Chapter Three - The Structure of Thought
3.4 PREDICTION3.4.1. Discrete Logarithms (*)
"For a simple mathematical example, let us look to the theory of finite fields[E1] [E2] . A finite field is a way of doing arithmetic on a bounded set of integers. For instance, suppose one takes the field of size 13[E3] (the size must be a prime [E4] or a prime raised to some power). Then, in this field the largest number is 12. One has, for example, 12 + 1 = 0, 10 + 5 = 2, 3 x 5 = 2, and 8 x 3 = 12. One can do division in a finite field as well[E5] , although the results are often counter-intuitive -- for instance, 12/8 = 3, and 2/3 = 5 (to see why, just multiply both sides by the denominator)."
the theory of finite fields .. a level reached .. literally a finite field .. the state of a system .. a system with far-wide implications .. society .. the level .. the levels .. the states reached .. the boundaries ..allowed .. extended to .. connections .. communications ..allowing .. language barriers .. geographical boundaries .. national boundaries .. fragmented .. the push that technologies afford .. boundaries surpassed .. the inclusivity parameter ..
the multiverse idea .. the universe .. a finite field ..of immense proportions ..yet a finite field in itself .. as all the finite fields that the systems comprising the universe .. are themselves .. too .. to the tinniest of systems within it.
the implication of the prime numbers .. a factor that seems ..uncanny .. yet inane .. cannot really one see .. appraise its significance in the whole case .. but taken in account the size of a field in every system in the universe .. the universe included .. that must involve huge numbers .. in terms of societies that must equate to the populations each society is made out of .. though in terms of individuals in societies .. this should be determined upon the participating individuals .. the ones that make use of what technology makes available for people ..
the arithmetic .. of the numbers within the finite field .. the extra twist .. when computing is taken place .. the fractal ..worlds ..produced within the finite field bounds .. numbers continue to be infinite .. but their extend do not exceed the boundaries of the field .. the implications of taken up the same values that the same process takes in other stages of their cycle .. and the size of their cycle as it is delimited by the size of the field .. intriguing .. remain the question whether this bears any significance .. in the derivation of outcomes .. or by being fractal .. this indicates that they are separate .. develop in fractional dimensions so they do not meet ..
In finite field theory there is something called the "discrete logarithm" of a number, written dlogb(n). The discrete logarithm is defined just like the ordinary logarithm, as the inverse of exponentiation[E6] . But in a finite field, exponentiation must be defined in terms of the "wrap-around" arithmetic illustrated in the previous paragraph. For instance, in the field of size 7, 34 = 4. Thus one has dlog3(4) = 4. But how could one compute the log base 3 of 4, without knowing what it was[E7] ? The powers of 3 can wrap around the value 7 again and again -- they could wrap around many times before hitting on the correct value[E8] ,4.
logarithm .. exponentiation .. more complicated ..computations .. included .. afford deep analysis .. how physical properties can be calculated measured .. examining their complicated associations .. associations where their ..expression ..surpasses .. simple addition .. complex ..parameters .. variables .. their connection ..association being of a greater degree .. multiplied instead of being additive ..
networks .. whole networks .. creating patterns .. adding networks .. their addition requires ..higher arithmetic operations .. where ..multiplication ..division .. exponentiation .. logarithms .. enter .. are required .. the continued spectrum of numbers .. require .. is the addition of networks .. afford us the following of their ..developments .. matrices .. how can a mind manage to follow .. the adding of networks .. how can observation fathom te addition of networks .. it can sense them .. intuition but not pinpoint their alternation .. from input to outcome .. there are too many in-between processes that is impossible to ..observe ..
without knowing what it was .. the implications of that ..phrase .. sentence .. the assumption .. brought forth .. imprinted in the human mind ..explaining our inability ..inherent inability .. mathematically expressed .. not knowing what we want .. intuition .. at bay ..
wrap around many times .. before hitting the correct value ..
The problem of finding the discrete logarithm of a number is theoretically easy, in the sense that there are only finitely many possibilities. In our simple example, all one has to do is take 3 to higher and higher powers, until all possibilities are covered. But in practice, if the size of the field is not 7 but some much larger number, this finite number of possibilities can become prohibitively large.
