Fibonacci, Fractals, and Inorganic Teleology

Among the most philosophically challenging scientific data of the last half century are those relating to the physical constants  of the universe (listed below) which allow it to be a cosmos instead of utterly disordered chaos. These constants were “finely tuned” to their present values when the universe came into existence out of absolutely nothing roughly 13.7 billion years ago. You cannot derive their values from something more basic; they simply occurred as “givens” from the first second of our universe’s existence. These values did not develop but were present full-blown at singularity. They did not evolve: they simply were.
Cosmologists Barrow and Tipler wondered what would happen if they were slightly different. Tinker ever so slightly with the values of any of the basic physical constants, and life would have been impossible, not just life of our kind, but life of any kind that involves complexity.

Because of their highly ordered nature, random origin of the constants has been widely conceded to be effectively zero probability (cf. physicist Donald Page has calculated the odds as 1 in 10,000,000^124. By comparison, there are 10^18 seconds since the creation of the universe and around 10^80 atoms in the observable universe).

Mathematician Emile Borel affirmed that anything with odds of happening less than one in 10^50 is impossible (Borel is best known for creating the the first effective theory of measuring sets of points beginning the modern theory of functions of a real variable). Random origin of the constants is well beyond this threshold -by orders of magnitude; selection by lottery would only overcome this statistical obstacle if there were an infinite number of unobservable universes from which ours was selected, yet a universe generating “machine” would also have to be exceptionally highly ordered too, and contemporary physicists have recently suggesting that multiple universes would be clones of one another rather than infinitely variable as the infinite unobservable multi-universes lottery selection theory requires.

Many physicists and philosophers have been attracted to similar arguments in the last thirty years (during which the ramifications of the delicately balanced physical constants first came to our attention; cf. the lecture by Dr. Francis Collins (PhD, & MD), first and long-time director of the Human Genome Project, here.  Collins’ PhD is in Quantum Mechanics, though his focus now is on genetics). It was this issue which former leading atheist and world famous philosopher Antony Flew cited as convincing him to abandon atheism for belief in God (many atheists claim it was rather because Flew must have become senile!).

Contemporary thinkers who remain atheists attempt to escape this conundrum in the only way left to them: they postulate a hypothetical infinite number of unobservable universes from which a highly ordered one could have been a random occurrence. Contemporary atheism is thus forced to argue for the actuality of an essentially zero-probability event by postulating something in principle unobservable (non-scientific/metaphysical).

“‘I can’t believe that!’ said Alice. ‘Can’t you?’ the Queen said in a pitying tone. ‘Try again: draw a long breath, and shut your eyes.’ Alice laughed. ‘There’s no use trying,’ she said: ‘one can’t believe impossible things.’ ‘I daresay you haven’t had much practice, said the Queen. ‘When I was your age, I always did it for half-an-hour a day. Why, sometimes I believed as many as six impossible things before breakfast.’” -Lewis Carroll, Through the Looking Glass

The slightest alteration of the following physical constants would result in a universe incapable of supporting life -not just life of our kind, but life of any kind that involves complexity, and in a universe that would be chaos rather than cosmos:

Gravitational Coupling Constant
Strong Nuclear Force Coupling Constant
Weak Nuclear Force Coupling Constant
Electromagnetic Coupling Constant
Ratio of Protons to Electrons
Ratio of Electron to Proton Mass
Expansion Rate of the Universe
Entropy Level of the Universe
Mass of the Universe
Uniformity of the Universe
Stability of Protons
Fine Structure Constants
Velocity of Light
Distance Between Stars
Rate of Luminosity of Stars
8Be, 12C, and 4He Nuclear Energy Levels.

An infinite number of unobservable universes -even were it the case- would not, of course, necessarily “belong” to our atheist friends who need it so badly to account for zero probability of random origin of the universe’s physical constants  at singularity; in fact it would be a perfect case scenerio of the ancient Augustinian cosmological theodicy of pleroma, which posited all possible varieties and ranges of entities might actually exist; we will leave that subject for a  possible future post; let us now move along to consider the central topic of this essay.

B. FRACTAL GEOMETRY:  ON FINDING ORDER IN UNEXPECTED PLACES

In what follows, we will explore one of the most astonishing and brilliant discoveries of the last century. Although initially a mathematical discovery, its implications can be seen to permeate our cosmos from the microsphere to the macrosphere to a degree that is nothing less than mind-boggling, in a manner that was utterly unknown just a few decades ago.

“This is how God created a system that gave us free will. It’s the most brilliant maneuver in the universe, to create something in which everything is free! How could you do that?! …exploring this set I certainly never had the feeling of invention. I had never the feeling that my imagination was rich enough to invent all the extraordinary things. I was discovering them; they were there although no one had ever seen them before. It’s marvelous! A very simple formula describes all of these very complicated things. Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity? We’ve all read stories of maps that revealed the location of some hidden treasure. In this case the map is the treasure!” -Benoit Mandelbrot, in Fractals: the Colors of Infinity (Arthur Clarke Documentary).

Why do certain patterns (Fibonacci/fractal patterns) constantly reappear throughout nature (organic and inorganic!) in phenomena as incredibly diverse as neuron firing patterns, trees and flowers, lightning, networks of veins in body, crystal structures, the inner structure of lungs, hearts, and other organs, hurricanes, spiral galaxies, viruses, most formations of plant life, cellular microtubules, chemical structures (e.g. platonic solids), family trees, snowflakes, even thoughts? (if you are not a mathematician or scientist, you will be able to more fully appreciate what all the things in this essay –indeed most things in our universe- have in common after you watch the video which follows

II. FRACTALS IN HUMAN PHYSIOLOGY (Yale University Biology Dept.)

“Some of the most visually striking examples of fractal forms are found in physiology: The respiratory, circulatory, and nervous systems are remarkable instances of fractal architecture, branches subdividing and subdividing and subdividing again. Nice pictures are provided in Goldberger, Rigney and West. Although no clear genetic, enzymic, or biophysical mechanism yet have been shown to be responsible for this fractal structure, few doubt this. Careful analysis of the lungs reveal fractal scaling, and it has been noted that this fractal structure makes the lungs more fault-tolerant during growth. The heart is filled with fractal networks: the coronary arteries and veins, the fibers binding the valves to the heart wall, the cardiac muscles themselves, and the His-Purkinje system, etc.

“In addition to falut-tolerance during growth, fractal branching makes available much more surface area for absorption and transfer in bronchial tubes, capallaries, intestinal lining, and bile ducts. Kalda has proposed a fractal model of the blood vessel system that achieves a homogeneous oxygen supply throughout the body. Also, the redundancy of fractal structures make them robust against injury. For example, the heart can continue to function even after the His-Purkinje system has suffered considerable damage. From his work on the ability of fractal drums to damp vibrations, Bernard Sapoval deduced another advantage of the fractal character of the circulatory system: “the fractal structure of the human circulatory system damps out the hammer blows that our heart generates.” “The heart is a very violent pump, and if there were any resonance in blood circulation, you would die.” Fractals may save our lives every minute. Here are some casts of animal lungs. Finally, we note the body exhibits dynamical fractals. For example, it is well-known that healthy heartbeats are chaotic rather than regular. A careful plot of heart rates over several time scales reveals self-similar scaling (Goldberger, Rigney and West). http://classes.yale.edu/fractals/Panorama/Biology/Physiology/Physiology.html ; On the fractal nature of the human circulatory system, see here.

III. BLOOD CELL DYNAMICS GOVERNED BY FIBONACCI RATIOS

Blood cell dynamics are also governed by Fibonacci ratios:  http://arxiv.org/abs/physics/0411169 “This paper demonstrates that the pattern of lipid spicules that emerge on the surface of red blood cells in the classic ‘Discocyte to Echinocyte’ shape change is a generative spiral, and presents a qualitative, fluid- driven mechanism for their production, compatible with the work of Douady and Couder. Implications for the dynamics of cell growth, plant cell phyllotaxy, programmed cell death and gravity sensitivity are explained in terms of a new qualitative model of cellular fluid dynamics.”

IV. LEAVES, BRANCHES, FLOWERS, AND BASIC STRUCTURES OF PLANTS

Plant Phylotaxis is the arrangement of plant elements as primordia on the shoot apex, e.g. branches, leaves, petals, stamens, sepals, florets, etc.).] The angle subtended at the apical center by two successive primordia is equal to the golden angle (137.5 degrees) in more than 90% of all plants studied worldwide. Such an arrangement allows incorporate maximum packing efficiency for fruits and seeds, maximal access to sunlight by leaves, etc. See further at http://www.math.smith.edu/phyllo//About/math.html

V. INFORMATION ENCODING IN THE BRAIN

Information encoding in the brain is also a Fibonacci process, as explained by Harald Weiss and Volkmar Weiss, “The golden mean as clock cycle of brain waves” in Chaos, Solitons & Fractals Volume 18, Issue 4, November 2003, Pages 643-652; cf. abstract and full article here; cf. also Weiss, Volkmar, “Memory Span as the Quantum of Action of Thought,” Cahiers de Psychologie Cognitive 14 (1995) 387-408 here.

