Monday, October 03, 2016

Science and Faith

Some fifty years ago, Briton C. P. Snow ignited a controversy by decrying the “gap” between science and the arts in the Western World – the separation of “Two Cultures.”1 Snow asserted that few scientists read Shakespeare and even fewer literary types understood the basic principles of physics. The “scandal” of Snow’s two cultures was, in his eyes, that the inferred gap of separation was a barrier to progress in developing countries; he pointed to the Soviet Union (!) as an example of a more balanced and integrated art-and-science culture. The absurdity of Snow’s socialist prescription for healing the perceived gap hasn’t affected the status of his diagnosis, which has reasserted itself in recent years.

There are all kinds of gaps in intellectual activity, even within scientific disciplines. In the earth sciences, for example, geologists and geophysicists often have difficulty communicating, reflecting different scales of observation and the largely qualitative work of the former in contrast with a more quantitative nature of the latter. However, what science, both contemporary and fifty years ago, has in common with the critical literary arts of the mid-Twentieth Century (Snow’s principal target) is adherence to the primacy of reason in search for truth within each domain. What has changed in recent years is the progressive deterioration of the search for truth in literary circles. As John Paul II writes of philosophy:
This has given rise to different forms of agnosticism and relativism which have led philosophical research to lose its way in the shifting sands of widespread skepticism. Recent times have seen the rise to prominence of various doctrines which tend to devalue even the truths which had been judged certain. A legitimate plurality of positions has yielded to an undifferentiated pluralism, based upon the assumption that all positions are equally valid, which is one of today's most widespread symptoms of the lack of confidence in truth. Even certain conceptions of life coming from the East betray this lack of confidence, denying truth its exclusive character and assuming that truth reveals itself equally in different doctrines, even if they contradict one another. On this understanding, everything is reduced to opinion; and there is a sense of being adrift.2
John Paul II’s observation applies not only to philosophy in relation to the search for religious and metaphysical truth; it also applies to the sciences. Even the fundamental theories which undergird contemporary science are under attack by some “post-modernist” philosophers, critics, and social scientists.

Despite the attack on science from part of the Left, a greater amount of attention is devoted to science’s apparent conflict with religion. The most obvious apparent conflict is between earth, cosmic, and life scientists on one side and certain fundamentalist Protestants on the other.  So-called creation scientists treat the first few chapters of Genesis as a scientific monograph: Creation took the same amount of time as seven contemporary rotations of the earth on its axis; plants and animals appeared on earth in the same order as the sacred text presents them; each species, including Man, was created separately; and the flood Noah and family survived affected the entire world. The “evidence” produced by the creationists is, frankly, an embarrassment of increasingly convoluted attempts to “prove” the youth of the world: 10,000 years maximum. In contrast, many, many independent measures by diverse scientific disciplines are consistent with the Earth and rest of the solar system having formed between 5 and 4.5 thousand million years ago, while the visible universe is on the order of 3 times older. Just a few of these lines of evidence are (1) isotopic dates of rocks and minerals from very different systems: uranium-lead, strontium, and potassium-argon, (2) thicknesses and fabrics of sedimentary rock sequences compared with contemporary accumulation rates of sediments of similar fabric (thickness divided by rate approximates duration, not counting hiatuses in deposition), (3) consistency of dated geomagnetic reversal chronologies in rocks at sea and on land, (4) shift in light frequencies with distance: the expanding universe, beginning approximately 13 billion years ago, (5) long-term modeling of the solar system in comparison with observations, (6) evolutionary sequences of stars (7), and organic mutation rates.

The gross inconsistency of creation “science” with observation (and, frankly, even Sacred Scripture itself, e.g., compare Genesis with the Book of Job) has led some scientists and others to dip a broad brush into creationist silliness. They then apply it to the rest of Christianity, thereby portraying a foundational conflict between science and faith. However, that perceived conflict doesn’t adhere to the canvas of traditional Christian theology.

