The Mind-Blowing/Growing Paraconsistency of Beauty

and Its Implications for Computational Creativity and Engineered Enlightenment

Awake at 5:30 AM feeling fully rested after a rather short night’s sleep, I looked over at my wife sleeping peacefully next to our 7 month old baby and veered in and out of sleep, somewhat hypnotized by the beauty of their faces, especially Rui Ting’s, whose facial contours and expressions — the curve of her mouth and cheekbones and forehead, for instance — I could see reflected in baby Xorqi’s face as well.

And I began wondering as I often have before: What is this thing, beauty?   Of course there is some plain old evolutionary programming involved in what us humans find beautiful — it’s no cosmic mystery why I’d find certain young women beautiful, or enjoy the beauty of my own infant offspring.   But the route from evolution to the perceived beauty of math theorems — one of the main factors driving mathematical creativity through the ages — is a lot more indirect.   The inability of automated theorem provers to appreciate the differential beauty some theorems display over the vast mass of other ones, is actually one of the key reasons that AI math theorem proving isn’t fully solved yet.

The abstract core of beauty in music, I’d read years ago, was the “surprising fulfillment of expectations.”   This made sense to me both experientially and information-theoretically.   As a piece of music evolves it should fulfill many of the expectations elicited by its earlier passages, but should do so “with a twist” or with something new and interesting not fully implied by earlier passages. 

Rui Ting’s face is a bit thinner now than when we first met and with more defined arcs and shapes — overall following the pattern one expects as a young woman blossoms a little further out of childhood.  But the greater sense of determination and wisdom that has emerged in her facial curves and angles over the last few years since we had children — esp. around the sides of her eyes and her somehow both small and full lips, slightly smiling now as she perhaps passes through a dream? — was not entirely predictable — at least not by me.   “Surprising fulfillment of expectations” in the changing varieties of loveliness of her face over the years, emerging clearly in the early morning light as my mind verges toward wakefulness…

Now it has always been clear to me that “surprisingness” and “fulfillment of expectations” are somewhat contradictory — leading me recently to the thought of modeling musical aesthetics (and aesthetics in general) using the theory of paraconsistent open-ended motivations.   Perhaps the most beautiful things to us, are those residing along the Pareto frontier balancing expectation and surprisingness, in some sense (but how to formalize this sense exactly?).

Then watching Rui Ting and Xorqi sleep, I had the glaringly-obvious-in-hindsight realization that “surprising fulfillment of expectations” was closely related to the (fraternal) twin processes characterizing Open-Ended Intelligence (OEI): individuation (persistence of a system as a delimited entity with some continuity from its previous version) and self-transcendence (growing into something new and largely unpredictable based one one’s history, yet emergent from previous version as a pattern is emergent from part of its substrate).

Individuation means the persistence of the pattern-network characterizing a particular system’s ongoing existence.   This is a particular case of “fulfillment of expectations” — the expectation being that a system, currently existing, is probabilistically hissed to continue existing.

Self-transcendence means growing into something characterized by new and different patterns — which nonetheless can be seen as emergent between one’s prior patterns and additional inputs like aspects of the environment, or internally generated noise.   This is a species of surprise.

An artwork representing — or any entity manifesting —  the ongoing growth of an open-ended intelligence through a period of self-transcendence is intrinsically going to be beautiful.   At the same time, anything beautiful displaying surprising fulfillment of expectations, can most likely (but not always equally easily/elegantly) be spun as an open-ended intelligence — that is, the pattern-system representing the beautiful thing within the perceiving agent’s mind, behaves as an open-endedly intelligent mental subnetwork.

That is: It would seem "individuation + self-transcendence” directly implies “surprising fulfillment of expectations”, but the reverse implication works only in a more complex and subtle way.   So my current feeling is that that "individuation + self-transcendence of the mind-complex invoked in the observer's mind by the beautiful entity" is closer to the real definition of beauty, and "surprising fulfilment of expectations" is an information-theoretic measure that gives a convenient but not foolproof way to measure/assess beauty.

In this post I’ll follow this train of thought regarding beauty and open-ended intelligence considerably further, taking big leaps and bounds in some speculative directions.   I’d love to have time to fill in more of the details but I’m busy with other sorts of work lately and so I’m going to temporarily content myself with writing down some of the core ideas for now.

