( 1 - v²/c² ) where v
is relative velocity between observers and c is the constant light speed.The post-script, on an electoral model of the Interval, was on my page about an electoral model of special relativity. The Interval is about extending Euclid's geometry of three space dimensions to a fourth dimension, being time. Pythagoras' theorem is then extended to measure the square of the hypotenuse as equal to the sum of the squares of the other four sides, where the time squared is a negative sum, being subtracted. The Interval hypotenuse is of a 4-D space-time ( "hyper-cube" ) instead of a 3-D cube of space.
The point of the Interval for Special Relativity is that the Interval hypotenuse is an agreed measurement for observers of a given event, in constant velocity with respect to one another, however their chosen times and positions of observation may differ.
Altho I hope my post-script on the electoral Interval was a fairly useful discussion, it didnt really get to grips with the problem of a four-dimensional measurement applied to elections. The same may be said of the following but again I hope it may be useful. Perhaps, I sensed this inadequacy when I next put up an old essay on the Laws of Motion and Elections. This went back to thinking in terms of dimensions of proportional representation, with respect to constituencies.
There is PR within constituencies, PR across constituencies, and PR between constituency systems. These would be analgous to three dimensions of space, the x, y, and z co-ordinates of a cube. My page on an electoral model of special relativity actually was based on only one of these dimensions of PR, namely across constituencies. I could get away with that, because, for simplicity, explanations of a geometrical problem commonly only discuss it in terms of one or two dimensions. Extensions to more dimensions is often straight-forward.
In the case of my electoral analogy, extension of the problem to another dimension, PR within constituencies was not straight-forward. Back in the early 1990s when I was writing the page, Laws of motion and election, I was also trying to do an electoral arithmetic of the Lorentz transformations, or more particularly, the Michelson-Morley experiment calculation. Essentially, that means I was applying the Fitzgerald-Lorentz contraction factor to proportional representation.
The contraction factor tried to explain away an estimated difference ( actually maximum and minimum expected time lags ) in an interference measurement of a light beam split at right angles and returned over the same distance journey. It was presumed that light waves waved in a universal medium called the ether, with its own velocity, to which all other velocities were relative. When one split beam happened to travel in line with this ether, the head wind should have the greatest slowing effect, still apparent even after averaging with the tail wind journey the other way. The other beam's exactly cross-wind journey, both back and forth, should produce the least slow-down.
The Michelson-Morley experiment showed no difference between the two experimentally observed light beams' speeds. The speed of light was shown to be constant, which Special relativity took as a basic postulate. In effect, Einstein abandoned the absolute velocity of a supposed ether and made light speed the new absolute frame of reference, to which all other objects' velocities are relative. At speeds that are a significant fraction of light speed, for observers moving respectively ( more or less ) with or against the light's direction, some fairly drastic adjustments are needed to the respective observers' space and time measurements to make them consistent with both observing the same light speed.
These adjustments are most simply made by the Fitzgerald-Lorentz
contraction factor and more generally by the Lorentz transformations. Tho, to
make the observers' measures actually agree can only be done in terms of a
four-dimensional space-time called Minkowski's Interval. The contraction
factor is ( 1 - v²/c² ) where v
is relative velocity between observers and c is the constant light speed.
Proportional representation ( produced by the transferable voting system ) is the Droop quota, which gives the ratio of representation R/(R+1), that is 1/2, 2/3, 3/4, etc of the votes for one, two, three etc seats per constituency.
Back in the early 1990s, I supposed a compensating ratio of representation, (R+2)/(R+1). The arithmetic mean of these two ratios would average as full or unitary representation. That is R/(R+1) plus (R+2)/(R+1) equals 2(R+1)/(R+1). And dividing the two ratios by two, for their average, gives full representation. But I could not see where the compensating ratio of representation could come from a consideration of PR within a constituency. That is why I long gave up this notion. And when I wrote the page on an electoral model of special relativity, I still didnt know.
The model used only PR across constituencies. In that case, variations in local population densities, are reflected within limits, by allowed variations in the number of constituents about a required average. Then a permitted over-size constituency might be greater than the average votes, V. So, for PR across constituencies, a supposed compensating ratio of representation could come from the largest permitted voters per constituency compared to the smallest permitted constituency electorate. That is with respect to the required average constituency.
Notice that the constituency with an above average number of constituents is under-represented. It turns out that the compensating ratio applied to PR within constituencies is also a mark of under-representation. Recently, I developed the single transferable vote with a method that avoids excluding candidates prematurely in the count. That is before one knows, later in the count, that the excluded candidate might have picked up enough transferable votes to win an elective proportion, or quota, of the votes in a multi-member constituency.
