“My Goedel Is Kill Me” by Resident Skeptic, James R. Cowles

James R. Cowles is a member of the diverse Bardo Group Beguines, publishers of The BeZine, which I manage and edit. James also regularly contributes to The BeZine’s sister site, Beguine Again. James isn’t shy of controversy and while you may not always be in agreement with him, you will always be encouraged to revisit and rethink … and, the man is endlessly entertaining. / J.D.

Have you ever had the experience of noticing a certain pattern in a wild variety of contexts, a pattern that occurs so consistently that you feel it simply has to mean something … but you have no idea what? I say “in a wild variety of contexts” to rule out cases of patterns that occur within the same context, even though, at the time, you may have no idea of the cause. I remember back in the early 1960s, when I was in junior-high school, I went on a “geology jag”. I spent several months reading books on geology, geophysics, and volcanology that noted with perplexity the mysterious – in the early ’60s – pattern whereby volcanic activity tended to be concentrated around the circumference of, e.g., the Pacific Basin, what we today call the “Pacific Ring of Fire,” and similar places. Given the context, it was very reasonable to suppose that the pattern had something or other to do with the physics of the deep earth. Several years later, along came tectonic-plate theory and suddenly the “Ring of Fire” pattern made all kinds of sense:  the dots were connected. Well … those are not the kinds of patterns I mean, i.e. patterns that are so closely associated with a common context that inferring a common context-related cause is almost unavoidable.

Kurt Goedel

M. C. Escher

Rather, what I do mean are patterns like the recurrence of the irrational, transcendental number pi in contexts that, at least on the surface, have nothing to do with the calculation of, e.g., the circumference of a circle, the area of a circle, the volume of a sphere, the period of a pendulum describing a circular arc as it swings, etc., etc. I have written about this elsewhere. I still find this pattern mysterious, enticing, almost an invitation to some kind of Platonic or Pythagorean mysticism.

I recently had an “Aha!” moment about similar patterns that are … hmmm … well … rather than attempt an abstract definition whose generality would probably render it unintelligible anyway, I will start by citing a specific example:  the enigmatic drawings of M. C. Escher. (The context in which I encountered Escher is also important, but more about that a little later.) Many of Escher’s drawings are conventional enough, distinguished by an austere, draftsman-like precision of line, geometry, and perspective. Others, however, are anomalous, counterintuitive, antinomic. The antinomy is especially pronounced in e.g., drawings in which two hands sketch one another, a spiral staircase where the uppermost landing coincides with the ground floor, etc., etc. In all these drawings, there is a kind of pseudo-hierarchy, “pseudo“ in the sense that ascending through the various echelons of the hierarchy ultimately leads back to the lowest level thereof. I show several examples in the images that accompany this column.

Penrose Staircase (by Roger Penrose), after M. C. Escher

Ascending and Descending … M. C. Escher

I had been familiar with Escher’s work for some time, first encountering it in Douglas Hofstadter’s fascinating and challenging book Goedel, Escher, Bach – An Eternal Golden Braid. Hofstadter termed “strange loops” Escher’s work, much of Bach’s music (e.g., The Musical Offering), and for technical reasons I will gloss over for now, Kurt Goedel’s monumental Incompleteness Theorem of 1931.(Goedel fled European anti-Semitism, emigrated to the United States, and took up a research position at the Institute for Advanced Study in Princeton, NJ, where he became an intimate friend of Albert Einstein, who had fled Europe for the same reason. Fascists in the Europe of the 1920s and 30s were much like Republican conservatives today, believing that too many smart people, especially really smart Jews, constitute a liability, not an asset.) Strange loops — I am pretty sure the term was coined by Hofstadter — are structures that appear to be hierarchical, but that are structured such that following the hierarchy up ultimately — after a perhaps large but finite number of steps — terminates in the lowest level, the “ground floor”, of the hierarchy, much as if one climbed the Washington Monument — and exited back on the Mall.

