through a melting temperature. The microscopic atomic arrangement in glasses, in
contrast with the crystalline state, exhibits no spatial periodicity or long-range order.
In contrast with fluids, the friction of passage of molecular elements of glasses past
each other, their shear viscosity, is large enough such that their macroscopic
shapes are maintained in the very slow flow for very long times. In-between the
crystalline and glassy states their exists a multiplicity of possible unstable
arrangements which result from what physicists call frustration, the inability of a
system to find a unique, lowest energy, ground state. The generic example of a
ferromagnetic crystal has two types of ordering principles: (1) The mutual alignment
of the atomic magnetic moments, visualizable as the lining up of dipole, positive to
negative, magnetic arrows; (2) The geometric crystalline low energy ground state
described above.
When the symmetry of these two ordering principles are incompatible,
imagine an arrangement of neighboring atoms that prefer anti-alignment of the
magnetic moments which are placed on a geometrically triangular rather than a
square lattice, there is no single arrangement that can satisfy both magnetic and
geometric principles. What emerges in this state of frustration is the potential for a
multiplicity of nearly equal energy states. Water has the potential for both geometric
ice crystal symmetry as well as arrangements of hydrogen proton (+) to oxygen
electron (-) magnetic moments (with well-ordered oxygen lattices but disorder
among the hydrogen positions). It is therefore not surprising that a multiplicity of
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indirectly by my sons and church elders about joining a study group for personal
conversion.
I was surprised to learn that discussions of current political topics were a
regular part of these discussions as well as the Sunday and Wednesday night
services. We received a weekly political action committee report. Their issues
involved abortion, school vouchers, sex education in schools, family planning,
school prayer and carefully chosen Christian elected officials for school boards and
the Congress. As a congregation, we frequently held hands in small circles and
prayed for the electoral success of our issues and candidates. Twenty years later,
this movement has evolved into the public political morality play of the Republican
base of George W. Bush.
Laying on of hands, dying in the Lord, speaking in tongues, dancing in the
aisles and praying with up stretched arms were routine in the hymn dense services.
The goal for all was the spiritual transformation of mind as in Romans, “…be not
fashioned according to this world, but be ye transformed by the renewing of your
mind that ye may prove what is the good and well-pleasing and perfect will of
God…” The pastor told us that the world ruled mind could not grasp spiritual things
as in Corinthians “…they are foolishness unto him and he cannot know them,
because they are spiritually understood.”
My research took me to a collaborative project at a European mathematics
institute for three months. I returned to our town very late on a Saturday night. I
planned to surprise my sons by appearing at their usual choice of the middle service
the next day. I drove up to the warehouse church fifteen minutes before the service
was scheduled and found that the parking lot of the strip mall was nearly empty.
There was no Cadillac parked at the front door. I banged on the double door when I
found it locked. More then a little surprised, I called my eldest. He told me that four
weeks before, the pastor disappeared, I later found that his disappearance
accompanied that of the congregation’s bank account, and no one knew where he
had gone. He had not warned or informed anyone in the congregation about his
plans. Calmly and without apparent awareness of my surprise and distress, my
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eldest asked me if I would like to attend the late Sunday morning service at their
newly chosen Charismatic Christian church. He gave me its address and told me
that the service started at 11:00 AM. There still was enough time for us to meet
there. I wondered how the Pastor Carl Austin would use this incident in sermons
about sin and redemption to his next congregation.
Further Readings for Pentecostal Phase Transitions
Religious and Spiritual Groups in Modern America. Robert S. Elliwood, Prentice-
Hall, Englewood, N.J. 1973.
The Name of Jesus. Kenneth E. Hagin, Rhema Bible Church. Tulsa, Oklahoma.
1979.
War on the Saints. Jessie Penn-Lewis, Robert Lowe, N.Y. 1973.
Discipleship, David Watson, Hodder and Stoughton, London, 1981.
Mysticism. Evelyn Underwood, Dutton, N.Y. 1911.
A Nation of Believers Martin Marty, Univ. Chicago Press, Chicago. 1976.
Introduction to Percolation Theory. Dietrich Stauffer. Taylor and Francis. London.
1985.
Modern Theory of Critical Phenomena. Shang-Keng Ma, Benjamin/Cummings.
Reading, MA. 1976.
A Modern Course in Statistical Physics. Linda E. Reichl, Univ. Texas Press, Austin,
1980.
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Manic-Depressive Illness. Fred K. Goodwin and Kay R. Jamison, Oxford Univ.
Press, N.Y. 1990.
The Pharmacological Basis of Therapeutics. Louis S. Goodman and Alfred Gilman,
MacMillan, N.Y. 1975.
Statistical Mechanics of Phase Transitions. J.M. Yeomans, Clarendon Press,
Oxford. 1992.
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CHAPTER 7:
AMPHETAMINE ROLL-UP AND SPLITTING
We try to understand the metaphysics and inner dynamical life of the
committed, judgmental, fundamentalist believer. In these sacerdotaly rigid and
faithful, disenfranchisement and righteous intolerance toward other denominations
are simultaneous with spiritual compassion, mercy and forgiveness for the members
of their own. This splitting between the good people and latent evil doers is seen by
psychoanalysts and dynamically oriented brain scientists as an all too common,
sometimes psychopathological, solution to the inevitable ambiguities of living. I am
certainly not alone in being fearful of Fundamentalists: Jewish, Christian, Moslem
and Hindu. From the overpass above the freeway, bearded Jewish Orthodox men
rained rocks onto the roof of my rented car because I was driving on Sabbath. A
research project had taken me to Jerusalem Mental Health Center’s neurochemical
laboratories for collaborative work with mostly secular Jewish scientists. Halachic
considerations, those of Jewish lawfulness, comparable to the constraints of Muslim
shirah, forbids working, even driving, on the Sabbath. Orthodox Jews live walking
distance from synagogues or benefit from a rabbinicaly blessed, network of
symbolically covered walkways for going longer distances on the Sabbath. This
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Sabbarian grid of permission obviously did not cover driving on the free way to the
mental health center.
It is the splitting of us from them that leads to the breakdown in empathy and
compassionate identification with others. Studies of the dominance of direction of
rotation within a closed space in small mammals have shown that amphetamineinduced
intensification makes the choice of right versus left (or left versus right)
rotation, broken symmetry, more statistically significant. In contrast, the Hefner
Foundation of Switzerland has shown that entheogenic drugs such as psilocybin in
man facilitate seeing both of the conflicting, simultaneously presented, right eye and
left eye images in place of the usual dominance of just one of the two
representations. A precondition of compassion might be that a person’s brain be
able to see and comprehend both or several sides of apparently conflicting points of
view at the same time. The Fundamentalists do not see things that way. In the
Koran, Mohammed says, “…give sustenance to the poor man, the orphan, the
captive…and for the unbelievers We have prepared fetters and chains and a
blazing fire….” In the New Testament’s Mark we find the final words of the risen
Jesus, “…whoever believes and is baptized will be saved but whoever does not
believe will be damned.” The Crusaders’ claimed scriptural support for their
murderous marches to reclaim Jerusalem.
Carl Jung wrote about the New Testament’s Revelations in his Answer to
Job: “…a terrifying picture that blatantly contradicts all ideas of Christian humility,
tolerance, love of your neighbor and your enemies and makes nonsense of a loving
father in heaven and rescuer of mankind. A veritable orgy of hatred, wrath,
vindictiveness and blind destructive fury that revels in fantastic images of terror
breaks out…overwhelming a world which Christ endeavored to restore to the
original state of innocence and loving communion with God…” As Princeton
University philosopher, Walter Kaufman, has noted in his Religion in Four
Dimensions “…compassion for unbelievers is implicitly condemned and
proscribed…Augustine argued expressly against compassion for the damned and
Luther used invectives against his (religious) enemies…” How can this be God’s
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setting for the spiritual work toward that promised in John “…that you love one
another; even as I have loved you, that you also love one another.”
In contrast with what has been described in previous chapters as the
entheogenic drug-induced transitions to a spiritual mind, one is tempted to describe
these Fundamentalists’ states as the amphetamine religions. The Los Angeles
Ram’s Hall of Fame defensive end, on very high doses of amphetamine (125
milligrams compared with the diet dose of 5 milligrams) taken four hours before the
Sunday games, the Baptist minister, Deacon Jones, used his famous and
consciousness annihilating head slap to daze the opposing offensive tackle in order
to gain access to and injure the other team’s quarterback. Before taking the handful
of Dexedrine spansuls, he would tell me, “See you on Tuesday.” Along with the
Deacon’s destructive aggression was the other invariant feature of the actions of
high doses of amphetamine, compulsive stereotypy, the fixity and driven repetition
of over simplified actions and thoughts along with the loss of breadth of vision and
adaptive flexibility. Deacon consistently rushed inside, took the inside lane, in spite
of offensive linemen, who having studied previous game films, being set up to
expect his route. They used this knowledge to take him out of the play. In modern
theological parlance, judgmental rigidity and thinly veiled disapproval take the place
of the more flexibly curious and lovingly humane feelings of the participants in the
evolution in spiritual understanding of today’s liberal Protestant process religions.
These are the ones that believe that the properties of God evolve along with our
biology, our brains and our growing scientific understanding of ourselves and the
world.
Angry splitting is not just a stimulant drug effect. Recall my experience of the
sudden emergence of a first second wind after a mile or so of my daily ten miles of
running. It was frequently accompanied by inner bursts of obsessive, paranoid
thoughts. Taking five milligrams of amphetamine felt much like the first second
wind. I am full of energy with arrogant feelings of power, mind fixated in grand and
simple ideas that I believe to be absolute and correct. I feel irritably intolerant about
anyone or anything different. It is my virtuous duty to set everyone straight.
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In the 1980’s, Moishe Zar, a desert castle dwelling, settlement organizing,
ardent Orthodox Jewish Zionist, now 65 years old, was the leading vigilante of the
West Bank He planted bombs in the cars of Arab mayors and plotted to blow up the
Dome of the Rock. Buying up farmland from the Palestinians beginning in 1979,
many of whom were then killed by their own because they were seen as
collaborators, Zar and his group of young volunteer settlers took over harvesting the
Palestinian’s olive trees and shooting rifles over the heads of those that would take
them back. Fundamentalist Christians share his vision that the coming of the
Messiah, the second for Christians, the first for the Jews, is dependent upon the
complete return of all of the land of Israel to the Jews.
