Extending the chaos to the endth degree through a fourfold exposure for enhancing the personal view of things, when every possible sense of the whole dissolves into senselessness.
Yes, James, nobody understands QM - in its mapping of mathematical formulae to... to what? We calculate and we make machines (like your computer, or camera) based on that theory, but we don't understand it.
The assumptions for creating the theory are simple, though also surpricing. You know, let's assume that a particle is best described by a wave function. And then... then the theory makes statements that do not correspond always completely with the assumptions.
Concidering the formalism of QM itself, and without any regard to any "described" natural reality, might be more fruitful than one thinks. A formal theory that is strong enough to answer any question it can state, *will* all state contradictions. Perhaps some of the answers of QM (the calculated results, or the theorems that it can prove) *do* contradict the assumptions or other theorems that it can prove. For example, starting with a particle that moves over the continuum of space you can prove that it can be here and/or there, but nowhere inbetween - quite in contradiction to the very continuum of real numbers that is used as an underlying system of coordinates. But it is hard to find out if this is really a completely formalizable theory.
Anyway, it could also be that QM just tries to find out the properties of matter as results of a much much more universal reason. Perhaps space and time are quantized themselves, and so any phenomenon taking place in there has to exhibit a non-continuous behavior. A behavior that looks only continuous if observed at scales directly visible with our eyes.
In other words, perhaps the differential equation is only a good approximation for a universe of tiny quanta of space and time, that should be "re-designed" using the much more "uncomfortable" discrete mathematics. (Awful, I can tell you.)
Take care and keep on diving into those dark waters.
The individual of a human society (if there is something left like that) cannot be considered to my cells or DNA or even my atoms, since those things do not possess any property of their owns that would build up such a society. Exactly as the atom is not an individual unit for temperature or pressure.
The example of helium is impossible in thermodynamic equilibrium, and it is only there where such a thing like temperature is defined. The nin-equlibrium thermodynamics do also know temperature but "more or less" local. You can't calculate with temperatures that way. The experiment you proposed simply doesn't work - and it has been tried many times. Macroscopic quantities do not do that - while of course things are different at the atomic scale.
The measurement of your height doesn't have to do that much with QM and Heisenberg if we assume a macroscopic treatment of matters. But of course, making the measurement by counting atoms and their positions makes things different *in theory*. The thing is that physics rely on experimental data, and believe me, there is no way to measure errors of your height due to wavefunction amplitude of your most top particle when the experimental error itself is magnitudes bigger. Such uncertainties are not for "this world of ours". You can't measure it, so just forget it, or else science would become astrology.
Exactly this was also one of main reasons for coming to QM and also to Heisenberg's relation. You can't measure *exactly* position (*and* impetus at the same time), so there is no absolute location-speed certainty, so forget it. Dealing with wavefunctions does however provide *some* way of locating... the "main" part of a particle, whatever that might be.
Now, this kind of uncertainties is what we find out when follwing the formalism of a theory, like for example QM. But nobody ever said that this theory is the last truth about anything. Its results have some (very big) similarity with what we measure, and so the theory is... very well usable. But the underlying modelling *is* a product of human mind, which of course *is* limited. This way, perhaps the wave perticle dualism is only there because we aren't able to construct some kind of other entity that delivers even better theoretical explanations.
Now I have to scratch my head - what would be no wave and no particle and not a mixture of both?
Ah, what I mean when I questioned an indivual v a collective was... Society is a collection of people but am I an individual? Perhaps my cells, which enjoy a certain autonomy, are the real individuals? Or my DNA? And in the case of a mole of helium, is it the atoms? Or their constuent particles (which is where QM tends to measure, particle by particle)? Or is it in the constituent particles' constituent particles (the quarks which comprise the electrons)?
You do, however, prove my point about temperature. Take a mole of helium and measure it at 50 degrees C. But take half that mole and it might be 45 (while the other half is something like 55 so it averages out). Then divide those each in half and you get a larger spread of disparate temperatures (possibly, though it is possible too that they would happen to all yield the same number). Temperature only tells us about averages over a system and not about the behavior of any given atom (or particle).
Uncertainty is now built into knowledge. It has reached out from its humble beginnings. True, it was there to remind us that we cannot know both the position and momentum of a thing, that there is an inverse relationship between knowing those two (kinds of) facts, but what about measuring my height? My height, surely is the distance from my bottommost atom to my topmost atom. Since I cannot exactly locate those two atoms, how can I measure my height exactly? I can only come up with an estimation which is sufficient on a scale of things. To my mind these are the kinds of consequences imposed by Heisenberg.