So, what if one defines the dynamical system nk = dlogb(nk-1)? Suppose one is given n1, then how can one predict n1000? So far as we know today, there is better way than to proceed in order: first get n2, then n3, then n4, and so on up to n999 and n1000. Working on n3 before one knows n2 is essentially useless, because a slight change in the answer for n2 can totally change the answer for n3[E9] . The only way to do all 1000 steps in parallel[E10] , it seems, would be to first compute a table of all possible powers that one might possibly need to know in the course of calculation. But this would require an immense number of processors; at least the square of the size of the field.
the wrap around feature complicates computations .. how can this be interpreted .. the numbers .. their sequential nature ..is disrupted by the nature of the wraparound field .. unfit .. unable to take into consideration the wraparound feature of the finite field .. they do not stop where the size of the field stops .. continuous in nature .. maths prohibitive .. could not have been otherwise if the computations would have some sense .. make sense .. ought to .. should be of that nature .. beyond ..human intervention .. wish .. will .. that is how the universe requires them to be .. the added twist in its ..ongoings .. words .. symbols .. thought processes .. underlying .. are fulfilled by the same premises ..
out of field of vision .. visual field .. and yet there .. the essence of .. finite fields .. levels .. the wraparound feature .. fundamental ..considerations ..
1000 steps in parallel .. what people are doing .. parallel processing .. all that are presented as problems .. all that is required to be solved .. society a parallel processing computer ..
This example is, incidentally, of more than academic interest. Many cryptosystems in current use are reliant on discrete logarithms. If one could devise a quick method for computing them, one could crack all manner of codes; and the coding theorists would have to come up with something better.
3.4.2. Chaos and Prediction
More physicalistic dynamical systems appear to have the same behaviour[E11] . The classic example is the "logistic" iteration xk = cxk-1(1-xk-1), where c[E12] =4 or c assumes certain values between 3.8 and 4[E13] , and the xk are discrete approximations of real numbers. This equation models the dynamics of certain biological populations, and it also approximates the equations of fluid dynamics under certain conditions.
It seems very, very likely that there is no way to compute xn from x1 on an ordinary serial computer[E14] , except to proceed one step at a time. Even if one adds a dozen or a thousand or a million processors, the same conclusion seems to hold. Only if one adds a number of processors roughly proportional to 2n can one obtain a significant advantage from parallelism.
behaviour .. out of all the properties of the unit .. all .. should be taken into account .. the enormity of the combinations .. and how much little have been afforded .. by the systems .. the filters being put .. the laws .. the perspectives ..open .. inevitable .. to seek out .. their instantiations .. to fulfil all .. what can be called aspirations .. out of their very own properties that struggle to confirm their existence .. societies in turmoil .. joy ..happiness .. the fulfillment ..sought after .. the integration of these properties in the whole they comprise .. the desire to incorporate .. these attributes .. into the whole .. to blend in an environment that is deemed suitable .. the properties of the environment .. the level ..that .. c .. represents .. the finite field .. the nature itself ..
c .. represents the level reached .. and the ..fluctuations afforded ..out of that level .. the level .. being the finite field .. the wraparound feature .. the fractal development .. that affords self-similarity .. what chaotic systems are prone to exhibit ..
c ..assumes certain values between ..3.8 to 4 .. the small range out of the level at bay .. that produces much larger outputs from the given ..tiny inputs .. a chaotic systems feature .. the dependent variables .. one must determine the associations of the variables involved .. the abstract .. mathematical relations ..out of the ..physical .. instantiations .. of the abstract mathematical properties .. out of the quantity .. the measured abstract quantity .. the quality .. how the features are coming into existence .. dependent variables ..derived variables .. being able of quantification .. multiplicity .. relationships .. relate to one another .. the unit bearing properties are added .. emergent .. emerging out of the unit .. what the unit can do ..
serial computer .. its comprehension ..out of .. serial and parallel circuits .. electric bulbs connected .. what governs their function.. serial connection.. from the same source .. when the source is off ..all bulbs are off.. parallel connection.. is not the source ..it is the connection with the source .. the route it takes .. the source is common in both cases .. the route differs .. in serial ..the source is connected through the same route .. each unit is making up the route .. the output diminishes .. sequential .. parallel ..different route .. for each unit .. output is un-diminished by the other ..
In general, all systems of equations called chaotic possess similar properties[E15] . These include equations modelling the weather, the flow of blood through the body, the motions of planets in solar systems, and the flow of electricity in the brain. The mathematics of these systems [E16] is still in a phase of rapid development. But the intuitive picture [E17] is clear. To figure out what the weather will be ninety days from now[E18] , one must run an incredibly accurate day-by-day simulation [E19] -- even with highly parallel processing[E20] , there is no viable alternate strategy.
chaotic ..systems of equations .. similar properties ..
mathematics of these systems .. chaotic systems .. in a phase ..of rapid development ..
the intuitive picture .. intuition ..at bay ..tuition .. inner .. being tutored by inner self ..being taught by inner self ..
now .. the now .. which it should take aboard .. what was on .. in the appropriate unit of time ..before now .. a cycle .. a complete cycle ..
simulation .. accurate ..day-by-day .. cycle-by-cycle ..
highly parallel processing .. simply .. being processed ..on its own .. independent of other processes .. but in the ..computing .. the completion of the algorithm at bay .. .. simultaneously .. without meaning that the processes are truly independent .. that they are not ..being affected by the conditions .prevailing ..