The application of conjugacy and adjugacy is in this wise: Shunya shall be the periphery of a circular disc! That is to say that in my mind there is only a circular periphery. Clearly, this is a pragmatic nonsense, and a misuse of the word Shunya!
However if I explain it thus: "Let my meditation be the periphery of a circular disc. fill my mental process with just the periphery of the disc, so that my subjective experience of Shunya is the periphery of this circular disc…"; then perhaps you may contextualise the "nonsense" as a shortened communication inviting a hypnotic , mystical trance.
Astrologers have always behaved in this way, and mathematicians, so called by Pythagoras are but qualified astrologers!
One famous quote is that " one should attempt to believe 3 impossible things before breakfast!" and Astrologers regularly practice this rule! The point here is the meditative experience leads to a creative state of mind which may or may not have pragmatic significance for the rest of the day. The possibilities are infinite!
Descartes "praxis" he believed was his greatest contribution to mankind. Mankind of course refutes this by ignoring his praxis and baudlerizing his results, and attaching his name to its concoction. The Cartesian coordinate system, revered by modernist mathematicians, takes little room in either his philosophy or his geometry. When examined, I found the greatest similarity between his presentation using the algebraic style, and th Greek more verbally dense rhetoric, and the diagrams followed in the main the Greek presentation of a ordinate line and a point of reference in describing the locus of many curves. The introduction of the Cartesian reference to a particular point has to be set against this background. It was necessary for Descartes to demonstrate the efficacy of his new notation, and indeed it was Pierre De Fermat who fully draws this out as an advantage in studying, that is meditating on many curves. In this light Fermat's last theorem has a very simple solution that is intuitively obvious when you examine the curves plotted on a Cartesian reference plane. All odd powers do not form a closed disc, all even powers form a closed "disc" that tends from a perfect circle to a perfect square as the powers increase. It is therefor intuitively obvious that only the tower 2 will produce the desired result.
This king of intuitive demonstration would not satisfy the Russelians of today, who therefore have to make do with Andrew Wise impenetrable "proof". This is simply because over the corse of time, the gentlemanly( but nevertheless cutthroat ) agreement to accept a demonstration of a point as establishing it has moved to a narrower concept of proof. Popper, in the end smashed this mindless Russelians myth of " logical" proof by adequately demonstrating its contradictions! The best we can do is agree as gentlemen and women to assent to what may be false but has not yet been demonstrated as false, but only if the notion is falsifiable.
We regularly accept tautologies as definitions,nthis is unavoidable, but as we recognise the tautology openly we may then precede to exposit relations and consequences that make sense of our experience. This experience of being in a system that ultimately makes sense is dependent on the Tautologies we accept or assent to. I cannot therefore convince you of the veracity of any damn thing! You must find that out for yourself, but I can give you a push, a suggestion, a persuasion, a hypnagogic experience that takes you on a fellows path next to mine, if you assent to it!
Are not the things we agree on real? We do not work that way ultimately. I individually accept my experiential continuum as reality or as an illusion, but how I interact with you is by accepting customary mores, standards and notions. For you this may in equal measure appear to be real, or an elaborate act that we both engage in never apprehending any sense of reality because we ceaselessly suspend belief or assent to any set of principle denoted as reality. "reality" is of your own choosing!
So therefor Shunya is everything, but I can choose that everything to be a nonsense like the periphery of a circular disk!
Let my Shunya be the periphery . Then let my focus be a part of this periphery. By conjugacy if I call my focus c then notc I may call K.
My terminology by conjugacy is that I have factorised the periphery. So to be brief I will label the periphery P. my action of conjugacy I will denote as conj(). Again this is a label for me conjugating .
The process of conj() relies on me focusing so the act of focussing on something I will denote by focus(). When I focus I focus on and in a region, so the focus action delineates or describes a region c
Thus focus(P) is c. This I may write as
Now if I apply the definition s of conjugacy
We can now observe that focus(P) = notK = c
The logical not used here is not defined here but in the definitions of logical statements which underpins this notational précis form.
