June 08, 2004
Body-snatching Lorenz equations
Posted by shonk at 04:47 AM in Geek Talk, Words of Wisdom | TrackBackFirst off, a bookkeeping note: Curt will be traveling around Europe for the next month, so he likely won’t be posting much, if at all.
Now, I promised last week, in my review of Strogatz’ Sync, that I would devote an entire post to an extended quotation from the book that I found very interesting. Reading the comment thread associated with John Sabotta’s post denouncing evolutionary psychology at No Treason, I was reminded of that promise, so this is that post. In the pertinent passage, Strogatz is discussing a chaos-based encryption system first envisioned by Lou Pecora. Pecora was trying to figure out a way to devise an encryption system based on Lorenz equations, wherein three variables are related to each other in a particular way (specifically, by way of a system of differential equations).
(As a side note, I would point out, apropos my earlier comments on Strogatz book, that this passage serves as an excellent example of both the strengths and weaknesses of the book. The primary strength, aside from Strogatz’ obvious depth of knowledge of the material, is his ability to describe complicated technical mathematics by way of metaphors that make it highly accessible to a lay reader. The primary weakness, at least in my view, is that he doesn’t ever give any of the technical details. Admittedly, systems of differential equations are a bit intimidating, but the fact that the entire book, basically, is written in metaphor is a bit grating)
Anyway, on to the quotation, followed by one or two of my own thoughts:
In technical terms his scheme can be described as follows: Take two copies of a chaotic system. Treat one as the driver; in applications to communications, it will function as the transmitter. The other system receives signals from the driver, but does not send any back. The communication is one-way. (Think of a military command center sending encrypted orders to its soldiers in the field or to sailors at sea.) To synchronize the systems, send the ever-changing numerical value of one of the driver variables to the receiver, and use it to replace the corresponding receiver variable, moment by moment. Under certain circumstances, Pecora found that all the other variables—the ones not replaced—would automatically snap into sync with their counterparts in the driver. Having done so, all the variables are now matched. The two systems are completely synchronized.
This description, though technically correct mathematically, does not begin to convey the marvel of synchronized chaos. To appreciate how strange this phenomenon is, picture the variables of a chaotic system as modern dancers. By analogy with the Lorenz equations, their names are x, y, and z. Every night they perform onstage, playing off one another, each responding to the slightest cues of the other two. Though their turns and gestures seem choreographed, they are not. On the other hand, they are certainly not improvising, at least not in the usual sense of the word. Given where the others are at any moment, the third reacts according to strict rules. The genius is in the artfulness of the rules themselves. They ensure that the resulting performance is always elegant but never monotonous, with motifs that remind but never repeat. The performance is different from minute to minute (because of aperiodicity) and from night to night (because of the butterfly effect), yet it is always essentially the same, because it always follows the same strange attractor.
So far, this is a metaphor for a single Lorenz system, playing the role of the receiver in Pecora’s communication scheme. Now suppose that time stands still for a moment. The laws of the universe are suspended. In that terrifying instant, x vanishes without a trace. In its place stands a new variable, called x’. It looks like x but is programmed to be oblivious to the local y and z. Instead, its behavior is determined remotely by its interplay with y’ and z’, variables in a transmitter far away in another Lorenz system, all part of an unseen driver.
It’s almost like the classic horror movie Invasion of the Body Snatchers. From the point of view of the receiver system, this new x would seem inscrutable. “We’re trying to dance with x but suddenly it’s ignoring all of our signals,” think y and z. “I’ve never seen x behave like that before,” says one of them. “Hey, x,” the other whispers, “is it really you?” But x wears a glazed expression on its face. Just as in the movie, x has been taken over by a pod. It’s no longer dancing with the y and z in front of it—its partners are y’ and z’, unseen doppelgängers of y and z, remote ones in the parallel universe of the driver. In that faraway setting, everything about x’ looks normal. But when teleported to the receiver, it seems oddly unresponsive. And that’s because the receiver’s x has been hijacked, impersonated by this strange x’ coming from out of nowhere. Sensitive souls that they are, y and z make adjustments and modify their footwork. Soon, all becomes right again. The x, y, z trio glides in an utterly natural way, flowing through state space on the Lorenz attractor, the picture of chaotic grace.
