Interesting read. I was going to bring up mathematical chaos and strong attractors but then you already did, so I have nothing more to add now.

I can ask a question you might be able to elaborate upon though. I like to think of time in terms of paths through the configuration space of the universe, visualizing for simplicity a 2D configuration plane with points in it extruded downward in proportion to their entropy, so that "forward in time" = "downhill" in this metaphor. I'm not as versed on strong attractors and chaos as I'd like to be, so I'm not sure what relationship they have to something like the slope of entropy in such a configuration space, and so if they would have any kind of visible effect on such a metaphorical representation of time.

I would like to guess that chaos would be rendered as "roughness" in that configuration plane (such that otherwise similar paths through it can easily be deflected in many radically different possible directions) while strong attractors would be something like a single well-defined groove in the configuration plane (so that even radically different paths are all inevitably diverted into it), but I don't know enough to justify that guess.

Along similar lines, Philip K. Dick has parallel histories struggling with each other and individuals able to traverse between them. [Man in the High Castle]
One of my favorite books is Lem's Solaris where an alien entity keeps trying to replicate humans to see what they are about.

Thus, to expand a bit, ergodic theories of history (ET) imply modest alterations of a causal process do not change the final outcome substantially, whereas butterfly theories (BT) imply even very tiny alterations in a causal process produce dramatically different results. — John Gill

I'm only superficially familiar with Ergodic Theories, and what little I know comes from the Information Philosopher instead of sci-fi authors. As far as I can tell, Ergodicity is equivalent to Enformy in my own theory of Enformationism. Both terms refer to an observed, but often denied, trend in evolution that works counter to Entropy to bring order out of chaos, and patterns out of randomness. When applied to history, these ideas may be related to Hegel's causal force that he called the "spirit of history". FWIW, here are some links to related theories of Negentropy, or to directional evolution.


Ergodicity : https://www.informationphilosopher.com/value/ergo/ergodicity.html

Enformy : http://blog-glossary.enformationism.info/page8.html

Extropy : https://en.wikipedia.org/wiki/Extropianism

Nevertheless, a general way to move into a mathematical setting is to view the XY plane (or complex plane, its equivalent) as points representing events, in a rough sense. A time dependent path through the plane means moving seamlessly from past to present, event by event. When a SA is present these paths converge toward it regardless of where one starts nearby. And when SDIC exists even the slightest shift at the starting point produces bewildering trajectories. — John Gill

Could you list examples of "strong attractors"?

I'm not sure what a configuration space is, but your idea is intriguing. I had a friend, a physics prof, years ago who would take a cigar box with a lid to class and open it, revealing a set of neat stacks of coins. The he would shake the box, reopen it and say, "That's entropy!" When notions are a tad vague and it's difficult to put them in some sort of numerical context, I'm out of my depth.

Thanks for the links. I hadn't heard of some of those ideas. "Ergodic" is difficult to pin down, but the statistical idea in a dynamical system is not hard to see. Roughly it can be explained as taking averages of a phenomenon two or three different ways and getting more or less the same result. But it's not entirely clear cut.


This is a mathematical concept, so I'll give a simple example. Start with the complex plane (corresponds to the XY plane) where z=x+iy, x and y real numbers and i^2=-1. An attracting fixed point is a point in the plane (or complex number) that attracts for some function f(z). It doesn't exist as an attractor by itself. The simplest and most powerful attractor is f(z)=a, where a is some point in the plane. Then no matter what value you use for z, the function takes you instantly to a.

A slightly more complicated example is f(z)=.5(z-a)+a. Starting with a particular value of z, let z1=f(z), z2=f(z1), z3=f(z2), .. . . .Then this process of iteration, with let's say each step taking one second, moves the point z as close as we wish to a in a finite period of time. This limiting process is the heart of the branch of mathematics called analysis. And when analysis concerns complex numbers it is complex analysis, my specialty.