VI. THE FINE STRUCTURE CONSTANT AND THE STRUCTURE OF SPACE-TIME
The fine structure constant and the structure of space are also according to Fibonacci. See Carlos Castro, “Fractal Strings as the Basis of Cantorian-Fractal Spacetime and the Fine Structure Constant,” Chaos Solitons Fractals 14 (2002) 1341-1351 here.

VII. FIBONACCI AND FRACTAL GEOMETRY: GENERAL OBSERVATIONS
All fractal self-similar structures (whereupon parts resemble the whole object in shape) incorporate in their geometrical design numbers which are functions of the golden mean.

In case the reader is unfamiliar with fractal geometry, the relationship between Fibonacci and fractals is illustrated simply in the photo above which shows phi (which represents the average of the Fibonacci series) in the Mandelbrot set. The Mandelbrot set is used to generate fractals and fractal art. As anyone knows who has viewed this type of art, amazingly complex and at the same time strikingly NATURE-LIKE and LIFE-LIKE PATTERNS (like insects, leaves, flowers, rock formations, faces, etc,) are quite common with the Mandelbrot “algorithm” (google fractal art!). Random generation is not contra “structure” given an algorithm which “guides” the unfolding; hence chance itself may be ‘governed’ in such a way as the gorgeous symphony of life may still be to a degree “orchestrated”

It is critically important to note that the “algorithm” which generates the ‘similarity in infinite diversity’ we see in nature is not a specific number (phi), but a range of Fibonacci ratios, e.g. from 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 32/21 and so on infinitely. Phi, the average of these ratios, is not a ‘magic number,’ but merely the average of all such ratios. In nature it is not so much phi that is important as it is the SPECTRUM OR RANGE OF VALUES which phi may embody as an average. It is the fact of a Fibonacci range rather than a specific uniform value that especially allows infinite diversity within the algorithmic similarity to be maximized. Thanks to God not all flowers or faces look alike! It is the startling diversity of our universe, in spite of its unfolding within the fine-tuned confines of specific physical constants etc. that never fails to take our breath away.

VIII. MAGNETIC FIELD BORDERS AND ELECTROMAGNETISM

“Also Mandelbrot curves have been discovered in cross-sections of magnetic field borders, implying there is a 3-D mandelbrot equivalent that is closely tied to electromagnetism and therefore a deep structural and fundamental aspect of life, and physical space/time…

“…we see the Mandelbrot relationship to the period-doubling ‘chaos’ equation which is used to describe population expansion, plant growth, weather instability and a host of other physical processes. This relationship also has a habit of popping up unexpectedly in other dynamic non-linear equations (fractals made from Newtons method of deriving a cube-root being the most obvious)

“Think about that for a moment – Take any slice of the magnetic field of the earth, sun, a plant, the data on audio or video tape, and there is our old familiar Buddha looking mandelbrot! This suggests an unknown, yet-to-be-clarified fundamental importance of the Mandelbrot Set in many physical processes. Clearly this is something far more significant than a means of generating visually pleasant mathematical abstractions.

“Many things previously called chaos are now known to follow subtle fractal laws of behavior. So many things turned out to be fractal that the word “chaos” itself (in operational science) had to be formally defined as following inherently unpredictable yet generally deterministic rules based on nonlinear iterative equations. Fractals are unpredictable in specific details yet deterministic when viewed as a total pattern – in many ways this reflects what we observe in the small details & total pattern of life in all it’s physical and mental varieties, too….” -http://www.miqel.com/fractals_math_patterns/visual-math-mandelbrot-magic.html

IX. NANOTECH BOMBSHELL: RANDOM MOTION OF FERROFLUID DROPLETS ON SILICON

Physicists, Stéphane Douady and Yves Couder from the Laboratory for Statistical Physics in Paris in 1992 demonstrated tiny magnetized ferrofluid droplets in pool of silicone oil ultimately form spirals described by the golden angle (see the video here; an abstract in Physical Review (Douady, S. and Y. Couder. 2002. “Phyllotaxis as a physical self-organized growth process,” Physical Review Letters 68 (March 30):2098-2101) can be found here).

X. FIBONACCI AND THE HUMAN FORM:  ADDITIONAL OBSERVATIONS
We have already seen the amazing extent to which Fibonacci inform us about our neural structure, thought structure, organ structure, blood dynamics, circulatory system, etc. etc.; the golden ratio also explains human perceptions of beauty to the degree that standard practice for plastic surgeons and dentists is to carefully employ these ratios in reconstruction of the face and teeth, branching patterns and proportions of skeletal structure, and many other applications beyond the scope of this brief essay.

XI. A “DNA” DOUBLE HELIX NEAR THE CENTER OF THE MILKY WAY?!!

In March of 2006, an incredible news story was released about a double helix nebula found near the center of the Milky Way. “We see two intertwining strands wrapped around each other as in a DNA molecule,” said Mark Morris, a UCLA professor of physics and astronomy, and lead author. “Nobody has ever seen anything like that before in the cosmic realm. Most nebulae are either spiral galaxies full of stars or formless amorphous conglomerations of dust and gas — space weather. What we see indicates a high degree of order.” See the original article at the UCLA website here.

The double helix, mysteriously mirrored in the stars and the DNA of living creatures, is another Fibonacci structure:

XII. ANCIENT ISRAEL’S HOLY ARK OF THE COVENANT AND BRAZEN ALTAR

Intriguing also is the presence of this precise ratio in the Tabernacle of Israel. If it was only there once, it would be more reasonable to attribute its presence to chance, and we are reminded of the exhortation to Moses to ensure he adhered exactly to the heavenly pattern (cited below). This “signature literally written all across the face of the cosmos” appears in two critically important structures which have different dimensions, in the most important artifacts known to ancient Judaism: the Holy Ark of the Covenant, and the Brazen Altar where all Israel’s sacrificial offerings were brought.

2.5/1.5 = 1.6666667 (=5/3)
Exodus 25:10:”And they shall construct an ark of acacia wood two and a half cubits long, and one and a half cubits wide, and one and a half cubits high.”
5/3 = 1.6666667
Exodus 27:1: “And you shall make the altar of acacia wood, five cubits long and five cubits wide; the altar shall be square, and its height shall be three cubits”
Exodus 25:40: “And see that you make them after the pattern for them, which was shown to you on the mountain.”

If one already considers Christian theism of reasonable warrant it becomes difficult not to see the very signature of God in almost every direction one can possibly look -from the heavens to the mirror.

XIII. LARGE SCALE STRUCTURE OF THE UNIVERSE, AND NEURAL CONNECTIONS
From Constantine, David, “They Look Alike, but There’s a Little Matter of Size” (New York Times, Aug 15, 2006): “One is only micrometers wide. The other is billions of light-years across. One shows neurons in a mouse brain. The other is a simulated image of the universe. Together they suggest the surprisingly similar patterns found in vastly different natural phenomena. Mark Miller, a doctoral student at Brandeis University, is researching how particular types of neurons in the brain are connected to one another. By staining thin slices of a mouse’s brain, he can identify the connections visually. The image (below left) shows three neuron cells on the left (two red and one yellow) and their connections. An international group of astrophysicists used a computer simulation last year to recreate how the universe grew and evolved. The simulation image (below right) is a snapshot of the present universe that features a large cluster of galaxies (bright yellow) surrounded by thousands of stars, galaxies and dark matter (web).” (Source by Mark Miller, Brandeis University; Virgo Consortium for Cosmological Supercomputer Simulations; http://www.visualcomplexity.com).

How can the large scale structure of the universe and a mouse neuron have the same structure? The degree to which the Fibonacci ratio is present in both inorganic structures and the bodies of living beings suggests there is something more than genetics and selection alone involved in the latter. Despite a stunning difference in scale there a key similarity: neural networks, like the fine structure of the universe, are (and the large scale structure of the universe) are fractal (http://www.fractal.org/Life-Science-Technology/Publications/Fractal-Neural-Networks.htm)

XIV. WHAT DOES IT TAKE TO BUILD A WORLD?

“What does it take to build a world? This is the central question of my research. My overarching goal is the creation from first algorithmic principles of an entire planet, well-defined everywhere and at all scales, with visual complexity, appearance, and beauty similar to Earth, and to bring that model to real-time performance. Needless to say, this undertaking subsumes a large number of interesting and challenging elements. These include developing our capabilities in visual realism, models of natural phenomena, computational efficiency in such models, and algorithmic art. I am confident that there are enough challenges involved to keep me busy for the rest of my days.

