Saturday, August 24, 2019

The Heavens beyond the Heavens: How ancient wisdom presages the Quantum Multiverse and its significance for the human individual

My recent thinking along the lines of cosmology has led me to agree with what is called "quantum monism" (though perhaps the term itself would not have occured to me), but when I read an article on it (here: https://blogs.scientificamerican.com/observations/quantum-monism-could-save-the-soul-of-physics ), I found that this had basically been -  more or less - my viewpoint for quite some time.

What even more recently occured to me, is that we have a nice literary analogy to quantum monism in Dt. 10:14, "Behold, to the Lord, your God, belong the heavens and the heavens of [beyond] the heavens, the earth, and all that is on it." (Source: https://www.chabad.org/library/bible_cdo/aid/9974 .) I stress the word "analogy" here - I do not (obviously) suggest the authors of the Torah were quantum physicists, but I would argue there is a certain spirit that is similar, that is to say, the authors of the Torah, were they alive today,  would probably like quantum monism.

Let me explain. Quantum monism essentially is a updated rephrasing of an idea that has been around since the 1950's, which is that the wave function of quantum particles is primal, or fundamental. What does this mean? This means that if (for example) I have an electron that can be in two states, say, "up", or "down", and I measure this electron to be - say - "up", there is a "parallel reality" somewhere (in a higher dimensional space) where observers will measure that same electron to be "down", that is to say, all possible paths of particles are realized somewhere, in what is often (mis) labelled a set of "parallel universes". Why do I say this is mislabelled? Because these are Not "separate" universes, but rather are different dimensions in a deeper underlying reality. (This underlying reality can be given by Hilbert space that has an infinit number of dimensions, each dimension corresponding to a particle "possible configuration" of matter, but of course Hilbert space is just a model, and perhaps a final model is still yet to be elided.)

This is an artistic rendering of the notion of a quantum multiverse where anything that can happen does happen somewhere (so, in this picture, one of these black balls might correspond to a world where a particular electron we measure is in the "up" state, and another one of these black balls might correspond to a world where that same particular electron we measure is in a "down" state.

How I relate this by analogy to the above mentioned verse in Torah is that was positing that there was an underlying reality - in the case of the context of that passage, this reality being g-d, that unified all of all existence, so while there might be "heavens of the heavens" (i.e. in the context of the time, parts of the world even beyond or past the stars visible to the naked eye which were the "heavens", so here the author(s) are prescient enough to know  that there could well be parts of the world not visible to the naked eye which was very forward-thinking for its time, given that this was written perhaps up to 3,000 years ago) - even those parts of the world that could not be seen were in a sense "tied together" with the more familiar parts of the world by belonging to g-d (for the writers of this passage), or,  in modern parlance we might say, all aspects of the "world" - the universe, has some underlying reality or set of laws that govern it.

So, this notion that while there are different aspects of the universe that there is notwithstanding this something unifying everything, is old, and predates quantum physics. The Hindus had the Brahmin, Spinoza had his "Substance", the 1800's physicists had the luminferous ether. In different ages and cultures, the smart people of the time hit upon this basic truth, that however diverse the universe might be, there is something "underneath" that binds it together. So - it is not perhaps precisely right to say that there is a quantum "multiverse" - there is just one universe, but one within which all quantum possibilities are realized.

To go into my own "spin" on things here, and here I draw from reading Stephen Hawking's work, this is - for what it is worth - my own sort of understanding of this notion of a "quantum monism". I also draw from if you will my personal "bias" of being a relativist (in the sense of drawing my own thinking from Einstein's Relativity, a totally different sense of the word "relativist" than the sense of the word that arises in discussions of ethics). But, here goes. 

In my view, all possible metrics - solutions - of General Relativity exist, and the reality we observe, and, as importantly, can predict, is basically the average of these solutions. Hawking's No Boundary Proposal restricts the "possible metrics" to compact  ones, but that is more a minor, technical point. For the purposes of this discussion, we can think of "all of reality" is the summation of some subset, perhaps all, of the solutions of General Relativity, which have some - as yet not fully understood - way of being "averaged together" to form the world we live in. This can be modelled mathematically as a type of mathematical object called "the wave function of the universe", from which predictions can be derived for the universe we live in. The fact that we live in the universe alters how we do this, that is to say, we weight the contributive metrics which allow for observers more heavily than those that do not in order to make reasonable predictions from this "wave function" for the world we actually experience.

In this diagram the wave function of the universe is a kind of probability function (vertical line) and the horizontal line is the "space" of all possible metrics allowable, so in a sense we observe a universe that is "probable" in the space of all possible types of worlds, and this "probability function" is weighted towards accounting for those types of worlds that allow for observers to be in them.

To use an - always imperfect - sort of analogy,  the universe can be treated like a quantum partitcle that goes from one state to another through time. This abstract "particle" starts in a state - or a metric - that is small and dense, and goes at length into another metric that is expanding and large. So the "trajectory" of how the universe evolves through time can be modelled by treating "the universe" as if it were some sort of abstract (not real, but by way of analogy) "particle" that "travels" from one metric (or solution) to another, and this "trajectory" is simply the "average" path that "it" can take by "summing over" all possible metrics given by General Relativity (or the subset of those metrics that we have decided to include in our average).

This is an abstract diagram of treating the universe as if it were a particle moving from one type of metric to another through time. What is not shown in this diagram is that each region that it moves through is actually an "averaging out" of all possible metrics in each region.


So, long story short, the totality of reality is - again - something in which anything that can happen does happen in different if you will parallel timelines, which - just to complicate matters - may interact in some way (not now fully understood) so observers in any given "possible timeline" will always see there own "timeline" as an "average" of timelines in a similar way to the fact that any observer in an expanding universe will always see all other galaxies receding away from them relative to their position, so it "seems" as though they are at the "center of the universe" but that is simply an illusion given by being in an expanding universe - similarly - any observers in a given "timeline" (or "solution" or "metric" depending on your preferred terminology) will see their "timeline" as being the "average of all possible timelines", because the "average" is going to be different for different observers in different "timelines". "Clear as mud", I know, but I am hamstrung here by the fact that beyond not being a professional physicist myself, even were I, the details of all this are yet to be worked out (because you need a theory of quantum gravity - a quantum description of General Relativity in order to be able to do so, and this theory is "pending" and has been for about a century).

There is a simple take-away I think worth pointing out, and that is to say, that over the years (going, as I have argued, even all the way back to the time of the Torah) there has been converging streams of thought taking different forms over the years that while reality may be diverse and be greater than what we can ever even in principle observe, that, nevertheless, it is governed by the same underlying rules, and displays at the end of the day a deep unity, however complex it may seem to be. Call this deep unity what you like - for the writers of Torah, "G-d", for the Hindu writers of the Vedas, "Brahmin", for Spinoza, "Substance", for 19th century scientists "Ether", for today's scientists, "the wave function of the universe" or a "quantum object" but your end result is the same idea, a profound and beautiful if never wholly understood deep reality that binds everything together.