I'm not sure what a configuration space is — John Gill

It's also called a phase space. It's an abstract space wherein each dimension represents one variable feature of some system, and so every point in the space represents some complete configuration of the system. So if you had a two-variable system, say a ball that can change size and color but that's it, then you could represent every possible state of that system in a two-dimensional space. A complete universe, of course, has way more variables in its configuration space, but that's hard to visualize.

In my conception of time, every moment in time is a point in the configuration space of the universe, and because more-entropic configurations are definitionally more plentiful than less-entropic ones, a random walk through the configuration space (making random smallest-possible changes to the state of the universe) will necessarily lead you into more and more entropic parts of the configuration space, so time inevitably marches in the direction of more entropy, wandering away from points of low entropy in diverging paths, so futures diverge while pasts converge. The "start of time" is the local entropic minimum in the configuration space of the universe, away from which all directions are "forward" into more entropy.

Great introduction to what is probably a very complex subject. Thanks.

I remember watching a TED talk on statistics where the speaker brings a contraption onto the stage and demonstrates the "logic" of Pascal's triangle with balls being rolled from the top of an arrangement of pegs in the shape of Pascal's triangle. The final result of the experiment is in accordance to Pascal's triangle with a few balls at the edges and most balls falling in the middle. The explanation for it was that there are more paths towards the middle than towards the edges.

I think the ergodic theory of history can be understood in these terms. Some points in our common history simply have multiple paths that approach it and that makes such points strong attractors.

As for the butterfly effect, I think it makes an appearance when there's enough deflection in the initial state to nudge the causal chain onto another path with a vastly differing endpoint.

Another thing I find troublesome is the definition of differences in initial conditions in the butterfly effect. As I understand it chaos theory basically claims that negligible differences in initial conditions lead to chaotic behavior but negligible in terms of a difference (subtraction) may not be the way to go. For instance the difference (subtraction) between 2 and 1 is a "negligible" 1 but actually 2 is twice the amount 1 is. I'm not a mathematician and all what I'm putting into words my be ignorant gibberish but, taken the way I suggest above, chaos theory seems to be a point of view too.

So, what is perceived as a negligible difference in the butterfly effect may actually be significant viewed in another, mathematical, sense. Imagine a pingpong ball balanced on the tip of a needle. The force required to tip the ball left or right is assuredly small and negligible but going left or right may result in hugely different outcomes. This implies that "negligible" as a description has the quality of a subjective feeling too.

As to how chaos theory, as generally understood, applies to history, I think it makes sense in terms of cumulative contributory causation. If we have enough men we can push a giant boulder even though the force applied by each is minimal. In addition, which way the boulder rolls will probably depend on that one negligible extra force applied by one man.

As I understand it chaos theory basically claims that negligible differences in initial conditions lead to chaotic behavior — TheMadFool

I think the word "negligible" should not be used in the context of chaotic dynamical systems. Clearly if a slight variation at the beginning of an iteration process leads to bizarre behavior that variation, no matter how small, is not negligible. Just the contrary. Your notion of "cumulative contributory causation" is well put.

It's also called a phase space. — Pfhorrest

That I am familiar with as I play with dynamical systems in C that involve velocity, etc. But configuration space looked a little strange. Just me. Thanks for the explanation.

"In my conception of time . . " I like this. Well thought out IMO. Of course it depends upon the idea of "moment" - I am sure this has been chewed over in this forum sufficiently. In real and complex analysis in mathematics most practitioners work with the real number system as it's described by Cantor and others and time is simply a real variable. But set theorists and math logic people go off to abstractions quite readily. At the end of an introductory course in set theory in grad school in the early 1960s the professor said, "If you don't intend to spend your careers in mathematical logic, I recommend you never take a course in that subject." I heeded the advice.

