To Penrose dodekahedrons (1) in preferens permutations.
( Kapital-i, in 'I, myself', now spels Il as in isle or aisle.
Leter y spels sym for seem or seam and partys for parties.
Leter w spels swn for soon. )
Links to sektions:
Quantum korelations apyr not to be in the natur of a yusabl signal, that kan transmit information faster than the spyd of lIt. They hav byn deskrib'd by Heinz Pagels as tw random sequenses of information, that in themselvs myn nothing. They only form a patern wen put together.
MIt not, then, quantum korelations form, in som sens, an elektion? In an elektion, kondukted by sekret balot, the voters' choises send no signal as to how they hav individualy voted. But in the kount, paterns or korelations of chois arIs.
As siens fiktion, tw inter-sekting galaxys mIt set up elektions by quantum korelations befor pasing to the other ends of the univers!
MIt quantum korelation ruls hav som-thing to ofer to traditional methods of kounting vots? Even the singl transferabl vot, at its most sofistikated, at som staij in the kount, usualy depends on having to eliminat som kandidat, transfering his vots, so the kount kan go on.
The problem with this is an element of presuming wat uon is supos'd to be proving. ( Voting methods other than STV do this much mor flagrantly and ar greitly inferior. ) Ther is an arbitrary element in exkluding som kandidat, hw hapens to hav the lyst vots, wen the kount hapens to hav run out of mor vots to transfer, til al the syts in the konstituensy hav byn fil'd by the kandidats hw rych an elektiv proportion of the vots.
Quantum korelations sujest an advans on traditional kounts' eliminating a kandidat by the hapen-stans that he has the fwest vots wen the transferabl vots run out. Insted, the elimination of a kandidat may be bilt into such ruls as quantum korelations.
The folowing 'elektoral' interpretation of the quantum ruls dos not solv this problem. Indyd, the interpretation is only for the smalest skail elektions. But the proses of interpretation did sujest a posibl alternativ prwf of non-lokality, to that given by Roger Penrose. And I amplify this, in the last sektion, without konfusing the prwf with 'elektions' in the sens we ar yus'd to them.
Kan we hav a human elektion by voters of kandidats yusing Penrose's formulation of the quantum ruls of non-lokality? Is it posibl to giv a konsistent or thoro-going elektoral interpretation of thys quantum ruls?
The previus paj show'd that a dodekahedron may represent al 120 permutations of 5 posibl choises. Say thys ar fIv kandidats. The voters hav a preferens vot, wich givs the voters a rank'd chois of kandidats: 1st, 2nd, 3rd, 4th, 5th. This is evident in the quantum rul, wich alows the observers as 'voters' to selekt any of the twenty dodekahedron vertises and then rank any of its thry ajasent vertises in uon's chosen order.
The first selekted vertex is equivalent to a first preferens. The thry ajasent vertises ar the sekond to forth preferenses. ( The kandidat, going by the label not amongst the first 4 choices, is the fifth or lyst prefer'd kandidat. )
But hyr we hav a puzl. The bel only rings for korelations betwyn the tw 'voters'' sekond, third or forth preferenses, not if they hapen to alIt on the saim selekt vertex, in the first plas. Wat kan this myn?
The selekt butons, to wich the ruls aplI, ar thos in the saim or oposit positions on the tw alIn'd but separat dodekahedrons. From the point of viw of labeling thys dodekahedrons for al the permutations of fIv choises A, B, C, D, E, similar and oposit positions always represent the saim kandidat. Ych kandidat is label'd at 4 vertices, to alow al permutations of fIv choises to be represented. ( See previus paj's figur 3. ) For instans, kandidat A is represented at vertex A and its komplytly oposit vertex A* and similarly at vertises a and a*.
Therfor, it is always for the saim kandidat that the selekt buton dos
not, in som sens, kount for a ring, even tho it wud sym to represent the
first and most important chois. Ther is a kind of elektion in wich this
mIt konsyvably be the kais. If al the voters ar also kandidats, then
ther first preferenses kan not kount towards the elektion, bekaus the very
fakt of them standing as kandidats is a vot for themselvs.
Then, only the sekond to forth preferenses kan be elektiv.
But this brings an other puzl. Ther ar tw separated observers akin to tw voters in a sekret balot. The quantum ruls only aplI wen they chws the saim kandidat. But the asumption has just byn mayd that al the voters ar also kandidats voting for themselvs in the first instans. So, how kan tw voters be uon kandidat?
Wel, a politikal kandidat typikly has an ajent to promot his kandidatur. An ajent has byn nown to suksyd his kandidat as an MP. Supos that tw kandidats ar ajents for ych other, or partners. Marij partners may enter the elektoral relationship in question. Som-tIms, husband and wIf tak turns to be hed of the family. In other words, either partner may vot for the other, as wel as themselvs.