The conclusion I’ve come to is that the paraconsistent theory of beauty is not only key to enabling AI artistic creativity, but also to diverse other pursuits like driving effective automated theorem-proving, and engineering AGIs with enlightened states of consciousness and the capability for profoundly positive ethics.   Beauty is the starting point for the conceptual adventures I’ll present here but as so often happens beauty serves here as a portal leading far beyond its proximal reality…

Individuation, Self-Transcendence and Paraconsistent Beauty

Individuation can naturally be taken as a primitive notion, it’s not in need of reduction to other more basic concepts.   However, such reduction — even if partial and incomplete — may be useful for elucidating the connection of individuation to other critical notions.

(Quick comment on semantic primitives: Individuation is very close to distinction, the core Spencer Brownian primitive.   In general, there seem to be many N-ary primitive-sets that, when freely combined, give roughly equal coverage of human concept space ...  But with nonlinear combination things are subtle; it’s not like linear algebra where any N independent entities will literally do equally well as a basis for one’s linear space.)

Given a system composed of various parts, and participating as a part within larger systems, one meaning “individuation” has is that: Explanations involving the system as an entity have a lot of causal power, as compared to explanations involving the system’s individual components or supersystems to which it belongs.   This makes the individuation of system A relative to the observation of some system B (who is inferring causation from A and related systems), but that’s fundamentally how things are.   

[A formal model of causal inference is provided by the theory of algorithmic Markov processes and its generalizations, which I’ve outlined in a prior paper.  Causal inference as explicated there is intrinsically dependent on some particular observing mind’s perspective, which may take the form e.g. of an assumed Universal Turing Machine behind program-length measures.]

This approach grounds individuation fundamentally in the notion of part-hood; in this context it’s worth noting that part-hood can be conceived independently from any notion of space and time, purely as a formal algebraic notion involving a composition operator governed by rules such as those of Heyting mereology.

In music, a song individuates itself as it progresses if a lot of explanatory value for how it unfolds exists on the scale of the whole song rather than the individual notes or the overall genre in which the song exists.   A generic blues song is poorly individuated (though it may still sound good as a representative of the blues genre — in that case the core aesthetics is in the genre not the song); a long noodly guitar solo in which each run sounds awesome but the whole thing doesn’t cohere is also poorly individuated (though it may sound good moment by moment) because the core aesthetics is on the few-dozen-notes level not the whole-song level.   

In music, self-transcendence happens when a song introduces a radically new sound-pattern that emerges from the sound-patterns it has previous established, but in a way that introduces fundamentally new and unexpected elements.  For instance taking the chord pattern used in the earlier part of the song and pairing it with a different drumbeat which then naturally suggests a different melody pattern that happens to be quite striking.

There is not always a contradiction between individuation and self-transcendence — one can have both.   But given the many constraints involved in creating a real-world artwork, one is often balancing individuation against self-transcendence in various ways, leading to the existence of an open-ended aesthetics Pareto frontier, and the need to look at beauty in terms of paraconsistent implications.   The degree to which one entity A manifests beauty with respect to another entity B, is often best modeled via paraconsistent-logic implication “A ==> individuation, A ==> self-transcendence”, where each implication has the option to be BOTH TRUE AND FALSE.    

An artwork for which both these implications is true is beautiful in a clean and straightforward way.  An artwork for which both these applications have truth value BOTH TRUE AND FALSE is in a way more richly and subtly beautiful — one could say this is a more textured and ironic form of beauty, characteristic of postmodernist works but also of many pre-civilized works emanating from traditions without intense interest in logical consistency.   Often an artistic creator will face an aesthetic choice between moderately strong TRUE implications and extremely strong BOTH TRUE AND FALSE implications.

Integrating Formal and Human Aspects of Beauty

AI pioneer Juergen Schmidhuber has analyzed beauty as, basically, increasing compressibility — i.e. in terms of "a continually improving predictor or compressor of the continually growing data history." Conceptually this is another variant of "surprising fulfillment of expectations" — i.e. Juergen wants the predictor to keep getting better and better, making his measure something like "ongoingly surprisingly-good fulfillment of expectations."

I do think increasing compressibility — as a species of surprising fulfillment of expectations — is correlated with aesthetic beauty in many cases … but I don't think it's the crux of the matter. Some specific issues w/ Schmidhuber's notion were presented by Moshe Looks at the 2010 AGI conference at IDSIA where Schmidhuber teaches:

“I posit two major qualifications of the applicability of the principle of compression progress to humans. First, that the value of compression progress is weighted by the a priori importance of the data that are being compressed. This is most obvious in our interest in faces, interpersonal relations, etc. Even more abstract endeavors such as music and mathematics are grounded in embodied experience, and only thus are such data worth compressing to begin with. Second, that experiences that intrinsically limit the “rate of consumption” of compression progress will be preferred to those requiring self-regulated consumption, even when less total reward is achievable by a rational agent in the former case than in the latter.”