Associated with ideas of Time, the concepts of infinitesimals, however, goes back in history. These are numbers in a sense that are both positive and smaller than any real number. Both Newton and Leibnitz had their ideas along these lines. And in the last century a mathematician named Robinson developed a mathematical model, making these strange little critters legitimate. It's called Non-Standard Analysis. One can teach a course in calculus from this perspective, avoiding the epsilons and deltas commonly employed. Time, it seems, can be Complex and/or infinitesimal. Go figure.:cool:

Strong attractors are fixed points (FPs) of functions that, upon iteration, draw a given initial point towards the FP. But there are other FPs that either repel the iterations or are neutral. In the example f(z)=2(z-a)+a all points close to z=a are pushed away from that point. What is the historical analogue of these points? And the most interesting FPs are indifferent fixed points, where the iterative behavior around the point is difficult to analyze. f(z)=(3z-4)/(z-1) has an IFP at z=2. This case, it would appear might be the most prevalent in evolutions of societies. Both attracting and repelling.

And then there are Strange Attractors (no, not a Netflix series) which generally are sets or clusters of points in C (the complex plane) that iterations are attracted to. Enough.

Time, it seems, can be Complex and/or infinitesimal. Go figure.:cool: — John Gill

Can you explain the "and/or" part?

Thank you.

I think the word "negligible" should not be used in the context of chaotic dynamical systems. Clearly if a slight variation at the beginning of an iteration process leads to bizarre behavior that variation, no matter how small, is not negligible. Just the contrary. Your notion of "cumulative contributory causation" is well put. — John Gill

Thanks. I missed that distinction. Important.

Look at how wikipedia describes chaos theory:

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. — Wikipedia

The impression I get is that chaos theory, as a mathematical field, is based in a very "big" way on the meaning of the word "small". What you described as slight changes and how they cause large variations in endpoints seems to be a source of astonishment and I suspect this astonishment has a much bigger role in the birth of the field of chaos theory than anything truly chaotic going on. Again I'm no mathematician and what I'm saying is probably complete nonsense. Educate me.


Also, another aspect of chaos theory is the dependence of chaotic behavior on the location of the viewing window. I can't think of a good illustration but how about history because it's easier to see the point I'm making. Open a window to history in 1914-1918 or 1939-1945 and you'll see war (chaos) but anywhere in between these periods you'll see relative peace (no chaos).

Coincidentally(?), the educational YouTube channel Veritasium just put out a short (12min) video describing the butterfly effect and strong attractors in terms of phrase spaces with some very edifying visualizations:

https://www.youtube.com/watch?v=fDek6cYijxI

Can you explain the "and/or" part? — Wallows

Complex time, t=x+iy , is productive in certain settings in physics, and its metaphysical interpretations seem immaterial to those who simply want answers that agree with observed reality and are predictive. If one posits time in infinitesimal increments then both x and y are infinitesimals. I am not aware of any applications involving these tiny critters, but who knows what's around the metaphorical corner? Most mathematicians in analysis don't use the Non-standard Analysis approach although textbooks on calculus have been written in that venue.

From a different perspective these ideas constitute interesting theories of history. — John Gill

In my brief online review of Ergodic theories, I noticed that they are mostly applied to abstract mathematical concepts, such as Riemann Manifolds and Markov Chains, that are far from my everyday concerns. Other than the sci-fi stories you mentioned, are you aware of any applications of Ergodicity to human cultural history? Are there any examples of historical trends and transformations that have been interpreted in terms of Ergodic Theory and the Butterfly Effect?

I suspect that history could be analyzed as a form of statistical thermodynamics, in the sense that changes (historical "energy") flow from Hot to Cold; such that, if cultural hot & cold spots could be defined ergodically, then future trends in history could be predicted mathematically --- at least in the short term. This might be presented graphically as a flow chart, and understood metaphorically like Hegel's "Spirit of History" following a zig-zag course from positive Thesis (hot) to negative Antithesis (cold), and thence to moderate Synthesis (lukewarm).

