## The nearness of the remote2000-07-15The discussion in this document is more involved than in the previous documents on these home pages. It discusses some deep issues connected with the foundations of scientific theories. It is much harder for me to write this in a way interesting to the general reader. Note later (2000-09-04):This now has the appearance of a pile of notes. It is a very difficult subject matter and it would need much more time to clean up and render it in a more pleasing manner. Still... Some topics concerned: - Time
- Hypothesis about the chaotic relation between the time of physics and an underlying framework.
- Poincaré cycles
- Isaac Newtons worries about the infinitesimals
- Infinity and infinities
- Cantorian set theory
- Some fragments gleaned from David Hilberts biography showing (as I interpret it), that mathemathicians, far from being guided by rigorous thinking, are motivated by a drive to increase the kingdom of mathematics (ie making it more interesting) rather than avoiding all uncertain paths. This means that they may have incorporated into the body of mathematical thinking, some ideas that could turn out to be blind alleys.
- if time doesn't exist on some level of physics, it may turn out to be possible to test the foundations of mathematics by experiments.
- Notation for exponentiation:10^x It is used, since HTML doesn't handle successive exponentiation. Eg 10^(10^100) means 10 raised to the very large exponent 10
^{100}
Example of state vectors: We consider as our system the entire universe as it is presently viewed by the majority of astrophysicists. I am picking the following numbers from the back of my mind and I may be in error, but here we won't need a greater accuracy. Very crudely it,(the universe) consists of say 10 ^{80} massive atoms
(and maybe something like 10^{110} nonmassive particles) Then the number of available states N of such a
system can be estimated to be around 10^(10^(10^x)) where x~2. These N states don't have exactly the same chance of
occurring, however we don't need to be very precise and it is an acceptable approximation that all states do occur
with the same probability.
We will further assume that this state vector undergoes some kind of random motion in its abstract frame of reference.
This means that the evolution of the universe is driven by some random microscopic processes.
From this it is possible to estimate the expected time before such a state vector will be expected to repeat any previous state. Due to the enormous magnitude of such numbers as N, the time unit isn't critical and neither is the precise formula tying N to the expected time. Eg N and N ^{½} dont differ in our approximation. We may consider N itself as
the time.
Such a time of return corresponds to the concept of Poincaré cycles. Henri Poincaré studied such problems
in classical mechanics.
The time estimated turns out to be very much larger than the estimated age of the universe. Even if we consider a
model universe consisting of maybe a 100 atoms the predicted time for a Poincaré cycle is still much larger
than the estimated age of the universe. Atoms is not a necessary choice of fundamental entity but it is chosen for convenience.
If atoms are chosen it means there are unavoidable long range interactions making it very hard to isolate smaller groups
fundamental entities. Therefore a background model of fundamental entities different from atoms without such long range
interactions might be of some interest.
The Poincaré cycle time, P, for the entire universe is so large that it is normally assumed that this is of no significance, since the whole of existence will have vanished long before P has passed. Note that the processes considered in calculating P, occur in the microscopic world, i e in a realm of existence which can be classed as belonging to the models, discussed above. What we refer to as the microscopic world is really only a model. Therefore we cannot be certain that the pertinent time concept is identical with the time in physics equations, when times like P are considered. - - - Inserted later: Therefore it could well be true that those extremely long times P should not be compared with ordinary time but should be interpreted as a meaningful measure idependent of time. And therefore it would make sense to explore all aspects of such long measures and see how it affects the general understanding of known laws. Eg how it might explain quantum phenomena or otherwise shed some new light on the foundations of science. - - - - - - Inserted later: The actual magnitude of the size of a system, whose Poincaré cycles are under consideration, is not necessarily a free parameter. You could in principle try to isolate any small-sized system to bring down the expected length of P and bring it within the reach of experiment. But there may not be any way to obtain the necessary degree of isolation for realistic systems, meaning systems consisting of some small number of atoms. Isolation seems to imply some kind of boundary and the boundary would have to be included in the analysis meaning the system would be much larger and hence the resulting Poincaré cycle time much longer. If atoms are chosen as fundamental entities, it is conceivable that all systems consisting of more than one fundamental
entity would be impossible to isolate and therefore Poincaré cycles might only apply to the universe as a whole.