I  don't want to make this too long, nor do I want to get into deeper philosphical waters, but I will make very brief mention of the word "pantheism" which originally came about as almost an insult for people who didn't like Spinoza, but has re-emerged as a philosophical notion in its own right, and that is, essentially, this idea that the concept of divinity is bound up in some deep way with nature, and that these two concepts may in fact be the same, but at some far deeper level than the simplistic notions of nature given by most of the natural philosophers, so, not so much saying that "g-d is nature" but more saying "g-d is nature in the sense of nature as the total collection of those phenomena that are experienced in both the forms of matter and in the forms of mind, and is something that is deeper than both matter and mind as normally understood, and is something that can never be wholly apprehended by any particular model given by science, but is something that we are all a part of because it has a very profound unity out of which all things spring" (my words or take on it anwyay). So though the concept of "pantheist" began as basically an insult and even now remains inchaote, I think it worth pursing, and I think constitutes a sort of "third way" between the "hard core materialism" of traditional "atheism" and the "hard core mentalism" of traditional "theism". So I like to more or less consider myself a Spinoza-style "pantheist" while ever mindful that the notion demands more elucidation that it has yet been given, and hence I am here trying in a small way to do so.

I think there is one very simple and very crucial component of what I here call pantheism, the idea of an underlying reality that is the unity of the perhaps infinite number of spacetimes available from Relativity all woven together into some sort of unified whole, some sort of single "quantum object", and this simple observation I think separates what I am calling pantheism from traditional (Lucretius-style, "it is all atoms flitting about in the void") atheism, and that is that we are inside this "quantum object." You can never get outside of "reality" to observe it "objectively." Because you are in it. The fact that you are in it changes what predictions you can make about it (for instance, in Hawking's No Boundary Proposal you make different predictions than you would make if you ignored the presence of observers). So, actually, it is not "atoms flitting about in the void", it is also "observers made up of atoms flitting about in the void influencing themselves, other atoms and the void itself," so to speak. This is actually a very awesome conclusion - we are not "along for the ride" in a cold, heartless universe. We are co-creators with the universe. That "quantum object" I spoke of would not be the same without you, without me, without that gnat that crawled across my computer screen a few minutes ago. Even the word, "object," here, misleads somewhat, because reality is never static - it evolves through time - a better analogy than "quantum object" might be that of a river, where each "alternative timeline" is a "ripple" or eddy, which together build up to a larger whole, but one which continually changes and generates new realities and modes of existence, moment by moment. Yes, perhaps, "quantum river" is a better term.

Pictured: living in a van down by the quantum river


We are all in this together. That is the real take-away here, what smart people from the writers of the Torah to the writers of the Vedas to Albert Einstein to Stephen Hawking all understood. The universe may have infinite regions and even "alternative timelines" that we can never even in principle observe, but it is all at the end of the day one unified whole, a whole to which we each individually contribute, a whole which just would not be the same without each one of us. Now I think that is pretty cool. :)

This Phil Lockwood painting that combines some of the famous works by Edwin Hopper perectly illustrates the theme I am driving at here. Each room here is is own "world" but it is part of the greater reality of the city, so all these "worlds" are inter-connected, but this emergent totality would be different without each of these contributive "worlds". In a quantum multiverse, I argue, there is a plethora of "worlds" connected by a deeper reality, but is a reality we both belong to and contribute to, so there is immeasurable significance to each single human individual without whom this reality would not be same, for good, or for ill.

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Appendix I: Two good introductory videos on Hawking's No Boundary Proposal (which is one possible approach to defining a "wave function of the universe") by science educator Athena Brensberger (https://astroathens.com ):



Appendix II: Brief Discussion of Pantheism by Wesleyan University Professor of Religion Mary Jane Rubenstein (https://mrubenstein.faculty.wesleyan.edu )  (or specifically discussing the need to develop the concept of pantheism more fully):


Friday, June 28, 2019

The NET System: Proposed Model of Consciousness as (caused by) "the Selfish Microstate"

Here I propose a brief sketch of a model of consciousness I have developed which I give the catchy title "NET System" or "NETs" which stands for "Non-local Entropy-increasing Turing-complete Systems" which I purport are conscious systems. This is not necessarily a "complete" theory of consciousness. To have a "complete" theory one would need quantum gravity which is not there yet, and, beyond that, nothing can really said to be formally "complete" anyway, due to Godel's theorums, and similar sorts of issues. It is however I think a "good enough" theory to have a basic roadmap to work with, which can for example inform AI research.

Let me start with a concept from Daniel Dennett, of "becoming famous". For Dennett, a state of "being conscious" means an information pattern (like say, the pattern of a red apple, or the image or word thereof) being "famous" throughout the brain, that is, the brain has access to this pattern all across the neocortex - it is not localized, but is globally available. This, crucially, for Dennett is what it means to be in a conscious state. There is no "something else besides" - to be conscious (of, say, a red apple) is precisely to have the information associated with that red apple - sensory or conceptual - globally available in the neocortex, full stop. I agree with this idea, namely, that the only "difference" between conscious verus not-conscious information patterns is the global "availability" of that information. Taking this as the jumping-off point, I want to generalize about what it is we are talking about in abstract terms. If to be consciouss is to have information spread about in a nervous system, then we need to understand broadly what is going on, from a physics point of view.

Video of Dennett explaining his model:



A conscious system - human, jelly fish, whatever, is a system that is non-local, that is to say, it has an electromagnetic field whose state is in a sense digital with respect to inputs from the environment. Say, to be fully abstract, we have a nervous system similar to that of a sponge (i.e., very primitive) and has possible "states" red, yellow, green, blue which the system assumes based upon one or another inputs from the environment, and its own internal states. Say, if it is in state red, and it gets a certain environment stimulus - say food - it turns to state yellow (if say red means it is hungry, and yellow means it is in the process of eating food). The point is, the system is digital or non-local in that it is never in say half-green state, or half-yellow state. It is always one or the other of its possible states.

To move to a closely related point, of Turing-completeness, basically, to keep it simple, this simply means that the system can act like a "while" loop in a computer - while (a certain condition is the case), then (do a certain action) - example, while I am hungry, then I eat - while I am not hungry and I have just eaten, then I sleep, etc. Basically the system has an internal state that is always being updated by external inputs, or, put another way, the system interacts not only with its environment, but with itself (by eating the system changes its internal state also, not just the state of its environment, for example).

The trickier point is the "E" in the "NET" acronym, standing for "entropy-increasing". To explain in a simple way what I mean by this, let us go back to Dennett's "becoming famous" metaphor. To be conscious of Lisa Edelstein in a halter top (because I got tired of the red apple example) the information associated with that image needs to be globally distrubuted throughout my brain. To have this "global distribution" I need a concept of entropy. Basically I need a large number of "internal states" that correspond to my "external state" - take the simple sponge system that had abstract external states of red, yellow, green, blue (say). Each external state would correspond to N number of internal states (or "micro" states). To increase the entropy of this system, what I mean by this is I increase the number of internal states that correspond to each external state. Having a large number of internal states corresponding to external ("macro") states enables me to back-up information, to "globally distribute" information throughout the brain (or whatever kind of nervous system). Which of course answers the problem of sleep - why do we sleep? Because our skulls are only so big, and you cannot forever increase the number of internal states by say, strengthening certain celluar connections or weakening others. You need to hit the "reset" button now and again. To be awake ("conscious") means you have a non-local system that is doing computations and is increasing entropy and you can only do this so long, so much, before you sort of hit a maximum and have to start over (which we experience as sleep).