As a test case, the flow of Socialist/Communist sympathies relative to Capitalist/Fascist hot spots in Europe, Asia, and South America could be tracked as they flipped back & forth (e.g. Russia and China) after WWII. What butterfly flaps caused those flips from Communist Prole-tocracies to Capitalist Oligarchies? Capitalism and Communism seem to be powerful social organizing forces that at first appear to be unstoppable as they spread out from Hot Spots. But it takes only a few generations for them to flip-flop, or zig-zag, if you prefer..

I think the intended application to history is something like the question of whether WWII would have still happened had Hitler died in infancy. If a strong attractor is involved in the human cultural system of the time, then the answer is probably yes. If Hitler was a butterfly, then no.

The most concrete efforts to apply ergodicity to 'history' have happened in Economics. Google 'ergodicity' and 'economics' and you'll find a tangle of approaches.

The difficulties faced in Economics apply to History in general:
- you can't do experiments, you only have post facto data
- most theories whether ergodic or not have a terribly poor ex ante record, i.e. prediction success is low
- the discipline itself is rooted in linear, equilibrium-based and 'rational' assumptions, and many of these assumptions insinuate themselves into ideas that purport to be ergodic, which need instead to be non-linear, freewheeling and arational/stochastic.

All the same there has been fruitful mutual work between economists and physicists/mathematicians, though it does not rest on a secure set of foundations.

The foundations would need a set of abstractions that could be agreed to be workable. Terms like 'Capitalism' and 'Communism' for instance come so laden with political baggage that they can be hard to use, e.g. was the Soviet Union a form of state capitalism rather than communism? It might be that one could begin with a narrow account of a relatively isolated State and make some sort of historical sense.

I think the intended application to history is something like the question of whether WWII would have still happened had Hitler died in infancy. If a strong attractor is involved in the human cultural system of the time, then the answer is probably yes. If Hitler was a butterfly, then no. — Pfhorrest

Ha! I suspect that Hitler was more of a Strong Attractor than a Flitting Butterfly. His "Make Germany Great Again" (MGGA) campaigns were obviously attractive to patriotic Germans after the humiliations of WWI, and his Aryan Myth was appealing even to many comfortable Americans & Britons, feeling besieged by pro-melting-pot Liberals. However, Churchill and Roosevelt were unpredictable butterflies flapping stubbornness and altruism. Perhaps we could put numbers on those leader's political baggage to make their contributions to bringing "order out of chaos", and "justice out of injustice" more objective.

I'm sure that any historical applications will have to begin with retrospectives. But eventually, once the definitions and inter-relationships are refined, the Ergodic theory should be able to put some numbers on trends projected into the near future. If so, then history would be able to look forward, and provide substance for our intuitive expectations. For example : is Trump enough of a Strong Attractor to swing the politically-divided US toward the stability of Fascism? Is his ergodic influence positive or negative, from the perspective of oligarchs or plebians?

It might be that one could begin with a narrow account of a relatively isolated State and make some sort of historical sense. — mcdoodle

Surely, some economists and historians have already begun to computerize social progress or regression.

I'm not aware of any real applications along these lines, meaning predictive power. And keep in mind, economics is called the "dismal science" because it most frequently looks back in time and not forward.

However, I have dabbled in dynamics of this nature for many years, and only recently has one of my theorems been employed in a predictive sense in a social context. Even then the authors overlooked the mathematical setting for the theorem, assuming it would apply to social reality. Me, I don't know.

Probably, I should not have used the word "ergodic" and would not have done so were it not Lem's appellation. It's a bit of a fuzzy notion and means slightly different things in different contexts. The Ergodic Theorem is a sophisticated result in math and for two forms of averages to coincide requires the time dependent form involve a highly restrictive function. One that is reduced to virtual triviality in the aspect of complex analysis in which I explore.