But what is so sinister here, and so eerie, is that y and z have now been turned into pods themselves. Unwittingly, they are now dancing in perfect sync with their own doppelgängers, y’ and z’, variables they have never encountered. Somehow, though the sole influence of the teleported x’, subtle information has been conveyed about the remote y’ and z’ as well, enough to lock the receiver to the driver. Now all three variables x, y, and z have been commandeered. The unseen driver is calling the tune.
— pp. 196-7
Unfortunately:
So far, chaos-based methods have proved disappointingly weak. Kevin Short, a mathematician at the University of New Hampshire, has shown how to break nearly every chaotic code proposed to date. When he unmasked the Lorenzian chaos of Cuomo and Oppenheim, his results set off a mini-arms race among nonlinear scientists, as researchers tried to develop ever more sophisticated schemes. But so far the codebreakers are winning.
— pg. 204
The obvious conclusion is that each variable in these specialized systems in fact encodes the entirety of the system. From a mathematical perspective, I admire the ingenuity required to come up with this sort of scheme. From a metaphysical perspective, though, it’s hard not to find this whole thing vaguely unsettling. After all, people usually respond in fairly predictable ways to outside stimuli. The point I guess I’m stumbling towards is this: the idea that a chaotic system in which the actors respond in predictable ways to the other actors could lead to the sort of lockstep mirroring described above seems, at least to me, to say something very pertinent to the whole determinism/free will debate. Admittedly, the eerieness is largely due to Strogatz’ metaphor, but the fact that the y and z variables, thinking themselves to be operating independently of the y’ and z’ variables, would ultimately end up mirroring those same variables simply because they followed the same rules in reacting to the x/x’ variable is pretty fascinating.
What I’m not trying to do is say that this sort of thing settles the determinism/free will debate. Rather, I’m just pointing out that these new mathematics give new insight into ways in which seemingly independent activity can yield identical results. And, although these are admittedly specialized cases, it’s somewhat surprising (at least on an intuitive level) that there are any circumstances in which variables seemingly reacting to another variable’s independent activities would, in fact, end up exactly duplicating the variables to which that wild variable was itself reacting. The result, of course, being a system in which all three seemed to be mutually reacting, while in fact one was paying no attention at all to the other two. In other words, one has to wonder, at least a little bit, in what direction causality points, exactly.
A very interesting post Mr. shonk …
shonk: In other words, one has to wonder, at least a little bit, in what direction causality points, exactly.
Doesn’t causality always point directly at the Truth?
What else are you suggesting that it could point to?
Posted by: The Serpent at June 8, 2004 11:51 AMI'm glad you enjoyed it. I posted it primarily for you.
Posted by: shonk at June 8, 2004 01:22 PMwhat does this have to do with the death of our greatest president?
For shame, Mr. Shonk.
what does this have to do with the death of our greatest president?
Well, nothing. I'll leave the eulogizing to someone else.
Posted by: shonk at June 9, 2004 12:25 AMSo shonk, are you also familar with Stephen Wolfram?
Posted by: The Serpent at June 9, 2004 09:27 AMFamiliar in the sense that I know who he is and have used Mathematica, but haven't read his book.
Posted by: shonk at June 9, 2004 09:34 AMYeah, his book (Wolfram) is over a thousand pages if memory serves me correctly. But I think he is another guy like Strogatz who just can’t throw off the yoke of Materialism completely.
They take 1000 pages to explain what Mandelbrot explains with one simple picture (image).
Look, instead of Strogatz’s dancers imagine some checkers on a checker board. You make some simple rules about how the checkers will propagate (or move), and maybe a rule that any checker surrounded on 3 or more sides is removed.
Then you begin to iterate this algorithm. It generates a pattern. Is the pattern random, or is it determined?