I recall that in the middle 1940’s, my father took me to a fund raising dinner
for the local chapter of the Jewish Antidefamation League. The whispered talk was
about blowing up a warehouse in which anti-Semitic pamphlets were stored,
planned for the middle of the night when it was unoccupied. Even at the age of 10, I
could tell that their quiet anger and firm commitment made these threatened men
feel less vulnerable. I understood a little more about the motivation for this proposed
nighttime property destruction when, the following year, my father explained the
reason for our being refused overnight rooms at several motels as we drove along I-
95 in Southeast Florida. It took us until late night to find a place to sleep. This was
America’s muted version of what Hitler and his legions were doing to Jews that, at
that time, was not generally known, except for Walter Winchell, in America.
Resonant with our chemical-cultural theme are the many reports that Hitler was
taking an amphetamine drug, Benzedrine, daily and in high doses for the last 20 or
more years of his life. One can hear the characteristic, amphetamine-induced,
higher pitched, rants in his recorded radio tirades. Compare the pitch and strained
voice quality of the singing of Bob Dylan in his early records made while he was on
speed with the gravely, much lower pitched voice, now that he is not. In our
behavioral neuropharmacology laboratory at the Brain Research Institute at UCLA,
Professor Charles Spooner and I used an audiographic oscilloscope to monitor the
sounds of baby chicks whose peeps became higher in pitch and rate following
injections of amphetamine. The earliest members of the methadrine-amphetamine
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chemical family were synthesized by the great organic chemists of the German
pharmaceutical industry in the early 1930’s.
The sequence of parallel streets in the neighborhood of my home and first
grammar school in Kansas City, Missouri were my street, Virginia, then Tracy,
Forrest and Troost. My school, Bancroft Elementary, was on Tracy and one block
down that street was the Lutheran Day School established by German immigrants
under the aegis of the Missouri Synod. Starting in the third grade in 1943, I was
intermittently and unpredictably chased by rock throwing, “damn Jew” and “Christ
killer” shouting boys from the Lutheran Day School. I had my choice of running for
safety directly from Tracy to my family’s half duplex at 4232 Virginia Street, or
moving away from school via Troost and then down several blocks and around to
sneak back to my home on Virginia without being spotted. One run-for-it afternoon,
my parents took me to the emergency room of the Menorah Hospital to have my
scalp sewed up where a sharp rock had landed.
When I asked my synagogue’s young people’s spiritual counselor, Rabbi
Kleigfeld, to explain the feelings and actions of these children of Martin Luther’s
Post-Reformation Christian Church, he answered that I already knew about similarly
difficult places and times of our Twelve Tribes’ like Rome, Medieval Europe, the
Spanish Inquisition, Persia (Iran) and, it was rumored, in Germany as we spoke.
“Conversion or death” was its most benign form, in places like Spain and Iran, many
Jews faked it, staying alive and practicing Judaism secretly. Kleigfeld told me that
the causes of this historical theme of persecution of Jews were complex.
Among the frequently unmentioned events recorded in the later part of the
worldly life of Mohammed, who lived from 570 to 632 AD was, ”…in the name of
Allah, the Compassionate, the Merciful…” his participation in the crushing of the
Jewish tribe of al-Nadhir in 626 A.D., the beheading of 800 Jewish men of the tribe
of Qurayza who refused to accept Allah as their God in 627 A.D. and putting to the
sword the Jews of Khaybar in 629 A.D. As in the section of the Koran called The
Cow, Mohammed proposed to “…fight against them (the infidels) until idolatry is no
more and Allah’s religions reigns supreme…” In contrast, the more entheogenic
spiritual orientation of the ecstatic followers of Mohammed in his earlier years
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speaks of the multiplicity of valid Ways to Deep Truth. The acceptability of many
ways is supported in the tales from the millennial oral tradition of the Sufi Masters in
their Teaching Stories. One of them, What Befell the Three, is attributed to the early
18 th Century Sufi teacher, the Dervish Murad Shami. In it, an apparition is mobilized
by the concentrated Truth seeking efforts of three Sufi Dervishes named Yak, one,
Do, two and Se, three. When this “…white smoke head of the very old man…” was
asked what he was, he answered “…I am what you think me to be…have you never
heard the saying ‘There are as many ways to the Deep Truth as there hearts of
man.’” In the narratives about the lives of the Mevlevi Islam dervishes called
Munaquib el-Arafin (1353), Jalaludin Rumi, the Sufi saint, instructs his ill and
troubled petitioner to ask forgiveness from the Christian he recently spat on saying
“…whether a ruby or a pebble, there is a place on His hill, there is a place for all…”
Cole Barks and Michael Green’s The Illuminated Prayer (2000) notes that the Rumi
follower, Bawa Muhaiyaddeen, a modern Sufi guru, was said to be keenly aware
how quickly spiritual entheogenic systems can become amphetamine-like and
“…develop rigid marching orders …which turn into a dumb obsession with other
people’s behavior…”
It appears that entheogenic and amphetamine spiritualities can coexist
contemporaneously, in Islam as well as in all the other of the world’s great religions.
One day, sneaking home from school, taking the long way around via Troost,
I was spotted and chased up some stairs into an apartment building’s dark hall.
Terrified, I swung hard and hit the leading angry and noisy head with a propitiously
found snow shovel that had been left near the apartment’s entrance. An ambulance
was called to tend to the twelve-year-old, transiently unconscious, Lutheran boy. He
recovered completely within a day and the chases after school and my desperate
escapes stopped suddenly, never to reappear. After several months, our family
crossed the socioeconomic divide in Kansas City to a more tolerant, upper middle
class, Southside neighborhood near Rockhill Road, to a suburban home, one block
from Missouri’s border with Kansas. There, persecution for my Jewishness took
more subtle forms such as not being permitted to play teen-age golf with my friends,
though invited, on their Blue Hills and Kansas City Country Club’s golf courses. It
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was decades later that the first Jewish member of the KCCC was the founder of H
and R Block. Unable to afford membership in the single all Jewish country club of
the region, I practiced for my high school golf team on Armour Hills Public Golf
Course, where, at the time, mostly white working class golfers played.
How can it be that spiritual states include both personal humbleness and
loving mercy toward some of mankind and judgmentalness, nonacceptance and
commitment to seduction, threat and even violence in the service of invoking
changes in the beliefs of others. How can the high energy calm of being home at
last in the born again condition with its new freedom from self assaults about sin,
most importantly that of disbelief, but also peccadilloes such as drunkenness,
promiscuity and familial abuse, be associated with readiness to judge, harass even
persecute others. Psychoanalysts would say that it is a riddance mechanism, the
projection of unwanted personal traits onto others. From the standpoint of rational
thought, this seems more like non-Aristotelian cognition, two, not either-or,
countervailing orientations toward mankind held simultaneously. The newborn
parishioners of these charismatic amphetamine churches express their fealty to
God with strongly held beliefs that diagram logically as contradictions. The
perception of the world’s peoples into believers and infidels, good and evil, our
people and your people, ourselves and the others. It is generally believed among
social psychologists that it is the perceived nonpersonness of others, which allows
the cruelty that empathic identification with them would never permit. Splitting feels
like resolution, its stereotypy reducing the complexity of spiritual thought as well as
true to life perception.
A concrete laboratory example of amphetamine conversion, the sudden
transition to a high energy, fixated, and delusional state called amphetamine
psychosis, is supplied by experiments in humans conducted by Professor John
Griffith at Vanderbilt University in the 1960’s. These experiments would not be
allowed by today’s human research committees or medical ethicists. Each one of a
group of psychologically screened-as-normal graduate student volunteers, at an
individually unique amphetamine dose, developed suddenly a personally unique
and peculiar system of new beliefs, obsessionally held as rational thoughts. Ten
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milligrams of amphetamine were administered to volunteer subjects every hour until
every subject crossed their particular threshold for personality change. The
graduate students underwent a global mind-brain-person transition at differing total
doses of amphetamine. The subject’s world was suddenly transformed into one of
enemies and friends. The syndrome dissipated over several hours when the drug
was stopped and the plasma levels of amphetamine and its metabolites declined.
As amphetamine makes memory formation and recall stronger, the subjects were
embarrassed when remembering what strange and forbidding yet uneatable things
they so strongly believed. These included such things as: they as good people were
caught in a network of bad person Russian spies; some threatening others arranged
for poison gas to be seeping out of the water faucet; the white coated scientists
were CIA undercover intelligence officers hoping to get information about their small
pornography collection. The subject’s world had become divided in, for each
person, a stereotyped way.
After a couple of weeks of return to normal living, the experiment was
repeated. Each subject again developed his or her individually unique set of goodguy,
bad-guy delusional beliefs and at the same dose of amphetamine as before.
Like those of strong faith, their ideas once again resisted the logical arguments
made by the professional staff: that the new realities were neuropsychological and
had an obvious pharmacological origin. While on the drug, all stuck to their story,
even while being shown the movie record of their first drug-induced episode. There
is reliable scientific literature describing kamikaze pilots on high doses of
amphetamine in an ecstatic state of Shinto nationalism. With their planes loaded
with explosives, they deliberately crashed their planes onto American aircraft
carriers in the Pacific Theater of World War II. One wonders if these drug-induced
states occur in the drug-free condition in today’s abstemious Muslim suicide
bombers.
A more abstract and general way of thinking about the sudden emergence of
fixation, repetitiousness and splitting in feelings and thoughts involves the
emergence of regular limit cycle oscillations in a complex system that was behaving
previously in a stable but flexible way. Locking up into a fixed, closed loop, is a
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common way for electrical circuits, computer programs, brain mechanisms and
other complicated systems, even cultural or spiritual movements, to behave when
one or more important control parameters crosses a threshold. Doyne Farmer of the
Los Alamos’s Prediction Company once said about this vulnerability in complex
system, “Those things can hardly wait to roll up.” The limit cycle lock-up occurs
most often as a sudden, discontinuous change, called a bifurcation, into
autonomous self-oscillations from an equilibrium state around which there was
some random variation. A bifurcation, a discontinuous change in outcome from a
smooth changes cause, characteristically occurs when the amount of an important
influence, a metabolic state, a drug, a psychodynamic conflict or level of emotional
stimulation crosses some critical value. The switch from one type of dynamical
behavior to another looks like the system has suddenly changed into something
else with an entirely new kind of life of its own. In the new life of rolled up, locked-up
repetitious motion, almost all new starting conditions follow pathways that lead into
the same limit cycle pattern. Evangelical Christians talk about all born again life
being in Jesus, fixed in a complete set of moral, social and political beliefs, ideas
and judgments. The limit cycle gets its name because the end state of the orbits of
almost all starting points of the dynamics winds up being drawn into the same fixed,
repetitious pattern of a stable cycle. Visualizing the simulation of one kind of
bifurcation to a limit cycle on a computer screen, we see a slightly jiggling point
explode suddenly into an orbit of ceaseless rotations around a circle.