A planet, at the scales of ordinary human experience, is defined by its landscapes. Landscapes are in turn defined by the form of the land, the lighting, the current state of the atmosphere, and by the life forms found within it. My research encompasses the first three, terrain, lighting, and atmospherics; peculiarities of taste and predilection lead me to eschew modeling life forms, leaving them to others to perfect. There is no accounting for taste, and “I love landscapes!” …All successful synthetic terrain models for computer graphics are fractal: That is, they feature complexity resulting from the repetition of form over a variety of scales. The complexity resulting from this repetition of form over many scales leads to the odd idea of fractal dimension: a spatial dimension which is intermediate between the familiar integer-valued (i.e., 1, 2, and 3) dimensions we’re used to dealing with.”

-Professor Ken Musgrave, Fall 1994 George Washington U. EECS Newsletter; http://www.wizardnet.com/musgrave/article.html Dr. Musgrave works with Benoit Mandelbrot, the discoverer of the Mandelbrot set.

XV. SEASHELL STRUCTURE, PLANETARY ORBITS, AND SPIDER WEBS?

All informed observers concede the undeniable and astonishing ubiquity of Fibonacci and fractals in nature. Even some basic natural phenomena which at first glance seem to depart from mathematization according to *phi* (an average of the ratios of the Fibonacci series) can be found, upon closer analysis to contain more Fibonacci than is first apparent to the uncritical eye. As we have emphasized, it is the Fibonacci series which is so prevalent in nature rather than *phi* (the average of said series) per se: phi is a mathematical abstraction of the range of things we see, it is but an average, a human methodological contrivance. There is an additional surprising point to consider regarding phenomena like seashells and planetary orbits, which are Fibonacci on the main, but with variations:  the manner in which variation from Fibonacci occurs in e.g. seashells and planetary orbits actually itself algorithmic according to precise Fibonacci exponents (see www.spirasolaris.ca/llight.html)

XVI. SOME SURPRISING CONCLUSIONS

There are some rather revolutionary conclusions from the recent revolution in fractal geometry stemming from Mandelbrot’s great discovery. First, “chance” itself is “governed” algorithmically to an extent entirely unsuspected just a few decades ago. Yet this algorithmic “governance” is such that infinite variety is not precluded, but rather enabled! The centuries-old philosophical presupposition that chance and governance are opposite has been effectively demolished by Mandelbrot’s amazing discovery. It is also interesting that the discovery was initially within the realm of pure mathematics, rather than the empirical (empirical, i.e. depending on sensory observation); as a video presented below emphasizes, some philosophers of science who prefer to methodologically restrict the “scientific” to the directly observable in a manner similar to that of the Logical Positivists of the early 20th century resisted acknowledging the vast extent to which nature embodies this newly discovered fractal geometry (the demarcation criterion between science and non-science is obviously of critical importance for the philosophy of science; significantly this issue has not been resolved and may be irresolvable in principle (cf. the Kuhn-Popper debate and beyond). Such skepticism, however, could not long survive decades of careful measurement.

Arthur C. Clark produced a fabulous series on fractals titled The Colors of Infinity which documents and describes all this and more, with interviews with such luminaries as Benoit Mandelbrot (who discovered fractal geometry), Stephen Hawking, and many other great pioneers of fractal mathematical applications (the videos are found below). We’ll close this brief essay with some fascinating quotations from some of these men taken from part 5:

“This is how God created a system that gave us free will. It’s the most brilliant maneuver in the universe, to create something in which everything is free! How could you do that?!”

“…Albert Einstein refused to accept the idea of a dice-playing deity. He wrote a letter to Max Borne in which he said ‘you believe in a dice-playing God; I believe in complete law and order.’ So he obviously felt that chance and deterministic laws were not compatible and he preferred the deterministic laws. Now what the Mandelbrot set and Chaos and related things have done for us is to show that you can have both at the same time. So it is not whether God plays dice that matters, it’s how God plays dice.” “I [Benoit Mandelbrot] can tell you exploring this set I certainly never had the feeling of invention. I had never the feeling that my imagination was rich enough to invent all the extraordinary things. I was discovering them, they were there although no one had ever seen them before.” It’s marvelous! A very simple formula describes all of these very complicated things…” “Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity? We’ve all read stories of maps that revealed the location of some hidden treasure. In this case the map is the treasure!” -from Fractals: The Colors of Infinity

© 2010 by katachriston.wordpress.com (text)

Excerpts from “Confessions of a Doe” by St. Justin Popovich

I have heard there are demons. Can those be worse than men?

…Between you, men, and us there gapes a broad abyss, and neither can we ever come to you, nor you to us. You do not have a sense for our worlds. If we, does, were to come over to your side with our heart, it would be as if we entered hell. Time ago, we used to be in paradise. But you, humans, turned that into hell for us. What for you the demons are, you, the humans, are the same for us. The birch-trees told us: “we saw where Satan fell from heaven onto earth; he fell among men, and he has remained there. He, the outcast of heaven, proclaimed: the pleasantest place for me is among men, and I too have my paradise, and this is them: the humans…

You men knowingly and voluntarily embrace sin, evil, and death, and in so doing drag us, without our consent, into them through your vileness and malice, for you had power over us, and so you shall account for us: for all our anguish, misfortunes, sufferings and death.

…I listened to the azure sky whispering to the black earth this eternal truth: on Judgment Day men shall have account for all anguish, for all sufferings, for all miseries, for all deaths of the earthly beings and creatures. All animals, all birds, all plants shall stand up and charge against the human race all pains, all insults, all evil, and every death men have inflicted on them in their proud love of sin. Because, coming along with the human race -both in front and after it -are sin, death, and hell.

Were I to choose from among all creatures, I would prefer a tiger to man, for not being as bloodthirsty as man; I would also prefer a lion to man, for not being as ferocious as man; and also I would prefer a hyena to man, for being less disgusting than man; I would prefer the lynx, too, to man, for not being as fierce as man; would prefer a snake, too, to man, for not being as cunning as man. I would rather prefer any monster to man, for even the most terrible monster is not as scary as man… O, I do speak the truth, and I speak from the bottom of my heart. Was it not man who invented and created sin, death, and hell? And that is far worse than the worst, more monstrous by far than the most monstrous, and more terrible than the most terrible thing ever in all my worlds.

I overheard the brook of tears warbling: “People boast of intelligence, their reason. I judge them for their major deeds: sin, evil, and death. And here is the conclusion I reach: if their creation of sin, evil, and death, is a function of their intelligence then that is no gift but a curse. Intelligence that lives and expresses itself through sin evil and death is a punishment from God. A great intelligence is also a great punishment. They would insult me were they to tell me I had reason as man. If reason is the only distinction of man then I not only renounce it but also damn it. Were my paradise and my immortality dependent on it then unto the ages I would have renounced both paradise and immortality. Without goodness intelligence is a punishment of God. And the great reasoning power without great goodness is an unbearable damnation. With intelligence, but without goodness and tenderness, man is a perfect devil. From the angels of heaven I have heard -while they were washing their wings in my tears, the devil is a great intelligence without a drop of goodness and love. Man too, is the same, if there is no goodness and love in him. An intelligent man -but one without goodness and compassion, is hell for my tender soul, hell for may sad heart, hell for my good-natured eyes, hell for my meek nature. My soul strives after one single yearning: that I would never live in this world or in the next, side by side with a man intelligent, but deprived of goodness and of compassionate tenderness. Otherwise, destroy me, O God, and convert me to non-being! -St. Justin Popovich, “A Doe in Paradise Lost: Confession of a Doe.”

The Laws of Thought: A Short Historiography of a Paradigm in Transition

From James Danaher, “The Laws of Thought,” The Philosopher, Volume LXXXXII No. 1.

Western philosophy to a very large extent has been founded upon laws of thought. We believe that our thinking should strive to eliminate ideas that are vague, contradictory, or ambiguous, and the best way to accomplish this, and thereby ground our thinking in clear and distinct ideas, is to strictly follow laws of thought. Ones like:

The law of identity (A=A),
The law of non-contradiction (A does not equal ~A), and
The law of the excluded middle (either A or not A but not both A and ~A).

In spite of how dominant these laws of thought have been, they have not been without their critics, and philosophers from Heraclitus to Hegel have leveled powerful arguments against them. But the issue does not seem to be whether the laws are applicable or not, but where and when are they applicable? Certainly, the laws of thought have a place, but what is that place?

Both the laws, as well as opposition to them, can be traced to the Pre-Socratic philosophers. It was Parmenides who first formulated the law of non-contradiction. “Never will this prevail, that what is not is.”