On the other hand restricting the argument to a single atom might change this interestingly. What if a single atom is an actual
example of a 'Poincaré cycle system'?. Ie it is seen like it is from a perspective of several Ps
in terms of some background model system. Poincaré cycles means that a system has reentered itself in some fundamental way.
Therefore such systems are expected to have fixed properties unchangeable with time as long as the background model world
is unchanged. Quantum mechanics already uses one kind of timeless abstract background framework known as Hilbert Space.
Atoms have some very distinguishing properties. Their constituents electrons and protons are the only known stable massive particles. No matter what intermediate states they may transmute into temporarily, the stable end product will still be such protons and electrons once things cool down a bit. Hermann Weyl suggested that their difference in mass might be a reflection of our asymmetry with respect to time Perhaps they represent limits of an extended present, ie where begins and ends the accessible interval of time at every instant. Like time horizons. Windows to the past and the present. The smaller particle ought to be the future, for which less is known. - - - A very striking property of electrons is their total indistinguishability. That is, they are really all exactly the same. If they had any persistent individual differences, it would show up in natural phenomena everywhere. Even if there are individual anomalies they must be extremely small and even if this cannot be outruled, it would produce another big dilemma. For if electrons were material individuals, but with extremely small differences, how would nature go about maintaining the smallness of those differences? The other option is to consider them to be more like numbers or something conceptually related, which is, roughly speaking, the way established physicists view them. One way to explain how it is that many apparent instances of electrons could carry exactly the same quantitative properties would be to view them (the electrons) as invariant properties of some Poincaré cycle system, ie systems far beyond their P. They get their sharply defined properties by exhausting all contingencies averaging them out and erasing them through the 'patience' of statistical averaging. The reason they have persistent properties is those properties derive from an averaging beyond time of contingencies. Doesn't it seem rather plausible that fundamental entities would be 'stabilized' in just this kind of way? If this is accepted, one might ask how the sizes of such microscopic systems are to be understood. Even though atoms and their constituent electrons and protons seem very small compared to the universe as we know it, the sizewise comparison may be misleading if they exist in different time contexts. What if they are kinds of transformed images of the average appearance of the ordinary universe as viewed over time? This is related to the ancient idea that the entire universe shows up in all of its constituent parts. It belongs to the general idea of a universe comprised of an abundance of caleidoscopic images, in this case including temporal separation. I guess some other term than caleidoscope would be pertinent to capture that aspect. - - - There are some odd results in physics related to time. P. A. M. Dirac, one of the founders of quantum mechanics, later experimented with a model aiming to find a so called classical electron theory. Classical in this context would mean less abstract and intangible than the very successful quantum theory. He arrived at an equation which he didn't really expect to correspond to anything in the real world. One strange aspect of this equation was that it made possible certain so called runaway solutions. In the limited context of the equation classical electric charges would accelerate so fast that the time relatively speaking would be frozen in. In the time t it takes light to travel across an atomic nucleus (as measured in the accelerated system) t'=billions of years go by in the other system. (the relation is roughly t'~exp(expt), where t>~1 and time unit ~ 10 ^{-23} sec)
Although the relevance of this particular result is disputable, it raises an important question: How would you recognize such phenomena if they show up in nature? If we don't know, then neither can we outrule that such things exist and could even be commonplace. Another odd result goes back to the time around 1909, when british physicists discovered a previously unknown aspect of electromagnetism. Namely that its equations are invariant when observed from certain accelerating reference systems. This means that observers travelling with such systems are unable to decide by electromagnetic means, who is accelerating and who isn't. These systems could be said to represent the harmony of the spheres of a kind. Spherical regions expanding with a constant acceleration. When Einstein began building his theory of gravitation he didn't know this and after Einsteins theory had gained acceptance the above mentioned class of