Ralph Abraham, the University of California at Santa Cruz pioneer in
graphical approaches to nonlinear systems, describes, cinemagraphically, the
emergence of limit cycles from a single point. He starts with a picture of an attractor
of water flow in the shape of a basin. All water that enters the basin, rolls down its
sides to the bottom, to what physicists say represents a potential energy minimum.
A little more technically, this attractor basin is composed of the set of all points such
that the orbits that flow from them tend to end up inside the basin as time goes
toward infinity, no matter where they start. Changing the value of a control
parameter of the system changes the shape of this basin-like landscape, of the
surface of the systems dynamical actions called a manifold, which can intuitively
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predict how the fluid will flow upon it. If we start with a simple bowl, a parabolic
basin, then the attractor itself is a point at the bowl’s very bottom. Changing the
value of some influential parameter may induce the sudden formation of a small hill,
growing at the center of the basin’s bottom. Now fluid flow in the attractor bowl runs
down to a path around the hill at its bottom. The autonomous motion of the fluid
flows takes place now in a circular orbit. The basin of the new attractor is the
original bowl minus the point at the top of the central hill. The fluid flow around the
hill at the bottom of the basin is circular and is called a limit cycle. Note that the
direction of the rotation of the limit cycle can circle in one direction or the other. In
some computational simulations, motion alternates between directions. This
suggests the aspect of the born again amphetamine religions, splitting. There is an
unstable and intermixed probability of right versus left turning directions and their
alternation. This vulnerability to directional splitting and often unpredictable
alterations in action themes can represent what seem to be paradoxical
combinations of both good and evil in the same strongly faithful, for example, the
apparent bidirectional morality of generous and loving, pederast priests.
These mathematically flavored images of the sudden emergence of a limit
cycle in complex systems was made biologically concrete to me by research
conducted by one of my first graduate students, David Segal. He is now a professor
of psychiatry at the University of California in San Diego. His program of work
involved the administration of very gradually increasing doses of amphetamine to
rats while their behavior was being monitored and recorded by a continuously
running video camera. He documented the behavior of rats in a walled rectangular
space within which, without drugs, they first wandered about randomly and then
settled down to rest in an individually selected, favorite home corner. Segal called
all of these phenomena, patterns of exploratory behavior. At doses of amphetamine
below 2.5 milligrams (mg) per kilogram weight (kg), the exploration of the entire
bounded space proceeded faster than was the case with their salt-water treated
controls, their paths being more uniformly distributed throughout the box. They
spent less time resting in their home corner. At almost precisely 2.5 mg/kg, the rat’s
behavior changed dramatically into an entirely new pattern of continuous circling. As
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was the case in the abstract manifold picture of bifurcations to limit cycles, some
rats tended to circle their chamber to the left and some to the right and switching
between them was often seen.
The influence of amphetamine and other brain dopamine neurotransmittermediated
drug manipulations on directional turning tendencies in rats, mice and
cats were the focus of brain and behavioral research of Professor Stanley Glick of
the University of Massachusetts. The asymmetry of dopamine concentrations in the
two sides of the brain, particularly in the medial prefrontal cortex and the brain
stem’s nucleus accumbens, predicted both the paw preference for pellet reaching
and direction of turning in several studies in rats. These findings were statistically
true over a population of rats, but not necessarily predictive for any single one.
Reminiscent of the conflict between good and evil in our human spiritual analogy,
naturally right turning male rats and left turning female rats, when compared with
the opposite paired group, were greater voluntary ingesters of alcohol placed in
their water bottles.
Splitting as a part of the phenomenology of limit cycle bifurcations, with
directional implications for good and evil, has neurological support in humans as
well. In the context of contrasting right versus left hemispheric temporal lobe
syndromes, recall that temporal lobe seizures with a right side excitatory focus
leads to the development of the Geshwind Syndrome, a high, softly energetic and
saintly state of spiritual preoccupation and voluminous writings, loving and generous
kindness toward all and the complete disappearance of sexual interest but not
sexual potency. A left temporal lobe excitatory focus leads to the development of
the Kluver-Bucy Syndrome of indiscriminate aggressiveness and hypersexuality.
Experimental simulations of this syndrome in cats lead to them mounting and
attacking living and nonliving things, even chairs. A variety of manipulations of the
symmetry of brain dopamine concentration and dynamics by its characteristic drug,
amphetamine, interact with lateral brain lesions such that we conclude that the
stimulant-induced limit cycle lockup remains a phenomena influenced by drugs, sex,
genetic predisposition and several other experimental conditions. This situation is
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perhaps not so different in variety and complexity from the range of representations
in art and literature of the left hand of evil and the right hand of grace.
Oscillations that appear spontaneously in nonlinear systems without external
periodic input were known to Henri Poincaré in 1882, and were systematically
studied and made accessible to non-mathematicians by early 20 th Century Russian
mathematicians and physicists, well represented by a 1949 book, Theory of
Oscillations by the Russian engineer-mathematicians, A. A. Andropov and C.E.
Chaikin. Another relatively early classic is Nonlinear Oscillations by Nicholas
Minorsky. The most common form of transition from a fixed point to a limit cycle was
pictured as changes in the surface of the action, the bowl-hillock manifold in the
paragraphs above, and is called a Hopf bifurcation. Recall that bifurcation means a
discontinuous change in an observable over a continuous change in what is known
as a control parameter, such as dose of amphetamine or intensity of an experience.
The mathematical mechanism resulting in circular directional motion represented by
the (eigen)vectorial states, was named for the German mathematician, Eduard
Hopf. His 1942 paper was a mathematical proof of its existence and was discussed
in the context of fluid flows that role up such that circling vortices arise from smooth,
called laminar, water flow, at a critical value of the flow rate. Hurricanes are another
example of these kinds of dynamics.
The Hopf bifurcation to limit cycles has been found in several, many
dimensional, physical, chemical and biological systems. The latter include calcium
conductance oscillations in the excitable membranes of muscle, heart and the brain,
cardiac arrhythmias such as ventricular flutter as well as oscillations in population
numbers in foxes and rabbits, predator-prey systems. California Institute of
Technology’s Professor, James Old and Johns Hopkins Professor, Joseph Brady
made experimentally obvious the potential for the rigid irrationality implied by the
brain’s inclination to be locked up into limit cycle behavior. They demonstrated that
animals, from rats to monkeys, could get locked up in apparent self torture,
repeatedly and endlessly pushing a bar to deliver current to pain systems in the
their brains. These pushes induced almost unremitting screams in monkeys and
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what appeared to be rageful biting and then immobilized resignation in behaviorally
depressed rats.
Freud’s last paper, Analysis, Terminable and Interminable (1939), featured
examples of what he perceived to be the unsolvable mystery of helpless
psychological entrapment in repetitious patterns of self-destructive behavior. He
blamed the Iliad’s and Odyssey’s villainous immortal, Thanatos, the everthreatening
spirit of death and destruction to contrast with the good, life giving Eros.
The Yiddish word for a personified Thanatos is Moloch ha-Moves. A range of
fixations in self-excitatory, repetitious, self-mutilating behaviors is documented in
domesticated animals. Dogs, particularly German Shepherds and Labrador
Retrievers, can lock up in compulsive grooming cycles of what is called acral lick in
which endless licking of paws or flanks lead to the break down of skin into seepingsore
dermatitis, which, in turn, stimulates more licking.
Mark Twain wrote a story about his getting stuck in ceaseless mental
repetitions of a catchy, clangy poem. He could not stop reciting it to himself even
after days of sleep loss and anorexia. He was finally cured by relating his problem
and the poem to his pastor who he then unwittingly heard creating a community
epidemic by including the rhyme in his following Sunday’s sermon. Psychologists,
who study this form of human mental limit cycle attacks, call this state of internal,
repetitiously recited, poetic stuckness, earworms.
There are additional invariants of sudden transformations into spiritual-mindbrain
bifurcations into a limit cycle lockups and, as discussed, one of them is
psychological splitting. In psychoanalytic theory, as first suggested by Freud in his
1937 written and posthumously published paper, Splitting of the Ego in the Process
of Defense (1940), splitting implies two simultaneous and contrary psychological
reactions, one can be conscious and the other unconscious. They can both emerge
in conflictual situations involving adaptive efforts of the personality to deal with the
opposition between some form of powerful instinctual pressure and attendant
perceived or imagined danger. Otto Fenichel’s Psychoanalytic Theory of Neurosis
(1950) elucidates multiple manifestations of splitting of the I (more technically, the
ego) into a conscious part that knows reality versus an unconscious part that denies
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it. In some situations, a logical view contends with a more irrational, magical one.
Today, the morning group praying, evening hymn singing, Christian Republican
Right Wing feed their feelings of being on the side of God by dividing people into
those that are like them and good and those that President Bush and Attorney
General John Ashcroft calls the evil doing “bad guys.” As noted previously,
psychoanalytic theory posits that the evil doing others may represent the projected
repository of our own unacceptable impulses and inclinations. It became quite clear
in my own psychoanalysis and psychoanalytic training that it is in healing our split
and knowledge of our own unacceptable things that will lead to our understanding
and forgiveness of others.
As we dig deeper into global brain-mind dynamics of emergent high-energy
fixation, stuck repetitiousness and splitting, we encounter their universality in the
structures of mathematical thought. Did we just make them fit? Do these thought
forms map onto internal and external physical reality? Are these abstract concepts
and operations simply products of our biological brains manifested as psychological
mechanics and used to explain to ourselves what we perceive and think? Does a
square have external reality or is it a universally imagined something, and, as such,
represented only in our minds and the pictures of it we draw? Is mathematical
understanding simply inborn perceptual skills combined with developed and
practiced logical cognition? Or, do we take the Platonic view of mathematical
relations: these abstractions are the ultimate realities, antedating and persisting
through the past, present and future of the universe and omnipresent.