Plato also refers to this in the Sophist: “The great Parmenides from beginning to end testified . . . “Never shall this be proved – that things that are not are.”" (Plato, Sophist, 237A)

It may seem strange that the principle of non-contradiction was not part of a natural way of thinking that had its origins deep in our prehistory, but rather was introduced by Parmenides in the 5th century B.C. Even more surprising is the fact that Parmenides’ law of non-contradiction represented a radical break from the Ionian philosophy of nature which preceded it. The Ionian philosophy was based on observation or experience in the ordinary sense. On the basis of such experience, Heraclitus argued that contradictions not only existed but were essential and the basis of a thing’s identity:

“Not only could it be stated that identity is the strife of oppositions but that there could be no identity without such strife within the entity.”

Heraclitus argued that since things changed, they had to contain what they were not. Only such contradictions could account for change. As Heraclitus says, “Cold things grow warm; warm grows cold; wet grows dry; parched grows moist.”

In direct opposition to Heraclitus, Parmenides claimed that identity involved the idea of non-contradiction. What made for the difference between Heraclitus and Parmenides was what they respectively believed were the proper objects of thought. For Parmenides, the things we encounter in our experience make for poor objects upon which to fix our thoughts. Indeed, the things we experience are not suited to provide the kind of knowledge that Parmenides, and so many others who were to follow him, wanted. The kind of knowledge they desired was a knowledge that was fixed and certain. Such knowledge would require objects of thought that were equally fixed and certain. Thus, Parmenides settled on the idea of being itself into which all change would collapse.

The Pythagoreans too desired objects of thought that were fixed and certain. For them, mathematics provided those kinds of objects. Plato too sought similar objects of thought and settled on otherworldly forms that were eternal and immutable. With the Platonic forms, as with Pythagorian numbers and Parmenidian being, the laws of thought are certainly applicable. Thus, Plato endorsed the laws of non-contradiction and excluded middle in the Republic when he has Socrates say,

“It is obvious that the same thing will never do or suffer opposites in the same respect in relation to the same thing and at the same time.” (4:436b)

Of course, Plato was well aware that in order for the laws of thought to work they needed to be restricted, for if left unrestricted they could lead to absurd conclusions. In the Euthydemus, Euthydemus’ brother Dionysodorus argues that Socrates must be the father of a dog, since the dog had a father, and Socrates has admitted that he is a father (298d-299). Since one cannot be a father and not be a father at the same time, Socrates must be the father of the dog. Although Socrates is obviously not the father of the dog, it was not so obvious in Socrates’ day where Dionysidorus’ thinking went wrong. Thus, Plato attempts to sort out where and when the laws of thought apply and where and when they do not apply.

The restrictions Plato places on the laws of thought (i.e., “in the same respect,” and “at the same time,”) are an attempt to isolate the object of thought by removing it from all other time but the present and all respects but one. Thus, although we are involved in many relationships, when we think about ourselves relationally, we must restrict our thinking to one relationship, at one time, in order for the laws of thought to be applicable. Thus, it is not only the Platonic forms that are abstract and apart from the world of experience, but any idea to which the laws of thought are to be applied must also be abstracted from the reality of our experience which is multi-relational and multi-temporal.

Like Plato, Aristotle also believed that the laws of thought, in spite of being controversial, were cornerstones of all right thinking. He argues for them in several places (Metaphysics G, 3&4; De Interpretatione 11, 21a32-33; Topics IV 1, 121a22-4; Sophistical Refutations 5, 167a1-6). It is, however, not so much that he argues for them as he sets them in a proper light. That is, he shows were they are appropriate and where they are not appropriate. Basically, what he says is little different from Plato. He argues that such laws apply only to attributes and attributes at a particular time and in a particular respect.

“The same attribute cannot at the same time belong and not belong to the same subject and in the same respect” (Metaphysics G, 3,1005b18-20).

By limiting the laws of thought in this way, Aristotle overcomes Heraclitus’ claim that identity contains contradictions because the attributes of a thing change over time. By isolating identity in one moment of time, Aristotle abstracts the objects of thought just as Plato had done. Thus, identity is set in a different light than it had been for Heraclitus who understood identity as dynamic and thus involving change and equally contradictions.

But Aristotle also introduces another element to further support the laws of thought. The principle he introduces concerns the way we formulate our concepts or ideas of kinds. According to Aristotle, our idea of a kind or species is best conceptualized by uniting the genus of a species with its differentia or the characteristic that differentiates that species from the other members of the genus (Metaphysics VII. 12. 8-40 and Post. Analytics II. 13). To establish a clear concept of the species “man,” we combine the genus “animal” and the differentia or that characteristic which distinguishes man from other animals * for example, that he is rational. Thus, the species “man” is conceptualized as, “rational animal.” It may be true that our clearest concepts are those which proceed from, and are members of, a single genus. Our desire for clear and distinct concepts has made Aristotle’s model for conceptualizing species enormously influential in Western thinking. In biology we classify and understand species under a single lineage whereby each concept or idea of a species belongs to only one genus. Every family of living things belongs to only one order, and every order belongs to only one class, and every class to only one phylum and kingdom. Such ordering gives us neat and clear concepts and satisfies our desire to conceptualize things in as simple and clear a way as possible. But the platypus does not fit neatly into a single genus or more precisely into the class designated as “mammal.” In fact, many species do not seem to fit such a neat Aristotelian model, and might better be conceived if we understood them to belong to more than a single genus.

This Aristotelian model for conceptualizing species has not only been applied to biological species, but we attempt to organize all of our experience in a similar fashion. In spite of the fact that many of our concepts might be better conceived if we understood them as descending from multiple genuses, the Aristotelian model of concepts which descend from a single genus is deeply entrenched in our thinking. One of the reasons behind its entrenchment is that such a principle allows the laws of thought to work consistently and appear universal. On another model in which concepts are thought to descend from multiple genuses, the laws of thought are not as applicable because, as a member of more than a single genus, a concept could contain contradictory attributes.

MATERIALISM AND THE CORPUSCULAR PHILOSOPHY
With the modern era, a mechanical view of the universe replaced Aristotle’s biological paradigm. With such a model, things were no longer organic wholes but composites of parts and, as such, more compatible with an analytic way of thinking that broke things down into ever smaller parts until all contradictions disappear, and the laws of thought prevailed. Basic to this mechanical view known as the corpuscular philosophy, was an apparent distinction between the kinds of qualities that we attribute to physical entities. Qualities such as shape, extension, motion, etc. were thought to exist within the objects themselves, while tastes, smells, colors, etc. were said to exist within us. The former kind were referred to as ‘primary’ and the latter kind ‘secondary’ (Essay Concerning Human Understanding, II. viii. 17).

The explanation the corpuscularians offered was that these secondary qualities were produced in us by the arrangement and motion of the insensible corpuscles which were made up of primary qualities and constituted the internal structure of a thing. So a physical thing like a strawberry, while not actually possessing anything that resembles the taste or smell of the strawberry, does have the power to produce those sensations within us because of the arrangement and motion of the insensible corpuscles that make up the strawberry’s internal structure. By contrast, when we perceive that the strawberry is extended, we are perceiving a quality that represents the thing itself, since the strawberry is made up of corpuscles and corpuscles are extended.

Thus, the claim of the corpuscularians was that primary qualities were more real than secondary qualities. Of course, what is meant by “more real” is that primary qualities seem to be more objective than the subjective, secondary qualities. But why should we privilege the objective over the subjective? One reason, perhaps the most important reason, is that the laws of thought are more applicable when subjectivity is removed. Subjectivity certainly undermines the laws of thought. While a thing can be sweet and not sweet at the same time, it cannot not be square and round nor in motion and at rest at the same time. Consequently, primary qualities make better objects of thought in the sense that the laws of thought better apply with them than with secondary qualities. What has in fact taken place, however, is that the objects of thought have been made ever more abstract and removed from the reality of the world we actually experience.

The idea of primary qualities, like the objects of mathematics, Plato’s otherworldly forms, or Aristotle’s idea of species that are members of single genuses, are enormously abstract and artificial notions and not like anything we actually experience. The pure objectivity of primary qualities is something we create rather than experience and we create it for the sake of having clear and distinct objects of thought to which the laws of thought might be consistently applicable. By the 18th and 19th centuries, such abstract notions of objectivity would come under attack from a phenomenological perspective which would take up the Herclitian theme and argue that the phenomenal world of experience is more real than the abstract world we have come to think about.

PHENOMENALISM
Berkeley’s phenomenalism privileged the world of experience over the abstract world of objective matter which the corpuscularians had introduced. Berkeley thought that abstract ideas of any kind were inconceivable (Principles of Human Knowledge, sect. 5), and that primary and secondary qualities were inseparably joined in the phenomenal world and could not be separated even in thought.

“I desire anyone to reflect and try, whether he can by any abstraction of thought, conceive the extension and motion of a body, without all other sensible qualities. . . extension, figure, and motion, abstracted from all other qualities, are inconceivable.”