Perhaps the most prominent of the scientific nonbelievers is Richard Dawkins3. In a different category is the late Stephen Gould4, who argued that there is no conflict between the two cultures because science and religion occupy different, non-overlapping domains. In short: to Dawkins, science and faith are incompatible (and, explicitly, science is superior because of its remarkable success in explaining the visible universe over the past four centuries); to Gould, two cultures exist, yes, but are separate and distinct, if not equal.

To many Catholics in the pew, the apparent conflicts between science and faith may seem distant, off in an irrelevant ivory tower. However, controversy fomented by the active nonbelievers is encountered in the evening news and leaks into their children’s classrooms, especially in college, challenging faith and its practice, weakening the fruit of the virtues, and replacing the moral strictures with relativistic ethics and morals. The vocal advocates of the superiority of science have no small role, whether conscious or not, in the prominent, growing culture of death (yes, another culture) that is progressively dominating the West.

Is there a conflict between science and faith, or do they occupy entirely separate, unbridgeable islands? Or, is there a third alternative?

Catholic teaching is all-encompassing: the Nicene Creed confesses that God made all that is, heaven and earth, the seen and unseen. The seen includes all that humanity can perceive through our own efforts. In the time of Galileo the seen included planets, moons, and comets, and microscopic views of plants and animals. In our own time the seen includes quarks and bosons, electromagnetic fields. In a not-too-distant future, the seen may include superstrings, Higgs’ boson, and multiple universes – objects of contemporary scientific speculation – if they exist at all.

Implicitly, the seen includes even the ability of humans to observe their own world, to make inferences, to postulate explanations, and formalize hypotheses into theories. The actually or potentially seen world even includes objects of logical inference such as the abstractions of higher mathematics in which virtual objects may have no analogy in the observable physical universe.

Orthodox Catholic understanding of the nature of the created, visible universe could be viewed by a Stephen Gould as itself distinct and separate from the world, except for one critical point – God’s intervention via revelation first to Abraham, culminating in the Incarnation. The object of Christian faith, the Son of God, entered his own creation, became man, and dwelled among us.

From the perspective of a Richard Dawkins, the scientist (not ideologue), is the coming-in-the-flesh of Christ Jesus destructive of scientific understanding? The Incarnation is beyond scientific investigation. Signs and wonders are similarly outside the realm of scientific law, although historical analysis of the Gospels provides persuasive evidence that the miracles of Jesus have a historical basis.5 If the natural law-giver is the Divine law-giver, natural law does not conflict with Divine Law, although the former is far inferior to the latter.

Perhaps a more relevant question might be: how can the fact of the Incarnation be compatible with the way the world works? It is a principle of theology that, beyond natural inferred knowledge that God exists and is the Creator of all, anything more we know of Him is that which he chooses to reveal. The Incarnation is the ultimate revelation of the Son of God come-in-the-flesh within his creation. That is, the Divine Law-giver, by emptying himself6, entering the natural, subjected himself to its natural laws – laws which he himself ordained.

How can this be? Consider a couple of parallel developments since World War II: information theory and fractals/strange loops.

The determinacy of early science, especially typified by Newton’s law of gravitation, bred a kind of growing hubris that dominated scientific advance well into the Nineteenth Century (and persists in another form even to the present day). This determinism led some early scientists to deny the existence of a personal God; rather if there is God, He created a deterministic universe with no room for freedom, no space for His interaction with humanity. With the development of thermodynamics and statistical physics in the Nineteenth Century, emphasis on deterministic “laws” faded somewhat, and was succeeded by intrinsic non-determinism of quantum mechanics in the early Twentieth Century.

By mid-Twentieth Century the underpinnings of thermodynamics, especially its Second Law, were shown to share a common foundation with what came to known as Information Theory. Even before Snow’s coining of “Two Cultures” the foundation of what became known as fractals7 – measures of multiple scale self-similarity -- was constructed on information theory, and first applied, not to natural phenomena, but patterns in word usage7 and, subsequently, to economics8,9, before recognition in Nature. Another formulation of recursive self-similarity, termed “strange loops”, was uncovered in music, art, and mathematical logic10.