I was so fascinated by strange loops that, shortly after reading Hofstadter’s book (hereafter GEB), I talked to a professor-friend of mine — I was a graduate student in math, physics, and philosophy at Wichita State University at the time — whose specialty was mathematical logic and Bob agreed to basically teach me Goedel’s great Incompleteness Theorem. Bob has passed now, but his legacy for me was a continuing fascination with the foundations of math and systems of inference — so much so that the semester after the independent study I read Goedel’s Proof, a semi-technical treatment of the proof by Ernest Nagel and James R. Newman. Anyway … the whole point of this paragraph is to give you some idea of how monumentally dense and dumb I was:  I understood almost all of what I read, but, lacking an appreciation of the “wild variety of contexts” I mentioned in the beginning, I saw only the individual trees and never the Forest.

Goedel and Einstein at Princeton

The reason Goedel’s Incompleteness Theorem — the biggest Tree in The Forest — qualifies as a strange loop is because, in the process of proving his eponymous Theorem, Kurt Goedel managed to mirror in the proof of the Theorem the Theorem itself. In fact, more than that, the proof of Goedel’s Theorem ends up being isomorphic, i.e., structurally identical, to the numbers and to the very statements about numbers that constitute the very subject of the Theorem. Goedel’s Incompleteness Theorem is actually about itself.  Now, for very deep reasons I simply haven’t the space to go into — hence the “hand-waving” tone of this column — strange loops, however different they are in other respects, all have in common this property of self-referentiality: in different senses, all strange loops are “about” themselves and lead back to themselves … except that there is no “back” because there is no movement. That is a common feature of the various species of contextual trees in the strange-loop Forest.

But I saw the Forest, in fact, I realized there was a Forest, only gradually as I began to reflect on other contexts — contexts radically “other” than mathematical logic and the foundations of math.  I remember the chill that ran up my spine — gradually and over time — as more and more of the Forest became visible, as strange loops manifest themselves in an increasingly “wild variety of contexts”.  Herewith a few:

o Goedel’s Theorem itself

Without getting lost in the technical “weeds,” suffice to say that Goedel’s Theorem asserts that, under certain very weak conditions (basically, you only have to be able to do elementary arithmetic in your system of mathematics), there are certain statements in any system of mathematics / inference / logic that are true but not provable. (Here “provable” means, essentially, producible by a “Turing machine” or “Turing algorithm,” i.e., an algorithm / recipe that just mechanically grinds out theorems for your system of inference with no admixture of creativity on the part of the mathematician / logician who is turning the Turing machine’s wheels.) That is to say, if your only way of proving theorems is via recourse to a mechanical, “paint-by-numbers,” recipe-like, follow-the-bread-crumbs prescriptive procedure, then Goedel’s Theorem says that there will always be certain statements that are true, but which cannot be proven.

If you want these unprovable theorems to be provable, you can always alter the axioms of your system — but then other statements, including statements previously provable, will end up being unprovable in the revised system. In any system of logic, there will never be a one-to-one, exhaustive relationship between statements that are true and statements that are provable. (Goedel proved his Theorem in response to Principia Mathematica, the monumental attempt by Bertrand Russell and Alfred North Whitehead to derive all of mathematics from logic alone. Goedel’s Theorem is a technically rigorous way of saying “Sorry, gentlemen! Y’can’t get there from here!”) A good visual metaphor for this is trying to trap a droplet of mercury under your thumb:  you cannot, because the mercury droplet will always find a way to squirt out. The “thumb” of any axioms and rules of inference will always allow certain true statements to escape.

Goedel’s Theorem is a “strange loop” because, even though the Theorem is a theorem about meta-mathematics, i.e., a theorem about all systems of mathematics as such, the proof of Goedel’s Theorem — you will just have to trust me here (though I do recommend Nagel’s and Newman’s book, as well as GEB) — relies on replicating the structure of ordinary, non-meta-mathematics. That is, you think you have climbed one round higher on the logical staircase from mathematics to meta-mathematics, but in reality, you are still on the ground floor. You have not actually gone anywhere in any hierarchy. In fact, there is no hierarchy. You have always remained on the ground floor of Escher’s mad castle.