motions was considered to be one particular case of the cases handled by Einsteins theory. However Einsteins theory is connected with the concept of mass, the nature of which still isn't convincingly explained by any theory. - - - No solution is in sight no matter what they say. (It is not even necessarily relevant to ask for the nature of mass, since the next level of enquiry might involve a change of perspective, making mass less fundamental. Mass originates from Newtonian mechanics as a parameter of a simple equation. Doesn't it seem unlikely, that such a parameter would generalize in a 1-to-1 fashion into the world of quantum phenomena?) The lack of innovative theoretical work is the main problem. The present situation in particle physics is such that scientists are under psychological pressure to come up with prestigeous discoveries, due to the economic context. That is, the nonscientific community would probably cut the funding unless something prestigeous comes out of it. This is a very unhealthy situation and no matter what size of the budget it would be preferable if the funding authorities granted the scientists full freedom to act without any such considerations. It would be great if scientists became completely outspoken about what they sincerely think about the present state of the theories. We need fresh air everywhere. Freedom. Questioning everything that isn't sound. Rebirth. I beleive we need that. Nothing sacred unless it's a real success. And the pioneering spirit in everyone engaged in science. Whatever you think about the priorities regarding science pure, the organization and the technological knowhow needed to make possible the experimental facilities used in modern particle physics is something very impressive. Its like the pyramids. It would be nice if all that knowhow could be put to use in some other context as well. - - - In addition Einsteins theory is not in agreement with experimental facts, with present knowledge as it has been presented by established astrophysicists. Electromagnetism however is very accurately described by the established theory. (In the 19th century the German mathematician and applied astronomer Gauss noted an anomaly regarding the Earths magnetic field. I haven't tried to find any modern discussion about it and I don't know if that is any reason to doubt the correctness of the theory of electromagnetism.) Therefore, with present knowledge, the invariance of accelerated systems mentioned above, may be of interest. If this is the case it could mean that very strange time relations may play an important role in natural phenomena. In some cases the time relations can be such that time appears not to pass at all as measured from another reference systems. Time frozen in again. In the usual way to interpret physics such strange time relations would be connected with very large amounts of energy and very drastic astronomical phenomena. It is normally assumed in that context that such things wouldn't play any important role in ordinary matter. It is this assumption that I am putting into question here. The general idea is to try and reinterpreting some of the properties of the already used models for the microscopic world. One element to try and find use for would be the Moebius transformation, (same as the harmony of the spheres referred to above) The reason why one might want to discover use for it there, is that it is part of an already well-corroborated theory and if it would be found to be important inside matter this would certainly be important to the understanding of the nature of matter. Next item and not obviously related to the previous one, would be to take seriously the idea that atoms are a kind of irreducible image or irreducible form related to the universe when taken over time. Primarily it wouldn't be expected that such an irreducible form would be causally related to the universe otherwise there might be paradoxes. The term irreducible is used informally here. Something that suggests itself as the most important properties independent of any contingency. - - - In speaking of such emergent regularities, lets pick an example from another problem area, such as human civilisation. In a book from the end of the 1980s, Gunnar Adler Karlsson wrote about the distribution of wealth in different types of societies and claimed that all examples studied showed the same basic distribution of wealth. Communist China and the Soviet Union were no exceptions to the rule. Incidentally Sweden scored among the lowest in Europe if not the lowest, regarding
equal distribution of wealth. I don't think the data were restricted to a short period but that they were very representative and that it included the
period when Sweden was considered by some observers to be some kind of semi-socialist country.
This sort of claims appear very paradoxical at first look. Of course the precise definitions used are important.
And I don't intend to debate the facts here. But basically if we decide to rely on that author we have an example of some kind
of fundamental law. Apparently independent of all efforts to change society.
