Simple Transposition

If we number the elements of a set with the symbols {a,b,c,d}, then a permutation of the elements [a,b,c,d] to [b,c,d,a] we simply may write in permutation notation as (a,b,c,d) (shorthand for a=>b=>c=>d=>a).

All permutations can be written as a product of transpositions (a permutation of two elements.) for instance, (1,2,3) is also the product of the two transpositions,(1,2)(2,3) (ie, 2 goes to 3 and 1 goes to 2. 3 goes to 2 and then as 2 goes to 1.) Likewise, (1,2,3,4,5) = (1,2)(2,3)(3,4),(4,5) = (1,5)(1,4)(1,3)(1,2) etc.

A cycle of length n+1 is represented by a product of n transpositions. We shall refer to a product of an even number of transpositions as "even" and that of an odd number of transpositions as "odd" when we refer to the elements of the group of all permutations of a set of (usually five for our needs) symbols.

It is clear then that we can also multiply cycles as with transpositions, so a product of two even elements is also an even element. Since the number of permutations of five symbols is 120 = (5x4x3x2x1) The group we call S5; of all even and odd permutations of five symbols is a closed set. Likewise the set of all even permutations is also a closed set, and happens to have 60 elements. We call this group in even permutations over five symbols A5.

So (1,2,3)(1,2,4) = (1,3)(1,2)(1,4)(1,2) etc. Its is also clear that disjoint cycles commute so (1,3)(2,4)(1,3) = (1,3)(1,3)(2,4) and we also see that (1,3)(1,3) = "e" where "e" represents the identity element,.. where no symbols are permuted, but are mapped to themselves.

So, how does this stack up with temptation?

When the senses are normally in employment, we may see something and touch it. This ordered sequence of (see,touch) becomes a familiar process, and we can construct a cycle (see,taste)(see, smell)(see,touch) where a person may see, then feel, then smell, then eat the item. We also note that we constructed that we had tasted the item "before we see the item." (taste, see) or as in a sequence where we had for example, (see,touch,smell,taste) where we saw, picked, smelled and ate something, along with its inverse in terms of "desire" (taste,smell,touch,see) where we desired to eat something with a good smell that is ready to hand and clearly in view. This is normal enough when we are familiar with the process of eating something that is good for food, as we would be driven by the need to eat for example. What is also true is that the consideration of each constructed sequence is time-independent when it is merely under consideration.

However, when there is an antagonism in the system, ie,.. there is one transposition, where we could state (see, lust) do we consider this transposition as if we "wish to touch an object because we greatly desire to feel its shape", or do we "feel its shape because we greatly desire to see more of it?" Comparing the object in question to something familiar like door keys in the dark is simple, as there is no antagonism,.. but when there is a fault involved, such as greatly desiring to caress someone else who is beautiful, then it makes no sense to consider the transposition, but rather the antagonism. For if the woman involved was your own wife, then it would be assumed to be familiar, but if it were some other man's wife, it would not be sensible to consider the transposition as familiar, but only the permutation of the symbols themselves.

It may make more sense to think of the presence of an n-cycle as a chain of events on a timeline, and for the antagonism to be the inverse of that cycle of events. Thus for three cycles such as; "I saw the fruit then ate it then tasted", would become as under antagonism, "I desired to taste the fruit so I ate it because it was there for all to see." Neither of these seems irrational if it were simply a case of raiding the fruitbowl - but if there was a commandment not to raid the fruitbowl we see that neither of these cycles justifies breaking the commandment. Both assume the freedom to obey the "lust of the eyes".

Thus whereas a set of two senses (our symbols in transposition) may familiarly be ordered, "see then touch" the antagonism is correctly ordered "desire touch, following seeing" so in this sense, the antagonism has been acted upon by the transposition. When something is familiar, the transposition will render some pleasure in the latter sense of these two cases also. However without the freedom to commute freely the symbols back and forth when there is indeed an antagonism, then the mind's logic steps in and should cease action, preventing us from "see then touch."

Without any great amount of pleasure involved, finding door keys in the dark may indeed be a great relief, but there is certainly no inner conflict as when there are circumstances against the latter case being permuted again, for (see, touch)(touch,see) = "e" which is familiar (we can easily find our door keys). But if there is an absolute such as "the door keys are indoors and can't be reached", or "thou shalt not covet" then we can not use the transposition, but only the permuted symbols. ie,. "first get the keys, then open the door" We consider it impossible to permute a fixed element, in this case "touch" with the door keys.