Where can the conceptual boundary be drawn between the physical reality of
the Babylonian surveyors use of the Pythagorean theorem to calculate distances,
that the sum of the squares of the lengths of the two legs of a right triangle is equal
to the square of the length of its hypotenuse, and its abstract, pencil-marks-onpaper,
algebraic development as in the definition of Pythagorean numbers, a,b, and
c such that a 2 + b 2 = c 2 . The dichotomy between the abstract and concrete,
consistently blurred in our work, is between a natural science with ideas that can be
disconfirmed, directly or indirectly, by experimental observation and the thinking of
mathematics as an a priori field in the sense of Kant. The modern Platonic view
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such as that held by Rene Thom is that once accepting a set of natural givens,
called the axioms, the rest of the knowledge of this reality grows in the form of
theorems that relate to the axioms and each other through their logical consistency.
Knowledge of reality is moved by the ever-forward mathematical refinement of a
priori conditions to do away with the theorems’ exceptions, called counter examples.
The Hebraic Bible’s view of signifiers such as words and symbols is close to,
but not identical with, the Platonic view of mathematical formalism. According to the
Torah, God made the word with words. God spoke and the world became real. The
Aramaic for “I create in speaking” is avara k’davara , or as the magician says, as he
waves his wand over an apparently empty black high hat, abracadabra. The
Hebrew word for word, davar, also signifies thing. This view contrasts with the
mathematical formalists, among them Hilbert, who considered the signifiers of
abstract mathematics simply symbols used in a game, the rules of which being
arbitrary, must include proofs of consistencies among them. Consistency from the
point of view of physics was addressed by Hertz, in Die Prinzipien der
Mechanik,(1894), where he expressed the formalist theoretical physicist’s work as
“…within our own minds we create images or symbols of the external objects, and
we construct them in such a way that the logically necessary consequences of the
images are again the images of the physically necessary consequences of the
objects.”
In another set of related contrasts, the constructionist mathematician will
argue that mathematical assertions are only true if they can be demonstrated, found
or constructed. In contrast, the classical school of mathematics can develop the
case for the truth of mathematical statements if they are consistent with field’s
network of theorems and proofs, even if, up to the current time, no specific example
of this truth can be demonstrated. The former can be thought of as a builder, the
latter as a discoverer. For example, suppose we try to make a proposition about
perfect numbers where a perfect number is defined as being equal to half the sum
of its divisors. Using the perfect number 6, we find that its non-identity divisors are
1, 2, and 3 and half of their sum = 6. Our proposition: either there exists an odd
perfect number, or else there exists no odd perfect number. An expression of this
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forced decision between yes and no is called the excluded middle. The
constructionist mathematician, an orientation without the excluded middle, asserts
that “an odd perfect number exists” would only be meaningful if one could show that
such a number had been found or constructed. The classical mathematician would
find the phrase “no odd perfect number exists” meaningful without a concrete
example, if the assumption of its existence would lead to a no (versus yes)
contradiction encountered in the proof-relevant network of established theorems
and their relations. The symbolic operations of these formal schools of mathematics
and their relationship to the objective and ideational realities of brain-mind-spiritual
life have been viewed by some as Western cultural products rather than
expressions of secular or spiritual Absolutes. Still others have argued that cultural
relativism is not relevant here because mathematicians worldwide constitute a
monoculture.
With respect to the real world existence of abstract mathematical structure,
our Platonic bias must be obvious. The thrilling experience of a new reality I get to
know from finally understanding how a theorem works and the rush of peering into
the grandeur of the Grand Canyon feel like the same kind of full-of-wonder high to
me. I blend them here without reservation. Perhaps this world of spiritual abstraction
is closer to the orientation of the school of intuitionist mathematics. Its founder,
L.E.J. Brouwer, required that every mathematical construction be so immediately
apparent to the human mind that no formal proof was necessary. This became my
form of spiritual transcendence, which led naturally to a mathematical, mystical
faith.
We carry the explication of this kind of reality further. Reflections of the good
and evil, right and left, moral directional biases and their relative weightings in born
again bifurcations to invariant circles called limit cycles, can be symbolically
represented in what are called the complex eigenvalues of matrices describing the
system’s set of orthogonal motions with changes in their control parameters. The
behavior of these complex eigenvalues underlies and characterizes the
mathematical mechanism of the Hopf bifurcation.
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The subject of complex eigenvalues brings up in me the emotionally
disturbing subject of imaginary and complex numbers. I can still feel a little of my
earlier anxiety. The episode started benignly enough. Our high school’s freshman
algebra class was studying how to solve quadratic equations, equations in which
the highest power of an expression was two. Told to work at the blackboard in front
of the class, I was given the problem of finding the two values of x that were the
roots of the equation, 5x 2 + 3x + 4 = 0. I had been taught to use the memorized
quadratic formula,
2
− b±
b −4ac
x = , in which a = 5, b = 3 and c = 4. I always
2a
calculated the square root part first and wound up with the expression,
9 − 80 = −71.
I can still feel the sinking feeling in my stomach as I looked at the
result. I anticipated the usual snide remarks and embarrassment as I contemplated
doing what I did not know how to do, take the square root of a negative number. Mr.
Kirby, the retired mechanical engineer who was my high school freshman algebra
teacher tried to help, but I did not trust him. It seemed to me that he had already
humiliated me in front of the class, several times. He asked, “… what number when
squared, multiplied by itself, would equal –1.” He then asked it another way: solve
the following equation for x: x 2 +1 = 0. Seeing something I could do, I wrote the next
line quickly x 2 = -1 and then, taking the square root of both sides, I wrote x = − 1 .
He then asked me what that meant. I answered by writing quickly, glibly and blindly
that that meant that − 1×
− 1=−1. He asked me to explain what that meant by
giving him an example from the real world. Not yet knowing about imaginary and
complex (combine real and imaginary numbers), I stood head down, ashamed and
silent, thinking that my smart friend Jerry Blau would get the answer immediately.
Mr. Kirby said he would go on with the class while I continued to stand in front of the
blackboard and thought about it. He told me to interrupt him when I was ready to
answer. Some classmates were smirking, others giggled aloud. They had seen him
do this to me before.
Mr. Kirby, a short, muscular man, an ex-marine with a military haircut and a
brusque manner, lectured that mathematical competence and obedience to
authority and class discipline were all of a piece. I asked him about mathematical
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creativity and he said that this class was certainly not about that. I disliked and
feared him. He seemed to feel (and wrote a note to my parents to the effect) that,
being “too arrogant” I needed to be “brought down a peg or two.” I had gotten the
best grades in the first two exams and was enjoying the role of after school tutor for
some of my friends. I suspect I was getting pretty egotistical. In class, I found myself
eagerly shouting out answers without holding up my hand, behavior that Mr. Kirby
met with his characteristic look of fatigued disgust. Twice I was thrown out of class
for my introjections. He then began to give me problems that I could not do, for
which I was not prepared. This left me standing at the blackboard until the end of
the hour, after all the rest of the students had solved theirs and sat down. On
parent’s night, Mr. Kirby told my father that I needed more “social and intellectual
discipline.”
Inspired and personally directed hard work and socially defined correct
behavior were not synonymous to this arrogant 13 year old who had already
brought chagrin to his mother, the conservatory classical piano instructor, with his
satirical pianistic jazzy composition called “How High the Moonlight Sonata.” I was
also a secret reader of the book on the top back shelf in my father’s library by Jack
Hanley called “How to Make Mary; A Gentlemen’s Guide to Seduction.” In Mr.
Kirby’s class, inspired by the book, I sometimes reached behind me, through the
crack in my desk seat, to caress the inside part of the long smooth legs and
sometimes moist panties of a well developed, tall and beautiful brunette girl behind
me. I was never caught and she pretended that nothing was happening. In fact, she
never talked to me outside of class. I felt then, vaguely, and now, more specifically,
that a content enriched, instinctually titillated and excited unconscious could lead
me to the solutions of intellectual challenges if it were both sufficiently indulged and
untrammeled, left alone in its work of being itself. Mr. Kirby did not see things that
way.
Since then, among my graduate and post-doctoral students in the
neurosciences, I have learned that the Mr. Kirby’s of modern American educational
practice have ruined generations of potential mathematicians and physical
scientists. Worse, they have created generations of very bright math phobics who
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run to other graduate fields such as biology and medicine and come to resist the
potentially humiliating incursions of new and potentially helpful abstract ideas and
operations from mathematics and physics into their fields. They do not want their
persecutory versions of Mr. Kirby to take up residence once again in their heads. I
can still feel his negative presence during long hours of struggle with the ego
deflating feelings of dumbness that an understanding of almost any new
mathematical concept requires of me. Holding Mr. Kirby’s voice off as long as I can
until, sometimes, the wonderful “aha!” experience arrives. I have tried to forgive him
since but forgetting him has not been possible.
It turns out that in the world of elementary, physically representative, real
numbers, the square root of a negative number has no meaning. Such a number
has understandably come to be called imaginary. Was this the answer Mr. Kirby
wanted? There was some conflict among mathematicians in the 17 th
and 18 th
Century about the arbitrary definition of −1
as an imaginary number. It was
symbolized by a letter, i, that is −1 ≡ i . The existence of i extended the range of
algebraic definitions so that a solution of the quadratic formula as above could be
found for the square root of a negative number. A further expansion of this idea was
to that of a complex number that can have both a real and an imaginary part. For
example, letting letters be generalized representations of numbers, a complex
number might be written, a + bi, real number a + real number b times i, the letters
such as a, b, c, d… symbolized real numbers. Consistent with membership in an
algebraic system, a + bi and c + di can be added and multiplied. This extension of
the real numbers into the imaginary realm permitted d’Alembert’s and Gauss’s
proofs (and many, more complete ones since) of the powerful Fundamental
Theorem of Algebra from which the faith derives about always being able to find at
least one solution to an algebraic equation. It was proven that any n th
algebraic equation (e.g.
has at least one real or complex root.
x
n
degree
n 1
+ x − + ... = 0 ) with real or complex coefficients always
Closer to an image that helps make intuitive connections with human born
again bifurcations, limit cycles and directional splitting is the geometric interpretation
of a complex number, let us now call it z. As above, algebraically, z is the sum of a
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real part a, plus b times the imaginary part, bi”; that is, z = a + bi. We can then set
up a geometric space to represent z by imagining a two dimensional plane with the
horizontal real axis extending from left to right, the usual x axis, and the vertical
dimension, called the imaginary axis, extending from bottom to top like the standard
y axis. These two axes, going from negative values to positive ones, left to right and
bottom to top, cross at the shared value of 0. Thus a and b can be visualized as the
rectangular coordinates of a point in the plane and the point locates the complex
number, z = a + bi. Since real parts and imaginary parts are like apples and pears
and for addition, like must be added to like, if two complex numbers, a + bi and c +
di are equal, then a = c and b = d and their sum is written (a + c) + (b + d) i.
