Математичар-кабалист во потрага по божанскиот број со кој би можел да ја декодира природата на реалноста (илузијата).
Seriously good shit.
216.
Закон! и јас го имам гледано
--- надополнето: 29 февруари 2012 во 10:51 ---
Does Mathematics Reflect Reality?
In recent years, the limitations of mathematical models to express the real workings of nature have been the subject of intense discussion. Differential equations, for example, represent reality as a continuum, in which changes in time and place occur smoothly and uninterruptedly. There is no room here for sudden breaks and qualitative changes. Yet these actually take place in nature. The discovery of the differential and integral calculus in the 18th century represented a great advance. But even the most advanced mathematical models are only a rough approximation to reality, valid only within certain limits. The recent debate on chaos and anti-chaos has centred on those areas involving breaks in continuity, sudden "chaotic" changes which cannot be adequately conveyed by classical mathematical formulae.
The difference between order and chaos has to do with linear and non-linear relationships. A linear relationship is one that is easy to describe mathematically: it can be expressed in one form or another as a straight line on a graph. The mathematics may be complex, but the answers can be calculated and can be predicted. A non-linear relationship, however, is one that cannot easily be resolved mathematically.
There is no straight line graph that will describe it.
Non-linear relationships have been historically difficult or impossible to resolve and they have been often ignored as experimental error. Referring to the famous experiment with the pendulum, James Gleick writes that the regularity Galileo saw was only an approximation. The changing angle of the body’s motion creates a slight non-linearity in the equations. At low amplitudes, the error is almost non-existent. But it is there. To get his neat results, Galileo also had to disregard non-linearities that he knew of: friction and air resistance.
Much of classic mechanics is built around linear relationships which are abstracted from real life as scientific laws. Because the real world is governed by non-linear relationships, these laws are often no more than approximations which are constantly refined through the discovery of "new" laws. These laws are mathematical models, theoretical constructions whose only justification lies in the insight they give and their usefulness in controlling natural forces. In the last twenty years the revolution in computer technology has transformed the situation by making non-linear mathematics accessible. It is for this reason that it has been possible, in a number of quite separate faculties and research establishments, for mathematicians and other scientists to be able to do the sums for "chaotic" systems where they could not be done in the past.
James Gleick’s book Chaos, Making a New Science describes how chaotic systems have been examined by different researchers using widely different mathematical models, and yet with all the studies pointing to the same conclusion: that there is "order" in what was previously thought of as pure "disorder." The story begins with studies of weather patterns, in a computer simulation, by an American meteorologist, Edward Lorenz.
Using at first twelve and then later only three variables in non-linear relationships, Lorenz was able to produce in his computer a continuous series of conditions constantly changing, but literally never repeating the same conditions twice. Using relatively simple mathematical rules, he had created "chaos."
Beginning with whatever parameters Lorenz chose himself, his computer would mechanically repeat the same calculations over and over again, yet never get the same result. This "aperiodicity" (i.e., the absence of regular cycles) is characteristic of all chaotic systems. At the same time, Lorenz noticed that although his results were perpetually different, there was at least the suggestion of "patterns" that frequently cropped up: conditions that approximated to those previously observed, although they were never exactly the same. That corresponds, of course, to everyone’s experience of the real, as opposed to computer-simulated weather: there are "patterns," but no two days or two weeks are ever the same.
Other scientists also discovered "patterns" in apparently chaotic systems, as widely different as in the study of galactic orbits and in mathematical modelling of electronic oscillators. In these and other cases, Gleick notes, there were "suggestions of structure amid seemingly random behaviour." It became increasingly obvious that chaotic systems were not necessarily unstable, or could endure for an indefinite period.
The well-known "red-spot" visible on the surface of the planet Jupiter is an example of a continuously chaotic system that is stable. Moreover, it has been simulated in computer studies and in laboratory models. Thus, "a complex system can give rise to turbulence and cohesion at the same time." Meanwhile, other scientists used different mathematical models to study apparently chaotic phenomena in biology. One in particular made a mathematical study of population changes under a variety of conditions. Standard variables familiar to biologists were used with some of the computed relationships being, as it would be in nature, non-linear. This non-linearity could correspond, for example, to a unique characteristic of the species that might define it as a propensity to propagate, its "survivability."
These results were expressed on a graph plotting the population size, on the vertical axis, against the value of non-linear components, on the horizontal. It was found that as the non-linearity became more important—by increasing that particular parameter—so the projected population went through a number of distinct phases. Below a certain crucial level, there would be no viable population and, whatever starting point, extinction would be the result. The line on the graph simply followed a horizontal path corresponding to zero population. The next phase was a steady state, represented graphically as a single line in a rising curve. This is equivalent to stable population, at a level that depended on the initial conditions. In the next phase there were two different but fixed populations, two steady states. This was shown as a branching on the graph, or a "bifurcation." It would be equivalent in real populations to a regular periodic oscillation, in a two year cycle. As the degree of non-linearity increased again, there was a rapid increase in bifurcations, first to a condition which corresponded to four steady states (meaning a regular cycle of four years), and that very quickly afterwards it was 8, 16, 32, and so on.
Hence, within a short spread of values of the non-linear parameter, a situation had developed which, for all practical purposes, had no steady state or recognisable periodicity—the population had become "chaotic." It was also found that if the non-linearity was increased further throughout the "chaotic" phase, there would be periods when apparent steady states returned, based on a cycle of 3 or 7 years, but in each case giving way as non-linearity increased, to further bifurcation’s representing 6, 12, and 24 year cycles in the first case, or 14, 28, and 56 year cycles in the second. Thus, with mathematical precision, it was possible to model a change from stability with either a single steady state or regular, periodic behaviour, to one that was, for all measurable purposes, random or aperiodic.
This may indicate a possible resolution to debates within the field of population science between those theorists who believe that unpredictable population variations are an aberration from a "steady state norm," and others who believe that steady state is the aberration from "chaotic norm." These different interpretations may arise because different researchers have effectively taken a single vertical "slice" of the rising graph, corresponding to only one particular value for non-linearity. Thus, one species could have a norm of a steady or a periodically oscillating population and another could exhibit chaotic variability. These developments in biology are another indication, as Gleick explains, that "chaos is stable; it is structured." Similar results began to be discovered in a wide variety of different phenomena. "Deterministic chaos was found in the records of New York measles epidemics and in 200 years of fluctuations of the Canadian lynx population, as recorded by the trappers of the Hudson’s Bay Company." In all these cases of chaotic processes, there is exhibited the "period-doubling" that is characteristic of this particular mathematical model.