Again, we scratch out collective head (that's my cells scratching my head)?
Temperature is more than a convenient way of talking about energy exchange within a system, James. It is (sometimes) a way of speaking about the energy of a system *and* at the same time its entropy. The individual of course remains out of control and even at a temperature of, say 20°C, there can be some individuals with the energy that would correspond to 2000°C, since the decline of certainty for them falls exponentially but never really reaches 0 - in the continuum approximation. (Diskrete is another story.) But in this case, well, what more would the individual treatment of each an every of the 6.023E23 individual atoms in a mole of He say to us?
Uncertainty is something different. It is generated because matter itself seems to be "diffuser" the closer you observe it. The result looks the same but the reasons are different. Taken to the endth degree of theoretization, even an elektron cannot be even "seen" if completely free of interactions in some hypothetical potential free space. (Renormalisation, you know.)
In addition, perhaps the concept of a wave or the one of a particle are themselves not adequate to describe such a thing. (Mathematically, perhaps we reached already the Goedel limitations, and we find inconsistenices - the most natural thing of the world.)
This question is very very related to your one. Indeed, the definition of the "individual" itself, as a "particle", is quite an approximation, and very problematic. Same goes for the other extremum, the wave.
So indeed, we scratch out collective heads, but perhaps the problems arise due to the answers that we are able to think of? Now let's scratch our collective head about the scratching of our collective head - metascratching, so to speak.
Well, in fact temperature is just a convenienct way of talking about states of energy exchange within a system (individual atoms interacting with one another). I'm not sure how bothered I should be if these artificial constructs (temperature, society) are highly predictable. Still the individual remains outside control. Consider again uncertainty.
Of course there are difficulties in attempting to discuss the universe in terms of indidividuals and collectives. What is an individual? What is a collective?
James, there are "collective" phenomena - those who only exist because a great number of "participants" exist, but do not exist otherwise. If you drive a car, or have a refrigerator, then you use them every day. I mean such things like for example the temperature of one mole helium gas in a chamber (or its pressure, or its (thermodynamic) gas volume). A single helium atom has no temperature at all - it only has energy. Temperature is not even definable for a single atom - or for two, three, and so on. It first comes when you have an immensely great number of helium atoms, like in one mole, but doesn't exist otherwise. Statistical thermodynamics calculate this temperature under the assumption of *total* chaos reigning over the motions of all single atoms of the ideal gas. ("Chaos" and thus "gas"). And the temperature obeys the applied statistical model very well. From all possible temperatures we measure the (stable) one that corresponds to the assumption of total chaos of the atoms. Now, of course even helium atoms *will* now and then undergo interactions that reduce the chaos a bit - but it is the model I am talking about, and that works also for many "not so ideal" gases. The temperature as a collective "property" obeys very very very very exactly the assumptions of statistical mechanics.
Same goes for example for the CD-absorption of aggregates like for example bacteriorhodopsin membranes. You see absorption bands that appear only when many of the absorbing molecules are close enough for sufficiently strong interactions. This time it is not chaos but rather ordered QM interactions that give rise to these behavior, but still it is there.
Already in pure mathematics we see very often that there are properties of (for example) categories, which can be only defined over the collection of functors we are talking about, but they are not definable for any functor for itself.
I don't know inhowmuch this can be applied to a society, since I do not have the knowledge to be able to judge. But in physics we have them countless times. So, we see that the whole can be really much more than the sum of its parts.
Yes, life being anti-entropic does seem to suggest that order is more random than disorder.
I completely disagree with your claim about collective behavior. There is no collective to be obedient. Society does not exist. It's a convenient way we have of talking about individuals. But it's no more real than a plane of focus. It's just a way of talking.
Thank you very much for the detailed answer, James! A real source of thinking to me!
It is not disorder that has to be discovered first but rather order. The idea of entropy might not be a purely human invention, but still its counterpart, information, is not always directly recognizable. That means, even if we know the entropy of a system in the sense of being able to give some analytically closed formula for calculating it, we are not at the same time always able to give some formula for the neg-entropy - the information. Strange as it goes, it seems that sometimes the surroundings of some information can be defined, but the information itself cannot, though it exists.