(Principles of Human Knowledge, sect. 10)

For whatever reason, Berkeley did not explore the consequence of his phenomenalism upon the laws of thought. Hegel, however, certainly did. Like Berkeley, Hegel’s phenomenalism attacks the idea of abstraction, but Hegel seems to have had more of an understanding of how much the idea of abstraction was at the very base of traditional logic and metaphysics. Hegel seems to understand that the focus of traditional logic was to make “abstract identity its principle and to try to apprehend the objects of reason by the abstract and finite categories of the understanding.”

By so doing, traditional logic secured a realm over which the laws of thought could sovereignly rule. Once proper objects of thought have been created through abstraction, the laws of thought certainly apply. Hegel would argue, however, that these laws of thought do not apply when the objects of thought are not such abstract entities. Thus, the laws of thought do not universally rule over all thinking but are only universal when the objects of thought are abstracted from the reality of the phenomenal world. If we turn our attention upon the world of experience, “everything is inherently contradictory.” Thus, Hegel posits the law of contradiction, rather than the law of non-contradiction.

As it was for Heraclitus, reality for Hegel is something that moves, thus making any fixed, abstract identity impossible. Things are always becoming and so they must contain within themselves that which they are not. Contradiction is the root of all movement and vitality; it is only in so far as something has a contradiction within it that it moves, has an urge and activity.

A little later, he says something that is even more shocking to those who strictly adhere to the traditional laws of thought and imagine them to be the basis of all right thinking. “Something moves, not because at one moment it is here and at another there, but because at one and the same moment it is here and not here, because in this “here”, it at once is and is not.” This is an obvious contradiction, and the laws of thought would say that something cannot be here and not here at the same time. Of course, what is behind Hegel’s statement is the matter of how we conceive of time. If we think of time and motion analytically, and the continuum of time moves from one fixed, analyzable point to another (i.e., t1, t2, t3 . . . ), thus constituting a present or here, then Hegel is certainly wrong. If that is the case, then something is here (e.g., t4) and not any other place. If, however, there are no fixed points on the continuum that is time, and time is continually moving, then it cannot be stopped and analyzed without making it something other than what it is. If the nature of time, like motion, defies arrest, then Hegel is right and analytic thinking is not suited to understand such things. To think of time as an ongoing continuum forces us to think contrary to the laws of non-contradiction and excluded middle, and understand that something is both here and not here at the same moment.

If motion defies the traditional laws of thought, then all living things violate the laws of thought in so far as they are in constant motion * not in the sense that they experience constant local motion but in the sense that all living things experience perpetual internal motion. This internal motion of all living things prevents them from having any fixed, analyzable point of identity.

“Abstract self-identity is not as yet a livingness…. Something is therefore alive only in so far as it contains contradictions within it.”

Hegel even attacks the law of identity and claims that the law of identity says very little in itself. The fact that A = A is no more than a tautology and has little meaning. It tells us almost nothing about the identity of a thing. The only way a thing truly takes on identity is through its otherness or what it is not. What a thing is not is as necessary to the identity of a thing as what it is in that what it is not is what gives boundaries, definition, and meaning to a thing. Thus, its otherness must be contained within the very identity of the thing.

What is at the base of all that Hegel has to say is a logic that is synthetic rather than analytic. With a synthetic logic which joins things into ever greater wholes rather than analyzing them into ever smaller parts, the laws of thought are not the universal principles they are with analytic thinking. With a synthetic logic that examines wholes rather than parts, contradictions are natural and to be expected. The way to eliminate contradictions is to employ an analytic logic which divides things into ever smaller parts until the contradiction disappears. When Plato and Aristotle qualify the law of non-contradiction and say “in the same respect,” and “at the same time,” what they are doing is breaking a thing down into its parts. If we focus on ever smaller parts, we can eventually eliminate all contradictions and thus preserve the law of non-contradiction and the law of the excluded middle. When, however, we deal with the whole, rather than the parts, we are treating all the respects or parts together and then we certainly may encounter contradictions and the truth is often both/and rather than either/or.

When we say that life is full of joy and sorrow, we can eliminate that contradiction, or any such contradiction, by analyzing life and dividing it into joyous parts and sorrowful parts. That is, in one respect, it is joyous and in another respect it is sorrowful. If, however, we leave life (or anything else) whole and do not analyze it into this respect or that respect, we see myriads of contradictions because that is the nature of the reality in which we live. We have been taught to think analytically about abstracted parts of our experience in order that the laws of thought can be neatly applied, but that is only one way of thinking. We can also think about wholes rather than parts and when we do, the laws of thought do not always apply.

This does not mean that there is no place for analytic thinking and the laws of thought. Analytic thinking is a mode of thought we use all the time. The problem lies in the fact that Western intellectual history has been intent upon creating an understanding that is founded upon universal laws, and, in order to create such universal laws, we have attempted to eliminate all objects of thought to which such laws do not universally apply. This, however, is irrational since there obviously are dynamic and holistic objects of thought to which the laws of thought do not universally apply.

Twentieth century science has discovered the bicameral nature of the human brain, and although pop psychology might be too quick to draw hard and fast lines between the two hemispheres of the brain and assigns analytic thought to the one and synthetic thought to the other, there certainly is something to the fact that the physiology of our brains allow us to think in different ways. Analytic thinking, based upon one hemisphere of the brain, has dominated in the West. Thus, it is no wonder that the laws of thought seem so absolute to so many.

But to strictly apply the laws of thought to all of our thinking is perhaps to use only half our wits.   James Danaher, “The Laws of Thought,” The Philosopher, Volume LXXXXII

Illustrations are by M. C. Escher.

The Paradoxical Justice of God (St. Isaac the Syrian, Homily 60)

“The watchfulness of discernment is superior to every discipline of men accomplished in any way to any degree. Do not hate the sinner. For we are all laden with guilt. If for the sake of God you are moved to oppose him, weep over him. Why do you hate him? Hate his sins and pray for him, that you may imitate Christ Who was not wroth with sinners, but interceded for them. Do you not see how He wept over Jerusalem? We are mocked by the devil in many instances, so why should we hate the man who is mocked by him who mocks us also? Why, O man, do you hate the sinner? Could it be because he is not so righteous as you? But where is your righteousness when you have no love? Why do you not shed tears over him? But you persecute him. In ignorance some are moved with anger, presuming themselves to be discerners of the works of sinners.

“Be a herald of God’s goodness, for God rules over you, unworthy though you are; for although your debt to Him is so great, yet He is not seen exacting payment from you, and from the small works you do, He bestows great rewards upon you. Do not call God just, for His justice is not manifest in the things concerning you. And if David calls Him just and upright (cf. Ps. 24:8, 144:17), His Son revealed to us that He is good and kind. ‘He is good,’ He says, ‘to the evil and to the impious’ (cf. Luke 6:35). How can you call God just when you come across the Scriptural passage on the wage given to the workers? ‘Friend, I do thee no wrong: I will give unto this last even as unto thee. Is thine eye evil because I am good?’ (Matt. 20:12-15). How can a man call God just when he comes across the passage on the prodigal son who wasted his wealth with riotous living, how for the compunction alone which he showed, the father ran and fell upon his neck and gave him authority over all his wealth? (Luke 15:11 ff.). None other but His very Son said these things concerning Him, lest we doubt it; and thus He bare witness concerning Him. Where, then, is God’s justice, for whilst we are sinners Christ died for us! (cf. Rom. 5:8). But if here He is merciful, we may believe that He will not change.

“Far be it that we should ever think such an iniquity that God could become unmerciful! For the property of Divinity does not change as do mortals. God does not acquire something which He does not have, nor lose what He has, nor supplement what He does have, as do created beings. But what God has from the beginning, He will have and has until the end, as the blest Cyril wrote in his commentary on Genesis. Fear God, he says, out of love for Him, and not for the austere name that He has been given. Love Him as you ought to love Him; not for what He will give you in the future, but for what we have received, and for this world alone which He has created for us. Who is the man that can repay Him? Where is His repayment to be found in our works? Who persuaded Him in the beginning to bring us into being Who intercedes for us before Him, when we shall possess no  memory, as though we never existed? Who will awake this our body for that life? Again, whence descends the notion of knowledge into dust? O the wondrous mercy of God! O the astonishment at the bounty of our God and Creator! O might for which all is possible! O the immeasurable goodness that brings our nature again, sinners though we be, to His regeneration and rest! Who is sufficient to glorify Him? He raises up the transgressor and blasphemer, he renews dust unendowed with reason, making it rational and comprehending and the scattered and insensible dust and the scattered senses He makes a rational nature worthy of thought. The sinner is unable to comprehend the grace of His resurrection. Where is gehenna, that can afflict us? Where is perdition, that terrifies us in many ways and quenches the joy of His love? And what is gehenna as compared with the grace of His resurrection, when He will raise us from Hades and cause our corruptible nature to be clad in incorruption, and raise up in glory him that has fallen into Hades?