He also discussed the distribution of power and the tendency of all of humanity to develop hierarchies and to gather as many subordinates A as possible (formulated as a universal law 1/A always decreasing), with levels such that each level multiplies the numerary in geometrical progression. Examplified by Djingis Khans armies were the geometric progression went like 1 10 100 1000 etc The optimum choice of multiplier would depend on psychological factors. How many things you can keep in your mind simultaneously. Like 7 different names of subordinates in the level below. I think the idea was that the total wealth on each level was roughly the same. There is a paper by the physicist Elliot Montroll where it is shown that the distribution of wealth is even more unevenly distributed. - - - The idea that atoms and microscopic properties of the world might derive from a kind of unfamiliar time perspective of the universe, sounds like mysticism, that's undeniable. I don't mind if you think of this as nothing but mysticism. However surprises are to be expected if any real progress is to be made. It is unlikely that concepts existing in the prescientific period would all generalize smoothly, in a 1 to 1 fashion with later ideas. Written before the previous paragraphs We will now make a digression on the time t of physics and some other things. t occurs only in terms of a differential dt. I e as an infinitesimal quantity. For this reason any hypothesis regarding a possible difference between microscopic time and macroscopic time cannot avoid the problems of the ever-present infinitesimal quantities in physics. Isaac Newton,(to me) the creator of modern physics, had worries about these infinitesimals. He invented mathematical physics (apparently there was a dispute over priorities regarding some of it with Leibniz). And he saw that the formalism that he had created worked, but he didn't like that infinitesimal quantities seemed to find a place in the foundation of a scientific description of the world. After Newton, engineers have used his formalism without ever worrying about the infinitesimals. Most students studying math at the university level, are being exposed to infinitesimals, the use of which is seldom presented as any problem. We now know, through the successes of certain parts of quantum theories, that the physics formalism, still keeping those infinitesimals, works extremely well. However the success was obtained through some clever workarounds involving the manipulation of infinities turning up everywhere in the calculations. Infinities and infinitesimals are inversely related and both seem foreign to the observable parts of the real world. There is no obvious, nontrivial, connection between the mentioned infinities and the infinitesimals, but they both connect to the foundations of mathematics. In the 19th century, Cantorian set theory was criticized by some but became incorporated into the body of established mathematical truth. One spectacular aspect of Cantors work was the suggestion ( I use that word) that there are different kinds of
infinities, which can be ordered in a certain manner.
I want to comment that the proof of this is dependent on the, usually unstated, fundamental assumption that infinities
are valid concepts in the first place. This fundamental assumption is much less discussed by mathematicians, than the
subsequent theoretical steps, which are worked out with the usual rigour of skilled mathematicians.
Also note that the words 'infinite' and 'infinity' are negatively defined concepts.
'Infinite' can be interpreted as 'without limit' in two different senses. 1)the open sense: like an unended straight euclidean line. 2)closed sense: like a circle, reentering itself. When mathematicians examplify infinities, they might say something like 'the set of all natural numbers'. Presently nobody has suggested any way to show that such statements are meaningful. Set theory is, as far as I know, isolated from the rest of science, and if it turns out to be in error somehow this would have no consequences. The alternative 1 above is conceptually related. There is no proof that any meaningful interpretation can be given to infinite in that sense. The famous mathematician David Hilbert is known in applied science for his contributions to the solution of an important class of previously intractable problems of great importance. We don't need to go into that here, but it is interesting to note that his reputation was in no way limited to highly abstract works. In a biography about Hilbert, (I have forgotten the reference), there was mention of a discussion between mathematicians, differing in opinion about Cantors ideas. The one in favour, said something like 'it would be a pitty to loose such interesting parts of
mathematics', meaning, as I interpret it, that he would be willing to risk introducing uncertain ideas into
mathematics rather than missing something that seemed interesting.
I am not sure it was Hilbert himself who said this but I got the impression that he wouldn't be completely foreign to
that kind of thinking.
Note that the concept of 'number' may be very natural and commonsensical, when finite numbers are considered, but this doesn't mean that the concept of 'number' can be generalized to the infinite. I suspect that the whole apparatus of logical thinking, the human language with its collection of concepts, is simply not sufficiently precise, for that kind of generalization to be reliable. - - Appart from infinities, even very large finite numbers cannot be assumed to be without conceptual problems. They cannot be stored anywhere in the universe, even if all matter is used to build a computer. Therefore, in a sense, mathematics for such numbers doesn't exist. No range of such numbers can be operated upon. Only special cases can formally be written
down. For instance you can write down the number V=10^(10^(10^(10^(10^(10^(10^(10^(10^(10^(10^(....)))))))))))