The eigenstates that you refer to, do often underline exactly the impossibility to find the complement to disorder, though they also allow to find the disorder itself. So the question arises, why the hell can't I determine (construct) the complement of a set, when I am able to determine the set itself?
Any pattern - even the most complicated one of the jazz musicians - *is* constructible a posteriori using the simplest means. You only need discrete interpolation, that's all. The set of tones they produce is not anything special. For an a priory construction... well, that's another story. ;-)
Statistics do not claim obedience of the individual but rather of the collective behavior. They do leave space for "outsiders". But I didn't mean the classical statistics but rather statistics in the sense of QM, like Fermi-Dirac, which do also predict expectation values but also not claim absolute certainty.
Certainly, most random systems are also predictable to some degree, but there are indeed many random systems that deny any predictability at all. They are (up to now) most of the time completely abstract systems that do not "describe" any discovered physical reality, but the mapping of physical realities to pure mathematical ones is only a matter of time.
Pure mathematics actually *are* up to now the only applied language for any physical "reality" in the sense of observed phenomena. And it *is* a human language too, which makes it very hard at the end to separate discoveries from inventions. Animals may like "order" but they don't try to comprehend why they perceive it *first*, in order to like it. Humans do try to do that.
What you say that order must first be recognized... I'm not sure if I can agree with that, that the idea of entropy is purely human and an independent fact of the universe. There are those who will talk about the necessity of an observer to collapse an eigenstate, but even this I regard with caution. In part because of what it takes to qualify as an observer. Is it any animate creature capable of perception? Does it require consciousness or self-awareness? Or does one have to know what an eigenstate is in order to cause the collapse? In which case perhaps this long chain of eigenstates leading back to the beginning of particle interaction have not yet collapsed (since we really don't know what the states mean). Again, animals besides humans understand and appreciate order. Introduce disorder into a living situation (and again here we are likening disorder and that which is a challenge to predict) and you will see stress levels rise and the consequences there with associated.
Does a formula ensure predictions of patterns or are there some patterns (and very ordered patterns at that) for which the formula is too dynamic to articulate? Consider a group of musicians playing jazz music. They are clearly making predictions (extremely accurate ones) about the future of the patterns in which the are involved. I should think you will find it profoundly difficult to create any such algorithm to even define that pattern a posteriori.
Obedience to statistics? Epidemiologists make statistical predictions about the spread of disease. Will you assert that individuals are obedient to those statistics? The predictions disregard any one individual. The same sort of thing holds true in quantum theory. It seems to me that those working in QM would call a system of gas within a chamber--or say the positions of the particles within that system--random. I think what is random can be predictable to some extent. The flipping of a coin is random, but we know that about half the time it's heads and about half the time it's tails (and maybe once in a hundred lifetimes it'll land on edge).
Math is difficult to talk about directly in terms of order and disorder. It is a human language. An orange is an independent object regardless of human existence. One is a representative of the concept of quantity of oranges, but one itself isn't out there. But when giving off a string of random numbers, the statistical outcome of the next number is not influenced by the last number (or at least should not be). In other words it should come as no surprise if I flip heads a hundred time in a row, though I would begin to suspect nefarious activity.
James, thank you very much for the great answer, which brings me some additional thoughts here. I didin't consider it from the point of view of life in the sense of thermodynamics/entropy, and still I I don't. Perhaps I am saying at the end, let it be also perfectly dead. I consider the rather "naked" mathematical conequences, especially in the case of "pure" chaos since nobody can give any last and eternal "formula" about the nonexistence of any kind of order in it, because the existence of any order has to be "recognized" by the observer first. In this sense there might be some kind of order in any photo, no matter if some particular mechanism of perception makes it impossible to perceive it - exceptof course in the case of total homegenity and isotropy.
Quantum mechanics does not predict anything about "random" systems ( "random" in the sense of unpredictable) but rather about systems that obey some kind of statistics - random in this case is something different than unpredictable.
Your example with the dog and the fridge is exactly what I mean using the natural number sequence. There we can always say that it is "ordered" - I guess because it just corresponds to some "built-in" property of our thinking and perception. But take it the other way around. Why is a finite sequence 324, 3455, 232, 345 "disordered"? Definitely not because there can be given no formula to calculate its numbers *after* it has be generated. Still it can be random in the sense that we can't say what the numbers will be *before* it is generated.
Since any photo can be seen only *after* it is generated, it is always possible to find some "formula" that clarifies its "order" - and thus... yes, thus what? This is something I have to think about.