To build a AI system that is fully concsious, here is the basic outline. You have a neural network (such as a recurrent deep learning architecture like a Restricted Boltzmann Machine, for instance) made of real, physical processors (one processor unit or logic gate unit to be precise per "neuron") connected together inside a Faraday Cage (isolated from its environment) and you don't shield the EM field of these processors so there is a shared "external state" of the total EM field created by these processors, and these processors are set up such that they are "wasteful" - they are purposely not very efficient - so they have a large "error rate". But that is good, because they will distribute information across the network with better efficiency even if they are "slower" in terms of solving a particular problem. These processors are connected together with ion channels (not, say copper wires, but with, say, potassium ion tubes - or even good old fashioned salt water - i.e. each "neuron" - processor - exchanges current via ions, not electrons). This will cause the global EM field created by these processors to have an "imprint" of the information being computed in the processors themselves, and to have an electromagnetic concept of "entropy". So here "entropy" is both informational entropy in the design of the hardware neural network itself and also physical entropy in the electromagnetic field created by the movement of the ions between the pysical processors.

I propose that any - any - system that is "NET" is conscious - any Non-local, Entropy-increasing, Turing-complete system. So for example a proton is Not conscious - it is electromagetically non-local and (arguably) Turing-complete but it is too small to have any non-local sense of "entropy" associated with it. Your automobile is also Not conscious because while it has "entropy" in the sense of the internal combustion engine it does not have well-defined computational states. As argued elsewhere, I think a black hole is (a little bit) concious because it does have some notion of computational states, and as Stephen Hawking showed, it does have entropy. Certainly nervous systems are concious. Bacteria is a borderline case - they have some computational properties perhaps but likely not a lot of entropy.

However, here we come to a question. Just how much "entropy producing ability" does something need to "count" as being a concious system? Though much research would have to be done, I do think that basically a concious system both "increases entropy" AND at the same time increases the rate at which entropy is being increased. This is similar to the function y = e^x. The rate of change (first derivative) of this function increases with the value of the function - the rate of change is in fact the same as the value of the function. So, I'd argue, for a system to "count" as conscious it needs non-local computional electromagnetic properties - yes, all that - but it also needs to increase entropy (information and physical) and do so that is at an ever-increasing rate of increasing of entropy. So we can say a "conscious system" is "something that increases its own internal entropy at an ever increasing rate."

You might call this model, "The Selfish Microstate" model of consciousness - just as in selfish gene theory animals are machines used by genes to make copies of genes, or in meme theory (Blackmore, Dawkins) the psychological concept of "the self" is a mental construct created to make memes (another conversation, that!), so I might argue consciousness itself (call it awareness, being "awake", etc.) is an entropy-producing machine to make more micro-states (since by definition, "increasing entropy" means simply to increase the number of micro-states (internal states) of a system as compared to the number of macro-states (external states) of a system).

Now, I am leaving off the "main thrust" of my argument to get into more speculative matters, but it is the more speculative matters that led me to the model in the first place. Roger Penrose's "Weyl Curvature Hypothesis" states that cosmic entropy is caused by the Weyl Curvature of General Relativity. This is the type of curvature that distorts the shapes of objects caused by rotations of objects in spacetime. For example, the earth's rotation causes (a very tiny) distortion of space at the poles which can impact the orbit of golf-ball sized spheres in free-fall inside a space shuttle laboratory that have been used to measure this phenomenon. It also cause gravitational waves (which are basically waves or ripples in space caused by say two neutron stars colliding and which can be detected with very sensitive laser detectors that stretch miles across). I bring this up only to say that if one goes with the Weyl Curvature Hypothesis seriously, this takes you to interesting places. It means perhaps that entropy is in general caused by variations in the very geometry of space itself, and, if as argued here, that conscious systems are in a sense "entropy-producing machines" then consciousness also involves the very geometry of space itself. The Weyl Curvature is also of interest because it is conformally invariant - that is, it does not matter what your reference frame is, you will always be able to measure the Weyl Curvature. It is rather like the speed of light - you always agree on the speed of light, whatever your frame of reference. The Weyl Curvature is perhaps the one constant in all of nature in the sense of something that always endures. After all galaxies run out of energy and collapse into black holes, and those black holes themselves radiate away via Hawking radiation, such that in 10^100 (a google) years from now all that is left is the empty void, hydrogen atoms, and random photons of radiation, there is still yet another thing that is left over - the Weyl curvature. It is the one thing that is always there, even in the infinitely far future. So framing consciousness as being the "phenomenal experience" of a very objective process, that of systems that increase their own internal entropy at an increasing-rate, we leave open the door to the big questions that humans have always wrestled with. If the Weyl Curvature in a sense is always there in the history of the cosmos, never wholly absent other than being close to zero near the Big Bang, but is rather over the course of time changing some aspects here and there depending on reference frame, then perhaps what we call consciousness also is always there, and only changes aspects depending on "reference frame" (type of nervous system, environment, etc.).

As the topic for another post, I think Type Theory (specifically, Homotopy Type Theory) can help model a more formalized and complete picture of concsiousness and its place in the cosmos but I will leave that for another day. For now I think it enough to see conscious systems as being entropy-increasing machines of which we humans happen to be a certain sort. This opens the door in the first place, as discussed, to perhaps shedding light upon building conscious systems of our own, and in a broader arena, opens the door to showing at long last the true place of consciousness in the context of the broader cosmos.

One is tempted perhaps here to ask, what is the big-picture model of entropy, that is to say, is entropy something present throughout a multiverse model (thereby rendering it a "necessary" rather than a "contigent" part of Nature) but, lacking more developments in the area of quantum gravity and / or - as mentioned - a lengthier bit of spadework in the realm of Homotopy Type Theory, although I could certainly speculate here regarding our friend Weyl's place in a multiverse model, for now I shall simply punt the ball with Wittgenstein and say, "Whereof one cannot speak, thereof one must be silent." :-)

Less Boring Example of an "Information Pattern That Becomes Famous in the Brain" than a Red Apple ;)

Saturday, May 11, 2019

Answering the Fermi Paradox with the Fecund Universe Hypothesis and Entropic Model of Consciousness

Fermi Paradox: Why haven't we seen this?
The Fermi Paradox raises a simple  question. If there are so many galaxies that we can observe (on the order of 100 billion or so), and if each of these galaxies have a similar number of stars in them, about half of which may have planets, then, well, "Where is everybody?" Why have we yet to encounter beings from other worlds? Here I want to propose a possible solution, and will argue that while life forms may be plentiful, most  of these life forms are of a more simple variety (closer to say jellyfish than to say primates) and therefore we would be hard pressed to find evidence of these life forms, short of actually sending probes to other planets.