I can now properly define conj(P) as focus(P)notfocus(P)
Written as conj(P)=focusj(P)notfocus(P)
From combinatorial theory one may recognise this as a product form, but here I am defining conjugacy as raw factorisation of a form into its parts described in logical statement form. I will then go on to define the product form as reflecting this logical statement form of the factorisation, and define the scalar reciprocity of the conjugate form as part of this product form. In so doing, the notion of factorisation will extend its meaning to describe the product of the logical parts.
This is sophistry. I concatenation a well defined idea into a more extensive but vaguer notion. Clarity only comes with focus! Once I restrict myself to the focus region I discard temporarily other concerns and notations to achieve clarity and focus!! But I have taken you through this loop to show you how our mental processing works, ad the potential pitfalls I have to be careful to avoid.
Hence I can write conj(P)=cK
Now the next process is the adjugacy of c and K
In this case c is simply not adjugated, but in general conjugation will produced regions which have many parts called adjugates. We may also say K is not adjugated by the same reasoning.
We now introduce the process of metrication.
In general we need to metricated, because the number of adjugates may be many, the difference between them huge, the relations between them complex.
Metrication is introduced in book7 of Euclid as standardising the form of any object. This is a Eudoxian idea which is much used today. The standards are called arithmos. The arithmos is constructed or synthesised in a standard way from an ultimate standard called Monas or monad in the predicate parts of speech, thus within a few lines we have a standard unit(Monas), A standard Form(Arithmos) constructed using standard units, and many many standard forms(Arithmoi) for the myriad of forms and shapes in our experience. What Eudoxus proposed is that every form had a standard representation. This standard representation is derived from the fundamental unit adopted. Thus every form is a multipleform of this standard unit. Before we get too carried away, the common notion undepinnings an arithmos was an epipedos, that is a mosaic floor covering. In general, these floor coverings were found in the houses of the very rich or the shrines to the Muses. We call them mosaics for this very reason, but in Eudoxus and Pythagoras day they were ornate floor coverings of various material and illustrative of many patterns. Epipedos may or may not have been the term for them, but certainly in book 7 Euclid lifts them from underneath our feet to the pinnacles of our minds in standardising the synthetic construction of the universe. In so doing he establishes the Medtronic rules we all have used ever since. This is how Eudoxus is said to have saved the Pythagoreans from internal strife, but I hardly think so. The strife I think was not about unity or Monas, but about the veracity of Pythagoras Theorem, as we call it nowadays. The challenge was that it could not be true because it leads to a contradiction!
The contradiction is not about even and odds, either, it is about precision and approximation. If the theorem is correct then we can find precisely what the three quantities must be if any 2 of them are precise. Instead we find that precision comes in triplets! Thus the theorem is not trustworthy because we cannot always know what these triplets might be in general. Of course, pragmatically we can generate these triplets till the cows come home, but realistically, when we measure space using a metron we cannot guarantee a precise answer!.in this sense Eudoxhs has fixed nothing.even if every form is a multiple form, standardised and neatly categorised as arithmoi, there are still combinations of triples that do not match Pythagoras theorem!
This was never a problem before and after the challenge to Pythagoras leadership. It became a tool misused by the challenger to sow doubt and discord about Pythagoras himself nd his fitness to lead the group! After the challenge Pythagoreans were quite happy to accept what we call irrational numbers, not as a new discovery, but as a mystery of our universe. We can only count in units, but the universe fractalises our units to distraction and exhaustion! Somewhere along the way we have to say "Enough! This will be my Monad!" when we do Eudoxus is waiting to take our hand and help us to synthesise our metric from this metron. But by preserving Pythagoras theorem we have preserved the most fundamental metric of our reality. By altering this one theorem we alter our entire spaciometry.
So therefore I take c as my metron.
Applying it to K I find that K is a precise multiple of c or not. If it is not I then use Euclid's algorithm to find the highest common factor! Thus Euclid's algorithm teaches that units (Monads) are obtained by factorising, that is conjugation!