The repetitive patterns observed in fractals and strange loops, whether natural or human-created, share at their bases the same two abstractions: analogy and identity. A mathematical fractal involves deterministic repetition of a pattern or constraint, or iterative solution of particular problems, over a range of scales. A statistical fractal, for example a cumulative distribution of earthquake magnitudes from Southern California, is remarkably simple and implies a common stress-strain regime at a spectrum of scales. Word frequency distributions possess a similar pattern, as noted above.

Near the end of the last century, a plethora of studies investigated the existence of fractals in natural phenomena and, to a lesser extent, human creativity. However, what most of these studies lacked was a sense of the “why”. Some fields are characterized by sterile arguments as to whether a particular phenomenon is indeed fractal, without consideration of what a fractal pattern really means. While a partial answer occurs at the foundation of the concept7, it wasn’t until the last few years of the Millennium that an answer was provided: fractals represent a repetitive pattern (implicitly produced by a recursive process) that manifests maximum entropy over a range of scales11.

Consider the significance of maximum entropy?12 Given a very large number of similar, if not identical, objects (could be particles, atoms, earthquakes, words, etc.) and a large number of states they, collectively, could be in, that state which is most probable has maximum entropy. Persistence of a pattern across multiple scales or through numerous different domains manifests a fractal. It is also an indication of the ability of each domain to accommodate that particular pattern or bit of information. For example, within a single chemical system, the possibility of a particular reaction depends on the reactive elements present and their proportions, sometimes the presence of catalysts, relative acidity (pH), temperature, and pressure. Chemical equilibrium is achieved as reactions occur and achieve equilibrium – maximum entropy.

Entropy, a measure of the dispersion of heat in thermodynamics, is at the core of the Second Law of Thermodynamics. Ironically, Snow1 presented ignorance of the Second Law as an example of literary ignorance, without himself understanding its relevance to not only science but also artistic creativity, as discussed below. Entropy governs a wide, wide variety of natural phenomena, such as chemical reactions in the earth and planets, the evolution of the Cosmos, nuclear reactions in stars, and processes in the growth and evolution of living organisms. Further, maximum entropy, more generally a measure of information, produces fractals across scales and domains, which may or may not involve chemical processes. Examples of both chemical and morphological maximum entropy are complex inorganic crystals which show fractal structure, such as quartz and calcite geodes and manganese oxide dendrites. These plantlike structures are actually inorganic crystal precipitates forming self-similar, fern-like crystals along fracture and bedding surfaces in rocks.

So far this chain of reasoning has gone from the structure of human language, patterns in economics, “fractals” and “strange loops”, and their presence in nature and the arts. Maximum entropy appears in nature and human communication by propagating kernels of information from one level or scale to another, from one domain to another domain. And, each level or domain must have the capacity to receive, contain, and propagate the same, or component parts of, the information.

What is the information? It can be a particular chemical molecule that creates self-similar structures as it forms. It can be a word pattern in a text. It can be a stress complex along a tectonic zone subject to spontaneous fracturing, accompanied by release of mechanical energy – and earthquakes. It can be a full-fledged scientific theory applicable in a wide variety of domains.

If we liberate the term fractal from strict quantitative scale similarity, and allow a more qualitative application, we can see that analogous information propagation in poetry, for example, is fractal-like. The rhythm of a sonnet, with rhyming couplets, draws attention to itself, to the individual similar words, in enunciation, in syllabic parallelism, and in meaning. Music, with repeated, inverted, compressed or expanded themes, sometimes overlapping (as in a typical fugue10) may have formal strange loop or fractal character; with the addition of lyrics with rhyme, double entendre, irony, and/or reference to other musical pieces, fractal character may be intensified and permeate an entire work.