Now that we’ve set up a point z on the plane, located with a complex number
at z = a + bi, we can then draw an arrow, called a vector, from the intersection of the
imaginary and real axis at 0 to this point z. Its length from 0 to z, 0z, we’ll call that
length ρ , is the size or amplitude-like modulus of the complex number, z = a +bi.
The angle this 0z vector makes with the real, 0a-axis, lets call this angle φ , is called
the argument of complex number z = a +bi. ρ is a length that can grow or shrink, φ
is an angle that can rotate. We imagine vectorial movement like that of a variable
length hand of a clock. This geometric explication of complex numbers prepares us
to visualize complex numbered eigenvalue solutions to matrices representing the
relevant equations that bifurcate to limit cycles and directional good and evil
splitting. ρ represents the dilatable clock’s radial amplitude of circular motion and
φ , the angle of vectorial turning from the 0a-axis.
The complex conjugate of the complex number, a + bi is the complex number
a – bi in which the sign of the imaginary part is reversed. Geometrically, this means
that a pair of complex conjugate numbers with the ρ ’s of both having below zero
values relative to the 0a-axis, that is negative real parts, could be imagined as the
points indicated by two same sized, mirror image, clock hands pointing at 8:00 and
10:00 o’clock. Note that the φ , the angle of vectorial deviation of the arrow pair from
the 0a-axis, turn in opposite directions in these mirror image moving clock hand
vectors. Without going deeper into the representation of the actions of the system in
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question (its differential equation) in the form of what is called its Jacobian matrix of
partial derivatives (a matrix representation of the differential equation indicating
orthogonal directional velocities of change of locations of the components of the
motion with respect to changing values of the control parameter), we know that
when the ρ of the matrix’s set of two complex conjugate eigenvalues is less than
zero, ρ < 0, the orbit representing the system, spirals into a stable fixed point. This
is analogous to going to the bottom of the parabolic attractor basin as described
above. Values of the invisible eigenvalues and their changes constitute the abstract
mathematical mechanisms underlying the observable dynamics of the system
observable physically.
The mathematical mechanism underlying the Hopf bifurcation of fixed points
into limit cycles (associated with bi-directional splitting that accompanies the
amphetamine transformation into limit cycle stereotypy of rigid ideas and equally
likely mirror image motions in the directions of good versus evil) is the crossing of
the systems real valued parts, ρ ’s, of its complex conjugate eigenvalues into
positive territory, ρ > 0. The mirror image of clock arrows is transformed from 8:00
and 10:00 o’clock to the clock locations of 4:00 and 2:00. At a Hopf bifurcation, a
pair of complex conjugate eigenvalues crosses the imaginary (vertical) axis such
that is real parts have positive value. In the orbit representing the motions of the
system itself, the fixed point disappears to be replaced by the action spiraling out to
an invariant circle. This is analogous to our manifold image of the disappearance of
the central attractive point and the sudden appearance of a small hill at the bottom
of a parabolic basic of attraction.. The new attractor is an invariant circular path
around the hill, with the spiraling out to the invariant circle being a two dimensional
picture of the disappearance of the bowl-bottom and appearance of a missing point,
hill top fixed point and a spiral flow to the path circling the hill. Underlying the
transition from a fixed point to a limit cycling, invariant circle, are a pair of mirror
image complex conjugate eigenvalues that turn in mirror image, we could say, good
versus evil, opposite directions. The Hopf bifurcating system inevitably has both.
The implications of this very abstract metaphor for the emergent limit cyclesplitting
style of spiritual transformation can be made deeper by considering the
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common practice of Rumi’s Mevlevi (and other) orders of Islamic Dervishes that
facilitate the onset and maintenance of their ecstatic states by an improvisational
dance which goes from rocking to irregular whirling. The Dervish teaching tales
place a symbolic emphasis on the power of the rotating wheel, the circling of the
heavenly bodies, the mill wheel and the millstone. As Rumi said, “The mountain of
the sun I’ll fashion to a mill. And as my waters run, I’ll turn thee at my will.” Note that
their work toward spiritual transformation results in neither the emergence of the
involuntary and rigid limit cycles of invariant circles or the associated divisive
internal eigensplitting of good self from evil other. The Sufi compass points to an
integrated field of divine consciousness, which contains the appearance of the
world’s multiplicity. In this profound unity, all humankind is perceived as one family.
The singular direction of all prayer, Salat, five times a day, at dawn, high noon,
afternoon, sunset and an hour after sunset, turns the entire world into a unified
directional field of prayer. At its center, the Islamic pilgrims wander round and round
the black cube of the ancient shrine of Kaaba,
This leaves one with the speculation that we started with: that the simple,
authoritarian rules of the amphetamine, roll-up and splitting religions may be
intrinsically more vulnerable to unpredictable breakouts into morally inconsistent
actions and that the righteously rigid limit cyclists are invariantly split into
ambivalence. In contrast, the more free form, chaotic turns of the entheogenic
dervish define us all as belonging to one unified ecstatic field.
Further Readings for Amphetamine Roll-Up And Splitting
Psychology and Religion. Carl G Jung, Princeton Univ. Press, N.J. 1938.
The Faith of a Heretic, Walter Kaufmann, Meridian, N.Y. 1959.
Nightmare Season. Arnold J. Mandell, Random House, N.Y. 1976.
165
The Rabbinic Mind. Max Kadushin, Bloch , N.Y. 1972.
Coming of (Middle) Age. Arnold J. Mandell, Simon and Schuster, N.Y. 1978.
Introduction to Islamic Theology and Law. Ignaz Goldziher, Princeton Univ. Press,
N.J. 1981.
Tales of the Dervishes. Idries Shah, Dutton, N.Y. 1970.
Open Secret; Versions of Rumi. J. Moyne and C. Barks, Threshold Books, Putney,
Vermont. 1984.
Amphetamine Psychosis, P.H. Connell, Oxford University Press, Oxford, 1958.
Amphetamine Use, Misuse and Abuse. David Smith, Hall, Boston. 1979.
Long-term Administration of D-Amphetamine. David S. Segal and Arnold J. Mandell,
Pharmacology, Biochemistry and Behavior. 2:249-255. 1974.
Amphetamine Enhancement of Reward Asymmetry. S.D. Glick, L.M. Weaver and
R.C. Meibach, Psychopharmacology 73:323-327, 1981.
Hopf Bifurcation and Its Applications, Appl. Math. Sci. Vol. 19,. Springer-Verlag,
N.Y., N.Y. 1976.
Dynamics, The Geometry of Behavior, I-IV, Aerial Press, P.O. Box Office 1360,
Santa Cruz, CA 1982.
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. John
Guckenheimer and Phillip Holmes, Springer-Verlag, N.Y. 1983.
166
Psychiatric Aspects of Neurologic Disease. D. Frank Benson and Dietrich Blumer,
Grune and Stratton, N.Y. 1975.
Drives and Reinforcements. James Olds. Raven, N.Y. 1977
Neurobiology of Stereotyped Behavior. S.J. Cooper and C.T. Dourish. Clarendon,
Oxford, 1990.
Mathematics Unlimited—2001 and Beyond. B. Engquist and W. Schmid, Springer,
N.Y. 2000.
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CHAPTER 8:
FAITH AND RATIONALITY
It was my belief that, without subjective evidence of Holy Spirit Energy, the
rush of reconfiguring transcendent experience, some glimmering of grace no matter
how fleeting, an experience of intoxication with God, Martin Buber’s self
authenticating I-Thou encounter, the many good citizens of this world, without these
moments of illumination, must be attending church or temple to negotiate a better
now and hereafter. Attending synagogue or church without the promise of a
mystical high seemed like a superstitious rabbit foot rubbing for personal health and
safety and a sharing of propitious contacts for social and economic advantage. Why
else?
I have had the good feel of what Jews call Tzedakah, the sharing of supplies
by the haves for the betterment of the have nots. I have known the quiet calm of
human right action as in the Unitarian Universalist’s serving the needy, open and
flexible, intimate mindfulness of others and their needs. Considering E.O. Wilson’s
brand of brain herd biology of altruism gives me a warm feeling about the potentially
intrinsic goodness of man. But compared with the Jamesian brands of ecstatic
transcendence, minds blown in Sufi twirling, Orthodox Jewish chanting, rocking and
dancing, hands-in-the-air praying and hands-on-the-head healings of Wednesday
night Pentecostal services, the soberly serious social engagement and
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responsibility sermons of Reformed Judaism and the Unitarians as well as the 19 th
Century hymns and high I.Q. apologetics of some Presbyterian and Methodist
clergy, are like near beer. Formally equivalent but without the rush and the delicious
risk and promise of life long addiction.
National opinion polls have found my preference for churchly fireworks in
religious experience quite common. My Charismatic Christian sons are among the
many with a preference for and loving labeling of these kinds of houses of worship
as rock and roll churches. In a recent survey of Americans, 46% of respondents
claim to be twice born, Evangelical Christians. Perhaps unfortunate with respect to
their children’s academic and professional ambitions, 48% do not accept a
Darwinian view of biology. Fifty million American readers are now buying books with
plots taken from the Babylonian prophecies and anticipate the Rapture of Return
with weekly, joyful, mini-rehearsals. They include praying in tongues as the Spirit
moves them like Peter, John, James and the rest of the one hundred and twenty in
the upper room on the day of Pentecost.
Those of us with two or more available cable religious networks can, on any
given Sunday morning, choose a smiling, kind, Proverbs quoting, rational
Presbyterian liturgical stylist. In his seventies, standing tall with a full head of white
hair and in a quietly resonant voice, he delivers a sermon about seven ways to
avoid growing old. His list includes learning new things and continuing to work. His
spiritual proposal was about personal faith, always leaning on the Lord. On another
network, the three hundred pound, restlessly pacing preacher of the Cornerstone
Assembly of God Church of San Antonio, Texas, stood in front of large maps of Iraq
and the Middle East. He preached from Ezekiel about the refleshing of dry bones
and a return of all Jews to Israel. He said that contributions to his church over the
past year helped finance the return of 4000 Russian Jews to Israel. He reiterated
the promise that, when the return was completed, there would be a massive Islamic
attack on Jerusalem and “we will all rise up to Heaven” in an ecstatic
disappearance. Jews, as long as they accepted Jesus as their Savior, were
welcomed along on the ride. More then two thousand parishioners erupted into loud
applause along with shouts of “praise Jesus.”