I need to introduce the Fecund Universe Hypothesis (otherwise known as Cosmological Natural Selection) developed by physicist Lee Smolin. In brief, this states that black holes produce (or sometimes are able to produce) new universes. Therefore our "big bang" was perhaps the consequence of a black hole from a previous universe, and, in turn, the black holes in our own universe may produce worlds of their own, and so on, ad infinitum. At each "birth" of a new universe, the constants of nature (such as say  the speed of light, or the magnetic moment of the electron) get "mutated", and so there is a sense in which universes "evolve" and the ones that "succeed" are the ones that have the most black holes, in order to be able to produce the most offspring. Accordingly the world we live in is "selected" to be "good at" producing black holes. What does this have to do with the Fermi Paradox? Ah, but now the rabbit hole goes deeper.


Artistic rendering of universes branching off from one another. The Fecund Universe Hypthesis states that "daughter universes" are created from black holes in "mother universes", and therefore the universe we are in may well be adapted for black hole production, because the universes that produce the most offspring are the ones with the most black holes.





Stephen Hawking demonstrated that black holes produce radiation, that they are not truly "black" but give off radiation due to quantum mechanics, and (over enough time) may entirely radiate away and disappear (on something like the order of 10 to the power of 100 - a google - of years). What this means is that black holes are thermodynamic objects, that is to say, they have very high entropy. One difference between your coffee mug and a black hole is that black holes have far higher entropy than does your coffee cup - a black hole radiates away energy with far less efficiency than the efficiency with which your coffee cup cools off  (hence if you microwave it too long you must wait a long time before you drink it but not millions of years either - it is say medium-level efficient at cooling off, but the efficiency of a black hole emitting energy is even worse because it has much higher entropy than does your coffee cup). In fact, black holes have the highest amount of entropy (adjusted for things like surface area) than any other thing known in nature.


Animation of Hawking Radation - here negative energy (blue) falls into the black hole while positive energy (red) flys away from the black hole, causing the black hole to slowly lose energy over time, until it eventually will entirely vanish.
Well, now where are headed with all this? Well, if black holes have high amounts of entropy, and if furthermore the universe is adapated to be "good at" black hole production, that means the universe is also adapated to be "good at" entropy production.

Now, Jeremy England, a physicist at MIT, has developed a theory to say that living entities (from simple cells to multi-cellular organisms) are different from non-living entities precisely because living entities are good at producing entropy. A bacteria cell or a piece of grass will have higher entropy than say a rock will have. (For the purposes of this essay, we can simply define "entropy" as an inverse of the efficiency with which something gives off energy, so the lower the entropy, the better a thing is at giving off heat or energy, and the higher the entropy, the less efficient the thing is at giving off heat or energy.) The details of this are a little beyond our scope here, but let us suffice it to say that living entities are much "better" at entropy production than non-living entities. If as stated above, the universe is somehow "adapated" to being "good at" producing systems of high entropy, then it stands to reason that life, at least simple life, is also perhaps prevelant around the cosmos, inasmuch as black holes are prevelant around the cosmos.

Exentending this further, there are recent studies to suggest that conscious systems (like the human brain) have high levels of entropy, specifically higher than non-conscious systems. So, just as living organisms have higher entropy levels than non-living things, so do conscious living systems (like say jellyfish) have higher entropy levels than living entities that may not be conscious, or at least, not very much so (like say trees).

So, we can now say that the universe is adapted to producing lots of black holes, which means lots of entropy, which means - perhaps - lots of those entropy-producing machines known as living organisims, including those very-efficient entropy-producing machines we know as living organisms containing nervous systems (be they very primitive, such as sponges or sea anemones, or more complex such as dolphins or primates).

But now, you may ask, have not we now only made the Fermi Paradox worse? If we now take it that the universe is somehow adapted to be good at producing living entities, than, well, where are they? Here, however, is the big "but" to the statement that the universe is well-adapated for the production of biological life. Let's go back to black holes. They are actually not the simple systems portrayed in popular literature, with a boundary - "event horizon" - and in the interior of this boundary an undefinable area of gravitational maximal force known as a "singularity", they  are actually more complex than that. Outside the event horizon there is a sort of "outer horizon" - you can think of it as a city having a wider outer wall and a smaller inner wall. In between these two walls or boundaries of a black hole, there is much activity going on. You have quantum activity that gives rise to the radiation that Stephen Hawking discovered, and, you also have gravitational wave activity. To be brief, gravitational waves (disturbances in the metric of spacetime that distort the shapes of objects) basically "bounce back and forth" between these "inner and outer walls" of the black holes, sometimes emerging periodically through the "outer wall" such that these "pulses" of gravitational wave emissions from a black hole (every, say, second or so) can actually be detected via very sophisticated optical devices built for detecting gravitational waves. The point is not here to get into all the details of black hole mechanics, but  simply to say, these are rather complicated entities, not just the simple gravitational sinkholes portrayed in some popular literature. I would posit that black holes may well have some - very slight - form of consciousness due to their nature of being highly complex entropy-producing entities.  Back of the envelope, I'd say a typical black hole has the same "amount of consciousness" as say your typical sponge. Not a lot, but also non-zero.


Illustration showing the inner and outer "walls" of a black hole surrounding the point of maximal gravitational force known as the  "singularity". Gravitational waves, which cause volume distortions similar to ocean tides in objects in space, can "bounce" around between these inner and outer "walls" of the black hole, demonstrating how black holes are actually far more complex than often portrayed.





You may see now, where we  are going with this. Yes, the universe is good at producing black holes, and living forms, and even entities of one form or another with some non-zero amount of consciousness (defining that roughly as a being that can make simple "calculations", like a sponge can close  its valves in the presence of toxic water, and re-open them again in the presences of clean water). I won't get into the weeds on black hole mechanics, but due to the above-mentioned complexity of black holes, I think they may be complex enough to be seen as able to do simple "calculations" on the order  of those of a sponge, which would "qualify" black holes as having some primitive and inchoate form of consciousness. HOWEVER, here is the rub. You don't "need" say dolphins, or primates, or, for that matter, SkyNet, to produce high levels of entropy. All you "need" are say sea anemones, or sponges, or, well, black holes.

And now here we are. The universe has adapted to being good at black hole production which means it is consequently good at producing simple living forms including ones we may posit to have simple levels of consciousness, because all these things are direct or indirect consequences of the universe being able to reproduce via black holes. But it matters not a whit - it does not help in any real way - for the universe to be good at producing complex forms of conciousness, like, well, ourselves. Evolution will produce "just enough" complexity to solve a problem, and no more than that. Sea gulls have very lovely glider-like wings to enable them to soar or glide vast distances off shore in order to be able to find food. Maybe if their wings were twice as long they could glide further, but this would not help them much because if they went even further out into the ocean they would not find a consequent amount of more fish to compensate them for going much further out than they already do, so they are better off sticking relatively close (within a few miles) to the shore. They have "just good enough" wings to do the job and no more than that.