The kernel of a fractal is self-similarity. And similarity, or analogy, is as old as human communication. For a mathematical fractal, similarity is present in attainable scales, virtually by definition. Further, the algorithm, the sequence of steps by which a particular fractal object is calculated or generated, can be itself fractal.

Gödel’s famous incompleteness theorem is proved iteratively with explicit translation of each previous symbolic step into an equivalent numerical symbol which is then inserted into the next step; the iterations form a strange loop14, i.e. fractal. Fractal structure is not explicitly part of Gödel’s proof algorithm, but is inherent in almost any such recursive algorithm.

A statistical fractal shows self-similarity over a range of scales, but the similarity breaks down at either extreme. Cumulative earthquake magnitude distributions are self-similar only over a range of magnitudes. In Southern California, for example, the maximum Richter magnitude observed while records have been kept is between M7 and M8, while below about M1 to M2, the self-similar pattern is lost. The maximum regional magnitude represents a combination of the finite strength of the earth’s crust combined with the rate at which stresses are applied to the crust. For lesser magnitudes, the distribution of seismometer stations is not dense enough to detect all the events in the region but only those close to the stations; thus the detection of an earthquake of a particular size is dependent not only on magnitude, but on distance from the earthquake to the seismometer. Analogous maximum and minimum limits apply to other statistical fractals.

The very working of science, hypothesis generation, testing, modification, and growth into theory and paradigm, is itself recursive. Paradigms, such as plate tectonics continue to be elaborated, expanded, and clarified in a fractal-like manner.

Within the realm of conscious human creativity, the idea of a fractal might seem to be of remote relevance. Similarity and analogy typically involve comparisons across only two or three levels or domains. Exceptions include the classic structure of a fugue in which four levels are present in the different “voices,” excluding the implicit self-similarity of musical harmony.

Virtually all forms of classical English poetry involve some degree of self-similarity, from simple rhyme and meter, to more sophisticated internal rhyme and the echoing or expansion of an idea from one verse to the next. Modern poetry, while attempting to escape the strictures of classical form, nevertheless evoke a kind of external similarity by intended or accidental reference to other works or experiences of the poet, reader, and/or hearer.

Shakespeare’s Sonnet 130 invokes echoes of other sonnets, including those not his own, by contrast and irony rather than comparison:

My mistress' eyes are nothing like the sun;
Coral is far more red, than her lips red:
If snow be white, why then her breasts are dun;
If hairs be wires, black wires grow on her head.
I have seen roses damasked, red and white,
But no such roses see I in her cheeks;
And in some perfumes is there more delight
Than in the breath that from my mistress reeks.
I love to hear her speak, yet well I know
That music hath a far more pleasing sound:
 I grant I never saw a goddess go,
My mistress, when she walks, treads on the ground:
And yet by heaven, I think my love as rare,
As any she belied with false compare.

When the resonance of similarity is observed or experienced, a kind of maximum entropy process is at work. A core symbol or idea, or other informational object is realized in different levels of meaning or memory; the presence of analogous structures or the capacity for their presence to be realized is like rapid formation of an elaborate cluster of crystal, whose form propagates in a self-similar manner. As ions “float” in the liquid surrounding the crystal, they either attach to existing atoms or molecules already in the crustal, nucleate with adjacent ions forming new, proximal crystals, or, as defects in the crustal occur, the geometry of new growth is, perhaps, displaced or rotated relative to older crystals. The process involves millions of ions and atoms metaphorically jostling around, bumping into one another repeatedly, until just the right configuration is achieved for addition to the growing structure.

Ideas, words, thoughts, impulses, smells, sounds, and taste continually impinge on the person. Those that stick or stimulate do so because of a capacity for realization of similarity or complementarity. What may be particularly striking is the experience of similarity in the unexpected. The resonance of the experience of novelty may be so striking as to almost totally engage one’s attention. The mathematician who proves a theorem for the first time, the scientist who produces a new explanation for an observed phenomenon, a museum visitor who sees a particularly stunning work of art: all know the bracing effect of discovery. Something outside of oneself exposes something within, because of the manifestation of a part of the self previously hidden.