o Christian theology

The ancient world conceived of the Universe as a vast hierarchy spanning unformed matter at the ontological bottom up to God at the top. According to St. Paul’s great hymn in the second chapter of Philippians, Jesus, the Second Person of the Trinity, descended to earth, and even under the earth, and as a result God the Father — who was also Jesus, by the way — exalted Him to God’s right hand. So, in a celestial sense, by following the Hierarchy of Creation, Jesus ended up back where He started. This is usually described in terms of kenosis, but it is also a grand, cosmic strange loop:  Jesus, while remaining God, descends from God and returns to God without for all that ever ceasing to be God. Jesus’ kenotic Journey is a Journey back to where He “came from”, i.e., where He always “was”.

o Literature, in particular, T. S. Eliot’s “Four Quartets”

As I have said elsewhere, I have come, after 40-plus years, to believe that the key to understanding the “Quartets” is the celebrated passage from Heraclitus that is the preface to “Burnt Norton”: The way up and the way down are one and the same. The “Quartets” comprise a literary embodiment of this maxim:  Eliot’s experiences during the London Blitz convinced him that the Journey into exaltation just is the Journey into pain:  the fire and the rose are one. So (“Burnt Norton”): At the still point of the turning world … Neither from nor towards … there the Dance is … But neither arrest nor movement … ,  so that (“Little Gidding”) the fire and the rose are one.  Thus we arrive where we started and know the place for the first time. Mystical spirituality is a strange loop:  a non-ascent through a non-hierarchy.

I could cite other examples of strange loops until you seriously consider slitting your wrists in a tub of warm water, e.g, fractal phase spaces of chaotic / non-linear / “far-from-equilibrium” phenomena, many short stories of Jorge Luis Borges, many paintings by Jackson Pollock, holograms, et al.  All these involve another characteristic of strange loops: scale invariance, whereby a piece of the strange loop, no matter how small, looks just like the entire strange loop, e.g., magnifying a small area of a Pollock paint-dripping painting.

But the most provocative, even uncanny, maybe even “spooky” aspect of strange loops is Hofstadter’s compelling argument in GEB that strange loops constitute the essence of consciousness.  Human consciousness has evolved as a strange loop:  from organic molecules, to single cells, to multicellular life … etc., etc. … finally culminating in human consciousness — which now “turns around” and contemplates itself and its own origins.  But beyond even this — which is momentous enough in its own right — is that the sheer ubiquity of strange loops, which are everywhere once you become sensitized to seeing them, invites the speculation that consciousness is not confined to the space in the skull between one’s ears. Consciousness may be a kind of ontological “field,” not unlike the old lumeniferous ether, that pervades all space and time like an ocean, and that individual consciousnesses are local waves in that vast expanse, Braham to the individual Atman.

Maybe strange loops suggest that Hindu mystics are right:  Tat tvam asi … “That art Thou”.

James R. Cowles

Image credits

Goedel quote … QuoteFancy … Public domain
Picture of Kurt Goedel … Getty images … CC by SA 3.0
Picture of Einstein and Goedel … Katachriston blog … CC by SA 3.0
Collatz fractal … Photographer unknown … Public domain
Penrose stairs … Sakurambo … Public domain
“Ascending and Descending” … by M. C. Escher … Fair use
“Autumn Rhythm” … Jackson Pollock … CC BY-SA 2.0
Escher photograph … Hans Peters — Dutch National Archives … Creative Commons Attribution-Share Alike 3.0 Unported

 

FROM RESIDENT SKEPTIC, JAMES R. COWLES: Weird Comics and the Topology of Non-orientable Manifolds

James R. Cowles is a member of the diverse Bardo Group Beguines, publishers of The BeZine, which I manage and edit. James also regularly contributes to The BeZine’s sister site, Beguine Again. James isn’t shy of controversy and while you may not always be in agreement with him, you will always be encouraged to revisit and rethink … and, the man is endlessly entertaining. / J.D.