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An inkling of something entirely different, neither human psychology nor
frenzy, was an unanticipated benefit of being at England’s Warwick University in
sabbatical residence in Math House #2. This large, round, many windows and black
boards, study with a small upstairs bedroom was one of the apartments for visiting
professors behind the Warwick Mathematics Institute in the English Midlands. I
attended a variety of churches and synagogues on the weekends. The perspective
that emerged for me at Warwick was that rabbinic Haggadah, inferences to be
drawn from imaginatively spawned narrative, isn’t the same thing as Halakhah, the
law dictated by Jewish legal tradition; that geometric insight and other intuitions
aren’t the same as mathematical proofs; that the mystical visions of the English
romantic poet and illustrator, William Blake, were not necessarily consistent with the
scientific observations and logical arguments of the contemporary Scottish
philosopher, David Hume. Paul Tillich wrote that the wisdom attendant to primary
spiritual experience that was without the unconditional character of sensible moral
obligation was not to be trusted without critical analyses. I learned that among High
Episcopal and Reformed Jewish English academics, God is not a hallucinogen, but
more like a spiritually based, social contract.
In his 1929 essay, Mysticism and Logic, Bertrand Russell noted mysticism’s
preference for: (a) Insight over discursive analytic knowledge; (b) Belief in the unity
of all things over oppositions or divisions in representational thought; (c) The denial
of the reality of time, even in the divisions of past, present and future; (d) Belief that
evil is unreal, manufactured by the innate divisiveness in some analytic intellects. In
modern brain hemispheric and other neuropsychological philosophies, these
countervailing descriptions of external observables can grow naturally out of the
brain’s abilities to maintain logically incompatible perspectives simultaneously.
Right-brain aesthetic holism in contrast with left-brain categorical analytics recalls a
popular example. Would one chose Blake or Hume to better explain how the time
dimensions of memory disappear with the scent of a past lover or the hearing of his
favorite music for lovemaking.
In the inevitable mix of primitive instinct with high purpose, the visiting
professors’ Math House #2 had an aura of infamy. It was the one in which, by the
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accidental intrusion of a campus security officer, the brilliantly eccentric Northern
California mathematician, Ralph Abraham, was famously arrested for pot smoking.
The campus officer told me that, late one night, thinking he had smelled fire, he
used his master key to make an unwelcome entrance. The incident became part of
the record in House of Commons hearings about the intellectual and moral decay of
English Universities. Apparently, even among English intellectuals, there were trivial
and politicized definitions of virtue.
Christopher Zeeman, the head of the Mathematics Institute was a worldclass
topologist who, among other things, demonstrated biological and socialpsychological
applications of Rene Thom’s Catastrophe Theory. I was invited as a
brain person and amateur mathematician, to see what might result from mixing me
with members of his fine mathematics faculty. In addition to learning some
bifurcation and lots of ergodic (statistical) theory, my chats with Christian and
Jewish mathematicians on Saturday and Sunday morning visits to the synagogues
and chapels of Oxford and Cambridge introduced me to an English intellectual’s
religious tradition. The spirit of C.S. Lewis was still very much alive. Surprising,
however, was that more than a few of these scholars had the elements of Christian
faith in full menu: virgin birth, incarnation, crucifixion, resurrection, original sin and
the promise of salvation. I was disabused of my belief that these elements of
Christian belief were incompatible with high mental capacity and intellectual
sophistication.
Yet, the spiritual climate of these English intellectual Christians were different
from today’s post, post Vietnam return of the religious themes of the turn of the
Twentieth Century, big tent revivalism and Billy Sunday’s brand of Christian patriotic
America. Today’s religious patriotism infuses George W. Bush’s Republican base,
National Security Adviser Condoleezza Rice’s after dinner hymns and Attorney
General John Ashcroft’s early morning bible study groups for his Assistant Attorney
Generals. Even the most religious of my English math buddies are without what
seems like adventitious baggage of today’s faith based Republicans: the belief in
the immorality and godlessness of teaching evolution in schools, what has been
called the massacre of the innocents in stem cell research and abortion clinics, the
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right to bear machine guns and the intrinsically venal sinfulness of a man’s
commitment in love of another man. Was the clustering of these apparently diverse
concerns the accidental result of a sociopolitical-religious short circuit, a classresentment-driven
spiritual split in geographic, socioeconomic and educational
class? Tim LeHay is selling millions of books, whole tables full at Wal-Mart’s, which
come packaged with these assumptions.
Surely higher-level theists would make today’s evil more subtle, abstract and
pervasive, perhaps involving inner life themes of envy, vengeance and aggression;
goodness implicating empathically made moral choices involving interpersonal
kindness and evidence of caring about the well being of others. My contact with
some English academicians taught me that even the mathematics of hard science
can be viewed as a gift of grace and belief in the possibility of a continually
emerging, Christ-centered, evolutionary process. Protestant philosopher
mathematician Alfred North Whitehead in his 1926 Religion in the Making, Catholic
anthropologist priest, Pierre Teilhard de Chardin in his The Phenomenon of Man
and the more modern process theologists of New York’s Union Theological
Seminary do not exclude Christ’s involvement in evolving science and other new
knowledge. They see Him participating in a spiritual evolutionary progress which
does not gather the barnacles of irrational ideas about the murder of less than
hundred-cell blastula or the psychoneurohormonally determined sexual partner
preference. They know about the ever-changing cultural and political appearances
of faux and real evil. Nonetheless, what I learned from my Christian and Jewish
friends at the mathematics institute was that, though the definitions of evil may
change, evil as a construct and spiritual mechanism is an apparently essential
component of the Christian experience. On Rosh Hashanah, even the reformed
Jews commit themselves to Teshuvah, making up for past evil deeds. The good
versus evil dichotomous view of man’s existence is true in the lives of Assembly of
God Fundamentalists of Georgia as well as the sophisticated Readers, Professors
and Dons of the high Episcopal churches and university chapels of Oxford and
Cambridge.
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Finding high-level mathematical thinkers at home in metaphysical surrounds
and metaphysicians diligently practicing mathematics are certainly not new. Some
instructive examples include, famously, the Pythagoreans, the 15 th Century Catholic
Cardinal Nicholas Von Cusa, who used geometric symbols to record his spiritual
philosophy, and the Talmudic-Cartesian style of argumentation of Nicholas de
Spinoza. This approach to an examination of metaphysical systems, sometimes
called mathematicism, exploits the machinery of the mathematical mind to evaluate
the consistency and completeness of thoughts, to create representative axiomatic
structures and to operate within them using syntactic calculus. The practice of the
rational dialectic of mathematicism, working for moral purity of heart, develops a
brain-somatic discipline much like the exercises of Yoga.
This approach flies in the face of the major premise of these essays, my
belief in the necessity of what William James and others have called the primary
religious experience in order to know God. Recall that my father’s favorite Jewish
mystic, Abraham Abulafia, said this experience gives birth to an activated mind that
can then immediately and completely inform the Spirit. Among the religious English
mathematicians, I learned that it doesn’t have to happen this way. One can
apparently think oneself to It. A well known example of a modern theistic Oxford
type, the Magdalene College English tutor and Don, C.S. Lewis, in his introduction
to St. Athanasius’s The Incarnation of the Word of God, wrote, “…I believe that
many who find that nothing happens when they sit down or kneel down with a book
of devotion, might find that their heart would sing unbidden while they are working
their way through a tough bit of theology with a pipe in their teeth and a pencil in
their hand…” In contrast, without my personal experiences with joyful
transcendence, the direct feeling of His presence, I would not have known about the
goals of his more analytic efforts. It was a struggle for me to use a rational mind to
share the meanings of the poetic ruminations in his BBC lectures, Mere Christianity.
This Reader from Oxford with two firsts in Latin and Greek followed by another first
in English Literature, described the world as “…enemy occupied territory…” the
omnipresence of the Good Power turned Dark Power of the Prince of Darkness and
the Christian as “…a man who is enabled to repent and pick himself up…”
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For C.S. Lewis, religious faith came from intellectual hard work. He was put
off by spirituality that arrived by thoughtless fiat. He rejected the idea of living in
simple and loving direct conversation with the God within, as described by Brother
Lawrence. Lawrence was described as the simple “great awkward fellow who broke
everything.” Lewis had little faith in what he perceived as the mindless spiritual
methodology of this selfless, silent, hard working Parisian monastery cook for a
hundred fellow monks who was also their dedicated smelly sandal repairer. Perhaps
reflecting his place in the British intellectual class system, Lewis wrote that
Lawrence’s conversations and letters in the brief pamphlet, Practice of the
Presence of God, “…full of truth… but unctuous and repulsive.” At the same time,
Lewis spoke of his own experiential evidence for God in Surprised by Joy in which
he admits, “I am an empirical theist. I have arrived at God by induction.” It is likely
that Brother Lawrence did not know and did not need to know the difference
between an inductive and deductive argument.
For most of my years, I have been a subject of Jamesian transcendent
experience, LSD expansive visions, Sufi moving meditation, long distance running,
Black Baptist shouting, Tantric orgasmic withholding, Yiddish Labovicher dancing,
Charismatic Christian Church rock and rolling, Hindi meditative rising Kundalini,
almost any ecstatic crisis inducing, God type. Recall that I am from a generation
that a Donovan song inspired to smoke bananas. I did not personally access
Brother Lawrence’s calm, work-a-day, devotional, quietly persistent, perspective
yielding, inner conversations with God until my sixth decade. The opportunity came
from my growingly severe, unfixably chronic, pain. The counter-intuitive insight and
helpful identification was gained from reading about Joseph de Beaufort’s
conversations with Brother Lawrence. Beaufort said Lawrence was born with the
name Nicholas Herman in 1611 and renamed Lawrence in honor of his parish
priest. As young soldier in the Thirty Years War of the 17 th Century, he was severely
injured. He was left with both sciatic nerves trapped between bone spurs and tissue
scarring from his early twenties. These injuries, involving the two biggest painconducting
nerves in the body, left him crippled in gait and in chronically severe
lower back and leg pain from which he would never be free. It was after this time
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and a few years of looking for God in what he called “wondering in the wilderness”
that he began his 40 years of monastery service as cook and sandal maker. He was
described as amazingly selfless and a “…gentle man of joyful spirit…” who
“…continually walked with God…not from the head but from the heart…” Doing long
hours of selfless work with such painful disabilities, how was it that he maintained
his joyful, loving and calm contact with God and his fellow man? How did he do it? I
found that, as with all miracles of God contact for me, it happened by itself.