We may freely permute four senses and the logic we use in a cycle. Consider (logic,taste)(logic,smell)(logic,touch)(logic,see)

...Or (taste,smell,touch,see,logic) where we apply logic in solving the problem of seeing an object, touching smelling then eating it. Logic can be used to solve the problem of how to defeat a fixed element in the senses. For instance. If one can not "hear where the wife says she put your chocolate bar". Then logic could help with that. Also, we can use the problem solving of logic to crack the door key problem by opening the window and reaching in. Fixed elements may be to some degree "solved" in order to solve an antagonism.

Likewise we may equate the cycle (logic, taste, see, touch, smell, hear) with (logic, hear)(logic, smell)(logic, touch)(logic, see)(logic, taste).

The cycle is also equal to (taste, see, touch, smell, hear)(logic, hear)

Therefore logic may step in to solve the antagonism or itself be considered as part of a cycle reduced to transpositions, i.e. "think then eat" (read; consider, then eat) in all cases where logic is part of the cycle. (To reason that the fruit of the forbidden tree will kill me, or whether another is good to eat, etc) In every n-cycle involving logic may reduce to a series of transpositions where each sense is considered logically to see whether there truly is any "harm" or "pleasure" - whether it would be illogical to pluck the coal from the fire etc. Likewise all such cycles may be written as a product of logical and carnal cycles where a cycle may be justified by one sense and then that sense by logic.

I.e. (logic, hear)(logic, smell)(logic, touch)(logic, see)(logic, taste) = (taste, see, touch, smell, hear)(logic, hear)

which is equal to (hear, smell)(hear, touch)(hear, see)(hear, taste)(logic,hear)

So cycles decompose into equivalent cycles that would appear to carnally justify the antagonism (as with hearing above), if logic would then justify the sense that does so.

Then also the cycles constructed by transpositions that look like this make sense in terms of "function composition" or by "chaining".

(a,b,c,d,e) = (a,b)(b,c)(c,d)(d,e)

We note that the five senses and logic form the group S6. The elements in permutations of the five senses only form the elements of the subgroup S5 (which is not normal in S6). That S5 group has 120 elements, and we consider the action of factoring down within that subgroup to A5, which is normal in S5.

It make little sense to consider the transpositions acting on the senses within the time-line itself - rather the time-line enables the transpositions to be formed in a single "quanta" or "packet" where their cycles may be truly said to be instantaneous (as on discovery by the mind) so that (a,b,c,d,e) actually cycles "e to a" as well, rather than restricting what is "mental judgement" to sensation that is merely "surviving" the time over which transpositions supposedly could occur. There is no sense of "e to a" last if the cycle occurs in the timeline itself, rather we must consider (a,b,c,d,e) fully formed followed by "decide last, then act."

Then every cycle or product of cycles does not occur in a real interval in time, but as a single packet that is linked to the environment upon which the mind is able to judge within a real interval. (This system would include I suppose, mental conditioning as for (logic, taste) in our cycles (as from the example of Pavlov's methods) to be decided rationally by the mind before the act to eat is made - there is still the instinct to salivate, even if the mind overrides the desire to eat or obey the ringing bell.)

So, in the case of lusting to touch a beautiful woman, logic should attempt to step in with one of two outcomes,.. introduce yourself or leave well alone. If one is already married then the latter is the assumed norm, but how may we permute the elements with (see,touch) when "see" or "touch" is fixed and we are only left with (logic,touch) (when we can't see) or (logic,see) when we cant touch. The answer is that logic must take the fall when we desire to sin, and we must replace logic with a permutation of (see,touch). However what is the manner whereby we may logically decide to do this?

It would appear then that if we were to discount logic we must in some manner test the lust of the eyes (see,touch) for example, on a basis that does not include logic. If a person in their "time-line" simply wraps products of permutation cycles with their inverses, all is become familiar and inverses cancel as there is no antagonism. The fixed element (or the minds logic) is without need of application in what is familiar, and there is no sense of "factoring down" using this fixed element "e", which is as the identity element. We will move on to the mechanism of justifying lusts with the senses.

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