“Come, men of discernment, and be filled with wonder! Whose mind is sufficiently wise and marvelous to wonder worthily at the bounty of our Creator? His recompense of sinners is, that instead of a just recompense, He rewards them with resurrection, and instead of those bodies with which they trampled upon His law, He enrobes them with perfect glory and incorruption. That grace whereby we are resurrected after we have sinned is greater than the grace which brought us into being when we were not. Glory be to Thine immeasurable grace, O Lord! Behold, Lord, the waves of Thy grace close my mouth with silence, and there is not a thought left in me before the face of Thy thanksgiving. What mouths can confess Thy praise, O good King, Thou Who lovest our life? Glory be to Thee for the two worlds which Thou hast created for our growth and delight, leading us by all things which Thou didst fashion to the knowledge of Thy glory, from now and unto the ages. Amen.

-St. Isaac the Syrian, Homily 60.

Freud’s Critique of Religion

Sigmund Freud

The thesis of God as a “father-wish” originated with the publication of Sigmund Freud’s main critique of religion, The Future of an Illusion (1927). This thesis has been widely discussed and analyzed; only some highlights will be reproduced here.

1. It is impossible to deny that “wishing for God” is a perennially recurrent phenomenon; even atheist Friedrich Nietzsche displayed this tendency: “I hold up before myself the images of Dante and Spinoza, who were better at accepting the lot of solitude… [and] all those who somehow still had a “God” for company… My life now consists in the wish that it might be otherwise… and that somebody might make my “truths” appear incredible to me” (Kaufmann, trans., The Portable Nietzsche, p. 441).

2. The deductive argument for atheism from Freudian Illusion Theory may be pronounced effectively dead, reduced in current thought to an instance of the genetic fallacy (the claim something is false merely by virtue of its ostensible origin). Something is not ipso facto illusory merely because it is wished for, but only if it is a mere wish. As Hans Kung observes:

“…is a faith bad and its truth dubious simply because –like psychoanalysis itself– it also involves all possible instinctual inclinations, lustful inclinations, psychodynamic mechanisms, conscious and unconscious wishes? …Perhaps this being of our longing and dreams does actually exist… the psychological interpretation of belief in God is possible and also legitimate. But is the psychic aspect itself the whole of religion? It should be observed that Freud has not in fact destroyed or refuted religious ideas in principle, and neither atheists nor theologians should ever have read this into his critique of religion. For, by its very nature, psychological interpretation alone cannot penetrate to the absolutely final or first reality: on this point it must remain neutral in principle. From the psychological standpoint, then, the question of the existence of God –and even the positive force of the argument must not be exaggerated- must remain open.” -Hans Kung, Does God Exist, p. 302

3. The God of Judaism and Christianity is as much an *anti-wish* as he is a wish. If it were not so there would be no human sin. Frequently God is precisely what no one wanted, or would be likely to want; he frequently demanded, and does demand, precisely the opposite of what any sane person would have placed on their wish list. It is not hard to imagine Abraham saying “You want me to cut off what part of my body?!” “You want me to kill my son Isaac?! The one you promised to bless the whole world through?!” Jeremiah had absolutely no desire to preach. Jeremiah’s audience had absolutely no desire to hear. God was not what anyone wanted; he was what everyone didn’t want. If Dietrich Bonhoeffer was correct in saying “When Christ calls a man, he bids him come and die” (Bonhoeffer, The Cost of Discipleship), that is precisely what no one wants unreservedly. C. S. Lewis was mortified when he became a Christian; it was the last thing he ever wanted to be; he describes himself as having entered the kingdom of God “kicking and screaming.” Similarly once upon a time the last thing I ever wanted was to become a Christian. It seemed silly and false to me. Yet I have found it to be a pearl of great price…

4. Not only the existence of God, but also the rejection of God, or the rejection of a particular kind of God, can have a psychological origin in wishes. Mircea Eliade brilliantly articulated the nature of Freudian ideology as a cultural fashion:

“…a cultural fashion is immensely significant, no matter what its objective value may be; the success of certain ideas or ideologies reveals to us the spiritual and existential situation of all those for whom these ideas or ideologies constitute a kind of soteriology” (Mircea Eliade, Occultism, Witchcraft, and Cultural Fashions: Essays in Comparative Religion, p. 5).

Thomas Paine’s opposition to the ancien régime in France, for example, undoubtedly gave additional impetus to his desire to assault the Christian faith so virulently. Professor Eliade’s observation clearly suggests the alternate possibility, from a Christian point of view, of:

5. A Christian Psychology of Atheism. Fr. Andrew Angiorus presents a Christian view of this whole question:

“‘Therefore God gave them over in the sinful desires of their hearts’ (Rom 1:24). Generally, atheists and agnostics are talking about themselves when they talk about the absence of God. They simply express their personal subjective truth (that their souls are empty) in an objective way and try to generalize their experience. In other words, there is no theology, or even philosophy here, it is just their own ill or deficient psychology, which is what atheism is… In the Scriptures Christ says clearly that only the pure in heart will see God. In other words, intellectuals, examiners and professors will never understand God, if their minds are not pure… How do we know if someone has a pure heart? The pure heart is evidenced by the way we live. As Peter says, a person devoted to the Lord “does not live the rest of his earthly life for evil human desires, but rather for the will of God” (1 Peter 4:2); “Who may ascend the hill of the LORD? Who may stand in his holy place? He who has clean hands and a pure heart, who does not lift up his soul to an idol or swear by what is false. He will receive blessing from the LORD and vindication from God his Savior” (Ps 24:3-5).” +Fr. Andrew Anglorus

Cf. also the extensive documentation by Paul Vitz of New York University of socio-psychological correlates to affirmation of atheism in Vitz, Paul, The Faith of the Fatherless.

© 2011 by katachriston.wordpress.com (text)

Prof. Danny Shectman Receives Nobel Prize for New “Impossible” Form of Matter: Infinitely Variegated Quasicrystals

Quasicrystals (cornell.edu)

He was fired by two time Nobel Prize winner Linus Pauling. As of today he is the recipient of his own Nobel Prize for defending the very “impossible” form of matter he was fired over. His story of ridicule, ostracism, hostility, and rejection by the mainstream scientific community for affirming something anomalous/discordant to the predominant scientific paradigm reads like something straight out of Thomas Kuhn’s The Structure of Scientific Revolutions. But he was right! The scientifically impossible in this case was not only possible, it was actual.

“For a couple of years I was alone, I was ridiculed, I was treated badly by my peers and my colleagues, and the head of my laboratory [two-time Nobel Prize recipient Linus Pauling] came to me smiling sheepishly and he put a book on my desk and said ‘Danny, why don’t you read this and see that it is impossible, what you are saying.’ And I said, ‘I teach this book… I know what it says, and I know it’s impossible, but here it is! This is something new!’ That person expelled me from his group  He said ‘you are a disgrace to our group; I cannot bear this disgrace,’ and he asked me to leave the group. So I left the group. And he was a good friend of mine! But he could not stand that people would say ‘this nonsense came from your group!’ This was the atmosphere. People not only did not believe in what I said: people were hostile! The community of non-believers was very large in the beginning. In fact it included everybody! The leader of the group was Linus Pauling, a two-time Nobel laureate. He was a very important figure, and the idol of the American Chemical Society, and to his last day he was standing on stages and publishing papers saying that Danny Shechtman is talking nonsense.

Hologram of a Quasicrystal (esrf.edu)

“Two years later I came back to Technion, and here I met professor Ilan Blech who was the first person to believe in my observations, and we linked forces and he proposed a physical model that explains how these crystals could form. And the two of us for publication in a journal, the Journal of Applied Physics, and the paper was rejected on the grounds that it would not interest the community of physicists. And so that summer of 1984 I went back to NBS and met another colleague of mind, John Cahn who was the chief scientist there. Then he invited another scientist, from France named Denny Gratius, and the four of us published another paper (D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “A Metallic Phase with Long Ranged Orientational Order and No Translational Symmetry”), which was published very quickly, and then hell broke loose, because it did interest the community, and many scientists around the world started to work on these materials, and they called me from around the world: ‘I have it! I have it! I have it too!’” And so the community of believers grew, slowly; the community of unbelievers shrunk. The new form of matter which had different symmetries than known before, which is called quasi-periodic crystals, or in short quasicrystals, is accepted into the community of crystal. So the definition of crystals was broadened to include crystals which were not known before.” -Dan Shechtman

Penrose Tiling (geom.uiuc.edu)

The Geometric pattern of Penrose tiling which occurs physically in quasicrystals was completely unknown even to mathematicians until the 1970s when it was presented as a mathematical model by Roger Penrose. As the tiling pattern expands over larger ratios it converges to the ever present Fibonacci/Golden Ratio known as phi[1] intrinsic to a vast swath of diverse yet similar structures including spiral galaxies, snowflakes, neural connections, flowers, electromagnetic field borders, circulatory and pulmonary structure, basic chemical arrangements (tetrahedron, icosahedron, etc.), and so on (see my article Fibonacci, Fractals, and Inorganic Teleology for further details and implications).