Does this mean the mathematical theorem or scientific model was latent in all of its wholeness and beauty in the mathematician or scientist? No, but what was latent was the experience and knowledge of parts – acquired through recursive experience, practice, and study – suddenly assembled into a coherent whole.

Such experiences can also be seemingly trivial or even absurd. The “lowest” form of humor, the pun, explicitly capitalizes on artificial contrast of meaning in a word or phrase, resulting in absurd juxtaposition of contexts or domains which have otherwise nothing in common. Some delight in such surprising absurdities while others groan.

In contrast to a pun or double entendre, authentic similarity is pervasive and deep, even if incomplete. Deep similarity develops from level to level, elaborating and refining, expanding and contracting. Such analogy may bifurcate into complementary bits of information, branching like trees, but with each child branch resembling or elaborating only part of the parent branch. Such deep similarity finds its analogy in multifractals – fractal similarity of several informational trends across a range of scales yet within the same dynamic system, whether natural or human-generated.

The plays and sonnets of Shakespeare manifest deep similarities, both within each play, between works in the corpus, and with previous and subsequent creative works. Identifying sources for each of his plays has become a cottage industry. The mid-Twentieth Century musical comedy Kiss Me Kate, with its play-within-a-play-within-a-play15, extends the Bard’s humorous story into a Broadway milieu, with elements of Vaudeville thrown in. The plot of Taming of the Shrew is mirrored in the backstage relationship of the principal characters. In the backstage plot, two parallel love triangles are present, with the male lead at the common apex of both triangles. In Taming Petruccio has but to “compete” with Katherine’s succession of unsuccessful suitors, and the younger sister must wait for Katherine to wed before she, too, can marry her preferred suitor. Music, illuminating plot and character, is present in all three “plays.”

Love triangles, apparent or contrived, recur in even Shakespeare’s dramas, such as Hamlet (although the triangle preexists and is broken prior to the first line of the play: the collapsed triangle a result of murder of the king by his brother, who subsequently marries the widowed queen). In the movie Casablanca the primary love triangle arises from misunderstanding: Ilsa believes her husband Victor to be dead, freeing her to love again: enter Rick. When Victor suddenly reappears in Ilsa’s life, she cannot explain the truth to Rick. Subsequently thrown together in the northwest African outpost of non-Vichy France, Rick and Ilsa are tempted to escape together. However, Rick’s latent conscience is aroused as he rescues a young couple from entrapment in a cynical triangle of necessity, leading him to send Ilsa and Victor off to safety as he and Captain Renault fade into fog.  

A ground-breaking musical play, Oklahoma, consists of a primary dramatic and secondary comic triangle. The two primary triangles (Curly, Laurey, and Judd and Will, Ado Annie, and Ali Hakim) are linked by a minor female character (Gertie via Curly and Ali Hakim), thereby creating a trimesh with two additional, but transient triangles. The working out of the triangles via parallel plots, is elaborated via dialog, song, dance, costume, and set.    

Love triangles repeatedly appear in Greek myth and drama through the Old Testament to the present. The tragedy of Oedipus arises from a combination of ignorance and “fate,” as the hero kills his father, only to marry his mother. David allows lust to lead to the wrongful death of one of his own soldiers in order to claim the man’s wife for himself.