But if you want to use it as the word length in a hypothetical computer, there is no way to implement it even in
principle.
If you have a lot of numbers similar to V eg V+23.87, there is no way to make calculations similar to calculations
we are used to.
It's out of reach operationally.
The example earlier about the number of states of the universe is an example of quantitative properties of a system, which cannot be completely represented mathematically within the system itself, the universe. These problems are encountered already in the realm of the finite. - - To me it seems very presumptious to beleive that we are capable to handle infinities without any way to check it experimentally. I doubt any human being is capable to think clearly about it without having some kind of feedback. Another famous mathematician, Kurt Goedel, discovered some fundamental weaknesses ( I use that word) in
mathematics. I don't think the conclusions are affected by the precise nature of infinities. As I remember it,
ideas connected with set theory was part of the reason for Goedels choice of problem, leading to his discoveries.
Even if set theory becomes rejected by future mathematicians, in retrospect it may turn out that it was a beneficial
type of erroneous idea since it provoked Goedels important discoveries.
Now that we have given a little background, we are ready to put forward a hypothesis. We let time in the underlying model be called w-time to tell it appart from macroscopic time t in dt. The hypothesis is that very long w-times like the P discussed do play an important role in physics. The time dependence in physics equations, through the differentials dt, makes them invariant under translations. According to the hypothesis, under some conditions states which are separated by w-times like P may be able to interact, provided there is no violation of causality. Lets oppose two cases: one where only the state vector is physically important and the other where a sequence of consequtive states are needed to completely characterize the situation. In both cases the following holds: When studying greater and greater intervals of w-time, much larger than P, the probability becomes near unity for finding any kind of replica of any given microscopic state(=state vector). Hence the title The nearness of the remote.
Infinitely far off along the w axis in the underlying model world there are infinitely many near replicas of any
given reference state. My suggestion is that time in the real world is something of an illusion, (which I am not the
first to suggest). I beleive it is a common opinion(?) among physicists that atoms are somehow beyond time. If the
models widely separated near replicas can really interact, then it would seem to mean that many changes, perhaps in
particular quantum events, might correspond to giant w-time jumps in the underlying model world. Since there is a
much larger probability for finding near replicas of the world for infinite w displacements, there ought to be an
accordingly much larger probability for such infinite jumps/interactions to occur. Now if this kind of basic argument
could be refined to make it a falsifiable theory, it seems possible that the precise nature of infinities, might have
measurable consequences. The reason why it would be possible is that ordinary time would have a very complex relation
to the underlying w-time. They would be very strongly chaotically related, since infinite w-time jumps are the rule.
A more detailed argument would involve a discussion of how quantum jumps would differ from so called adiabatic changes. In what way would the second, 'non-quantum' evolution in the real world correspond to something in the underlying model? Would it perhaps correspond to a similar continuous motion in the underlying model world?
Ie suggested correspondence:I) Real world quantum jump <==> ~infinite time jump in model world Real world non-quantum process <==> continuous motion in model world - - II) Alternatively the w-time belongs to some kind of closed geometry, like a torus or a Moebius strip, in which case, the term 'infinite' can be kept but with its second interpretation. The idea of explaining some properties of quantum mechanics by way of such periodic geometries has occurred to many physicists before and it would seem surprising if the crude variant II hasn't been tried before. - - Do note that the model world considered is not some variant of quantum mechanics, but some more fundamental model world aimed at explaining quantum mechanics and maybe bringing it closer to the conceptual frameworks used in classical statistical mechanics. In Model I, the infinitely many infinite w-jumps, when characterized more than just qualitatively should provide numbers, maybe like those in chaos theory or something. Perhaps the numerical data in atoms would find their explanation in some such context. Electrons ought to be a particularly simple aspect of such fluctuations/jumps. When I thought about this some years ago I liked to think of electrons as some such giant fluctuations. (I haven't explained this carefully above. The Poincaré cycles above would be associated with the system having undergone extremely improbable fluctuations. For comparison, it would be like all the air of the room where you sit would fluctuate into another room leaving you in a vaccum. Such events are allowed by the laws of nature but so improbable that as long as only finite time is considered, the probabilities can be disregarded. The idea put forward here, is that such improbable events do play a very important role and that
this comes about because in the context in which those events happen, the time w-time is not restricted to
finite values. We simply consider the underlying model world to be a kind of mechanical system beyond time.