Decagonal Quasicrystal

_________

[1] “For icosahedral quasicrystals [see image below], each approximant may be characterized by a rational approximant T,, = Fn,+1/Fn (Fn is a Fibonacci number) to the golden mean T. As TnT the lattice constants and the number of atoms contained in a lmit cell tend to infinity. The lattice constant of an nth order cubic approximant to an icosahedral quasicrystal may be expressed where aqc is a quasilattice constant – the edge of the golden rhombohedral tile. Each next approximant therefore has a lattice constant that is approximately 1.62 times larger. The volume of the next approximant, and correspondingly also the number of atoms in the unit cell, increase by a factor of T^3 = approximately 4.24. In practice, a third or fourth generation approximant is essentially indistinguishable from the infinite quasicrystal” (J. B. Suck, Michael Schreiber, Peter Häussler, Quasicrystals: An Introduction to Structure, Physical Properties, and Applications, p. 397).

Icosahedral Quasicrystal

Continue to Fibonacci, Fractals, and Inorganic Teleology

What is the Largest Number Representing Something Empirical?

"Infinity" (2002, Acryllic), by Geoffery Chandler

What is the largest number representing something empirical? I am not asking a question concerning immense hypothetical numbers that do not actually represent some *thing* …such as googol (10^100), googolplex (10^googol), Graham’s Number, and Moser’s number.

If you guessed it’s the total number of atoms in the entire universe which we can currently observe through our most powerful telescope (10^80 atoms), good try; sorry, that’s not it.

As it turns out, the largest number that actually represents something empirical relates to our brains. As John D. Barrow relates (John D. Barrow, The Constants of Nature (2002), pp. 116-118):

“…astronomy is not the place to look. The big numbers of astronomy are additive. They arise because we are counting stars, planets, atoms and photons in a huge volume. If you want really huge numbers you need to find a place where the possibilities multiply rather than add. For this you need complexity. And for complexity you need biology. In the seventeenth century the English physicist Robert Hooke [1635-1703] made a calculation ‘of the number of separate ideas the mind is capable of entertaining’ (the estimate was reported in Albrecht von Haller’s Elementa Physiologiae, vol. 5, London, 1786, p. 547). The answer he got was 3,155,760,000. Large as this number might appear to be (you would not live long enough to count up to it!) it would now be seen as a staggering underestimation. Our brains contains about 10 billion neurons, each of which sends out feelers, or axons, to link it to about one thousand others. These connections play some role in creating our thoughts and memories. How this is done is still one of nature’s closely guarded secrets. Mike Holderness suggested (in Holderness, M., “Think of a Number,” New Scientist, 16 June 2001, p. 45) that one way of estimating the number of possible thoughts that a brain could conceive is to count all those connections. The brain can do many things at once so we could view it as some number, say a thousand, little groups of neurons. If each neuron makes a thousand different links to the ten million others in the same [neuron] group then the number of different ways in which it could make connections in the same neuron group is 10^7 x 10^7 x 10^7 x … one thousand times. This gives 10^7000 possible patterns of connections. But this is just the number for one neuron group. The total number for 10^7 neurons is 10^7000 multiplied together by 10^7 times. This is 10^70,000,000,000. If the 1000 or so groups of neurons can operate independently of each other then each of them contributes 10^70,000,000,000 possible wirings, increasing the total to the Holderness number, 10^70,000,000,000,000. This is the modern estimate of the number of different electrical patterns that the brain could hold. In some sense it is the number of different possible thoughts or ideas that a human brain could.”

Fine and good you say. So why can’t I remember where I put my keys?
If we can’t blame the equipment, perhaps it comes down to operator error…

For more on large numbers, cf. Wolfram Mathworld’s article “Large Number,” and Scott Aaronson, “Who Can Name the Bigger Number?“
——————–

The video below is a computer representation of the outer (pial) surface of a mouse’s cortex through all six layers and subcortical white matter to the adjoining striatum:

Addendum: “The Human Brain Has More Switches Than All the Computers on Earth” by Elizabeth Armstrong Moore

“The human brain is truly awesome. A typical, healthy one houses some 200 billion nerve cells, which are connected to one another via hundreds of trillions of synapses. Each synapse functions like a microprocessor, and tens of thousands of them can connect a single neuron to other nerve cells. In the cerebral cortex alone, there are roughly 125 trillion synapses, which is about how many stars fill 1,500 Milky Way galaxies.

These synapses are, of course, so tiny (less than a thousandth of a millimeter in diameter) that humans haven’t been able to see with great clarity what exactly they do and how, beyond knowing that their numbers vary over time. That is until now.

Researchers at the Stanford University School of Medicine have spent the past few years engineering a new imaging model, which they call array tomography, in conjunction with novel computational software, to stitch together image slices into a three-dimensional image that can be rotated, penetrated and navigated. Their work appears in the journal Neuron this week.

To test their model, the team took tissue samples from a mouse whose brain had been bioengineered to make larger neurons in the cerebral cortex express a fluorescent protein (found in jellyfish), making them glow yellow-green. Because of this glow, the researchers were able to see synapses against the background of neurons.

They found that the brain’s complexity is beyond anything they’d imagined, almost to the point of being beyond belief, says Stephen Smith, a professor of molecular and cellular physiology and senior author of the paper describing the study:

One synapse, by itself, is more like a microprocessor–with both memory-storage and information-processing elements–than a mere on/off switch. In fact, one synapse may contain on the order of 1,000 molecular-scale switches. A single human brain has more switches than all the computers and routers and Internet connections on Earth.

Smith adds that this gives us a glimpse into brain tissue at a level of detail never before attained: “The entire anatomical context of the synapses is preserved. You know right where each one is, and what kind it is.”

While the study was set up to demonstrate array tomography’s potential in neuroscience (which is starting to resemble astronomy), the team was surprised to find that a class of synapses that have been considered identical to one another actually contain certain distinctions. They hope to use their imaging model to learn more about those distinctions, identifying which are gained or lost during learning, after experiences such as trauma, or in neurodegenerative disorders like Alzheimer’s.

In the meantime, Smith and Micheva are starting a company that is gathering funding for future work, and Stanford’s Office of Technology Licensing has obtained a U.S. patent on array tomography and filed for a second.”

 

Did Luther Get It Wrong? Most Major Contemporary Pauline Scholars Say “Yes”

Luther’s understanding of the law was a keystone of his theology, and of much Protestant theology until the present century. Was Luther simply wrong? That he was in fact wrong  has become a central axiom for a majority of contemporary Pauline scholars in our generation (see further below) -a development which took place primarily in the last 44 years (1977-2011), gathering momentum like a veritable  academic snowball.

The mainstream academic view today is that Luther wrongly assumed the ancient Jews held essentially the same theology of law and/or merit as Roman Catholicism (this interpretation itself first appeared in medieval  Roman Catholicism, and which is absent in early patristic/post apostolic Christianity of the first Christian millennium, i.e. paleo-orthodoxy; the medieval paradigm continues to encounter repudiation as late theological innovation by Eastern Christianity). Luther’s paradigm of Jewish law wasn’t gotten from early rabbinical Jewish and other early extrabibllcal Jewish literature (though not long after Luther it was being widely, but -according to contemporary scholarship- illegitimately, read *into* especially rabbinic literature by many writers), but from the controversy of his own time with Roman/Latin Catholics with their “storehouses of merit,” indulgences and so forth. Luther wrongly assumed the Jews taught some key particular errors he perceived in medieval Latin Catholicism, and also read the same NT on the basis of that assumption in a powerful and majorly influential way. Locked in the struggle of his own day Luther assumed the view he opposed actually belonged to ancient Jews and was reflected in the NT. Pretty much all Protestant theology followed suit for the next 400 years.

If one can accept the notion -and it is difficult on academic grounds not to- that primary sources by Jews describing what their own religion was like before, around, and after the time of Christ are a better index of what Jews actually believed than majorly conflicting Roman Catholic, Lutheran, and later accounts which first appeared in the middle ages, the conclusion might well appear on the order of a no-brainer.

The following brief excerpt from the article “Law” in Gerald F. Hawthorne, Ralph P. Martin, and Daniel G. Reid, eds., Dictionary of Paul and His Letters: A Compendium of Contemporary Biblical Scholarship (1993) by F. Thielman tells the basic story:

“It is easy, when reading Luther, to concentrate on the theological argument with the Roman Catholic Church in which he is so energetically engaged and to miss a subtle hermeneutical impropriety in which the great Reformer and theologian has indulged. Especially in his lectures on Galatians, but elsewhere as well, Luther assumes that the Jews against whose view of the law Paul was arguing held the same theology of justification as the medieval Roman Catholic Church. This hermeneutical error would be perpetuated over the next four centuries and eventually serve as the organizing principle for mountains of Protestant scholarship on the Old Testament and ancient Judaism.