Not too long ago, I published a column on “weird”, X-Files-ish phenomena, the kinds of events and (alleged) experiences that are regularly recorded in Fate magazine. My original intent in writing and publishing that column was, quite frankly, to break my addiction to Donald Trump, Trump-ism, Russia-gate, and what was, and often still is, my unhealthy incipient addiction to the raw sewage that has flooded the White House and the Executive Branch, by getting my mind onto a different track.  But writing that column also had the unintended and unforeseen side-benefit of prompting some persistent reminiscences of the kinds of comic books I used to read just before and just after I entered puberty.  During that time, in addition to Fate, I read three comics published by the American Comics Group (ACG):  Forbidden Worlds (hereafter FW), Adventures into the Unknown (AITU), and Unknown Worlds (UW). I realize in retrospect that, just as the earlier column got me out of the rut of Trump and American para-fascism, the reading and remembrances of those four magazines got me out of what would have been the rut of unalloyed skepticism. Maybe the following will do the same for you. At the very least, maybe it will give us both a good laugh after having suffered through the passing of the fiscal kidney stone of the Republican tax “reform” bill, a.k.a. “No Multi-Billionaire Plutocrat Left Behind”.

The writing in the three latter magazines was almost unrelieved schlock, and the art on the covers — comprising buxom, scantily clad young women dressed in tight bodices and levitating hemlines, rather risque for that day — was often worse. (Fate was the most blatant offender, though in a different way, as a glance at today’s over-the-top-lurid Fate web site will attest.) But, at least as far as the writing is concerned, the operative word in the above is almost:  ” … almost unrelieved schlock“. Almost. But not always. Occasionally, i.e., often enough that I kept buying the “Big Three” ACG comics, the writing rose to the level of The Twilight Zone, Alcoa Presents One Step BeyondScience Fiction Theater (which was appearing at about the same time), and The Outer Limits. That the writing attained even occasional brilliance is all the more remarkable when you reflect that all the stories — every last one — were written by one man, the managing editor of ACG, Richard E. Hughes, who wrote all the stories under a series of five pseudonyms:  Lafcadio Lee, Zev  Zimmer, Kurato Osaki  (!), Shane O’Shea, and Pierre Alonzo, drawings of whose purely fictitious faces prefaced each story. To this day, I consider Richard E. Hughes to be a literary diamond in a cattle feed-lot. Consider …

o Forbidden Worlds

Hughes wrote what I consider 3 classics in the “weird-comic” genre, the first of which is “The Train that Vanished” in a May-June (cannot recall the year) issue of FW. The story centered on a brilliant, avant-garde subway design engineer who, working on his own time, discovers a way to enable 2 subway trains to run on the same subway track at the same time. (Think of Albert Einstein working in the Swiss Patent Office.) As a proof-of-concept / “beta test,” this genius engineer designs a black box and installs it on the track. When train 1 passes the black box, it is shifted to dimension A; when train 2 passes that black box, it is shifted to dimension B, and the 2 trains then alternate by trading dimensions, each time they pass the black box, so they never occupy the same track in the same dimension at the same time. Subway senior management discovers what he has done, and, perhaps because the engineer had not filled out the “goldenrod” copy of his time-sheet correctly in quadruplicate, fires him, whereup0n the engineer boards a subway train, waits for it to shift dimensions, and then leaps from the car into a community of dimension-B beings, who do value his creativity and genius.

When I first encountered this issue of FW and the subway story, I was coincidentally getting interested in the topology of what I later learned were called non-orientable manifolds, intuitively, surfaces like Mobius strips and Klein bottles in which concepts like “up-down”, “in-out”, “top-bottom”, “inside-outside”, etc., cannot be defined. (Hence the term “non-orientable”.) Going into detail about the “weeds” of non-orientable surfaces would eat me alive. So suffice to say that, if a way could be found to alter the local topology of spacetime into a non-orientable manifold, then, with other, even more technical tweaks, what the subway engineer did with the subway trains would be possible. I am astounded that Richard E. Hughes understood such a recondite subject even well enough to write a — rather brief! — comic-book story around it.

o Adventures Into the Unknown

The second “Hughes classic” is “The Man Who Couldn’t Sleep” in a November issue (again, I cannot recall the year) issue of AITU. Larry Keith — I still remember the character’s name after 50-plus years — is a neurochemist who becomes fascinated with what human beings might achieve if they no longer needed to sleep … and thereby waste roughly one-third of their lives unconscious. So he formulates a drug which, he thinks, will perform all the functions of sleep and yet leave the person fully awake, conscious, and alert. He violates the canons of science, however, and tests the drug on himself.