My development uses the quota not only to elect the most prefered candidates but also to exclude the least prefered candidates. Essentially, the ballots of preference are reversed, starting from the last prefered, for an exclusion count. These two-way counts can be extended in combinations logicly given by the bi-nomial expansion ( in non-commutative form and usually not beyond the second order of expansion ). The idea is elementary tho prohibitive to work-out without computers.
The Binomial STV ( single transferable vote ) is essentially a controlled experiment in a series of counts using different combinations of election and exclusion of candidates to determine the best over-all representatives. In some counts, candidates may be definitely elected or excluded on a quota. Another way of considering their election or exclusion is by the candidate's keep value. This is a measure of how many of the votes, he receives, are to keep for himself, as against how many are surplus votes transferable to next prefered candidates. If a candidate has a keep value of just one, that means that he is just elected on a quota of votes with no surplus votes to give away to next prefered candidates. A keep value of less than one, say .8, means that eight-tenths of the votes he received are enough to make up his quota. The other two-tenths worth of his votes are transferable to help next prefered candidates reach their quotas.
Suppose a candidate, with .8 in one of the counts, does not achieve a keep value of one or less in another of the counts. To do my new version of STV, there had to be a way to measure the keep values of candidates who dont reach an elective keep value. The normal rule is that the keep value plus the surplus transfer value equals one vote. If a candidate doesnt have enough votes to reach a quota, that means that he doesnt have a transferable surplus of votes. On the contrary, in so far as he falls short of the quota, he has the negative of a surplus, in other words, a deficit.
The normal rule is keep value equals one minus ( surplus ) transfer value. The extended rule is keep value equals one minus a minus, or deficit, transfer value. That is one plus deficit transfer value. Say, that in a further count, instead of having two-tenths of votes in surplus of the quota, the candidate now is three-tenths of votes in deficit of the quota. The candidate's keep value for this count is one plus .3 equals 1.3.
If the election filled all the seats on the two counts, in which the candidate gets keep values of .8 and 1.3, then that candidate would have an over-all keep value of .8 times 1.3 equals 1.04. This is just above one and so not quite elective of the candidate.
Also, there might be two exclusion counts, these would work in the same way as the election counts but with the preferences reversed, starting with last preferences. There would be exclusion keep values, just as above, but their meaning would be reversed. So an exclusion quota of .8 would mean that a candidate had achieved and surpassed a quota of unpreference or dislike. But an exclusion quota of 1.3 would mean that a candidate had "failed" to achieve the exclusion quota. An over-all exclusion quota of 1.04 would mean that the candidate had just "failed" to be excluded.
If the candidate had the same value exclusion keep value as his election keep value, then the two would cancel, giving a final over-all quota of 1.04/1.04 equals one, which would be just sufficient to elect the candidate. This illustrates the rule that final over-all quotas are achieved by multiplying the over-all election keep value by the inverse of the over-all exclusion keep value.
My idea of a deficit transfer value, giving an unelective keep value of more than unity, gives the answer to how my initial notion of a compensating ratio of representation might be applied to PR within constituencies ( as well as across constituencies ). The Droop quota, V/(R+1) is the number of voters in the multi-member constituency divided by the number of representatives plus one. This gives a proportional representation as the quota times the number of representatives or VR/(R+1).
Suppose an artificially simple manner in which this proportional representation might be realised during the conduct of a count using the single transferable vote. Suppose all the voters give their first preferences to one candidate. He only needs a quota, so all the votes minus the quota are transferable to the next prefered candidate. That is the first elected candidate keeps the proportion of the vote given by the quota divided by the total vote received, which ( only in this simplified example ) happens to be all the votes. This is {V/(R+1)}/V which equals the ratio 1/(R+1).
The second preferences of all the votes, for the first elected candidate, transfer at a value of: 1 - 1/(R+1) = R/(R+1). Suppose again the simplest possible case that all the voters now prefer the same second choice of candidate. Then all the votes are transferable at a transfer value of R/(R+1) to that candidate - normally, that transfer value would be the going rate for second preferences split between several candidates.
The initial transfer value was one. ( No-one could have a keep value before the first round. ) The first elected candidate received votes at full unitary value. The second elected candidate also gets to keep a quota of the vote. His keep value is 2/(R+1). So, the remainder of the votes' value is transferable, to third preferences, at a correspondingly reduced value of (R-1)/(R+1) for the transfer value.
If all the votes then went to just one third prefered candidate, that person would keep a third quota's worth of votes, giving a keep value of 3/(R+1). This is the same as R/(R+1) if there are just three seats to be filled in the constituency by three representatives or R = 3. This simplest case scenario shows proportional representation by the Droop quota expressed as a keep value.