I would say that the pattern appears random, but in reality it is actually determined. I would say that there is no possible conceivable way the pattern can be “random”, and that “random” is essentially a meaningless (incoherent) term. “Random” is what you call something when you are unable to perceive the pattern (or the underlying logical mechanic which is generating the pattern).
Posted by: The Serpent at June 9, 2004 10:51 AMHmmmm …
So did you read Micha Ghertner’s reply to my post over at No Treason?
Were you (personally) satisfied by his response shonk?
I get the distinct impression that Mr. Sabotta and Mr. Ghertner dread the thought of living in a reality that isn’t “random” and “magically incomprehensible”. In fact, I don’t believe they can even tolerate the thought.
Posted by: The Serpent at June 11, 2004 11:25 AMSo did you read Micha Ghertner’s reply to my post over at No Treason?
If you're talking about the comments associated with the article I linked to in this post, his reply seemed to be directed towards Sabotta, not you.
Posted by: shonk at June 11, 2004 01:33 PMMicha Ghertner: Nearly everyone, including, most likely, yourself, does not believe in absolute free will - a will that is completely undetermined by external factors, whether those be nurture or nature. Our preferences are largely determined by external factors, our ideas about right and wrong are largely determined by external factors, and most of our decisions are made according to habit or instinct.
Most decisions are Determined, but Some are actually “free will” (random, magical)???
Isn’t that the same as saying that everything is random and magical?
Or perhaps he doesn’t have any idea what he is talking about?
Micha Ghertner: Falsehoods do not serve libertarianism, except to weaken and destroy it.
Yeah, but apparently “Anarchist” like weakness and destruction?
Micha Ghertner: I don't find religious nonsense as spiritually satisfying as secular philosophy, evolutionary biology, and cognitive science. But let's assume that wasn't true. Let's assume I found religious nonsense more spiritually satisfying than all these other fields of inquiry. Is that a good reason for me to believe in religious nonsense? Of course not: I should believe in theories based on things like empirical evidence, probable likelihood, reasonableness, etc. - not spiritual satisfaction.
What a load of horsecrap! Ghertner believes in plenty of spiritual, mystical, religious nonsense (so does Sabotta). They just want to pretend that their religion is more magical and special (“random”) than all of the other “religions” combined.
Ghertner has already determined he is “smart”. He knows enough to follow the “One True Faith”. He’s delusional.
Posted by: The Serpent at June 11, 2004 03:20 PMWhat a load of horsecrap! Ghertner believes in plenty of spiritual, mystical, religious nonsense (so does Sabotta).
I don't deny that, but am not at all sure that you don't do the same.
For example, I agree that it seems duplicitous to assert, on the one hand, that "most of our decisions are made according to habit or instinct" while, on the other, giving no specific justification for the use of the term "most" instead of "all".
On the other hand, your assertion that we must trust our perception when it asserts that all that we can observe behaves deterministically yet must deny our perception that we do make choices and have some semblance of free will is also duplicitous. Perhaps your mind works differently than mine, but I perceive, at least some of the time, that I am actually making true choices, that my actions are not strictly determined. I assume such is the case for most people. Of course it is impossible for me to independently verify this, as I know of no way to go back in time and re-insert myself into the same situation with precisely the same knowledge I originally had and then observe myself either making the same choice every time or making different choices.
And so I am left with what my perceptions tell me. This certainly includes the perceptions that lie at the foundation of science, the apparent fact that, given the same objects in the same situations, the same results occur, which makes a strong case for determinism (leaving aside, for the moment, the apparently randomness that seems to occur at the quantum level). This certainly does seem to point to determinism. Given a particular brain in a particular physical state responding to a particular stimulus, one would expect it to produce the same physical result. This perception, indeed, is a strong argument in favor of determinism. However, I would argue, one cannot embrace this particular subset of perception and reject the perception that I (and I assume others) have that real choices can indeed be made, at least not without having good reason for doing so.