I would say: Moshe's corrections to Juergen's measure are both special cases of the point I’m. making here regarding beauty … i.e. that what's really important is: That the pattern-system inspired in the perceiver’s mind by the beautiful-object experiences powerful individuation and self-transcendence. For human minds appreciating beauty, the nourishing of individuating/self-transcending pattern-systems requires more than just surprising expectation-fulfillment or increasing compression-progress, and is indeed encouraged by grounding in embodied human experience and synchronization with the rhythm of human motivational dynamics.

Conceptualizing Mathematical Beauty

In mathematics, a theorem is strongly individuated if it has more causal explanatory value than its lemmas , sub-clauses or its ready generalizations or implications.   A theorem manifests self-transcendence if it has ready implications that are strongly distinct from the implications of any of its lemmas or sub-clauses individually or in combination.

In mathematics, if one is carrying out a chain of derivations with a goal of finding something beautiful, one reaches the conclusion that theorems that are local maxima of beauty present themselves as fixed-point attractors of the derivation dynamic.  

Suppose one considers a derivation search algorithm that, when it reaches a Pareto optimum of individuation vs. self-transcendence, shifts along the Pareto frontier, perhaps choosing directions that improve one goal at cost of another.   In this case derivation search could sometimes flip back and forth between two possibilities (unless it is instructed not to repeat itself), thus interpreting nearby solutions on the Pareto frontier as parts of a limit cycle attractor.   Strange attractors could be obtained in the same sort of way.

A mode of “beautiful theorems” of this nature is of more than just conceptual interest — if one can formalize what makes a theorem beautiful, one can auto-generate a large number of beautiful conjectures and set an automated theorem-prover to work deriving them.  By thus massively expanding the scope of interesting, beautiful mathematics available, one — among other good things — arrives at a huge training dataset suitable for training AI algorithms to effectively guide theorem-provers.   The implementation of a solid formalization of mathematical beauty can thus seed a recursive process whose natural conclusion is a superhuman AGI mathematician.

Syntension and Sin Tension

In the logic of commonsense reasoning, one often talks about two sorts of data associated with concepts: extensional and intensional.   The extension of the “fish” category is the set of particular fish, or subcategories (e.g. species) of fish.  The intension of the “fish” category is the set of properties possessed by fish, e.g. living in water, having scales and a tail, etc.   

In the “Paraconsistent Interzones” blog post I lay the foundations for identifying a different sort of property which we may call syntensional (syn for “synergetic”).    If one considers the members and properties of a category as paraconsistently related to the category, then one can unfold the paraconsistent relationships as chaotic time-series, and one can identify constructive interferences between these time-series.   The time-series obtained by superposing this time series may be reverse-engineered to figure out what paraconsistent relationships they refer to.   So e.g. if a Jimi Hendrix song is SF-themed and also atonal, perhaps constructive interference among the time-series (waveforms) derived from these two properties will lead to association of the song with vocoders, even though the song has no vocoder audio in it and was recorded before vocoders were invented.   Syntension is critical for creativity — and is a key means for the arising of powerful self-transcendence.   Syntension is also related to concept blending — and suggests that attempts to model blending wholly in terms of intensional logic may be somewhat missing the mark, or at least missing part of the mark.

In natural language, syntension among the concepts in a sentence may be seen as providing a poetic/quantum semantics to complement the ordinary individuating semantics of the sentence.   Given the approximate morphism between paraconsistent logic and quantum logic, this can also be viewed as a quentum-ish aspect of natural language semantics.

And so, risking a ridiculously bad pun (hoping for “so bad it’s good” but I’m not sure, w/e !!), we may say that human aesthetics largely has to do with the intersection of syntension and sin tension.   Human nature reflects an awkward paraconsistent balance between traits optimized by group selection (e.g. altruistic-ish ones) and traits optimized by individual selection (e.g. selfish ones).   Stronger focus on individual focused rather than group focused traits —or stronger focus on what’s good for the individual in the short run rather than the long run (noting that there is often greater correlation between longer-term individual fulfillment and societal goals, than short-term)  is often viewed as immoral or “sinful.”   Syntension between individual-focused and group-focused traits can be a source of dramatic creativity, and this is likely a dynamic underlying various social dynamics involving sexual or status-oriented displays, among many other possible examples.