Note that there is a remaining transfer value of 1/(R+1) that has no-where
else to go. This represents a remaining proportion of votes that go
unrepresented. ( It is possible, tho most unlikely, that four candidates
might all tie for the quota. In that case, the one who got the short straw,
or other form of random exclusion, would break the tie. )
Binomial STV may to some extent redeem this short-fall of representation
because it also takes into account unelective keep values of more than
unity.
Suppose a system of controlled counts which permits of both positive and negative final transfer values of 1/(R+1). That is a surplus transfer value and also a deficit transfer value. Suppose just two such controlled counts in which an average candidate ends up with their corresponding keep values, namely the traditional proportional representation credited to transferable voting, R/(R+1), and also (R+2)/(R+1). The two values are multiplied for an over-all keep value, giving a proportional representation of R(R+2)/(R+1)². My reform of STV suggests that the proportionality of transferable voting may not be as limited as has been assumed.
Going back to my efforts in the early 1990s, this was the notion I was toying with, namely an increased proportional representation from R/(R+1) to R(R+2)/(R+1)². But I didnt know where the compensating ratio (R+2)/(R+1) could come from just in relation to PR within a constituency, until I thought of it as an unelective keep value ( from a correspondingly deficit transfer value ).
If the Binomial STV count does increase the proportion of representation, thru the augmenting of ( elective ) keep values with unelective keep values, this would be a significant answer to critics of STV that the Droop quota is not proportional enough for them. ( This likely advantage is one I omitted to mention from those given towards the end of my page on the Retransferable Vote. )
My page on an electoral model of special relativity shows that the square root of R(R+2)/(R+1)² is equivalent to the contraction factor if light speed constant, c, is taken at unity ( as is usual in physics calculations ) and the relative velocity is 1/(R+1). This would be the ratio for the Droop quota if c was total votes in the constituency set equal to one. The contraction factor, electorally considered, contracts the proportional representation R(R+2)/(R+1)² to a value equal to itself, namely the square root of R(R+2)/(R+1)² which is slightly greater proportional representation.
It was explained how the Michelson-Morley calculation gives a simpler form of the Lorentz transformation which essentially relates two observational times by the contraction factor. These times ( t' and t ) were the greatest and least delays in times estimated as caused by light speed under supposed ether medium "wind" or "stream" effects more or less slowing down light. But the times were always the same in experiments, so light speed had to be constant.
Also was explained how the two times could be given electoral equivalents
in the inverses of the proportional representations R(R+2)/(R+1)² and its
square root. In which cases, they were considered as maximum permitted votes
per constituency across a uniform-seat constituency system. The latter was a
smaller maximum implicit to a smaller variation permitted about the required
average votes per constituency in a uniform constituency system.
The contraction factor could contract the larger maximum to the smaller
maximum, for equality of representation between two constituency systems,
with their larger and smaller permitted variations in proportional
representation across constituencies.
These two ( greater and lesser ) maximum constituencies could be considered in terms of ( more or less ) unelective keep values. The just elective keep value of unity relating to the required average votes per constituency, V, set at unity. Permitted less-than-average constituencies would then have keep values of less than one, being more or less over-representative.
The same two ratios of representation ( R(R+2)/(R+1)² and its square root ) were related to the contraction factor to equate PR across constituencies between two systems with greater and lesser permitted maximum constituencies. This relation can be interpreted also in terms of proportional representation within constituencies ( another dimension of PR ) since Ive extended the keep value in meaning to include unelective keep values greater than unity.
An average representative might have elective and unelective keep values, of R/(R+1) and (R+2)/(R+1) respectively, which are multiplied for a final keep value of R(R+2)/(R+1)². This is elective, being less than one. As this is the average, it represents the degree of proportional representation in the constituency. Suppose another constituency similarly arrives at an average keep value of the square root of R(R+2)/(R+1)². This happens to be the same as the contraction factor, which would have to be multiplied by it, to equate to the former value. This means an equation between differing proportional representations within two ( otherwise equal ) constituencies.
A constituency with an elective keep value of R/(R+1) reflects the PR given by the Droop quota. And there is the corresponding unelective keep value of (R+2)/(R+1). As a multiple of the former ratio it suggests a higher PR for the constituency. Applying the contraction factor to these ratios suggests another constituency with a higher PR still. But this "relativist" treatment of PR within constituencies raises a question, because the elective keep value is no longer the PR corresponding to the Droop quota: not R/(R+1) but the square root of this quantity. If R was four representatives, the four-fifths PR of the Droop quota would be contracted to a ratio of about 2/2.25 or about 4/4.5.