And this, I think, is where I struggle to follow your arguments. Why should I base my conclusions on certain of my perceptions while rejecting others out of hand? One could argue, I suppose, that my perception in this realm is flawed, that in fact I'm being deceived by my perception in this regard. But this would require an explanation as to how and, more importantly, why I am consistently deceived in this way, an argument I know that you have not made rigorously, because such an argument, to avoid be self-contradictory, would have to avoid psychology and philosophy completely and be based purely on physics, biology and chemistry.
Of course, I will admit that the notion I mention above of going back in time and "re-doing" a particular choice to see if it might have been made differently would not necessarily be sufficient proof for or against determinism, either. After all, if free will exists, it should not be random. In fact, one would expect that, in a particular situation and given a particular set of beliefs, values, knowledge, etc. that one would consistently make the same choice. After all, only the insane make choices detrimental to their scale of values, choosing other than the option which is most beneficial to themselves (with benefit, of course, perceived subjectively rather than absolutely).
And so, in this sense, determinism is tautologically true. In any situation, a sane person will act to maximize his perceived benefits and minimize his perceived costs. And, faced with exactly the same situation, we would expect that he will always make the same choice that he actually did. In a sense, we would say that his actions are determined. However, this isn't particularly useful, as it simply pushes the argument back a step. Instead of arguing about whether men can make real choices, we would instead argue about whether they have any volition over the values and beliefs that determine their actions or whether, in some sense, they choose their value structures or whether those structures are determined completely by external factors.
Of course, this line of thinking can be taken quite a bit further, and I see compelling arguments in favor of both alternatives, but if you follow it far enough, I think the debate ultimately comes down to whether consciousness can be said to have a meaningful existence or not. If it does, then it can, presumably, exert influence on the physical world, can reconfigure the physical state of the brain with volition, yielding a particular desired state that will thereby act according to whatever apparently deterministic physical laws govern the universe. On the other hand, if consciousness does not meaningfully exist, if it is a mere delusion, then it cannot influence the physical world and therefor human action would indeed be ruled by those same apparently deterministic physical laws.
Unfortunately, I don't see how one could coherently argue that consciousness is nothing more than a delusion, because all arguments and all reasoning, it seems to me, accept the viability of consciousness as a fundamental principle (similar, though perhaps not identical, to an axiom). This is what makes the opposite more appealing: consciousness cannot be argued against without contradiction, and so the alternative is the default.
However, I admit that this is not particularly satisfying and this use of the law of the excluded middle would certainly leave the mathematical intuitionists and other assorted constructivists howling. To me, the existence of consciousness, and therefore some approximation of free will, seems a reasonable working hypothesis but is not something I would consider demonstrably true.
Posted by: shonk at June 11, 2004 07:13 PMshonk: Your assertion that we must trust our perception when it asserts that all that we can observe behaves deterministically yet must deny our perception that we do make choices and have some semblance of free will is also duplicitous. Perhaps your mind works differently than mine, but I perceive, at least some of the time, that I am actually making true choices, that my actions are not strictly determined.
Perceptions are your inputs. I am talking about processing (cognition).
I am not saying that Determinism is True because we perceive Determinism, I am saying that Determinism is True because there is no other Logical option.
Its either Determinism (Fatalism) or it’s the result of Magic (the Supernatural).
If you prefer the Supernatural (at least some of the time) then there isn’t really a mechanism for me to convince you otherwise.
shonk: I assume such is the case for most people. Of course it is impossible for me to independently verify this, as I know of no way to go back in time and re-insert myself into the same situation with precisely the same knowledge I originally had and then observe myself either making the same choice every time or making different choices.
Just out of curiosity are you familiar with the EPR paradox? (we talked about that before didn’t we?)
shonk: And so I am left with what my perceptions tell me.
I still don’t see how your perceptions could be telling you that there is no discernable pattern? I thought patterns were all that you perceived?
shonk: This certainly includes the perceptions that lie at the foundation of science, the apparent fact that, given the same objects in the same situations, the same results occur, which makes a strong case for determinism (leaving aside, for the moment, the apparently randomness that seems to occur at the quantum level).