One might ask whether a mathematical fractal is applicable by analogy to recurrent themes in secular and sacred literature. The examples above apparently don’t “scale” mathematically – an intrinsic property of exponential fractals. To the extent that triangles range from tragic to dramatic to comedic and from primary to secondary to even tertiary, a different kind of “scaling” is present. In both Casablanca and Oklahoma! the secondary triangles illuminate and, perhaps, even motivate the collapse of the primary triangles.
The musical play My Fair Lady is based on G. B. Shaw’s comedy of manners, Pygmalion, which is, in turn, based on the Greek myth of Pygmalion and Galatea. Henry, Eliza, and Freddie comprise the obvious triangle in both My Fair Lady and Pygmalion), but is there a secondary triangle, as in Oklahoma! and Casablanca? In a sense there is: Alfred (Eliza’s father), Alfred’s unseen woman, and his luck or “freedom”. When Alfred’s involuntary entry into the middle class induces him to finally marry, giving up that freedom, the event foreshadows, pserhaps even motivates Henry’s recognition of his latent affection for Eliza. (Do they marry after the curtain falls? The play is situated in Edwardian England.) Within My Fair Lady, substantial satire is manifested in Henry’s acute lack of self-awareness, comically projected on those around him, especially women, and most especially Eliza. “A Hymn to Him” (“Why can’t a woman…”) should make every adult male nervous, as the depths of irrationality and insecurity beneath surface self-regard are progressively unearthed as the plotline advances.

The novels of Jane Austen revisit distinct themes, even similar types, while producing deeper questions, if not insights, into the nature of male and female relations – always from the perspective of the principal female character. Elizabeth Bennet, Elinor Dashwood, and Anne Eliot are cut from the same cloth. Sensible, balanced, and solid, their varying degrees of self-deceptions are amplified in Emma Woodhouse’s misadventures, although Emma’s eventual May-December marriage to George Knightley mirrors that of Marianne Dashwood to Colonel Brandon, rather than of similarly aged Elinor and Edward.16 Is there a comparable set of novels written by a man from the perspective of a principal male character, which similarly probes the mystery and necessity of courtship and marriage? Similarity in characterization, the drama of the single woman finding the right man, and disordered familial relations are found in each of the Austen novels.

From a Christian perspective, the greatest depths of self-similarity are found in Sacred Scripture. The Church Fathers emphasized the presence of anticipatory types of Christ Jesus in the principal personages of the Old Testament: Adam, Noah, Abraham, Isaac, Jacob (Israel), Joseph, Moses, David, and the Prophets, types referenced in the Gospels, Letters, and Book of Revelation of the New Testament. The prophets, especially Elijah, anticipate both John the Baptist and Jesus. Similarly, key women anticipate Mary: Eve, Sarah, Hannah. Sarah and Hannah also are echoed in Elizabeth. Numerous symbols in the Hebrew Scriptures were also recognized as fulfilled in Mary: ark (of Noah Moses, and the Covenant), burning bush (aflame within without consumption), Jerusalem, Israel, bride, mother of the bridegroom.

Faithful and unfaithful marriage in the OT is a persistent reference point for the relation of the LORD to Israel, especially in Isaiah, Jeremiah, Hosea, and Malachi, as well as the Psalms. The Song of Songs acquires deeper significance when passionate human love is viewed through the lens of divine-human love: “deep waters cannot quench love, nor floods sweep it away” (8:7).

In the Catholic Rite of Baptism the water is blessed via invocations of Salvation History: the Spirit hovering over the waters at the dawn of Creation, the great flood, the escape of Israel through the Red Sea, and Baptism of Christ in the Jordan. Water or its absence is repeatedly encountered at other key points in the drama of Salvation History: Joseph is temporarily placed in a dry well before being sold into slavery. In Egypt, Joseph attains power by anticipating a famine – the same famine that brings his brothers and fathers to Egypt. The infant Moses is saved from death via an ark floated down the Nile. Moses brings forth water from the rock during the forty years of desert wandering. Joshua leads the remnant of Israel across the Jordan River into the Promised Land. Isaiah invokes water (“All you who are thirsty come to the water!”17). Jeremiah, like Joseph, is placed in a dry cistern by his enemies. The exiles weep by the waters of Babylon. Water pours forth from the new temple in Ezekiel’s mystical vision.