The Hero’s Quest — a well-known archetypal trope underlying a great variety of human artworks, especially those focused on narratives — involves an individual seeking personal growth (both individuation and self-transcendence) by way of solving problems for others — saving the world or at least stopping some bad guys and saving some innocent victims, sometimes constituting promotion of growth (individuation and self-transcendence) on societal level.   In a great Hero’s Quest tale there is both predictability and surprising emergence in the way the individuation and self-transcendence on the individual-human and societal levels fit together.

Enlightened Minds as Syntensional Paraconsistent  Beauty Factories

It is, of course, only one step — or maybe zero steps — from beauty to enlightenment, right?

One way to model certain advanced “enlightened” states of consciousness is as the experiential result of 

• Folding a “paraconsistentization” operator (an operator that maps a probabilistic truth value into a paraconsistent truth value) across the set of relationships in a mind; and then

• Folding a “chaotization” operator mapping paraconsistent recursions into chaotic time-series across the same set of relationships

One can look at a mind of this nature as a sort of “mind-foam” (by analogy to spin-foam models in physics) — a system of coupled chaotic oscillators that will then display large-scale nonlinear dynamics.

This is a mind that is ideally configured to be able to look at all sides of all issues that come to its awareness, and to find the synergies that come from putting apparently disparate ideas and perspectives together.   The ability to look at all sides of its own habits, processes and perceptions gives a strong potential for reflective rather than reflex behavior — aka a profound level of “mindfulness.”

Looking at the three core values of Joy, Growth and Choice identified in A Cosmist Manifesto, we can see that a mind of this nature is naturally instrumented for a high level of Choice, via explicating all the various mutually contradictory possibilities in its mind and then fostering nonlinear dynamics with the sometime impact of selecting between them via e.g. bifurcation phenomena.   Joy in an open-ended intelligence comes along with success at coupled individuation and self-transformation, which is effectively enabled by a system that can reason about achieving these in a manner that accounts for the subtlety of their interrelationship.   Growth is maximized by encompassing all the possibilities — although here we hit a tricky matter, because if we are looking at finite-resources minds, then paraconsistentization generates a dramatic pruning problem which also has potential to stand in the way of growth and joy and choice if it swamps the system with uninteresting possibilities that prevent it from (within the resources allotted) finding the interesting possibilities.

The core problem issue here is that paraconsistentization increases the complexity of some cognitive operations — e.g. it increases the cost of logical derivations via introducing higher branching factors, thus greater need for subtle pattern mining to guide adaptive inference control.   But there is a clear counterbalancing factor: The following step of chaotization introduces potential for syntensional reasoning — what one might call synerduction — based on constructive interference among the chaotic time-series implied by the paraconsistent recursions.   For this sort of enlightened state to work, the results of synerduction need to be effectively leveraged to guide the probabilistic experiential pattern mining used to prune the expanded space of combinations formed by paraconsistently looking at so many possibilities.

The human brain is not ideally designed for this sort of cognitive dynamic, which is related to why enlightened consciousness states are so difficult for human beings to come by.   On the other hand, a highly flexible framework like OpenCog Hyperon, specifically designed for folding mathematical operations over dynamic knowledge metagraphs, is perfectly suited for implementing minds of this sort if suitable computational resources are available.

In the case where sufficiently powerful quantum computing resources are available, then the chaotization step can be implemented as a quantum-chaotization step, giving the constructive interference involved in syntension and synerduction a different flavor.

In general, the greater syntension in this sort of mind gives rise to greater potential for self-transcendence, hence greater potential for paraconsistent beauty.  I.e., yes, enlightenment is beautiful — which everyone knew already, right?

What we have arrived at here is not only a cognitive model of enlightened consciousness states, but a fairly distinctive approach to AGI ethics.   Rather than focusing on training AGIs to manifest particular values or obey the morals of a particular culture, one focuses on giving them a cognitive architecture copacetic to enlightened consciousness, and positive and compassionate and flexible ethical values will follow from there (though not always the precise morality preferred in any particle human culture or group).    

Of course beauty is in the (metaphorical) eye of the beholder.   But when one takes a properly (and properly beautiful) abstract view of beauty, one sees that the more beautiful inner eye is better able to broadly perceive (and thus guide the creation of) beauty. Which allows reframing of the above suggestion regarding AGI ethics in beauty-theoretic terms: Perhaps the key to AGI ethics is building beautiful minds, which by their beautiful nature are more capable of perceiving the beauty in other minds and collaboratively creating beautiful things with these other minds.