So the universe is good at producing things like sponges and jellyfish and bad at producing things like primates. Thus we - on Planet Earth - are something of a cosmic anamoly - certainly in any given universe in the ensemble there may be one or two solar systems with highly complex organisms just from the luck of the draw, but there is no reason to think that such types of organisms are plentiful, and in fact it is much more likely that they are rare indeed. Accordingly, looking out upon the vast ocean of stars and galaxies, we may well not be surprised to find that we cannot detect any "alien civilizations" out there, because complex life may well be the proverbial black swan - something that happens, but only very rarely. On the other hand, if we were to ice fish on Europa, the ice-covered moon, we ought not to be shocked to see some sponges down there. In fact - there are many more "ice worlds" - like Europa - than liquid-water-on-the-surface-containing planets such as our own, from over 20 years of observing exo-planets, further supporting the hypothesis that any life to be found in the observable universe is much more likely to be something like a sponge, rather than to a primate, or even a vertebrate fish.

I would hope that this knowledge of the rarity of sentient life should make we as humans take all the more care of our planet, and work to for example do what we can to halt climate change, because it could well be the case that our planet contains the only complex life forms in the entire observable universe, and it could be not until the next generation of universes produced by the black holes in our own universe, before the ensemble (or a local part of the ensemble) of universes will see complex - or intelligent - life again.
'

Earth as seen from the Voyager spacecraft.

On a personal note, as a lifelong fan of shows like X Files and Twin Peaks, and as somebody who to this day has the SETI screen saver on my laptop (which parses signals from space looking for radio transmissions from other civilizations) I of all people would enjoy finding evidence of sentient beings on planets other than are own, however, taking all things into account, it may well be the case that while life may be plentiful, sentient life may not be.


Far from being chagrined at such a state of affairs, I think we should find this uplifting, because we on earth are unique in the vast cosmic trajectory of time - Earth, unlike so many other planets out there, hosts sentient beings vouchsafed with the ability to ask these kinds of questions. Knowledge of the solitary nature of sentient beings, esconced as they may well be, here, alone, on this our only planet, should empower us all to be partners, not conquerors, but partners with the ecosystem of Earth, for it is perhaps the only ecosystem which is the proscenium for sentient life, suspended in the illimitable void somwhere between that primal black hole of ages past, and that consummating black hole of ages hence.

Sunday, February 10, 2019

On the Isomorphism of Spinoza

Baruch Spinoza
In mathematics, an ismorphism is where you can map all the elements of Set A onto those of Set B, such that Set A and Set B can at least from a certain point of view be seen as actually being the same set. If (for example) Set A was {apple, orange} and Set B was {yabloko, oranzehevyy} I would be justified in sayingthat these sets were isomorphic to one another, since here Set B are simply the Russian words for the same items in Set A, "apple" and "orange."

Now, to go to Spinoza. Spinoza had a clever argument for why G-d and what we might call "The Universe" or Nature were in a sense isomorphic (although that is a more modern word than he would have used). Basically, Baruch Spinoza said that if G-d by definition is "the greatest thing" and if furthermore, G-d was seperate from Nature, then if we took the sum, G-d + Nature, we would have a totality that would be greater than G-d, contradicting the starting axiom of saying G-d is the greatest thing there is. Therefore (for Spinoza) G-d must in some way include Nature in Herself, perhaps in a sense establishing some sort of isomorphism between these two notions.

Now, this starts to get confusing, fast, and, indeed, has confused thinkers for going on four centuries. It would be easy (and, perhaps, arguably lazy) to accuse Spinoza of playing "word games" and simply using the "G-d" word as another word for Nature. However, I think what is going on here is a little more nuanced. Let us look at what Spinoza means by the word Nature or the Universe. He says there is one substance with infinite attributes. So, there is one "Reality" and what we have access to by experience and observation are these attributes of this one "Reality" or "Substance".

So, I think what Spinoza really means is that for him, "G-d" is this totality - the one substance that is, all that is, was, or will be, and Nature is if you will all the infinitude of attributes of this one substance. So this is not *quite* a perfect isomorphism. Rather, for Spinoza, "G-d" is "the whole thing", and "Nature" are those "attributes" which present themselves to empirical experience, which, in turn, are part of "the whole thing" or the "one substance" or what we might call "Reality".

Now, here I am going to get into more my own personal musing, which Spinoza may or may not have agreed with, but here we go. We have here, in a sense, a notion of "G-d" as if you will, the "Universal", and "Nature" as the particular attributes of this "Universal". But, by definition, I do not have access to experientially a "Universal", I can only "experience" particular things. So let us look more closely at the word "attribute". For example, say I am drinking chocolate milk. I "experience" the "attribute" of say the creamy texture of the milk, and the "attribute" of the chocolate taste of the milk. I may not directly "experience" say any additive vitamins that have been added by the manufacturer to the milk in terms of being able to taste them, but said additive vitamins are part of the totality of the chocolate milk that I am drinking. After the fact, I may look at the label and read about the vitamins, but I do not really "experience" them in the same way that I "experience" that chocolate flavor. So, without looking at the label, I would not list the additive vitamins as being among the "attributes" of the chocolate milk.

So, here I may say, that the "attributes" of the "One Substance" of Spinoza - Reality, say - are those aspects of this reality that I have some direct experience with and that furthermore I can label linguistically. But, I may not always be able to label all my experiences in terms of language. So, while I cannot (by definition) "experience" the "Universal", I can and do have experiences which sort of "call forth" this "Universal" by virtue of not being able to wholly pin down or capture these experiences in a linguistic manner.

Tersea of Avila
Now, the psychoanalyst Jacques Lacan referred to those experience of life which cannot fit easily or at all into language as what he called "The Real", and furthermore Lacan asseverated that this "Real" was the category of what are termed mystical or spiritual experiences. His canonical example was the mystic Teresa of Avila whose trances in which she claimed to encounter the Divine have long been cited as an example par excellance of the sublimity of the "other-wordly." Lacan's point was that the trances of Teresa of Avila could not be communicated by description, that they lay wholly outside the realm of language, but had a profound impact on her and her followers. Going back here to Spinoza's "one substance with infinite attributes," I would posit that Lacan's "Real" are those attributes which belong to raw experience and can never be wholly subsumed into a logical or mathematical framework, and, because of this, "point the way into" that Universal in which all resides, but is not (by definition) apprehendable in-and-of-Itself, anymore than a fish can "directly apprehend" the water in which it swims.

