The mathematical description of physical events is founded on set theory, and it is in set theory that we find strong dichotomous analysis. The emphasis is on set membership (is/is not - one:many dichotomy type) and developing sets of ordered pairs derived from other sets (refinement). Any relational operation is applied as if we are dealing with a dichotomy despite any present complexity. Note that, from Cantor's work with transfinite numbers, there is a sharp distinction between the characteristics of Cardinality and Ordinality.
In logic, for example, the expression 'A and B or C or D' is processed serially left to right with the first dichotomy being A/B and the relation being 'and'. This act forces the emergence of a dichotomy tree of at least one level manifesting 'TRUE/FALSE' for 'A and B'. This level them becomes the context for the next dichotomy of the A/B result with C and the relation being 'or'. Even if we introduced parantheses to dictate the order of evaluation, the same dichotomy process occurs:
+-------------------------------+
| T | F | T | F | T | F | T | F | or D
+-------------------------------+
| T | F | T | F | or C
+-------------------------------+
| TRUE | FALSE | (A and B)
+-------------------------------+
Fig 1 Bottom-up developing dichotomies tree in logic. Each level is dependent on
the previous result which becomes the context for the current level.
This same process occurs in Calculus where, for example, the first derivative is based on the context of position and gives velocity, and the second derivative is based on the context of velocity and gives acceleration.
Set theory forms the fundimental, although gross, context of mathematics, and, since it is rich in dichotomies, as is logic, the characteristics of dichotomy trees will be found in ANY mathematics derived from set theory, or logic, irrespective of complexity. For example, this can be found in the 'main' method of describing quantum states - the wave equation - which is basically formed thus:
Set A = set of all possible states of an entity.
Set B = set of all possible states of the universe.
Set C = A x (B - A)
= set of all possible ordered pairs of A and B.
Set C is equivalent to psi, and it is the collapse of psi that is reality; the point where possible (one of many states - ordered pairs) becomes 'real' at the moment of measurement. The treatement of this as a wave emerges from the number of possible states. Emphasis on context creates a dichotomies-tree that increases by a power of two for each level. If the time ranges from micro-seconds to seconds then the number of possible states is astronomical, and the whole level can be treated as if a wave; each state manifests a harmonic of the wave of the whole. This is a NATURAL function of serialization and I suggest it is derived from the hybridization of the human brain's association areas originally oriented to making auditio/vision distinctions. Note that language favours ordinality over cardinality; cardinality is expressed in the emotional tones encoded on top of the order..
Reflecting on the sets listed above, the huge number of possible states in set C can be contained by the observation of stability after time. By this I mean that, in most situations, any state is achieved after a short period of time and therefore we need look at a reduced set of possible states; overall stability is possible after six timeframes and therefore we find sixty-four possible states for a system that has a range within two values. As we saw in part I, to enable some form of analysis we have to stop somewhere; as noted in part I, in Quantum Electro-Dynamics we find the introduction of 'renormalization' in equations where both sides are divided by infinity to remove the emergence of infinity in the results. This may seem like 'fudging' but it works.
This process of mathematical 'fudging' introduces a question as to what is 'real'? However, there seems to be some confusion in understanding that the 'wave' nature of things is primarily mathematical. There comes a point where the probability wave (for that is what psi represents) collapses and we 'see' a particle in a specific place. As we saw in our dichotomy tree, although there are specific states at deep levels, there are so many of them that we can treate them as one - a 'wave' with each state being founded on harmonics. Furthermore, as we go to deeper levels, if we are analizing an extreme position, it's complement becomes apparently unknowable due to the increasing number of possible states inbetween but the dichotomy is still valid.
Thus, the collapse of the wave and the manifestation of 'reality' has the same functionality as the log/anti-log and a+bi/a-bi concepts described earlier where a real number 'emerges' from the '/' position.
There has been much speculation on where/when the wave collapses; the point where probability becomes 'reality' and we enter the rhelm of classical mechanics by seeing an electron hit a phosphor screen.
Evett went as far as introducing the many worlds hypothesis which ment that the wave never collapsed since for every possible state there is a possible universe. This seems to have been derived from Feynman's sum-over-histories model of QM. In this model we assume that everything is a wave and that at any moment in time everything tries to do what it wants and it is the wave interferance that causes reality. This model works. What is noteworthy is that there is a strong contextual influence here.