This latter ratio means that the quota in the multi-member constituency would not be one-fifth to be won by four prefered candidates but one over 4.5, to be won by four prefered candidates. The Droop quota gives the least number of votes that candidates need to win to be elected before any others. The latter quota would mean more votes have to be won by a candidate to be elected. This makes it more difficult for all four vacancies to be filled.
Also, the square root of (R+2)/(R+1) is an unelective keep value closer to one, the elective keep value, than simply (R+2)/(R+1). That is about 1.1 compared to 1.2 for R=4. Over-all R(R+2)/(R+1)² gives a keep value of 24/25 or .96 and its square root gives about .98. One can think of this as corresponding to a higher PR, because a higher final keep value means a smaller final transfer value of votes wasted for no-where else to go when all the seats are taken.
The same ratios could be interpreted a third time for proportional representation between constituency systems. This is for the third of three dimensions of PR with respect to constituencies compared to three dimensions of space.
The traditional formula for the transfer value used in transferable voting is total of votes received by a prefered candidate minus the quota, with the difference divided by those total votes. That is: {V- (V/(R+1)}/V. But that formula only applies to a surplus transfer value. The deficit transfer value is {V+ (V/(R+1)}/V. These give, respectively, VR/(R+1) and V(R+2)/(R+1). They can be considered as positive or negative changes, by one quota, in the required average number of votes per constituency, V, in a constituency system. Here the quota is the same size as the Droop quota, to keep the arithmetic the same with previous examples of PR across and within constituencies. But the quota is not here being used as the standard elective quota in STV. Here it is simply a way of decreasing and increasing the required average vote by a quota that is just a standard number of votes.
The change from the smaller votes per constituency, VR/(R+1), to the votes per constituency, V, to V(R+2)/(R+1) is an arithmetic progression by the same amount at a time, V/(R+1). The arithmetic mean of these three quantities is the smallest plus the largest, all divided by two, which equals the mid-term V.
But it is also possible to take another average of these end-points,
called the geometric mean. This assumes that the series is not in arithmetic
progression, as above, but in geometric progression.
The contraction factor is ( 1 -
v²/c² ) where v is relative velocity between observers and c is the constant
light speed. This can be considered in terms of a geometric mean, with
respect to end-points ( 1 - v/c ) and ( 1+ v/c ). ( Their arithmetic mean is
one, which is obtained by adding them and dividing by two. ) This is analgous
to the previous electoral ratios R/(R+1) and (R+2)/(R+1).
In mechanics, time ( and also mass ) are inverse to distance. Velocity is distance divided by time. That is distance multiplied by the inverse of time. In comparison, an election might consist of three dimensions of PR with respect to constituencies. The concept of the keep value might be extended from PR within constituencies to PR across constituencies and PR between constituency systems, as the analog to three space dimensions. The analog to time would be the inverse keep value derived from the exclusion quota count.
At first, I thought this was fanciful. One seeming problem was that exclusion counts with inverse keep values only seemed to apply to the Binomial STV count for the election of candidates in a constituency. That is the one dimension of PR within constituencies. That would be analgous to linking time to only one dimension of space. After a while, I decided that the Binomial STV election and exclusion procedure could be mimicked in the other two dimensions of PR.
For instance, for PR across constituencies, there could be a sort of
election and exclusion procedure for constituencies as "candidates" for
inclusion or exclusion in the permitted range of variation about a required
average number of constituents per constituency. Likewise for PR between
constituency systems. Each system has its average constituents per
constituency. Some of these constituency systems may be included or excluded
according to whether their average constituencies are inside or outside
permitted variations about an over-all constituency systems' average
constituency. This over-all average constituency would differ according to
the measure of the average, for example whether arithmetic mean or geometric
mean.
The truer average taken, the most representative item in a distribution,
depends on the nature of the distribution, arithmetic or geometric
distribution or whatever.
Having an exclusion as well as an election count for all three dimensions
of PR suggests three dimensions of time to three dimensions of space, which
could be a little awkward for an analogy with mechanics.
These confusions may sort themselves out, with the later more precise algebra
of the section below, from which this page takes its title. The discussion
below, in this section, suggests the "arrow" of time has a voting analog in
the direction of election and against the direction of exclusion. ( The
concept of time is notoriously difficult in physics. Even the far-fetched
possibility of more than one dimension of time continues to be investigated.