And let it's time span over any values. The poincaré cycles are then common place. Not only that, but since infinitely
long jumps (encompassing infinitely many giant fluctuations), lead to stronger resonances in a sense, the whole
quantitative basis of reality ought to be hiding in the mathematics of such infinities. It is in that sense that I
suggest
that the precise nature of mathematical infinities could be within experimental reach.
Usually giant fluctuations of the kind under consideration here are not considered as some abundant feature but instead as something very removed from everyday experience. The perspective taken here is that instead such phenomena are commonplace and even make the whole basis of reality. Considering the statistical properties of infinitely many giant fluctuations would undoubtedly lead to statistically invariant properties and hence fixed numbers lending themselves to comparison with the known properties of atoms etc. What does all this mean in terms of everyday experiences? Who knows, teleology might find some basis in this kind of framework. But there are several unanswered questions to be handled long before we arrive at such a high level. To begin with, how exactly does ordinary time emerge from the processes in the underlying model? From the above discussion time would seem to be a slow average of something extremely chaotic going on in the model underworld. However the higher level time is derived, I think it would be natural to assume that in terms of that time, the world doesn't make jumps over time in a manner resembling that in the underworld. Otherwise the present could interact with the remote future. Whenever an observer experienced change of any kind it could mean either that a continuous change took place inside of one particular epoch of the universe or alternatively, that a switch took place, where a very distant future state so much resembled the present that the observer moved into the future state, without noticing anything strange, since the future state was almost identical to the starting state. This would be interesting, however it would render the higher level time concept useless. It would be just like w time. Initially this is how I started to think about it. I figured quantum jumps would be that kind of thing. An extremely long time jump to another state of the universe many Poincaré cycles distant, where an almost identical state of the universe existed. Indefinitely closely resembling the initial state. It would be nearer to the initial state than any continuous change would allow, due to uncertainty. The longer jump however, due to the long time involved could make possible extremely sharp resonances and therefore no uncertainty. This seemed like an interesting possibility. But I then moved to the above idea where the long jumps are not directly connected with the high level time t, leaving unanswered the question of how, exactly t derives from w time, appart from noting that the relation is extremely chaotic. - - The problems discussed here are in need of much clarification. Perhaps there will be more about it some other time. Note added: In an earler version of this document I discussed the hypothetical correspondence between atoms and Poincaré cycling systems in more detail, trying to anticipate obvious counterarguments. For example, atoms can be rearranged and changed in many ways and how does that fit in with anything timeless? I never completed the explanation, because it would make it more difficult reading. That kind of complexity belongs to real science, but not necessarily in this context. I am trying to paint the broad contours. But since I have brought it up, I will mention that I suppose it is really the fundamental entities like the electrons that would correspond in the most simple way to the Poincare cycles. And although I never mentioned it above, I actually didn't mean Poincare cycles, but rather much longer times still, in order to have a high probability to get a fully recurring system. I could continue and bring up more and more detailed visions of how I figure dynamical processes taking place in a system for which the fundamental properties are derived from beyond time, but you see how much more complicated the whole thing becomes as you try to make it more convincing. And that kind of complexity is what I try to avoid here. It is not my intention to try to convince anybody about the correctness of any idea. The motive is to inspire people to look at the foundations of science from maybe a new angle that they came to think of after freeing their minds. Thats what I am trying to do. To free your mind. That's it. Return to Introduction |