The Stochastic nature of quantum mechanics is easily demonstrated to be a fantasy.
How often does 2 + 2 “randomly” not equal 4?
shonk: This certainly does seem to point to determinism. Given a particular brain in a particular physical state responding to a particular stimulus, one would expect it to produce the same physical result. This perception, indeed, is a strong argument in favor of determinism. However, I would argue, one cannot embrace this particular subset of perception and reject the perception that I (and I assume others) have that real choices can indeed be made, at least not without having good reason for doing so.
What is it (specifically) about the atoms that make up YOU that is different then the atoms that make up the MOON? In other words, how is it that your atoms have “free will”, while the Moon’s atoms do not have it?
If you are asserting “free will”, then I would think you would have some explanation (accounting) for it?
shonk: And this, I think, is where I struggle to follow your arguments. Why should I base my conclusions on certain of my perceptions while rejecting others out of hand? One could argue, I suppose, that my perception in this realm is flawed, that in fact I’m being deceived by my perception in this regard. But this would require an explanation as to how and, more importantly, why I am consistently deceived in this way, an argument I know that you have not made rigorously, because such an argument, to avoid be self-contradictory, would have to avoid psychology and philosophy completely and be based purely on physics, biology and chemistry.
I could argue that there are two kinds of perceptions regarding the nature of the Earth. One set of perceptions would lead me to believe that the Earth is flat and motionless, while the other would lead me to believe that the Earth is a sphere and is not motionless.
Now to someone who believes that the Earth is flat and motionless it may be very difficult to get them to perceive that the Earth is actually round and moving; but I don’t think it would be Logical to assert that BOTH beliefs were equally supported by observation (perceptions).
The Observations are observations of outputs. But the outputs are not the system itself. The outputs are only clues as to how the system itself functions.
Look, once upon a time the notion of a round, moving Earth was very frightening to many individuals. I believe that Fatalism is very similar, only much more frightening. But it isn’t a fear of Fatalism … it is a fear of perceiving things in a new way (a fear of change).
shonk: Of course, I will admit that the notion I mention above of going back in time and “re-doing” a particular choice to see if it might have been made differently would not necessarily be sufficient proof for or against determinism, either. After all, if free will exists, it should not be random.
I think you are on the right track here. Like you say, if “free will” is “random”, then what purpose would “free will’ serve? Wouldn’t determined will be better than (superior to) random will?
shonk: In fact, one would expect that, in a particular situation and given a particular set of beliefs, values, knowledge, etc. that one would consistently make the same choice.
Like I said, how often does 2 + 2 not equal 4?
shonk: After all, only the insane make choices detrimental to their scale of values, choosing other than the option which is most beneficial to themselves (with benefit, of course, perceived subjectively rather than absolutely).
Two points:
1) Insanity is definitely related to what we are discussing. Take the example of a Red traffic light. I would say that a sane (or “Good”) individual knows that Red light means stop, and thus they tend to stop when they approach a red light. An insane individual does not know that Red light means stop so an insane individual will “randomly” either stop or go. An “evil” individual knows that red light means stop, but the evil individual prefers the consequences of not stopping.
2) I agree that all actions are ALWAYS in the direction of maximum perceived benefit, and I comprehend what you are saying when you say that one’s sense of benefit is “perceived subjectively”; however, I would argue that this is only partially True.
What I mean is that while each individual has their own unique and inherent sense of what is beneficial I would argue that this “inherent (subjective) sense” is arrived at via a completely objective process in reality. We are all produced by the same objective system, but that system is designed to generate unique (subjective) outputs over Time.