( Uon exampl is a husband's komplaint that his wIf taks over al the plesant dutys towards the children and lyvs him to tak ker of disiplin. )
So, the nyr and far observers hav bekom fIv voting kandidats hyr, hw hav spesifik partners in fIv voting kandidats ther. In lIn with the quantum ruls, it is a random proses wether voting kandidat, say, A is selekted hyr or far away. He chwses uon of for vertises A, A*, a, a*, with ther respektiv thry ajasent vertises.
Vertises A and a ar ajasent, as ar ther oposits A* and a*. This
sujests that a given partner has the option to vot for himself first and his
partner sekond, or vice versa. How-ever, ther is no obligation to vot the
partner in sekond any mor than third or forth. It kan only be said that the
partner must not be a fifth or lyst prefer'd chois.
This raises questions about the natur of such a 'partnership', wich begins
to lwk lIk som stranj monogamus kum polygamus form of kompetition,
model'd on a quantum korelation. It mIt be beter to naim the partners as
'korelativs'.
Of mor imediat konsern is the obligation to mak vertex 'a' the chois for a partner of A. It has byn asum'd A* and a* wer uon partner's vertises and A and 'a' wer the other partner's vertises, thus giving the tw partners tw diferent sekond to forth preferens options.
An exampl, of how the 'quantum' voting prosedur mIt work, may resolv the inkonsistensy. Supos a voter, hyr, is randomly given his turn to vot. And he kan selekt either himself or his partner, far away, as an other voter. The konvention kud be that the hyr voter prefers himself by selekting either vertex A or a, depending on wether the thry ajasent vertises, to either A or a, konstitut his sekond to forth preferenses.
Alternativly, the hyr voter mIt selekt for first preferens his partner, represented by the A* or a* vertises, again depending on wether the thry ajasent vertises, to A* or a*, konstitut his sekond to forth preferenses.
Wether, say, the hyr voter selekts himself or his partner, an
inkonsistensy arises, wen he koms to order his chois of the thry ajasent
vertises. Bekaus uon of them stands for the saim chois as his first
preferens selektion.
This may be alrIt, provided we mak klyr, as a mater of prosedur, that
wen tw ajasent vertises both naim the saim kandidat, for selekting him as
first preferens, wich-ever of the tw ajasent vertises not so selekted, then
switches to bekom the naim of that kandidat's partner, for the purpos of
being the sekond, third or forth chois.
Uon may objekt that this prosedur involvs an inkonsistent klasifikation. And as far as a fix'd labeling of the dodekahedron vertises is konsern'd, this is sertainly tru. But the re-labeling, that dos tak plas, is akording to an orderly prosedur. From a statik point of viw, the labeling is inkonsistent, but dynamikly ther is a konsistent order of events.
Diskus'd on my first paj about 'sientifik method of elektions', S S Stevens' theory is of the 4 skails of mesurment, klasifikatory, ordinal, interval and ratio skails, ych uon mor akurat and powerful than the uon befor. With referens to this, uon mIt argu that the konsistensy of the ordinal skail over-ruls the inkonsistensy of the klasifikatory skail.
This argument mIt, or mIt not, korespond to an inkonsistent yus of the ordinal skail ( perhaps the doktrin against 'non-monotonisity' ) being over-rul'd by a konsistent yus of the interval skail ( exemplify'd in a weited kount of preferens vots, wich Laplace gav a prwf for ).
And som such over-ruling mIt okur of the interval skail with regard to the ratio skail. Indyd, the ratio skail mIt be konsider'd a 'kontrol' of the interval skail, as explain'd in the abov referens. Such posibilitys mIt explain away som aparent paradoxes of demokrasy put forward by social chois theorists, in ther lojikal krityks of voting methods.
How-ever that may be, the elektoral interpretation of the Penrose dodekahedrons, syn as preferens permutations, sujests som-thing of mor imediat interest. Roger Penrose's disprwf, of a klasikal explanation of the the ruls of quantum korelation, depends on not being abl to fix any of the vertises as bel-ringing positions, wer the order of chois stops.
But the abov elektoral interpretation, if yusful or korekt, gos further than saying you kan't pin down a final chois ( to a vertex ). It further says that yu kan't pin down a first chois ( to a vertex ) wich is only konditional on the subsequent choises ofer'd by the thry ajasent vertises to any uon vertex. Two ajasent vertises hav to stand for the saim initial selektion, to ofer the option of difering ajasent vertises with diferent subsequent choises.
The uon, of the ajasent vertises, konditionaly representing the saim first chois, that is not selekted, must bekom a chois for the first choise's partner or 'korelativ', wich may be taken up as a sekond, third or forth chois, in the order'd presing of thry vertex butons ajasent to the first chois vertex.
Klasikal asumption of no-influens fail'd to pin down a bel-ringing vertex, as final preferens. LIkwIs, such independens mIt be disprov'd by not being abl to pin down an initial selekt vertex or first preferens. Supos any vertex on the dodekahedron is pre-asIn'd as a first preferens. By the abov rysoning, it has an ajasent vertex that also represents the saim first preferens, but with a diferent option in posibl sekond to forth preferenses.