As about the circle, well, the second part - death predicts life - is sure for a limited time. We will be recycled until the very death of tzhe universe! ;-)
Let's take a different angle. Is disorder best when it is completely disordered? To the extent to which the random is the disodered, this is equally false. Imagine a system is total disorder. It is a dead system. Life is anti-entropic. The reinjection of order is a reinjection of life and dynamics into that system. It is precisely the interplay between order and disorder, between the random and the sequenced, between life and death, which make a system interesting. Pure life or pure death lose meaning.
Is predictability a test for something in the random? Quantum theory predicts with stunning accuracy events which are utterly random (mere rolls of the dice, much to the chagrin of Einstein).
I must grant you that if I start saying "one, two, three, four, five, six..."--under the defined laws of mathematics--there is no reason to assume that you can interject "eight, nine, ten". And yet it would be with total confidence that you would do so. But it would be folly to suggest that these kinds of predictions of the future are to be reserved for humans alone. Any dog can guess what's coming when the he hears the refridgerator door open.
Which I guess brings us full circle: life predicts death; death defines (perhaps a posteriori) life.
The concept of self-similarity under the fractal is not necessarily the only thing that deserves examination, since it is created by a mind that *has* to be able to perceive it, and perhaps bring ot into some definite form of description. But perhaps also, for some different intelligence, it would be completely "invisible", which... means that it is then nonexistent?
And thus, the question I posted is transformed to the next: What are those little tiny things that make chaos conceivable, or even describable - what is the "little order" that you already mentioned. This is my question - and I think that it might never have an exact answer, but still thinking about that can be very fruitable.
Thank you so much for the great comment that caused my threads of thinking to get so excieted again.
Now, wait, wait, wait, James - we shouldn't throw everything in the same salad, though on the other hand, it is exactly that mixture that leads to new concepts/discoveries/etc.
OK, where do I start... Well, randomness is *not* chaos! Randomness might create what we call chaos, but in itself it is not the same with chaos. Your very interessting example of the random number sequence shows that very very nicely! The numbers generated this way are way not "really random". They are *pseudorandom* since the *is* an alsorithm that can create exactly the same numbers out of the same seed. It is only that we can't find any analytic closed form of the sequence of those numbers, plus the fact they have equal probabilities to appear in the sequence, what makes them "approximatelly" the same with real random numbers. In this sense, even the sequence 0,1,2,3,...9 can be produced by a random number generator, *despiete* the fact that we are inclined to call this sequence completely "unrandom". Actually, the a posteriori possibilities to "predict" any number of such a sequence doesn't help us to predict it *before* its generation. In this sense, any specific seqience of numbers - even completely random ones - is exactly reconstructible, but only *after* it has been generated. Give me any number sequence and I *will* provide you a formula for calculating them, but I will not be able to provide you any formula for calculating what the *next* sequence will be.
We see here that chaos and order are subjects to our own thinking. The same algorithm that spits out 0,1,2,3,,,9, also spits out (for example) 876, 2542, 907373, and so on, but we consider the first to be the very familiar sequence of natural numbers and the second to be... chaotic?
As about fractals, well, quite the same! It is only opur specific "ability" to "see" some kind of order, some kind of pattern in such structures, but nobody can ever assure us, that some kind of order couldn't be found in what we perceive as complete chaos.
(Continuing on the next message due to size limitations of messages.)
Choas is best when completely choatic? I'm not sure I can agree with that. Consider two facts. First, if you ask a computer to generate a random string of numbers it will not appear to a human as random as it could be (the computer has no problem with repeating the same number several times even in a row). Second, the fractal like expressions emerging in the work of Jackson Pollock. Chaos, it seems, is best when it is driven by a little order.
As I already told Eren, James, I neither know what it does. A big question it is, to ask if chaos should still be for creating some kind of convergency. Still thinking!
Thank you very much for the encouraging comment, Eren!
I still have my doubds considering the... convergency of that chaos. On the one hand there has to be some kind of hook line, but on the other hand the creation of chaos is best when it is completely chaotic - isn't it? But this field of ambivalence might also lead to fruitable thoughts, so lets see.
nick..you've give a detailed perspective of caos..i don't think at all that you've exceeded the right amount..it's all perfect..and really well balanced!soooo congrats..another great work!
Yes James, and that's a main question to me here. How much chaos is "appropriate" for posing the question about an unknown locality of the own self? It seems I exceeded the "right" amount. :-/ Nick
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