At first, he only notices that he is up, out, and about ‘way past his normal bedtime. But as the night wears on, he notices that weird things begin to happen, the kinds of things that occur typically in nightmares:  his neighborhood is invaded by dinosaurs, including a troupe of great apes; a raucous Mardi Gras, New Orleans-style jazz band, hundreds strong, camps outside his window and begins to howl for human sacrifice, etc. Of course, they settle on Larry Keith as their victim. (I still remember their blood-cry from having read the story so long ago:  “Larry Keith! Let it be he!”) Finally, the drug wears off, and he awakes in his own living room unharmed, but splashed with mud and filthy water from his headlong flight away from the dinosaurs and the jazz-band musicians. The last frame of the story shows Keith, dressed in pajamas, and now in bed and remarking “I guess sleep is more important than I believed. So I’m going to get some. Good night!”

Aside from broaching the old conundrum about how one knows that the world one sees round about is the real world, and that one’s dream world is just a dream world, the story raises the unsettling possibility that, even if the waking world is the real world, perhaps the dream world would become real, were it not that it is just that:  the dream world. Maybe our dreams would come true in the absence of sleep, thereby, in a Platonic nightmare, releasing the visions of the id from the constraints of the superego and allowing them to become ontologically realized in what we are pleased to consider the actual world. If you are inclined to just smile indulgently at such a possibility, remember that Dr. C. G. Jung speculated that UFOs — phenomena with a demonstrably objective existence — were projections from within the mind’s collective unconscious. In any event, be careful what you wish for.

o Unknown Worlds

The third “Hughes classic” is a story that appeared in UW about an obscure, grey little man, much like Simon and Garfunkel sang about in “A Most Peculiar Man”, who keeps to himself in his basement apartment, has no friends, and who remains unknown to everyone. All that makes him conspicuous is that he has a prodigious talent for fixing all kinds of machinery. But not only does he repair it, he ends up improving it … without intending to or knowing how he does it. As the story unfolds, a young couple brings him a black-and-white TV to repair. They leave it with him, pick it up when he calls to say it is fixed, but immediately return, breathless with amazement. Their black-and-white TV now displays vivid color. (Remember: this story was published back when color TV was a high-tech luxury, unaffordable to anyone but the one-percenters of the late 50s / early 60s.) But notwithstanding, people still persecute and ridicule the little man because of his harmless eccentricities.

Some time before, the grey little man noticed he has a large hole in his apartment wall. He has never bothered  to fix it, and just hangs a curtain over it. But one day, especially depressed at being the pariah of his apartment building, he decides to explore. He climbs through the hole … and to his astonishment discovers an entire world on the far side of the hole. In that through-the-hole world, there are people of great compassion and discernment who, recognizing his genius, not only accept him, but accord him an exalted place in their society. The last frame of the story shows the grey little man as viewed through the hole, surrounded by his new adoring friends on the far side, who, like the people in our world, bring their devices to him, not only because they value his skill, but even more so, because they value him. As a kid who was a nerd before such a word had ever been coined, this UW story, for obvious reasons, resonated profoundly with me. Twenty-five years or so later, I found my own refugs — my own “hole in the wall,” if you will — in my wife and in my in-law family.

If there is a common motif in all three “Hughes classics”, it is that physical technology, especially when developed carelessly, can bite the hand that creates it. But the “technology” of compassion and dignity never turns upon and rends the one who practices  it. The former involves only confronting problems. The latter involves confronting Mystery. A mature skepticism always requires a recognition of one’s cognitive limitations. As the old Scholastics expressed it Omnia exeunt in mysterium.

James R. Cowles

Image credits:

“Adventures Into the Unknown” cover … Edvard Montz … Public domain
“Unknown Worlds” cover … American Comics Group … Public domain
ACG pseudonyms of Richard Hughes … http://www.a-zcomics.com/SCANS/UW.html … Public domain
Photograph of Richard E. Hughes … American Comics Group … Public domain
“The Train that Vanished” … American Comics Group …. Public domain


ABOUT

“Why I Write,” George Orwell

The pen name “George Orwell” was inspired by the River Orwell in the  county of Suffolk (England). Photo courtesy of Adrian Cable under C BY-SA 2.0 license.


