledged the evidence that other Jordan bloggers noted.

I assert that liberalism simulates fractality by recognizing apparent similarities at NO MORE THAN TWO LEVELS, and, such "similarities" apply only to non-liberals. The conservative looks for self-similarity (or dissimilarity) at multiple levels. What could be more reasonable? What could make more sense?

[Alternatively, liberals and conservatives see things through different fractals. But, since I can't see the liberal fractal, I'm not sure.]

Look at it in another way. Consider the apparent shortcomings of a recent President. It was asserted that his private actions, while clearly discordant with public "perceptions", were irrelevant. There need be no correlation. Missiles hitting pharmaceutical factories and camel tents have no connection with the underdesk of the Oval Office and a blue dress.

Then, there are the tapes of Mr. Wead. Curiously, there is really nothing new there (so far). What we see as the public President Bush is entirely consistent with what Mr. Wead managed to extract surrepitiously (while, simultaneously apparently attempting to acquire some advantage).

The authenticity treasured by the true conservative is a consistency in every level of life. No coincidence: this is a key component of Christianity, as derived from Judaism. That which is hidden will be revealed (therefore, make sure that that which is hidden is worthy of such revelation -- thus consistent with what is visible). And, that which was of value in prior ages does not lose value in the present. Whitewashed tombs can, conversely, conceal rot within.

Postmodernism (the implicit presumption of contemporary liberalism) declines to accept any kind of self-consistency in its art or science. Even the orthodoxies of previously contemporary liberalism, e.g., Darwinism, are not immune to attack.

Recent analysis from statistical mechanics and information theory shows that fractality -- self similarity at all relevant levels -- is the most probable stable state of a complex system (
link). Coincidentally (but not really), in Moral Philosophy, the person of integrity is the one who demonstrates consistency of goodness at every level of life.

Intuitive inference: The discordance between left and right is that between the unnatural and the natural, between the fractal and the nonfractal. Catholic interpretation: God's creation is good (at every level); the corruption due to the sin of the first humans is superficial, not pervasive. If we reclaim, through the great gift of the Passion, Death, and Resurrection, what was intended in the beginning, the goodness of God's creation permeates all of creation at every level.

All of this from a single blogger's post? Hey, Cap'n Ed is Catholic, isn't he?

And Catholics believe that all that scientists authentically discovery of God's creation will only enhance the his glory and our understanding of his goodness.

(Best wishes for the recover of First Mate.)