I suffered my first testicular cancer in my thirties. I felt the little hard rock by
accident while scratching. It was on the left side. Surgical removal was followed by
a five-hour radical abdominal lymph node dissection that left me with incidental
abdominal sympathetic nerve damage, urinary hesitancy and ejaculating backwards
into my bladder. The tissue diagnosis was of embryonic cell carcinoma with
chorionic elements. The U.S. Armed Service Pathology Department’s statistical
book gave me 5% chance of living beyond two years. My second testicular cancer
occurred in my fifties and on the right, two little joined lumps found by my wife. It
was a seminoma with cure rate of 85% but requiring four weeks of almost daily x-
ray treatment. The combination of radiation induced blood vessel scarring (they had
to blast widely since my earlier lymph node dissection confused the usual
radiological anatomy), a pre-existing laterally curved spinal column and the arthritic
changes resulting from fifteen years of running over 10 miles per day with this kind
of back led eventually to the degeneration and collapse of several of the bodies of
my vertebrate pinching several leg nerves between bone spurs and radiationinduced
scarring. I have been in increasingly severe back and leg pain for fifteen
years.
It was in this way that I fell heir to both Brother Lawrence pain syndrome and
what I now think was his strong inclination to live in the Spirit, as far as possible
outside the concerns with his own mental and physical body. In my experience, this
led naturally to a decreased in my life long narcissistic preoccupations, diminished
my ego-driven achievement desperation, setting up a more comfortable inner
seating for conversations about and with God. The choice was between fully
embracing a God-oriented place for most of my daily existence or the chronic use of
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enough narcotics to eliminate complexity of thought, real interpersonal feelings and
hope for meaningfully creative work. The remarkable thing to me was that people
began to talk about my “improved disposition,” an increase in out-of-mypsychiatrist’s
office personal empathy and kindness as well as a significant
decrement in my overweening, ego-stoking ambitious and competitive urges. Any
return to the earth body of tense readiness to competitively succeed, protect with
ego defensive anger, fantasies of assertive sexuality, stand tall grandiose notions of
intellectual superiority, even getting up for scientific combat, was accompanied by
the return to this world of pain. Only lovingly detached, unpretentious, other
directed, quietly calm inner dialogue with Him was a place that I could live. This was
an inner land of still another kind of God than I had previously known. I could even
read and struggle with theological ideas thoughtfully, without referencing personal
mystical, psychopharmacological, Holy Ghost-mimetic, experiences. I could enjoy
the rational, social responsibility valuing, spiritual peace of a white Protestant
Sunday morning service. I could attend Reformed Jewish Friday night services
about man’s responsibility to man without restless boredom. No longer seeking the
feeling of God’s thrill, I could think about it, even without being in the state of my
father’s and Abulafia’s activated mind.
If I had been benefited with a classical language education beyond the high
school and early college Latin of Julius Caeser and Cicero or matriculated in an
academic theological seminary, I would have already studied, maybe even worn
out, the deeper aspects of what seemed like a paradox of the consonance of faith
and reason. I would have been familiar with the rhetorical argumentation in the
patristic Latin commentary on sacred texts by Tertullian and other Fathers of the
early Christian Church, the Talmudic discussions (the Mishna in Hebrew and
Gemora in Aramaic) of the oral Torah by the Rabbinate, the Muslim explication of
Koranic Islam in the oral tradition of the Hadith. Robert Wilken in his recent The
Spirit of Early Christian Thought was in no doubt about the harmonic relationship
between rationality and faith: “…by putting itself in the service of truth, faith enables
reason to exercise its power in realms to which it would otherwise have no
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access…” It is perhaps strange to come to this common knowledge so late, but I
came to my life with my forbearers and father’s magical, mystical biases.
My father had parodied what he thought was the “wasteful time” spent in
rational, Talmudic discussion. He said that is what Jewish men spent their time
doing to avoid physical work while sitting near the city gates. It was the women who
raised the crops and cared for the cattle and children. He had a favorite conundrum
satirizing the village gate discussions. Jewish males, after the age of thirteen,
accompany their morning prayer of commitment to loving and serving God with the
ritual of wrapping scripture embedded animal skin, tefillin, and winding them seven
times around the left arm, near the heart, and around the head, symbolizing the
mind. This contextualizes how my father made fun of a typical topic of these all
male Talmudic seminars: “If one had seven arms, would one wrap the tefillin once
around each appendage or seven times about one of them. If the latter is the case,
how would one chose which one.” In fact, there remains an on-going debate about
the order with which the embedded four passages from Exodus and Deuteronomy
should be arranged and inserted in the tefillin such that some compromising
orthodox Jews wear two types of tefillin, each representing one of the theoretically
justified orderings. I know now that there is an implicitly positive confirmation of a
jointly held faith and feeling of ethnic belonging achieved by such apparently
abstract discourse and argumentation.
In truth, I had not come to Warwick to explore the relationships between faith
and rationality using the cognitive style of mathematicism, but rather to be saved by
the mathematical miracles of the Brain God. Not unrelated to what C.S. Lewis saw
as a prominent characteristic of spiritual experience, “wonder,” and what Philip
Davis and Reuben Hersh in their 1981 book, The Mathematical Experience, spoke
of as “beauty” and “surprise.” I know about the attack of excitement that comes with
the sudden emergence of counterintuitive conceptual connections while exploring
new mathematical ideas. In energetic high, I start skip reading, underlining the book
frantically, jotting commentary on the margins, copying the relevant equations into
my notebook. Was this the same break through to a glimmering of grace, everything
beautifully in order and precious, that I experienced on LSD while sitting for hours
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inside Paris’s towering, echoing, Notre Dame Cathedral, hearing Latin chants in the
dank sweet smell of old church and chained, swinging canisters of smoking incense
as the pipe organ roared? Those realities that George Berkeley, the 1721 author of
Treatise Concerning the Principles of Human Knowledge, the theist whose name
was given to a mostly agnostic Northern California city, saw as grounded in the
spirituality of God’s infinite mind and broadcast as universal ideas through our
derivative, finite minds. Rational religion and mystical religion joined in faith by the
presence of implicit and universal mathematical structure
I spent about two years at a mathematics institute in France, Institute des
Hautes Etudes, IHES, sitting at the guru feet of the mathematical great and
metaphysician, Rene Thom. His mathematical pallet was breathtakingly broad, a
taste of what in past centuries was called natural philosophy and what seemed to
me to be about the unapologetic geometrization of the Intuitive God of the Mind.
Natalie Angiers, erstwhile mathematician, now reporter and atheistic hard ass,
writing in the New York Times, called Thom’s ideas the talk of “…an Emperor
without clothes…” The Kantian theme of the personal a priori status of an intuitive
geometry, an already in us representation of all that’s out there, was implicit in his
Catastrophe Theory research program and was published first in his classical
Structural Stability of Morphogenesis (1977) and made more overt in his later
(1990) Semiophysics.
To get a feeling for the rational-logical versus mystical-intuitive spiritual issue
in a mathematical context, consider the following: most of us remember the struggle
to unify the strange and difficult cognitive duality of the high school geometry
experience. On one hand, shapes and their relations and rearrangements could be
intuitively grasped, even manipulated; on the other hand, we were taught that these
mental images and the results of their intuitive transformations were not to be
trusted.
In mathematics, as in my belief in the fireworks of primary religious
experience, seeing is not necessarily believing. In my high school geometry class,
what was to be believed was what followed from the proper practice of the tightly
organized, Euclidean system of axioms, postulates and the derivative logical
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operations resulting in the surety of proofs. The unresolved tension about what I
believed from intuitive experience and what I was allowed to believe from the logic
of theorem and proof, perhaps not unlike my belief in the transcendent experience
over logical theological argument as Reality, continued throughout my life. For
example, many decades later at IHES, I saw the world class dynamical systems
theorist and differential geometer-topologist, Dennis Sullivan, use a projector to
display a computer-generated, intricate and beautiful, mathematical object, the well
known, computer screen saver, Mandelbrot set. It represents the control parameter
plane of the well studied complex analytic map, z → z 2 + c. Sullivan, pointing to a
small, discrete complicated little part of it that looked like a little version of the whole
of it, from a distance looking like a point, said, “An important Ph.D. dissertation is
waiting to be done on the question: is this (pointing to the little object) really there?”
In the audience of about a hundred professional mathematicians and one amateur, I
was the only one that laughed.
Historians of mathematics point to the successful generalization of Euclidian
geometry via its abstract axioms, postulates and logical operations to a new, not
naturally intuitable, almost nonvisualizable, non-Euclidean geometry (with the new
geometric axiom, parallel lines do meet at infinity), as evidence against the Kantian
idea of the intuitively accessible, a priori status of geometry. This served as an
example of where mathematics naturally resides, and argues in favor of the thought
control imposed by the modern set theoretic and logical rituals of mathematical
theorem and proof. Thom, in a hereditary-evolutionary biological argument
developed in Semiophysics, said “Objections raised to the Kantian apriority of
Euclidean geometry after the discovery of non-Euclidean geometries, and the
theories of twentieth century physics (restricted and general relativity, quantum
mechanics) appear to me to be irrelevant…they deal with …the infinitely small and
infinitely large…which lies outside the usual cognitive activity of ancient man.”
In my discussions with him, Thom found equivalence relations between
mental and real world objects and their behaviors. He described what he called an
abstract physicalist truth that describes a psychic universe, which, in turn, simulates
outside things and processes. Much like the transcendent experiential God I have
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experienced, seek and think I know about, Thom was not after the logical proofs of
geometry but rather viewed mathematical theorem and proof work as activity
derived from intuitive experience with geometric relations as the thought forms that
represented real Reality. Though a Field’s Medal winner in mathematics (recall that
it is the Nobel Prize in mathematics awarded every four years at the International
Congress of Mathematics) and for his life time, one of the most brilliant and fecund
mathematicians in the world, so many mathematicians admit that they got the seeds
of their life work from his throw away remarks, Thom, with a little smile and his eyes
twinkling, admitted to me with apparent pleasure that “I have never proven any
theorem in my life.” All his discoveries came from insightful moments of grace and
the courage to pursue them. Riding back from Paris late one night on a train that
didn’t stop at IHES’s town of Bures sur Y’vette, I watched him use the red
emergency phone to call the train’s engineer to stop the train suddenly for our exit. I
loved him, in part, because he had the courage to believe in and act on my kind of
intuitively realizable, experiential God.