“It was frequently assumed among Old Testament scholars, for example, that at least from the period of the restoration of the Jews to Israel under Ezra, the history of Judaism was a story of spiraling degeneracy into legalism, hypocrisy, and lack of compassion. Similarly, when Protestant scholars discussed rabbinic Judaism they tended to assume Paul’s polemic against Judaism interpreted through the lens of Luther’s reaction against Roman Catholicism provided a sound basis for systematizing the religion of the Mishnah, Talmud and related Jewish writings of a later era. F. Weber’s “popular” description of Talmudic theology (1880) is typical. Keeping the many and peculiar commands of the Law, said Weber, was the means by which the rabbis believed salvation was earned. The ordinary rabbi, therefore, believed that the goal of rabbinic religion was the search for reward on the basis of merit, that God was a stern judge, and that the approaching death brought with it the fear of losing salvation due to a lack of merit.

“A large part of this portrait of ancient Judaism found its way into the interpretations of the NT generally, and especially into expositions of Paul’s writings. Widely used commentaries such as that of W. Sanday and A. Headlam on Romans (reprinted seventeen times fro 1895 to 1952), and influential books about the NT, such as R. Bultmann’s popular description of Primitive Christianity in Its Contemporary Setting (1949, ET 1956) used this picture of Judaism as a backdrop for their explanations of NT theology. In Sanday and Heeadlam’s commentary, for example, Paul’s struggle with the law in Romans 7:7-25, which they take to be a portrait of his preconversion existence, is interpreted as the natural consequence of the “stern” Rabbinic view of the Law, which, they claim, “was fatal to peace of mind” (Sanday and Headlam, 189). Similarly, Bultmann, in a section of Primitive Christianity titled “Jewish Legalism” claimed that the Jewish view of the Law in the first century made “radical obedience” to God impossible because it held that once a certain list of commandments had been kept, one was in the clear and was free to do anything (Bultmann, 69). In addition, said Bultmann, it taught that God would punish sins strictly according to the law of retribution, that salvation was never a certainty, and that even repentance and faith could be transformed into meritorious works (Bultmann, 69-71).

“The Lutheran picture of ancient Judaism now clad in the impressive robes of scholarship, did not go unchallenged among Jewish scholars. As early as 1894 the distinguished Jewish reformer C. G. Montifeore objected forcefully to what he viewed as the tendency of Christian theologians to paint rabbinic Judaism as a dark shadow against which Paul’s theology could brightly shine. The rabbinic literature, pleaded Montifeore, reveals a compassionate and forgiving God ready to lay aside even grievous infractions of the Law at the slightest movement toward repentance by the offending party. It portrays rabbis, moreover, as those who regarded Law as a gift and a delight, who placed a value on faith in God as high as Paul’s and whose daily prayer was “Sovereign Lord of all worlds, not because of our own righteous acts do we lay our supplications before thee but because of thine abundant mercies” (b. Yoma 87b). “I wonder,” Montefiore asked in an address before England’s St. Paul Association in 1900, “if there is the smallest chance that you, unlike the theologians, will believe me when I say that all this business of the severe Judge and the stern Law giver is a figment and a bugbear?”

“Montifeore’s critique of the Lutheran caricature of Judaism at first fell on deaf ears, but through the work of several influential scholars over the next seventy years began to gain the ascendancy not only in Jewish circles, but among nearly everyone working in the field. In 1927 G. F. Moore published a two-volume study of rabbinic theology which, in contrast to Weber’s work, emphasized the role of grace, forgiveness and repentance in the earliest literature of rabbinic religion. This was followed in 1948 by W. D. Davie’s detailed study of Paul and Rabbinic Judaism in which Davies argued that Paul’s doctrine of justification by faith apart from the Law was only one metaphor among many, probably developed first in the heat of argument (Davies, 71-73), and that the apostle’s letters revealed simply a Pharisee for whom the messianic age had dawned (Davies, 71-73).

“Without question, however, the pivotal event in bringing Montefiore’s complaint from the backwater to main stream was the publication in 1977 of E. P. Sander’s Paul and Palestinian Judaism. Sander’s book was so powerful not because its approach was original but because Sanders addressed pointedly and exhaustively the distorted view of Judaism which Lutheran scholarship, and those under its influence, had produced. Sanders made his way step by step through the most influential works of modern NT scholarship in order to show that they departed from ancient Judaism as a religion in which salvation was achieved by meritorious achievement. He then embarked on a lengthy journey through not only the rabbinic literature of the first 200 years after Christ but through the Qumran literature, the apocrypha and the pseudepigrapha as well to determine how those documents answer the question. What must one do to be saved?

“His conclusion was that in all of this ancient Jewish literature, with the exception of the atypical document 4 Ezra, salvation came not through achieving a certain number of meritorious works but through belonging to the covenant people of God. The proper response to the covenant was, of course, obedience, but means of atonement were readily available for those who failed to obey fully. This “pattern of religion” Sanders called “covenantal nomism” (Sanders 1977, 75; 1992, 262-78), and, he claimed, it bears little resemblance to the description of Jewish “soteriology” in most handbooks of Protestant biblical scholarship.

“Largely as a result of this important work, most students of Pauline theology now believe that Montefiore, Sanders, and other dissenters from the classic Protestant perspective have proven their case…” (F. Thielman, “Law” in Gerald F. Hawthorne, Ralph P. Martin, and Daniel G. Reid, eds., Dictionary of Paul and His Letters: A Compendium of Contemporary Biblical Scholarship (1993).

BIBLIOGRAPHY. R. Bultmann, Primitive Christianity in Its Contemporary Setting (Philadelphia: Fortress, 1956); W. D. Davies, Paul and Rabbinic Judaism : Some Rabbinic Elements in Pauline Theology (4th ed.; Philadelphia: Fortress, 1980); T. J. Deidun, New Covenant Morality in Paul (AnBib 89; Rome: Biblical Institute, 1981); J. D. G. Dunn, Jesus, Paul, and the Law: Studies in Mark and Galatians (Louisville: Westminster/John Knox, 1990); R. B. Hays, Echoes of Scripture in the Letters of Paul (New Haven: Yale University, 1989); M. Hengel, The Zealots (Edinburgh: T. & T. Clark, 1989); H. Hübner, Law in Paul’s Thought (Edinburgh: T. & T. Clark, 1984); H. Maccoby, Paul and Hellenism (Philadelphia: Trinity Press International, 1991); C. G. Montefiore, “First Impressions of Paul,” JQR 6 (1894) 428–75; idem, “Rabbinic Judaism and the Epistles of St. Paul,” JQR 13 (1900–1901) 161–217; G. F. Moore, Judaism in the First Centuries of the Christian Era: The Age of the Tannaim (2 vols.; Cambridge, MA: Harvard University, 1927); H. Räisänen, Paul and the Law (WUNT 29; Tübingen: J. C. B. Mohr, 1983); idem, The Torah and Christ: Essays in German and English on the Problem of the Law in Early Christianity (SESJ; Helsinki: Finnish Exegetical Society, 1986); W. Sanday and A. C. Headlam, A Critical and Exegetical Commentary on the Epistle to the Romans (ICC; 5th ed; Edinburgh: T. & T. Clark, 1902); E. P. Sanders, Judaism : Practice and Belief 63BCE-66CE (Philadelphia: Trinity Press International, 1992); idem, Paul and Palestinian Judaism : A Comparison of Patterns of Religion (Philadelphia: Fortress, 1977); idem, Paul, the Law, and the Jewish People (Philadelphia: Fortress, 1983); idem, Paul (New York: Oxford University Press, 1991); S. Sandmel, The Genius of Paul: A Study in History (Philadelphia: Fortress, 1979); idem, Judaism and Christian Beginnings (New York: Oxford University Press, 1978); F. Thielman, From Plight to Solution: A Jewish Framework for Understanding Paul’s View of the Law in Galatians and Romans (NovTSup 61; Leiden: E. J. Brill, 1989); P. J. Tomson, Paul and the Jewish Law: Halakah in the Letters of the Apostle to the Gentiles (CRINT 3.1; Minneapolis: Fortress, 1990); F. Weber, Jüdische Theologie auf Grund des Talmud und verwandter Schriften, gemeinfasslich dargestellt (2d ed.; Leipzig: Dörffling & Franke, 1897; 1st ed., 1880); A. J. M. Wedderburn, The Reasons for Romans (Edinburgh: T. & T. Clark, 1988); S. Westerholm, Israel’s Law and the Church’s Faith: Paul and His Recent Interpreters (Grand Rapids: Eerdmans, 1988); N. T. Wright, The Climax of the Covenant: Christ and the Law in Pauline Theology (Edinburgh: T. & T. Clark, 1991).