In a historical context, one thing that Spinoza was reacting to was the philosophy of Descartes, who was a thorough-going dualist, and so in a sense Spinoza's work can be seen as a rejoinder or counterpoint to Descartes, meaning Descartes held the notion that concepts such as "mind" or "consciousness" and "G-d" were wholly seperate from the world of empircal observation. Spinoza, working from the Jewish line of thought of the "one-ness" of the Divine, sought to work against this concept of dualism, as, full disclosure, do I. As a side-note, one of the many Jewish names for the Divine is "ha makom" ("the place") coming from the mystical tradition that the universe we experience and observe "resides" in some sense within the nature of the Divine, that is, the Divine is not "outside" the world we live in, but rather, the other way around, the world we live in is within the Divine. This I would  argue relates back to the ancient Jewish admonition regarding avoiding idol worship, an admonition that I think remains relevant to this day, where people make many things as central to their concern, such as, for instance, the economy, or a sports team, or some celebrity or other. What all "idols" or things in this world that retain people's central focus (or their "Ultimate Concern" to use the phrase of theologian Paul Tillich) have in common is these things all "live" in the world we experience, while Spinoza would have said that this approach has it completely backwards. The "Divine" is to be found in the Universal, not in a particular person or thing (like say the economy) within the Universal. "G-d" for Spinoza is not an object within the world, nor an object outside the world (as argued above) but is rather the Totality of the world, which always "over-flows" the ability of language to describe it. I would argue that Spinoza is the logical end point of the thousands of years of tradition regarding admonition against idols, namely, that the Divine is neither something within the world we experience, nor is it something apart from it (as say Descartes might have said) but rather is the totality of the world we experience, but this totality always is greater than the ability of humankind's most rarified systems of science or mathematics to wholly describe it.

On a personal note, I (independently) for reasons too long to get into in this essay had the notion some time back that entropy plays a role in human consciousness. Some time after I had this thought, Canadian scientists doing brain scans on epileptic patients found that indeed patients in states of "higher wakefulness" (as opposed to being in say a seizure state) do have brain states corresponding to higher levels of entropy. So beyond the validation that this was in line with my own thinking, I find this interesting because here we see in a sense a confirmation of Spinoza's intuition, because entropy is something that "ties together" subjectivity and the objective world. Entropy is found everywhere, including in the principles of physics that enable the internal combustion engine used in automobiles to run. If entropy is also involved in consciousness / subjectivity, as the evidence now seems to be suggesting, we have these two seemingly seperate realms being tied together. Entropy may form a "bridge" between the world of empirical experience and our innermost subjective feelings. This gives support I would argue to Spinoza's main thesis of monism, that there is but one Universal whose various as he called them "modes and attributes" form our experience, and what we call the Divine or "G-d" is in a sense this same "Universal" which (I would contend) is "encountered" in the experience of mystery, what Lacan in his analysis of the story of Teresa of Avila and elsewhere referred to as the "Real".

An artistic rendering of Lovecraft's "Azathoth" metaphor
To give a literary example here which may I think help explain what I am driving at, the writer H.P. Lovecraft had many fictional monsters or space-alien type creatures in his stories, the most famous perhaps of which was Cthulhu, an octopus-like entity living on the Pacific Ocean floor. One of his lesser known perhaps but more interesting creations was a being he called "Azathoth" that lies somehow outside of the known dimensions, which he dramatically writes "is that amorphous blight of nethermost confusion which blasphemes and bubbles at the center of all infinity." Scholars have often thought that Azathoth was different from Lovecraft's other characters, not meant to be taken as a literal space-alien type thing (like, say, Cthulhu) but was a metaphor for the cosmic Unknown. In some readings, the universe we live in is part of a dream which Azathoth projects. As a life-long Lovecraft afficianado, I take the view that Azathoth is indeed a metaphor for our limited understanding of the ultimate nature of Reality (and, indeed, in one story, "The Whisperer in Darkness," Lovecraft explicity states that Azathoth is indeed a metaphor for the "outside", or that which is unknown to the sciences but which may now and again enter human experience). Azathoth basically is a metaphor for the fact that the Totality of Reality is always greater than whatever models science can come up with, and the idea of our world being somehow "within" the dream of Azathoth further reinforces this notion of the limitations of human knowledge. So Azathoth perhaps can be seen as a metaphor for the Universal and specifically for the failure of language or mathematics to fully pin it down.

To summarize, I would posit that the isomorphism of Spinoza between the Divine and the Universe is more nuanced than it is often portrayed. When I think of the word, "universe", I often (instinctively) visualize in my mind's eye some sort of space-time diagram of (for example) anti-DeSitter spacetime, one of the solutions to General Relativity which is used often to describe the development and expansion of our observable cosmological horizon. How Spinoza would use terms like "Universe" or "Nature" is arguably different. By "Nature" he does not mean (say) anti-DeSitter spacetime, or any particular mathematical model of the world we live in, but rather, he means the totality of Reality itself, which always outstrips human ability (even in principle due to Godel's Incompleteness Theorums) to be able to comprehend it. Therefore the "Divine" for Spinoza is something like the totality of the mystery of being (one could phrase it thusly perhaps) and "Nature" is that part of being of which we can perhaps say something about, but we can never describe what we might call "Reality" completely. There is, I would argue, an ethical dimension to all of this. If indeed we as human beings are part of this one universal Substance we might call "Reality" then - by definition - we are each of us connected (literally in an ontological sense) to everything else, all other people, all rocks, plants, animals, stars, black holes, Higgs boson particles. Thus (as for example Buddhist or Quaker philosophies also say) to harm others is to harm ourselves, for we are all inextricably linked together in the cosmic nature of things. The lesson of Spinoza then is that at the end of the day, dualisms (of whatever sort) are deceptive for each particular thing is part of an infinite and indivisble Universal which is never static, but is always in a creative state of becoming, bringing new realities into being each and every moment, a Universal that can only be apprehended indirectly at the edges of language, and about which we can only say (with the prophets of old) that She will be what She will be.

Because it is my blog and I will picture Spinoza's Universal however I bloody well please :)

Saturday, December 1, 2018

Proposed Model of the Origin of Tensed Time and Entropy


Models of time come in two basic variants. The tenseless theory, given by modern cosmological theories such as General Relativity and Quantum Field Theory, holds that time is essentially another spatial dimension (with a negative curvature parameter, as opposed to positive curvature paramaters for normal space dimensions, in the anti-DeSitter solution to Relativity, which is basically the mainstream model of how the universe we observe developed over time). Therefore, if time is "tenseless", then the cosmos taken as a whole is in a sense a timeless mathematical object. On the other hand, less a part of physics, and a more a part of philosophy, the tensed theory of time holds that time is not another spatial dimension but is "real" in some sense, and thus the models of physics that given time as a type of dimension are just that, models, but are not accounting for the true nature of time. Here I want to give a possible account of how a tensed ("real") sense of time can emerge. First, I will digress to talk about Cosmological Natural Selection, for that is what I was thinking about when I came up with this particular accounting of the origin of tensed time (and, congruently with it, also entropy).

Lee Smolin published a book, Life of the Cosmos, which I read as a teenager and have recently returned to thinking about. Essentially he accounts for the constants of nature by a type of natural selection. To be very brief, in Smolin's model, black holes create new, "baby" universes, outside of our own. They "inherit" features such as for example the ratio of the electromagnetic force to the gravitational force, or, say, Boltzmann's Constant, etc., from their "parent" universe, with some variations (call them "mutations"). Over time, universes with lots of black holes dominate the "fitness landscape" because the ones with the most black holes reproduce the most. Thus, in terms of probability, one is likely to find oneself in a universe fine-tuned to produce black holes. A prediction of this is that neutron stars must be less than 2 solar masses. This has since been confirmed in the years since the publication of the book. So the theory, while of course not proven, has withstood scrutiny and falsifiability so far. 