Let us suppose that K is a multiple of c , then I may write that K is 5 lots or applications or copies of c. There are many linguistic forms used to describe this multiple relationship but the essential notion, the fundamental one is adjugacy defined in Euclid as Katameetresei and sugkeimai. Basically we just cover the space with the metron , being careful to ensure at least contiguity!
This I can write as K=c+c+c+c+c, where I use the + sign to indicate contiguos contact. This has been described as aggregation.
But what about Euclid's algorithm that states units are factors? The standard form in Euclid s the right gnomon, nd this is used all over the place. In particular, when dealing with the standard form it became customary to use not rounded pebbles or any form that suited in the construction of the arithmos, but little baked squares of coloured clay suitably glazed. Thes small squares are almost synonymous with the idea of a mosaic, but not quite.
In this case the gnomon matyials these squares Ito a standard gnomic form. From this standard form we may easily deduce the factors of multiple forms , if they are Artois, that is exact. This is why we must avoid the motion of even and odd! For 9 is an exact(Artois) form(arithmos)!
Thus I may now write in standard gnomic form
K=5c which is in the conjugate or factorised form.
Now here I must be careful to distinguish the conjugacy!
But in this context adj() has not yet been formally defined.
Adj(K)= K which defines the adjugacy of K
Similarly adj(c)=c which defines the adjugacy of c
But now c is taken as a metron for P
Thus m() is the notation for metron and
Thus I may also write
m( c,K)=c as a way of saying c is the metron for c and for K. This not a contradiction, but a shorthand for saying I am using a common metron for c and K which will be c
In this context I can now write mc,K() for m(c,K)() which is this common measure acting on something.
But now we recognise the standard gnomic form as a conjugate form what does this tell me?
At first one is tempted to think con(K)=5c, but referring back to the periphery this is clearly incongruent! What it does indicate is that conj(K) = c5!
This is a remarkable result!
We arrive at it by factorising K into its 5 adjugates. And now by a straightforward application of the rules of conjugacy we rename each part as l, m, n, o where l is 4c, m is 3c, n is 2c and o is c
Then conj(K) is cl
Conj( l) is cm
Conj(m) is cn
And finally conj(n) is co
Now because conjugates can be nested, that is I can conjugate within a conjugate I am able to write all of this as
Each of the forms on the LHS is a conjugate form so I have fulfilled that obligation.
The question is how am I going to notate this complex conjugate form?. The second is why is it different from adjugation?
I need to answer the second to justify the first.
Clearly this is a conjugation process not a metrication process! The fact is I need to do a prior metrication to establish this conjugation process. Thus adjugacy underpins this conjugation structure, but it is not adjugacy. At each stage of the conjugation the adjugates are diminished, and pass from one conjugate to another defined conjugate. Thus the conditions of adjugacy are not fulfilled which require a fixed conjugate context to be developed.
I have used a fixed conjugate context to define specific adjugates. I have then kept these adjugates fixed to define specific conjugates. Thus I have a factorisation structure which I may justifiably call co factors! To be specific they are cofactors of the conjugate K. Thes cofactors are also adjugates of K where they take on the role of structuring K and "establishing" or consisting K. Thus K is not the same as conj(K)! This makes sense as K arises out of conj(P).
Thus I think that it is unambiguous after one gets used to it that
But o is c thus
And consequently I may now write conj(P )= c6
This all relies on the metron being c for bothe conjugates, and for P to be adjugated into 6c!
This relationship between conjugacy and adjugacy exists everywhere, but a good analogue is the logarithmic ratios of John Napier.
Doing this terminology cal meditation, that is setting out the propositions and definitions in symbolic terms, reveals the hedonic bias I have been indoctrinated with, that is to start with the plane. In fact Euclid started with the sphere and analysed it down to the plane. He then synthesised it up from the point to the plane to solid space.
This conjugation would benefit from being done on the sphere!, especially when it comes to the nested conjugates and the notion of arithmos, which is inherently a solid form!