No need to withdraw the idea in my view, John. It's a very productive idea, even if fuzzy. And as soon as a fuzzy-in-itself idea becomes a metaphor, it soon finds itself proposing to solve world problems when the humble practitioner - you in this case :) - is still saying, 'But I'm not sure it actually works.'

Probably, I should not have used the word "ergodic" and would not have done so were it not Lem's appellation. — John Gill

Lem's association of Ergodicity with History was probably based on a philosophical, rather than mathematical, definition. Mathematical theories of dynamics-in-the-abstract may be too far removed from our experience of the dynamics-in-practice we call "history". But the Information Philosopher has applied the abstruse notion of thermodynamic Entropy and Enformy (my term) to the personal values of progress and retrogression in human culture. That's why your original post struck a chord with me. Like Hegel, and many others, I see evidence of a progressive "force" or trend in natural and cultural evolution. But a mathematical definition of that positive path within randomness might make the concept of an upward arc in history more palatable to skeptics, who view Randomness and Entropy as all-powerful. It could also help to explain how highly-organized Life & Mind emerged from the erratic path of evolution.


Ergodicity : Ergodic processes (in our new technical use of the term) are those that resist the terrible and universal Second Law of Thermodynamics, which commands the increase of chaos and entropy (disorder). Without violating that inviolable law overall, ergodic processes reduce the entropy locally, producing pockets of cosmos and negative entropy (order and information-rich structures).
https://www.informationphilosopher.com/value/ergo/ergodicity.html

Enformy : In the Enformationism theory, Enformy is a hypothetical, holistic, metaphysical, natural trend or force, that counteracts Entropy & Randomness to produce complexity & progress.
http://blog-glossary.enformationism.info/page8.html

As you noted, these what-if questions pop up in different contexts. In the 1980s paleontologist Stephen Jay Gould raised this question with respect to the history of life on Earth. He supported the "butterfly effect" view: replay the tape of evolution, and due to the accumulation of contingencies, life would most likely go on a different path, and there would probably not be anything like the human species. Others, including another eminent paleontologist Simon Conway Morris, took the opposing "ergodic" view: convergent evolution would lead to similar, if not exactly the same forms developing, assuming the environment is roughly the same.

As you might imagine, neither side could offer much in the way of hard evidence for their position. However, since then other paleontologists and evolutionary biologists have weighed in. There is some limited empirical support building for convergence (ergodicity), but generalizing and scaling these results is difficult (see for instance a recent Science paper Contingency and determinism in evolution: Replaying life’s tape).

I think that part of the difficulty here, in addition to the scale and complexity of the problem, is that we use different models for different scales and granularities, and these models are neither practically, nor in most cases theoretically reducible to each other. When we go up the scale and coarse-grain our analysis, what was deterministic at a smaller scale becomes random or altogether invisible. Thus the butterfly may flap its wings, but we wouldn't know it or wouldn't have the means to factor it into our analysis. Of course, the very existence of coarse-grained models implies some degree of robustness: if the world really did go askew every time some damned butterfly did something in China, then what would be the point of trying to predict anything on a larger scale? We would all be reduced to butterfly-watching.

The mathematics of analogous dynamic systems is fun to play with, but social sciences are messy, even economics and history where one looks backward rather than forward. I don't have much hope of seeing Lem's ideas quantified in any predictive manner. But it's fun to speculate.

Stephen Jay Gould raised this question with respect to the history of life on Earth. He supported the "butterfly effect" view: replay the tape of evolution, and due to the accumulation of contingencies, life would most likely go on a different path, and there would probably not be anything like the human species. Others, including another eminent paleontologist Simon Conway Morris, took the opposing "ergodic" view: convergent evolution would lead to similar, if not exactly the same forms developing, assuming the environment is roughly the same. — SophistiCat

In my personal Enformationism thesis, I hold to a synthesis of both views (BothAnd). Most materialist scientists & philosophers, assume that randomness (chaos) and Entropy are the dominant forces in evolution. But, if that were the case, the human species would be astronomically unlikely to emerge (e.g. billion to one odds). Yet, other eminent researchers & theorists have observed the recent rapid pace of evolution --- since Life, with its novel function Mind, emerged from eons of incremental physical & chemical aggregations --- and have concluded that logically there must be some kind of counter-balancing (Ergodic) force that serves to bring order out of chaos. IOW, thermodynamics has a thermostat.