George Orwell

Eric Arthur Blair (1903 – 1950)

Why I Write” is an essay by George Orwell detailing his personal journey to becoming a writer. It was first published in the Summer 1946 edition of Gangrel. The editors of this magazine, J.B.Pick and Charles Neil, had asked a selection of writers to explain why they write.

1EN-625-B1945
Orwell, George (eigentl. Eric Arthur
Blair),
engl. Schriftsteller,
Motihari (Indien) 25.1.1903 – London
21.1.1950.
Photo 1945., Public Domain

“What I have most wanted to do throughout the past ten years is to make political writing into an art. My starting point is always a feeling of partisanship, a sense of injustice. When I sit down to write a book, I do not say to myself, ‘I am going to produce a work of art’. I write it because there is some lie that I want to expose, some fact to which I want to draw attention, and my initial concern is to get a hearing. But I could not do the work of writing a book, or even a long magazine article, if it were not also an aesthetic experience. Anyone who cares to examine my work will see that even when it is downright propaganda it contains much that a full-time politician would consider irrelevant. I am not able, and do not want, completely to abandon the world view that I acquired in childhood. So long as I remain alive and well I shall continue to feel strongly about prose style, to love the surface of the earth, and to take a pleasure in solid objects and scraps of useless information. It is no use trying to suppress that side of myself. The job is to reconcile my ingrained likes and dislikes with the essentially public, non-individual activities that this age forces on all of us.” MORE


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When Sexual Violence Goes Public, an essay by Michael Watson, M.A., Ph.D., LCMHC

Regular Wednesday Writing Prompts will resume on January 3, 2018. This thoughtful piece is shared here with Michael’s permission. It was originally published on his blog, Dreaming the World.

Well, the weather turned warm again, with a bit of rain; now the temperature is dropping slowly and there are hints of blue through the overcast. There are rumors of a snowstorm next week and more before Christmas. We shall see.

Here in North America we tend to forget how pervasive sexual violence is, and how retraumatizing public conversations about sexual abuse and harassment can be for victims of sexual crimes.

This was brought home to me again yesterday while speaking with a colleague in Boston. She works with severely traumatized individuals and spoke about her clients’ experiences of retraumatization due to the recent flood of sexual assault accusations against prominent men. We agreed the resulting, much-needed, public discussion about sexual assault has resulted in a cascade of memories and fear for our clients. This adds to the retraumatization caused by the behavior of government officials who seem Hell-bent on glamorizing sexual assault while destroying the social framework. We also agreed we are experiencing much increased anxiety as we try to understand how to provide some sense of safety to our clients and ourselves in an increasingly difficult social environment.

Not surprisingly, our culture’s focus on sexual assaults and intimidation by males has felt isolating for clients who were abused or harassed by women. Somehow we as a society appear to have once again lost sight of the uncomfortable fact that women can also be abusive. Perhaps there is less attention to assaults by women simply because abuse and harassment at the hands of women appears to be underreported in general. In addition, men, particularly, report experiencing more shame when speaking of being abused by women and are, thus, more reticent to report being assaulted.

The sad truth is that people of all genders are capable of harming others when given the opportunity. Further, such abuses become more frequent when openly, or tacitly, accepted by communities. I’m sure we will hear much more about sexual abuse by persons with power in the days to come. How we respond is crucial.

© 2017, Michael Watson, essay and photograph, All rights reserved


Michael Watson

MICHAEL WATSON, LCMHC (Dreaming the World) is a poet of the spirit, if not of the pen, and a contributing editor to The BeZine, an essayist and a practitioner of the Shamanic arts, psychotherapist, educator and artist of Native American and European descent.

Michael lives and works in Burlington, Vermont,where he is retired from his teaching position in undergraduate and graduate programs at Burlington College. He was once Dean of Students there. He also had wonderful experiences teaching in India and Hong Kong, which he’s documented on his blog, Dreaming the World. In childhood Michael had polio, an event that taught him much about challenge, struggle, isolation, and healing.


ABOUT THE POET BY DAY