One issue with the theory has to do with information. If objects that fall into black holes "leave behind" their quantum information at the boundary (event horizon) of the black hole, then how is it that any "baby universe" produced by the black hole can "inherit" features from its "parent" universe? This was and remains a challenge, but I have a possible solution, which will then lead me into my main topic about an origin for a tensed model of time (and entropy while we are at it).

The nice thing about a part of General Relativity know as Weyl Curvature, which measures how objects are distorted or bent out of shape in a gravitational field (similar to how the oceans are affected by the moon's gravity that produce tides), is that Weyl Curvature is conformally invariant. This basically means that it holds on whatever scale we are talking about, and in whatever reference frame. To make a silly example: If I have a large basketball with very thin rubber, on, say, Jupiter, this basketball may be however slightly distorted in its shape due to Jupiter's gravity. Now, say I am on a NASA station on Jupiter, and I then put a second basketball into a rocket and launch it into orbit. The basketball in orbit (or, technically, free fall) around Jupiter will on the one hand be affected differently or distorted differently than the basketball at my ground station, but, if I know how the basketball at the ground station is distorted, and I know the velocity of the rocket, I can work out some math to tell me how the basketball in the rocket is distorted. This is conformal invariance. Another even more simple example is if I have a map of New Bedford that lies flat on my table, I can figure out how to get from my home to City Hall, just as well as if my map of New Bedford is on a globe. Either way I can get to City Hall. Conformal invariance means that if I change my reference frame, or how I create a map, I can still preserve some properties between these transformations. Thus, going back to black holes, a child universe produced by a black hole could "inherit" properties from a parent universe due to the conformal invariance of the Weyl tensor. So, even if much or most "information" about the parent universe is lost, a suprising amount could likely still be transfered via how space is curved or distorted in the vicinity of the black hole as measured by the Weyl Tensor. I digress to this subject to point out that I think Smolin's Cosmological Natural Selection may well hold up under scrutiny and the biggest critique against it, the issue of information transfer, may not be so big after all.

As an instructional aide, here is a smart-sounding fellow in a British accent explaining Cosmological Natural Selection:




Well now, this then raises the question of if the universe(es) or cosmos at the grandest scale is a Darwinian process of "successful" ones being the ones that "reproduce" the most (by creating black holes, specifically from stellar collapse of stars that are spinning, known as Kerr Black Holes), then does this process go on infinitely to the past, or must have it begun at some point? Technically, if as I believe, the "clock" of entropy is "reset" in each "baby universe", due to how the geometry of space looks like coming out of the black hole in the new "baby universe", then in a teneseless model of time, this process of cosmological selection can indeed extend into the past infinity. But if we have a tensed model of time, this cannot be so, for then you have the logical paradox of needing an infinite succession of causes and events to arrive at any one point in time. Smolin himself has sided with the tensed model camp so (one presumes) he would also see the need for the cosmological natural selection (CNS) process to have begun at some finite point in the past. Or if not then one must find a way to explain how anything happens at all, given that for anything to happen you need an infinite succession of causation in a tensed theory of time that does not have a starting point.

So now we arrive at the age-old conundrum. This (again) is not so much a conundrum a tenesless theory of time. You can simply posit (say) a Big Bang, and say that such a point is the t = 0 point and you cannot have points of time earlier than that anymore than you can have points south of the south pole. There really is not a problem because the spacetime metric is seen as finally a "timeless" geometrical object (for example the Hartle-Hawking No Boundary Proposal (1983) ). But if one insists upon a tensed theory of time the situation becomes murkier indeed. What I propose here I can certainly not "prove" in a Popperian falsifiability type of way, but it seems to be a logical starting point.

When we talk about "the origin of (tensed) time" we already are in an odd situation in terms of the language - the word "origin" pre-supposes time, so to speak of an "origin" of time we are in a meta-narrative or self-referential situation of asking for what is the "origin" of the notion of "origins". It is tricky, doubtless. But here goes anyway.

Let us say "in the beginning" there was an infinite expanse of a undefined (maybe infinite) number of dimensions of space anterior to the process of cosmological natural selection. The only "things" in this expanse were small, primordial pieces of curvature scattered about on this expanse, ripples, so to speak. As a thought experiment, think of it as perhaps an infinite Jackson Pollock painting, with ink blotches as these curved "ripples" or distortions (our friend the Weyl Tensor again). Some of them would be grouped together in collections or sets of ripples. So we could draw imaginary boxes around these "ink botches" of ripples and create collections or sets of ripples in the infinite Jackson Pollock painting of the cosmos anterior to the cosmological natural selection process. With me so far? Good. Now, there are two rules about making sets: Sets cannot contain themselves, and sets cannot have duplicate members. In real life, I can have a coin collection with say two Gold Buffalo Ounces. But in Set Theory this is not so. I can only have 1 Gold Buffalo Ounce - or one Jackson Pollock inblot of a given sort, per set. I also cannot contain the set as a member of the set itself. To explain that, say I got a list of books from my local library. Then I got lists of books in all the libraries in the world. Then say I made another list, call it the Master List, of all the lists of library contents in the world. You might say the Master List contains all the lists (of library books) possible, but one list the Master List does not contain, namely, the Master List does not contain itself. This is Bertrand Russell's Set Paradox.

As another instructional aide, here is a animated video further explaining Russell's Set Paradox, using the analogy of the Barber of Seville, a Barber who shaves all the men of Seville who do not shave themselves. If he shaves all the men of Seville except himself, he has left out one man in Seville who does not shave himself, namely, himself, and if he does shave himself, by shaving himself, he has violated the rule that he must only shave those men in Seville who do not shave themselves. This video of course follows the well-known Internet Rule that animation always makes math less intimidating:





OK, let us review. So we have an infinite Jackson Pollock painting where the canvas is space, the ink blots are gravitational "ripples" and we can take an imaginary pen and draw squares around these ripples to make sets of the ripples. Only rules are, we cannot make a set that has duplicate elements, and we cannot make a set that contains itself, for that would violate the rule that Sets cannot contain themselves, anymore than we can have barbers in Seville who shave themselves in violation of the rule that barbers in Seville must only shave those who do not shave themselves. Well and good.

But "time" has not entered the picture yet. We just have a static object. But hang on. The rule about no sets contain themselves? Well let's violate that. Boom. If a set, say, Set A, also contains itself (call it Set A1) as a member object, then we immediately have an infinite recursion. Set A1 then has its own self as a member, call it, Set A1.1, which in turn has - say - Set A1.1.1 ...). Let's stop that before we get a headache.

Time may very well begin so to speak as a "violation" of the first rule of Set Theory. As soon as you allow sets to contain themselves, you get an infinite sequence, which may in fact be time.