In other words, a counting program produces unique (subjective) outputs via an entirely logical and objective means (Loop: X = X + 1; Print X; Goto Loop).
shonk: And so, in this sense, determinism is tautologically true. In any situation, a sane person will act to maximize his perceived benefits and minimize his perceived costs. And, faced with exactly the same situation, we would expect that he will always make the same choice that he actually did. In a sense, we would say that his actions are determined. However, this isn’t particularly useful, as it simply pushes the argument back a step. Instead of arguing about whether men can make real choices, we would instead argue about whether they have any volition over the values and beliefs that determine their actions or whether, in some sense, they choose their value structures or whether those structures are determined completely by external factors.
I am having trouble articulating what I should be saying at this moment …
But suffice to say, I believe those structures are completely determined by external factors, and I would say that this is the case via logical necessity. In other words, it is impossible to conceive of any other possible way it could have occurred.
I also believe there is far more “utility” in this way of thinking than you may have perceived. Do you think it is possible to have two algorithms that perform the same function, but one algorithm performs that function objectively better (more efficiently) than the other?
shonk: Of course, this line of thinking can be taken quite a bit further …
Ohhh, I agree.
shonk: … and I see compelling arguments in favor of both alternatives, but if you follow it far enough, I think the debate ultimately comes down to whether consciousness can be said to have a meaningful existence or not. If it does, then it can, presumably, exert influence on the physical world, can reconfigure the physical state of the brain with volition, yielding a particular desired state that will thereby act according to whatever apparently deterministic physical laws govern the universe. On the other hand, if consciousness does not meaningfully exist, if it is a mere delusion, then it cannot influence the physical world and therefore human action would indeed be ruled by those same apparently deterministic physical laws.
But aren’t You (erroneously) assuming that “free will” is the goal when it is more likely that Individuality itself is the goal (or “a goal”) of the system?
You seem to be taking Individuality for granted. I would be a lot more leery of making such an assumption, as default observation would seem to lead one to conclude that Solipsism were True.
In fact, it would appear to me that with Solipsism as the default status, “free will” would be the price (the cost) of Individuality. Maybe there are some individuals who simply cannot accept the fact that they can’t have their cake and eat it too? Maybe there are some entities that always want something for “free”? (something for/from nothing)?
shonk: Unfortunately, I don’t see how one could coherently argue that consciousness is nothing more than a delusion …
It may interest you to know that dozens of “very smart” Atheists have tried to convince me of EXACTLY that – “consciousness is an illusion”.
That’s where the road of Materialism inevitably leads.
shonk: … because all arguments and all reasoning, it seems to me, accept the viability of consciousness as a fundamental principle (similar, though perhaps not identical, to an axiom). This is what makes the opposite more appealing: consciousness cannot be argued against without contradiction, and so the alternative is the default.
I agree completely.
The one fact that you can be most certain of is that Your Consciousness is real (exist in reality).
You have more empirical evidence (observations) of your own consciousness than for any other system.
Of course that is precisely why Atheists like to argue that consciousness is merely an illusion. It is the ultimate philosophical cop-out. It is a sure fire way to avoid all substantive discussion on the issue because obviously it is pointless for non-conscious entities to debate anything.
shonk: However, I admit that this is not particularly satisfying and this use of the law of the excluded middle would certainly leave the mathematical intuitionists and other assorted constructivists howling. To me, the existence of consciousness, and therefore some approximation of free will, seems a reasonable working hypothesis but is not something I would consider demonstrably true.
I would say that Consciousness is all that exist in reality. We are the ultimate True matter particle (Gravitons). But it is not “free will” that makes us who we are. It is individuality that makes us who we are, and individuality and “free will” are incompatible notions.
In fact, I would argue that “free will” is a self-contradictory notion. The moment you can abandon a belief in it is the moment that the pieces of reality begin to come together and make logical sense – kind of like perceiving that the Earth is actually round and moves, for the first time.
Posted by: The Serpent at June 15, 2004 12:46 PMWow, so there is actually a real, living Leibnizian out there. Wait until all the philosophers of the world hear the news: Leibniz is commonly assumed to have been the only Leibnizian in the world, past or present.
Posted by: Curt at August 16, 2004 05:16 PMPowered by Movable Type 2.661
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