Water and marriage are repeatedly linked, as well: the future brides of three Patriarchs are encountered at wells: Rachel (Isaac), Rebecca (Jacob), and Zipporah (Moses). The Samaritan woman, with her successive husbands and lovers representing the historical unfaithfulness of the northern kingdom of Israel, encounters the Bridegroom and the waters of eternal life at Jacob’s well.

Analogy and similarity, archetype and symbol, meme and theme, form the core of human communication, from simple conversation to classic literature and drama to scientific theory to Sacred Scripture. The realization of effective communication is a process drawing on the nature of the communication medium, the knowledge and experience of sender (author) and receiver (hearer, viewer, reader). The probabilistic basis of fractals and recognition of the processes that facilitate their formation have their analogy in each of the dimensions of human communication. In other words, the laws that govern Nature are also manifest in human interaction and creativity.

For the Christian, God the Creator created the substance and nature of all that comprises the physical universe, including its laws. And, the first chapter of Genesis, indeed the whole of Scripture, makes it clear that His Creation is distinct from Him while present to Him. But, Genesis also tells us that the first man and woman were created in His image and likeness; analogy and similarity not only describe the divine origin of Creation, but are explicitly invoked in our first parents. By implication, then, all children of Eve are created in that same image and likeness, but each one manifests a different dimension and quality in their specific image and likeness.

Consider Bonaventure’s medieval Franciscan-Scholastic information theory:

The entire sense world, therefore, in its three classes of objects, enters the human soul through apprehension. The exterior sense objects are the first which enter into the soul through the gates of the five senses. They enter, I say, not through their substance, but through their likenesses, which are first produced in the medium; and from the medium they enter into the organ and from the exterior organ into the interior organ and from this into the apprehensive faculty. And thus the production of the image in the medium and from the medium in the organ and the turning of the apprehensive faculty upon it bring about the apprehension of all objects that the soul grasps from outside.18

More than this, however, is recognition of the profound foundation of Christian belief: the Incarnation. The God in whose likeness man was created Himself became man. Image and likeness became, in one person, more than mere similarity; they became identity. The Son of God entered his Creation. The laws that governed the coming into being of the Universe and of humanity, also, from all eternity, were destined to welcome the Diving law-giver.

Applying Douglas Hofstater’s formulation14, the Incarnation is a “strange loop.” Indeed, it is the strangest of loops, God becoming man. Love of husband and wife is a sacramental realization of the sacrificial love of the Bridegroom for his Bride. The earth from which man is formed gives forth the fruit of wheat and wine, which on the altar become more than symbols, but realities of the continuing presence of the Incarnate Son to and within His Creation. The first-born son, spared from death by the staying hand of an angel, is replaced by a young ram. The first-born lamb is the Passover sacrifice, recalling the escape from Egypt. The Son of God is not spared, and by his sacrifice becomes the Lamb of God who takes away the sin of the world.

To say that Salvation History is fractal and the Incarnation a strange loop is not to trivialize either mystery, but to recognize the effect and ultimately the presence of the Law-Giver in His observable Creation. What science has discovered and continues to encounter is the real Intelligent Design. No, unaided science will never find Christ before the end of time, but the faithful scientist cannot help but see him reflected in every valid experiment, theory, and paradigm. God created the Universe and the way the Universe works for his own pleasure, to give a place for created body, spirit, and soul – man and woman – to dwell, for the Son of God to enter into relationship with man and woman, and, ultimately, to bring the crown of creation, the Church of His faithful people, into the presence of His Father, in the power of the Holy Spirit.