If uon of any pair of ajasent first preferenses is pre-asIn'd as the first preferens, then the other uon kud no longer serv as the first preferens ( with its partikular ajasent thry choises ). Therfor, the far voter's chwsing the koresponding chois of vertex or anti-podal vertex kud not elisit the tw quantum ruls.
I trI'd to mak a human elektion mimik Penrose's quantum ruls. But I had to supos that wen uon of tw ajasent vertises, for the saim kandidat, was mayd the initial selektion, the other wud automatikly chanj to represent the partner kandidat. I dont no wether this kud be mayd to work in quantum terms.
If it kud work, then mIt Penrose's tw ruls exchanj ther aplikation? This suposes a konservation prinsipl in the dodekahedron, wer-by the chanjing of a vertex representing a kandidat to representing the partner, has a revers efekt on its anti-podal vertex. ( That is to sujest that rul uon mIt aplI to figur tw and rul tw aplI to figur uon, on the previus paj. )
If the far voter also randomly given her turn to vot, in this instans, is not the hyr-voter's partner, then ther wil be no quantum korelation, that is no bel wil ring to the partner's presing vertises that korespond, as either in the saim or oposit positions on ther respektiv dodekahedrons.
If the dodekahedrons hav oposit position vertises taken for selekt first preferens, the bel stil may not ring for any of the sekond to forth choises of ther ajasent vertises. It is stil posibl that non of the partners' sekond to forth preferenses korespond. Even tho the thry choises must be the saim, the order the tw partners pres them may be komplytly diferent. Then, no bel rings and indyd neither partner kan no wether ther partner or som other voter, not a partner, mayd an unrelated selektion of vertex.
'Elektoral' interpretation of the Penrose dodekahedrons sujested a posibl nw prwf of non-lokality. It mIt be a gud idea to re-konsider its asumptions without konfusing social or politikal konotations.
The steps in the hypothetikal prwf started with a dodekahedron as the jeometry of al 120 permutations of fIv choises. Ych of the fIv choises hav for vertises ych. They ar al in tw diametrikly oposit ajasent pairs of vertises.
The ajasent pairs ofer tw alternativ sets of sekond to forth choises. The asumption is mayd that ajacent vertises, standing for the saim chois, ar never-the-les distinkt in the kontext of the diferent choises ofer'd by ther respektiv thry adjacent vertices. I trust this may be true of quantum mechanics, or the ensuing prwf dos not klaim to be valid for that siens.
This lyvs the problem of the diametrikly oposit pairs, wich apyr to be redundant in the way the ajasent pairs ar not. The opos'd ajasent pairs ofer the saim tw sets of sekond to forth choises, in ther respektiv ajasent thry vertises. ( Sy figur 3 on previus paj. )
How-ever, we dont wory about that for the moment. Supos the klasikal fysiks asumption is mayd that a selekt vertex ( or first chois ) is pre-asIn'd ( insted of a bel-ringing vertex, wich signals a final chois, as in the Penrose prwf ).
The pre-asInment of any vertex, on the dodekahedron, as 'selekt' automatikly myns that ajasent vertex, wich stands for the saim first chois, is not availabl for a quantum korelation betwyn its thry ajasent vertises and the koresponding or anti-podal vertises on the far dodekahedron.
Mor-over, uons a vertex has byn selekted, its ajasent vertex, wich stands for the saim first chois, kan not any longer hav that myning. This involvs a sekond asumption, wich may or may not aplI to quantum mekaniks, in this kontext. Naimly, that this vertex not only syses to be availabl as first chois. It also must automatikly tak on an other identity.
It is now that the puzl, of the symingly redundant oposit pairs, afords a klu. They ar redundant as to the suksyding choises they ofer. But ther very existens, as distinkt positions on the dodekahedron, implIs the similarity is not komplyt. Indyd, the distinkt ruls of quantum korelation, for koresponding vertises and for anti-podal vertises, tel as much.
Therfor, we wud sym to hav a redy-mayd nw identity for any vertex
that is the forgon first chois and bekoms a posibl sekond, third or
forth chois. It bekoms, in efekt, its diametrik oposit vertex.
That is to say, if any vertex is pre-asIn'd as selekt, its ajasent vertex,
for the saim first chois ( but diferent later uons ) mIt swop identity
with its diametrik oposit vertex, also standing for the saim chois, but
subjekt to a diferent quantum korelation with the other dodekahedron.
The swop is nyded to balans the bwks. By a konservation prinsipl, al the
propertys of the dodekahedron must stil be systematikly represented.
This prediktion, of a switch ( betwyn figurs uon and tw on the previus
paj ) of the tw quantum korelation ruls, betraying a pre-asIn'd first
chois, may stil be yusful in its implikations, even if it prov'd to be
fals, bekaus it kud entail alternativ explanations.
My
experiens of elektoral method tels me that a prwf involving first
preferenses ( or initial selektion of vertises ) wud probably be mor potent
than uon involving final preferenses ( or bel-ringing vertises ). I suspekt
this wether or not my own atempt at such a prwf is on the rIt lIns or
not.
Richard Lung.
10 April 2002.