Some truths are only known by inversion, by negation and through a proof by contradiction. Alan Turing’s proof of the undecidability of non-trivially complex algorithms, that is – the impossibility of analysis to arrive at certainty concerning whether a given computer program will terminate or continue forever, was just such a proof. Kurt Gödel’s proof of the incompleteness of non-trivially sophisticated formal axiomatic systems relies upon a similarly self-inflected logic – where algebraic statements are each themselves assigned (numerical, algebraic) values and these values are recombined in ways that demonstrate that there must always remain statements that, while they may indeed be true, can not be proven to be so from within the formal system with which they are constructed. Yet another fascinating proof, perhaps more explicitly by logical self-inflection than by pure negation or contradiction, was Georg Cantor’s analysis and demonstration that there are different sizes (or “cardinals”) of infinity – a proof demonstrable using a mere handful of real numbers and a simple manipulation of their decimal expansions. This last proof was no less mischievous and managed to ascend (i.e. count) an infinite sequence by demonstrating that there were always more numbers which demonstrably could not have been included in any list you have made and consequently, by diagonalisation, that there was always another way to extend the referential system and list of decimal expansions into, within and through itself.
The one fascinating thread which passes through all of these proofs – if each in different ways – is that of recursion, of self-reference and of the endlessly self-inflected foundations of logic, mathematics and (by extension) physics and – plausibly – of material reality itself. Never mind that an implied foundational discontinuity in logic suggests that the non-trivially sophisticated computer programs (and associated technologies) that we are so hurriedly building our world upon can never – ever – be completely reliable and infallible, nor that the endlessly proliferating security holes and data breaches from which our online world is constantly reeling are in some sense endemic and inevitable as direct consequences of the indefinitely extensible nature of logical systems. What is perhaps most profoundly interesting (and simultaneously, if subtly so – disturbing) is that this intrinsic extensibility and openness of algorithmic sequence, of logical structure and of the recursively self-inflected extensibility of any non-trivially complex or sophisticated pattern of matter or information is both a nod towards what life (and consciousness) constitutively is and, also, why we can not – ever – fully explain the world, the Universe, or ourselves.
The unfurling logic of life, whatever it may be and whatever limit of rational definition or knowledge we might aspire to approximate it to, is fundamentally a logic of this open, unbounded and indefinitely-extensible, recombinatory patterning principle in nature. It is curious that the systems of analysis, synthesis and explanation we use to attempt to explain the world (and ourselves) with have, themselves, emerged from the crucible and living womb of such a materially and structurally exploratory and unbounded logical entity. The logic of mechanism, empiricism and rational analysis is of necessity constructive, procedural and – regardless of even our most intelligent and creative assertions on the matter – intractably linear. It could perhaps hardly be otherwise – proofs and demonstrations must proceed, stepwise, to their goals and it is no concidence that our language, our common logic and the narrative psychological experience of our lives is cast in (or as) just such a linear progression – through time as much as through space.
These are, of course, very deep waters and the essence, in a nutshell, is that it is not what we know that provides us with sentient momentum anywhere near so much as it is precisely, inversely and counter-intutively what we do not know that drives us and moves us forwards. The unity we seek in our systems of explanation and theoretical models of reality does not, in fact, exist. This is not the unabashed pessimism or cynical solipsism it may at first appear to be. The distributed, endemic and omnipresent, logically abstract absence of unity and completeness or material, sentient, psychological and conscious wholeness precisely is the unifying, binding principle that we are seeking and the most sophisticated pattern or logical structure and programmatic compression of theoretical model will approximate to the randomness and probabilistic drift into uncertainty, maximally-complex information description or (possibility of) representation and randomness – not because they are superficially similar, but because they are fundamentally, foundationally and intractably the same thing.
Do not discard rationality but I urge you to intuitively embrace paradoxical complexity, the darkly-bubbling mysteries of an intimate entropy and it’s infinite extensibility, structural impossibility and logical emptiness – these are what we are and the highest sophistication of complexity, intelligence and (yes, the qualitatively indeterminate and intuitive insight of) wisdom will only ever arrive at this truth. The resonant dissonance of a mischievous absence at (and as) the living heart of our experience is both lock and key in this context.
Postscript: This was a first draft. I will return to it later for a remediation and refinement and that is really the whole point here.
2 replies on “Ascending An Impossible Self-Disassembly of Infinitely-Extensible Emptiness”
I remember reading Gödel’s proof back in college over 50 years ago, where he proved mathematics was incomplete, that he mentioned the set of all sets that can’t be contained in a set. Paradoxes may be symptoms of the incompleteness of systems, but I believe paradoxes are self-referential statements that lead to infinite recursion which, in algorithms, makes them unstoppable, thus violating Donald Knuth’s requirements for a valid algorithm. However, when we extract the self-referential statements from a system and insert them into a metasystem, then assertions about the contained system without paradox or recursion, because the assertion refers to something other than itself. Does that kind of make sense?
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Yes… makes sense, in as much as this class of entities ever can… the Gödellian point is that even in that metasystem, there will exist unresolvable antinomies as will there also exist in the next level up… and on it goes, recursively forever… beautiful, really… The strange loops and tangled hierarchies that Douglas Hofstadter extrapolated so well…