  1. Snow, C. P., 1960, The Two Cultures, Cambridge.
  2. John Paul, II, 1998, Fides et Ratio, On Faith and Reason, no. 5.
  3. Dawkins, R, 2006, The God Delusion, Bantam Books, New York, 406 p.
  4. Gould, S. J., 1999, Rocks of Ages, Ballantine, 256 p.
  5. Brown, R., 1997, An Introduction to the New Testament, Yale University Press, New Haven, 928 p.
  6. Philippians 2:5-11.
  7. Mandelbrodt, B. B., 1953 An informational theory of the statistical structure of language, in  Communication Theory, the Second London Symposium, Edited by Willis Jackson, Academic, New York, p. 486-504.
  8. Mandelbrodt, B. B., 1957, Application of thermodynamical methods in communication theory and in econometrics. Institut Mathématique de l'Université de Lille.
  9. Mandelbrodt, B. B., 1960, The Pareto-Lévy law and the distribution of income, International Economic Review: 1, 79-106.
  10. Hofstater, D., Gödel, Escher, Bach: an Eternal Golden Braid, 20 Anniversary Edition, Basic Books, New York, 832 p.
  11. Pastor-Satorras, R. and Wagensberg, J., 1998, The maximum entropy principle
    and the nature of fractals. Physica A 251, pp. 291-302.
  12. Grandy, W, T., 2008, Entropy and the Time Evolution of Macroscopic Systems, International Series of Monographs on Physics, Oxford University Press, Oxford, 209 p.
  13. Hofstater, D., Gödel, Escher, Bach: an Eternal Golden Braid, 20 Anniversary Edition, Basic Books, New York, 832 p.
  14. Hofstater, D., I Am a Strange Loop, Basic Books, New York, 436 p.
  15.  “We Open in Venice” implies a traveling troupe presenting Taming of the Shrew across Italy – the intermediate play sandwiched within Shakespeare’s play and the backstage plot.
  16. Elizabeth Bennet: Pride and Prejudice; Elinor and Marianne Dashwood, Colonel Brandon, Edward Ferris: Sense and Sensibility; Anne Eliot: Persuasion; Emma Woodhouse, George Knightley: Emma.
  17. Isaiah 55:1.
  18. Bonaventure, Itinerarium Mentis ad Deum, The Soul’s Journey into God, Chapter 2, ¶ 4, in Bonaventure, The Classics of Western Spirituality, 1978, Paulist Press, Mahway, NJ, translated by E. Cousins, 353 p.

Monday, January 13, 2014

In the appendix of the paper I reference below is a summary derivation of the relationship  between entropy and fractals (power laws). The appendix is also posted in another blog, Plate Frames. The introduction to the post reads...
In a paper published a couple of years ago (Pilger, 2012), I describe the application of a simple principle, transformed into a distinctive abstract object, to an optimization problem (within the plate tectonics paradigm): simultaneous reconstruction of lithospheric plates for a range of ages from marine geophysical data . It is rare that the relation of the principle, maximum entropy, with a particular transformation, power-series fractals, is recognized, since Pastor-Satorras and Wagensberg derived it. I'm unaware of any other application of fractal forms to optimization problems analogous to the paper. The following derivation is taken from the 2012 paper, with slight modification, in hopes that it might prove useful in other fields, not merely the earth sciences, but beyond. I'm investigating  applications in a variety of other areas, from plate tectonics, to petroleum geology, and, oddly enough, the arts.
Pilger, R. H., Jr. (2012) Fractal Plate Reconstructions with Spreading Asymmetry, Marine Geophysical  Research, Volume 33, 149-168. (rexpilger (at) gmail (dot) com.)

Thursday, December 05, 2013

My sister-in-law keeps posting photos whose beauty comes, in part, from fractal structure (a site with other examples of fractal art, natural or human-made). 

Here is her latest example:

Wednesday, December 22, 2010

Fractals and Plate Tectonics

Can fractal criteria be used in deriving plate reconstructions of asymmetrically spreading ridges? See: link.


Wednesday, November 17, 2010

Peer Review

An article in Physics World describes an "experiment" in peer review and its effect on the quality of published scientific research.
Just a small number of bad referees can significantly undermine the ability of the peer-review system to select the best scientific papers. That is according to a pair of complex systems researchers in Austria who have modelled an academic publishing system and showed that human foibles can have a dramatic effect on the quality of published science.