In my view, that organizing force, which I call Enformy, offsets the disorganizing force of Entropy, but only in a precious few pockets of biological and psychological phenomena. In fact, out of the whole universe, the only known examples of anti-entropy are right here on the "pale blue dot". Ironically, in this otherwise insignificant corner of the Cosmos, Physical Evolution has given birth to Cultural Evolution, which is accelerating at a neck-snapping pace, compared to the previous billions of years of Natural Evolution. As far as we know, only here on Earth is evolution being influenced by human intentions. Which makes us co-creators of the emerging universe. And which may indicate that the sudden appearance of conscious beings is not an accident, but an important phase of an overall plan with some higher hidden purpose, as presumed by meaning-seekers over several millennia.

So it seems that the Butterfly Effect has been in charge of creating minimal order out of Chaos up until now. And from here on out, Ergodic processes, including human ententions will be in charge of directing the progression of Evolution. Toward what end, you well might ask? I dunno, I could frankly answer. That's the perennial mystery of life, for curious humans who woke up in the middle of an ongoing story that fades into the barely remembered Past, and a future that can only be known by turning a page each day. Which perhaps indicates that the ultimate Author only wrote the outline, and left it to us characters to improvise the details. In that case, the meaning of your life will be written by you (the actor), instinctively or rationally (Ergodically) responding to the unpredictable surprises of the Butterfly Effect.. :cool:


Enformy : causal Energy plus directional Entention
http://blog-glossary.enformationism.info/page8.html

Wiki: "The mathematical universe hypothesis suggests a new paradigm, in which virtually everything, from particles and fields, through biological entities and consciousness, to the multiverse itself, could be described by mathematical patterns of information"

I don't subscribe to this, but it is an interesting perspective.

I do find a multiverse concept appealing, however. Each "instant" an uncountably infinite number of universes spring into existence with a sort of probability directing event patterns. Time travel without the grandpa effect might be possible.

See, I do metaphysics too. :wink:

This is a bit of a tangent, but seeing something about the movie Avengers Endgame just now make me think of this thread. With the way time travel works in that movie, time lines tend to branch chaotically if Infinity Stones are removed from them, but with the Infinity Stones in place changes to the past get somehow subsumed in the flow of history and cause the future to turn out unchanged. That seems to me like somehow the stones generate strong attractors that keep history converging to the same outcomes, and without them butterflies reign.

The familiar example of the latter is for a butterfly to behave differently in Asia means storms later in California rather than fair weather had it not changed its behavior. — jgill

Not sure if this is actually true for a dead weather system. It needs life for substantially changes. If the butterfly accidentally moved in the room where the people resided of the Art Academy in Wien, and the guy who refused Adolf Hitler to join in was distracted by the butterfly, and AH would have joined, history could have been quite different.

Ergodic theory says that in a system of particles every particle eventually visits all spots in the system. You can investigate the walk through the park of one stroller many times, or you can look at all strollers at once. Ergodic theory says you get the same distribution of walks.

Ergodic theory says you get the same distribution of walks — Jarjar

Surprised to see this thread resurrected. Yes, the math and physics definition of ergodic is technical about averages. Lem used this term to imply averaging out of incidents in strong social movements. More or less the opposite of the butterfly effect (SDIC) of chaos theory.

So, basically, we have to use x (a variable) for all (efficient?) causes of all events. The variable x can take on any value; the effect (the event in question) will occur regardless!

Cause then is not unique (I could lift the rock, you could lift it, Hitler could, Dick Cheney could, you get the idea).