But, hang on a moment, time is not just an infinite recursion of a set of objects. Time is change. Successive moments have different configurations of particles, etc. So what gives? Here is where we violate the second rule, we allow duplicates. Going back to our friend, Set A, say A has two objects which are mirrors of itself, A1_I and A1_II. So then if we go to the "next generation" of Set A, we have two child sets, A1_I and A1_II (each containing themselves, etc.). If we go to the "second level" or "next generation" in our recursion and randomly pick sets, we are more likely to pick sets that had duplicates, for the simple reasons that sets with duplicates birthed more recursive descendents. Allow this process to continue, N generations into the recursion, sets with duplicates dominate the ensemble, but, and here is the key point, "from the outside" of a set, we may not know which ones have duplicates. Pause here. Think. The further down the rabbit hole we go, at any given scale of sets containing other sets and so on, the more likely it is that sets will have duplicates, but unless we examine each one, a set with a duplicate will "look" like a set without a duplicate in terms of how it "behaves" as a set that is a member element of yet a higher set. Duplcates only serve to replicate sets with more frequency but (since they are not really part of Set Theory anyway) they  don't change anything else. Where are we going with this? 

Recall that entropy is simply a matter of objects that have internal states that are invisible from the outside. Think of an impressionist painting. If you change a couple of lines in it you won't notice a difference. The increase of entropy means that as you go along in time the more objects you run into that have internal states whose changes are not noticable. Allowing an arbitrary or infinite number of sets containing themselves at the start of this whole thing to have duplicates means that as you go down the recursive rabbit hole, the deeper you go, when you randomly sample sets at a given level of the recursion, you will find more and more sets having duplicates, analagous to the idea of entropy of "macroscopic" objects having an increasing amount of internal states the further in time you go.

So, example:

Level 1:
A = {Foo, Bar_I, Bar_II, A1_I, A1_II,}

Level 2:
A1_I = {Foo, Bar_I, Bar_II, A1.1_I, A1.1_II}
A1_II = {Foo, Bar_I, Bar_II, A1.1_I, A1.1_II}

Level 3:
A1.1_I = {Bar_I, Bar_II, Foo, A1.2_I, A1.2_II}
A1.1_I = {Bar_I, Bar_II, Foo, A1.2_I, A1.2_II}

etc.

At each generation you can have duplicate recursive elements spawing new levels, and as you saw I sneakily did, you can change the order of elements from level to level. [Sets need not have ordered elements, but traditionally most do, so by allowing ordering to not be enforced, we give them flexibility.]

If we allowed each level of recursion to have varying numbers of duplicate elements, we can change the number of "internal states" each Set has, without changing the Set's "behavior." Since entropy works by allowing macroscopic objects to have an increasing amount of internal states, we can model entropy by levels of recursive sets having increasing numbers of duplicate elements.

Furthermore, by changing the ordering of elements could - for instance - change the distrubution of matter/energy from moment to moment.

So we have violated two rules and one tradition of set theory. We violate the rule that sets cannot contain themselves, we violate the rule that sets must have no duplicates, and we violate the tradition that ordering is preserved in sets. By doing this we can model the passage of time and also entropy.

What does all this have to do with cosmological natural selection? Well this is just a lower level, mathematical description of how it all works at the higher level. At the higher level, you start with a blank canvas of nothing but energy ripples. Time chugs along (in a tensed manner, i.e., it is "real"), black holes form, making baby universes, that make more black holes, ad infinitum. All that stuff about sets was to give a mathematical description of how this gets going in the first place, because tensed time's boundary is trickier than tenseless time's boundary. In tenseless theories, the beginning of time is like the South Pole - you cannot have times "earlier" than the the beginning of time just like you cannot have points "more south" of the South Pole, because time is treated like a mathematical object. In a tensed theory of time you must have a beginning to avoid logical paradoxes, but then you are in the uncomfortable position of time "appearing out of thin air" as it were. If time comes finally down to violations of the basic rules of Set Theory, so much the worse perhaps for Set Theory, but so much the better perhaps for (tensed) time.

To try to give a non-technical stab at what I am trying to say, here would be an approach. The beginning of the cosmos at the start of Cosmological Natural Selection is indeed like a Jackson Pollock painting with random splotches of energy scattered about. You can think of it as chaos or darkness or void. Much like the creation myth in Beresheit (meaning "Beginning" in Hebrew), the first book of Torah - "there was chaos and darkness in the void, and the wind from G-d blew over the waters". The "void" here is space and the "waters" or "wind" if you will are the ripples in the fabric of space. Then - to channel again the poetry of Beresheit - "G-d said, let there be light." You can think of it as a disruption of the rules of logic (as we known them), and this disruption of the rules of logic, so to speak, caused time itself to come into being. What caused the disruption you might ask? Here we must go to Wittgenstein: Whereof one cannot speak, Thereof one must be silent. For the sake of conversation let us simply say that the cosmological process of Creativity disrupted the nature of things, a disruption that nucleated in the same instance both time and entropy. And following on from that began the never-ending cycle of universes birthing other universes, with each birth resetting the entropy clock so that each universe history would have its own unique trajectory of evolution and stellar (and consequently, black hole) formation, a process producing an ever-increasing ensemble of universes, into time unending.

As a disclaimer, even if this particular approach of using Set Theory to model the origin of time and entropy at the start of Cosmological Natural Selection proves to be a less than optimal approach, I think the basic intuition here is correct, and that is, that the cosmos is initially a blank infinite canvas of arbitrary dimensionality with energy in the form of metric disturbances scattered randomly about, and then a "disruption" occurs which instantiates both the flow of (real, tensed) time, and also the concept of the increase of entropy through time. If as is perhaps probable better technical models of this process come forth in the future, I think this essential intuition will hold.

[As an additional disclaimer, I do note that I do not take the view of mathematics, including Set Theory, as being some sort of timeless collection of truths, but rather I take the view that mathematics is a human-made project of finding ways to model the reality that we experience, so my approach here to use Set Theory to model the origin of the flow of time and entropy is simpy a technical model, and is not meant to be taken in a literalist manner, consistent with my viewing here of time as "tensed" or "real" and hence not seeing anything, including mathematics, as being in some way "timeless" or "outside" of the experienced reality that we try to find ways to describe. Hence for instance I would reject the idea of there being a collection "out there" of different types of mathematical worlds, one of which being our own, and rather I would take the view that there is simply the ensemble of universes generated in real time by the process of Cosmological Natural Selection, wherein we reside.]

It is instructive perhaps that we can think of how time and entropy nucleated to kick-off the process of Cosmological Natural Selection as a kind of disruption of the rules as we know them. This is perhaps the ultimate nature of the cosmological process of Creativity, a sequence of disruptions of the status quo to produce that which is truly new. Whatever the technical descriptions may be, the unassailable fact appears to remain that the disruption of the old to create the new is the most fundamental feature of existence.

As the late Debbie Reynolds memorably sang in the animated film, Charlotte's Web, "how very special are we, for just a moment to be, part of life's eternal rhyme."