In keeping with his characteristic style of generalizing mathematical systems
beyond their carefully defined specifics, Thom defined the concept of singularity
very broadly, speaking of them as distinctive and noteworthy things, points where
the usual or expected properties, laws and definitions fail, where smooth and
continuous processes become discontinuous. For Thom, these were the settings for
the unexpected and miraculous. He believed that his work and that of many others,
now and in the future, would indicate that the set of miraculous singularities were
finite, systematic, universal and describable. Most importantly for our purposes,
Thom believed them to be archetypal. It was through the structure of archetypal
singularities that he regarded inside and outside realities as mutually reflective.
I was blessed by hours of discussion with him during his car travels to
lecture around France. Thom often asked me to accompany him as he drove from
IHES to various branches of the University of Paris. He used these times to
exercise my geometrically flavored, mathematical intuitions. He used words to
create visualizable structures without the diagrammatic aid of a blackboard. He
used mental topological structures created by the properties of imagined motions,
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flows, which led to examples of some of his universal singularities that he claimed
could be found in all real physical, biological and psychological systems. For some
examples: One of his archetypal singularities was a boundary at x = 0 such that the
flow couldn’t spread from where it was in x ≥ 0 into x< 0 and was therefore like the
border, the membrane, between the inside and outside of a cell as well as the
hoped for sociopolitical functions of the Great Wall of China and the Maginot Line. If
we were to blow up the boundary line from two to three dimensions, R
2 →R 3 , the
straight boundary line becomes a cylinder for directionally organizing and
connecting flows as in blood vessels, oil pipes, cables and wires. Since production
and delivery need not occur at similar rates, temporary storage is required and may
take the form of a spherical blow-up in the vertical segment of R 3 leading to an open
bottle which may serve as a dead end storage branch of a network of connected
cylinders. In the conceptual reductionism of Semiophysics, Thom said, “…life is
essentially a question of embankment, canalization and the struggle to stem
dispersion.” These structures of mind and world are built and maintained.
Coagulation of blood is an example of a canalized fluid repairing gaps like a
tubeless tire. Thom considered apparent the problem of making something from
nothing, birth, that of finding the hidden sources: the bubbling spring emerges from
an unseen, underground network of canalized fluid flow converging on the apparent
source, birth being the invisible becoming visible. In contrast, a canalized flow
emptying into lake can represent disappearance as a flow.
Mathematicians from all over the world attended Thom’s 65 th birthday
celebration at IHES. His Field’s Medal winning work on the topology of differentiable
(smooth) manifolds, cobordism and related ideas, was mentioned frequently, and
great homage paid to him with respect to these areas of his work. However, in two
days of lectures of personal and professional tribute by the world’s great
mathematicians, his work relevant to Catastrophe Theory and Semiophysics was
not mentioned, even once. The form taken by mathematicians’ most severe
judgments is silence. As the New York Times’ Natalie Angier’s comments indicated,
this is not the time for the intuitive conduct of applied mathematics.
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It was upon Thom’s recommendation, that I spent the year in the
Mathematics Institute in Warwick, England. Using the Math House #2 s home base,
I made many trips to Oxford University and a few to Cambridge. It was in these
places that I learned first hand that belief in the Resurrection was not simply a
matter of socioeconomic class. I tried to schedule my trips to Oxford or Cambridge
to coincide with the weekend so I could hear the remarkably literate sermons at the
Universities college chapels. In these places, for hundreds of years, just because
one was a top-notch practitioner of mathematics or linguistics did not mean that the
Don did not have within him the full panoply of beliefs attendant to the Christian
God.
Maybe this easy combination of logic and Spirit derives from the character of
English mathematics. There are graduates with professorially enfranchising Masters
of Art Degrees in Mathematics from Cambridge University where the subject is
considered by many to be part of the culture of the humanities, closely akin to
philosophy and linguistics.. In the universities of United States, for example the
Massachusetts Institute of Technology, an academic degree of Ph.D. in
mathematics is seen by most faculty as an indication of the intellectual equipment
required for a life of scientific work in which disconfirmable experiments are the
ultimate criteria for knowledge. The field of pure mathematics (not ostensibly
relevant to the real world outside the mind) has itself evolved in this direction.
Recently, a physical scientist, a theoretical physicist, Edward Witten, was given the
mathematician’s ultimate award, the Field’s Medal. In American universities in
general, very few mathematics departments are in schools of the humanities. Most
are in the schools of science. This variation in bureaucratic, metaphysical, sorting
reflects our continuing struggle with the true nature of reality and the role of
mathematics in its knowing. The now emergent field of computer science removes
mathematics even further from intuition and Spirit. Difficult problems such as proofs
of theorems can be systematically examined for all possibilities quickly by trying
them out in what is now known as a computational proof. On the other hand,
pointing at this computation’s graphics, the theorem and proof, real mathematicians
can ask, is this really there?
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Mentioned briefly above was one of humankind’s beacons, Pythagoras, the
intellectual and spiritual progenitor of Plato. He taught the disciples of the
Pythagorean Brotherhood in Crotona, Italy, that reality at its deepest level was
mathematical thought. Studies there included philosophy, geometry, music and
astronomy, all at the service of achieving closer union with the Divine. Pythagoras
and his school, only his student’s writings remain, was said to be working at unifying
elements of the ancient tribal mystery cults with the observables of worldly events
through meditative, mathematical, philosophical mysticism. Knowledge was gained
through spiritual intuition made harmonious with formal systems of thought. As Plato
later said and as quoted by Thomas Heath in his 1921 History of Greek
Mathematics, about the study of the motion of stars, “…leave the heavens alone…”
because what one sees is only an approximation of the real and more perfect
mathematical structures involving points, lines and circles. To which Newton added
an elongated circle, the ellipse, and Nineteenth and Twentieth Century
mathematicians and physicists, the geometries of positively and negatively curved
space.
It is perhaps not an accident that debates about evidence for the existence
and location of God and where the ideas and structures of mathematics live and
breath generally involves a stand off between those that believe that both are out
there and can be seen, like thoughtful, humanistic actions and caring service for
needful others, versus those that feel the phenomenology of both are projections of
the psychobiologically intuitive Brain God and can be felt like an ecstatic rush of
insightful illumination.
Further Reading for Faith And Rationality
Introduction of Comparative Mysticism. Jacques De Marquette, Philosophical
Library, N.Y. 1949,
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Mysticism and Logic. Bertrand Russell. Norton. N.Y. 1929.
Sefer shel Devarium (The Book of Words). Lawrence Kushner, Jewish Lights,
Woodstock, Vermont. 1998.
Semiophysics: A Sketch, Aristotelian Physics and Catastrophe Theory. Rene Thom.
Addison-Wesley, Reading, MA. 1990.
Mere Christianity. C.S. Lewis. MacMillan, N.Y. 1952.
Sacred Geometry. Miranda Lundy. Walker, N.Y. 1998.
Fractals, Form, Chance and Dimension, B.B. Mandelbrot, Freeman, San Francisco,
1977.
Non-Euclidean Geometry, H.S.M. Coxeter, University of Toronto Press, Toronto,
1957.
Fundamentals of Mathematics, Vol. I. H. Behnke, F. Bachmann, K. Fladt and W.
Suss, MIT Press, Cambridge. 1983.
Religion Explained, The Evolutionary Origins of Religious Thought. Pascal Boyer.
Basic Books, N.Y. 2001.
Catastrophe Theory. Alexander Woodcock and Monte Davis, Dutton, N.Y. 1978.
It Must Be Beautiful; Great Equations of Modern Science. Graham Farmelo, Granta,
London, 2002.
Neurobiological Barriers to Euphoria. Arnold J. Mandell, American Scientist 61: 565-
573, 1973.
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Brain Physics and the Respiritualization of Healing. Arnold J. Mandell, Bulletin of
the National Guild of Catholic Psychiatrists. 28:19-24, 1983.
Toward a Psychobiology of Transcendence, God in the Brain. Arnold J. Mandell, In
Psychobiology of Consciousness (eds. J.M. Davidson and R.J. Davidson). Plenum,
N.Y. 1980.
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APPENDIX
AN INTUITIVE GUIDE TO THE IDEAS AND METHODS OF DYNAMICAL
SYSTEMS FOR THE LIFE SCIENCES
Arnold J. Mandell and Karen A. Selz
Biological Scientists Can Understand and Use Ideas and Methods of
Nonlinear Science
A yield of advances in computer hardware and software is that even quite
difficult applied nonlinear mathematics can become accessible to experimentally
oriented biological scientists. Before this time, the development and analysis of a
particular set of nonlinear differential equations, describing the actions of a
neurobiological system in motion, involved decades of specialty training, rare insight
and many hours of highly skilled, trial and error computations by hand. Since the
idiosyncrasies of each nonlinear system were considered unique, the results of their
analyses were thought to concern only the particular nonlinear system being
studied. Often a shift in hypothetical mechanism meant starting the long and painful
process all over again. In addition, these findings were usually communicated only
to a small and arcane mathematical community in the form of dense theorems and
difficult to follow proofs, insurmountable language barriers to biological researchers
wishing to use them to better describe and understand their experimental
observations.
For today’s neuroscientist with a desktop computer, an inclination to program
and access to computer algebra and numerical software such as Maple,
Mathematica or MatLab, operational definitions and computational empiricism can
replace the theorem and proof continuity required to do old style applied
mathematics. For those of us without sufficient facility in algebraic manipulation to
easily follow the arguments of professional mathematicians, a computer algebra
program such as Maple serves as a delightfully accessible consultant with which to
“check out what the guy is saying". Those motivated enough to write their own data
generating or analytic programs in C, Fortran, Pascal or Basic (though not
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essential) can find easy-to-use algorithmic help in Cambridge University Press’s
Numerical Recipes series (Press et al, 1991).
The conceptual and communication gaps between applied mathematicians
and physicists and the bench practitioners of the neurosciences, that inevitably lead
one or the other, most often both, to surrender their deepest intuitions to jointly
shared images that are inevitably more simplistic, are no longer inevitable. With her
own hands on both the quantitative conjectural and experimental machinery, the
motivated practicing neuroscientist can honor her own insights, read about and
construct symbolic representations from her intuitions and do her own quantitative
theory. Computerized numerical techniques have become so powerful and
accessible that, even in academic settings, there is debate about whether
fundamental analytic tools, such as series expansions, should be taught in
undergraduate courses about differential equations.
The practice of “try it and see what happens", with the current name of
experimental, computational mathematics, is accessible to all. In addition to the