Wednesday, August 4

My Goedel Is Killing Me

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


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