Von Neumann, after much contemplation on the wave-collapse problem, concluded that the actual collapse occured in the mind. But as we saw in part I, the abstract concepts of complex numbers, the form of number most used in QM, 'collapses' to reality when we combine number and conjugate to get a 'real' number, and thus the collapsing 'nature' is part of the most used methodology of symbolization.
Within the context of Physics, in mapping the Universe we have created two highly successful models - relativity and quantum mechanics.
Although highly successful, it has been difficult to form a unified theory, for there seem to be apparent contradictions between the two.
A major 'sore point' centres around the speed of light.
Relativity has shown that matter cannot exceed the speed of light. Using the wave approach, the matter-wave of an object cannot have infinite frequency, and this occurs in this universe at a speed we call the speed of light. Consideration of this suggests that the overall influence here is the concept that you cannot break the whole; in this context, to exceed the speed of light would break the 'boundary' that distinguishes mass, mass would become 'infinite' in all directions.
Quantum mechanics has shown, through Bell's theorem, that, when a photon is broken down into a positron-electron pair, and the pair is split and each element goes off in the opposite direction to the other, the influencing of one element can change the characteristics of the other element regardless of the distance inbetween. This implies some form of faster-than-light communication that relativity has shown to be impossible. However, recent experiments, such as the polarization experiment discusses earlier where, when used to test Bell's theorem, it seems to suggest that this behaviour does occur, need explanation.
Of note in the QM system is that we are dependant on dichotomies and that the influencing of 'correlated' particles suggests the SAME result as relativity - you cannot break the whole. But this result has it's roots in dichotomies - a supposed product of mind.
What is often not considered, or ignored, or just 'taken for granted', is the controlling context in which a dichotomy is made. This overall context represents the whole, and it is it that cannot be cut, since to cut it would imply that each unit is infact independant of the other. In life forms, to cut the right from the left leads to injury and possibly death. In mind this is also so; if I could cut the wave/particle dichotomy such that I could consider the wave part on it's own, then it would no longer be part of the overall context - quantum mechanics.
In QM, it has been observed that the probability wave for an electron can extend across a barrier. The thickness of the barrier can block an electron but below a certain value, about 1/2 (!) the wavelength of the probability wave, this blockage is overcome and electrons can be detected the other side of the barrier - as if they had tunnelled through. If the highest probability is near or on the edge of the barrier, the wave extends through the barrier allowing for the possibility of electron tunnelling. If we treat BOTH electron and barrier as waves, this is acceptable since waves can pass through each other.
Gribbin discusses an experiment (p119 "Schrodingers Kittens") to show both the wave and particle natures of reality in one. In this experiment light tunnels across a gap between two prisms. For the effect to take place the width of the gap must be half of the light's wavelength. This is the same as cutting the whole. Using probability we square the amplitude to get a form that is identical to the polorization model.
In any dichotomously-derived system, the application of probabilites of s specific paths occuring leads to the establishment of a probability wave at all levels. Using the refinement model, the probability wave is like any other wave with the extremes of at either end and in the middle are the symbols for 50/50 representations.
In the texts of many esoteric dichotomously-derived categorization systems (or 'typologies'), there is mention of forms of 'gateways' at specific locations, and when applied to the probability wave, occupy similar positions to the tunnelling concept. We here have a 'map' that has similar 'gateway' characteristics, and this map stems from dichotomous thought. It must be emphasized that we are not concerned with 'reality' here, as much as with the coincidence of description. However of note is the observation that the 'position' relates to the area linked to bonding - aspects of the whole, suggesting a specific relationship is required for these 'gateways' to work, rather than a specific type of whole or part being responsable.
The apparent strong relationships between mono-zigotic (MZ) twins can also be considered as an example of a dichotomy. MZ twins are examples of a walking dichotomy-tree with their level of 'purity' putting them at extreme positions.
Of note is that these areas are contained within the areas of apparent 'gateways'. (Note that ANY level of inpurity, either physiological or psychological will bring the twins in towards the relationship areas (RA) where 'tunnelling' appears to be common. (The 'pure' areas are highly single context). Again, we are not concerned with reality, but more with the ease in which the descriptions can be made. By looking at the mixing map for an element of a dichotomy, the area concerned seems to be the area of 'bond'; where we deal with the relationships of wholes to parts and parts to wholes (aspects of the whole).
Using the me/universe context, categorizations based on dichotomies allow for 'gateway' conditions. This seems applicable to electron/wave contexts as well. These gateways allow for (include possibility of) transformations.
One point of interest is that in QM we find that 'jumps' are in integer steps. Does this in some way reflect the indivisibility of the whole? Of note is that prime numbers are more symbolic of 'wholeness', they are the 'purist' integers, whereas all transcendental numbers are symbolic of ratios which are manifestations of dichotomies taken beyond level 1. Their 'infinite' characteristics reflecting the observed expansion of possible states in dichotomies:
= 3.14159265358979323846....
Fig 2 Pi taken to 20 decimal places.
In figure 2, the pi symbol - - is here transformed into number, but since it is the manifestation of a dichotomy (circumference/diameter), we dissapear into infinity. This suggests that all transcendental numbers are infact symbols associated with aspects of the whole.
How can these 'gateways' etc be symbolized? My analasis suggests that they are based on the harmonic series of 1/2^n.
In this series, the sum of all of the elements apprroaches 2 and the establishment of some form of 'connection' occurs as we approach 2. For the above mentioned twins, this is 1/1 + 1/1 since they are identical and thus the same 'whole' exists in two places at the same time. For family it would be 1 + 1/2 where the 1/2 represents the mother and father; thus a degree of connectivity between parents and offspring. We can extend this in considering the often 'chance' connections observed between close couples. This suggests that correlation is more important than distance in these sorts of relationships.
This is testable at the atomic level through the use of crystals. If we can split a crystal without losing any layer, then the two surfaces would be highly correlated. For each layer lost, any connection would drop-off at a rate of 1/2^n. This suggests a statistical messaging system would be possible but the degree of surface variations due to local environmental influences would lead to a quick decay in overall stability and thus a large expense in maintaining that stability. (Of note is the 'fact' that crystals that are cut along their X/Y plane are know for apparent random frequency jumping and thus this cut is not used in the creation of radio crystals. This property, however, could be the continuum effect at work in that a sudden change in part A ellicits a change in part B - irrespective of the seperating distance since we only deal with correlation. (Analogous to the chemical concepts of a covalent bonds rather than an ionic bonds)).
The level of integration within the associative areas of the brain is such that we can include the probability of sensory hybridization, where the audition/vision systems share neurons.
The question is, what are the connotations of this? At the psychological level, there exists cases of mental states that seem to resemble the joining of the senses. This is called synesthesia. Synesthesia may be an example of hybridization or an example of strongly linked, but seperate, sensory cues responding to a stimulus. However, the senses can be seperated in that I am aware of seeing and hearing.
For example, I know someone who teaches singing. One of her methods is to teach her students to imagine the notes as colours and to use these colours to 'paint' music. She has used this since she discovered herself doing it; when she sang she saw colours. As mentioned in part I, closer examination of this shows a link between colour, harmonics, and emotion.
The colour/harmonics link is based on sound frequency (harmonics) being represented visually as light frequency; colour. The emotion link is the fact that it is frequency that strongly elicits emotional responses. This is easily detected by the adding of harmonics to a single tone, or the presentation of differing colours. Both systems can easily elicit varying emotive responses and so emotion is the universal responder that links the senses of audition and vision.
In infants, where the senses have not yet differentiated, we find that any specific sensory stimulus ellicits a wholistic response in that the whole child turns towards the stimulus. As the child grows, so the process of sensory differentiation is refined and the wholeness is now split between a dichotomy of whole response vs part response; the latter allowing for a degree of multi-tasking where I can talk or listen to someone whilst watching something else (and comprehending both).
But what if hybridization creates sensory associative neuronal systems that are outside explicit associative awareness; systems that are unconscious? This could lead to the mixing of perception and thus what we seem to observe, wholes that at times appear as 'waves' - harmonics of a greater whole.
There does seems to be an overall concept that is challenging to our culture, and this is the apparent duality that we seem to observe as a 'natural' manifestion of nature.