)
The whole counting process could be considered the other way round, as designed for exclusion. Elections could be considered as exclusions, say of incumbents or the officials that voters wanted out of office. ( By the way, this is being considered only for purposes of speculation, not as a political platform. ) The proportional representation of exclusion could then be fairly determined over the three dimensions with respect to constituencies, as before. Three exclusion keep values would then be the analog of space. Then, one election keep value might compare to time.
Minkowski's Interval of the 4-D Pythagoras' theorem, also, can be written two ways round. The so-called "signature" is either the sums of three distances squared minus the time squared, or, the time squared all minus the sum of three distances squared. That is provided one does the same for different observers of the given event, with which they share the same space-time measurement or Interval, whichever way round they agree to take it.
Binomial STV even mimicks a peculiar character of physical laws, which dont distinguish our perception of the "arrow of time". Brian Greene's second popular work, "The Fabric of the Cosmos" includes a thoro explanation of how far physicists have rehabilitated the common sense direction of time into their science. Here, I just want to mention the most basic terms in which popularisers allude to the "arrow of time" difficulty, for the purposes of pointing out an electoral analogy.
On my page, Basic concepts of mechanics and elections, I compared the probabilistic nature of determining representation with the probabilistic nature of time's arrow. Voting methods have been treated like theorems for achieving certain results. Experience shows that electoral system design is less about achieving mathematical certainty and more like improving a machine whose efficiency gains at the expense of simplicity.
A film shows spilt milk, on the floor, gather itself like rocket fuel into a reassembling glass bottle rocketing right way up onto a table. From the view-point of physical law, this is a perfectly possible thing to happen. It is just that to engineer something so complex is practicly impossible. So, we know by experience that the film has been run backwards.
The bi- in Binomial STV refers to two factors preference and unpreference. Binomial STV, as a method, works equally well one way or the other, either as an election count or an exclusion count. And its possible progress in two directions does help-on a decisive result. But people, in the much greater complexity of their every-day lives, dont elect and exclude in equal measure. Consider how hopelessly impractical it would be to get-on with our lives by first excluding all possibilities but the one we intend to take next. We would never get anywhere. We have to make positive choices.
I suspect that physical laws resemble Binomial STV, in that it would work well for the well-defined and limited conditions of elections. But taking the human condition as a whole, the Binomial STV count's treatment of election and exclusion on equal terms, is not practical. Instead, there is an asymmetry between election and exclusion, with the balance of probability in favor of election as the mode by which people mostly operate. Time's arrow, so to speak, is with election.
New Labour's Britain has politicly correct campaigns against "exclusion" of minorities. And it must be admitted that in some respects, we are all minorities. ( STV systems allow all minorities - and not just party supporters, usually all more or less political minorities - to prefer their candidates for proportions of the vote, if they so desire. )
The traditional assumption, just criticised, is that for any given group
of candidates, a certain set of these will be the voters' chosen
representatives. Rather, an election system must be recognised as having
efficiency limits, so that more than one election may have to be held to
bring the number of candidates down to a required number of representatives.
Those limits may be provided conveniently by the condition of local
primaries. Tho, locality is relative and primaries could be conducted, where
appropriate, up to federal government level. Whereas, present practise still
uses the principle of local representation to stifle elections with
monopolistic single member constituencies.
Primaries, without being sorted candidate-wise by the locality principle,
pose logical problems about how to ensure all the candidates are fairly
pitted against each other, so that none are excluded without being proven
less popular than the eventual winners.
By simple algebra, the Lorentz transformations can be re-stated as transformations between geometric means of time ( and of space ) over related ranges of velocities. These can be modeled in terms of a prototype transferable voting method. This uses geometric mean, or some other average of election keep values and their inverse, exclusion keep values ( analgous to space and its inverse, time ). Thus, the count can be considered as a sort of elective and exclusive averaging of the votes ( appropriate to their form of distribution - the geometric mean averages a geometric distribution ).
To show how this is done, we start with the standard forms of the Lorentz transformations. Many writings, including pages on this web-site, have explained them. Take the Lorentz transformations of time. These are two complementary formulas by which two observers transform or translate their local times ( t' and t ), which will differ significantly with respect to an observed event at velocities significantly approaching the speed of light.
The observers' respective velocities are u' and u. In classical physics of the Galilean relativity principle, their relative velocity, v, is simply the difference between their two velocities. Coming to measure velocities of a significant fraction of light speed, a revised formula was needed to accomodate the fact that no combination of a velocity with light speed could ever exceed light speed, c ( for constant ). If light speed was set at unit distance traveled per period of time, ( c = 1 ) then a light beam carrier itself traveling at one-quarter that distance per time period, could no-how effect a combined speed of one and one-quarter, or any speed in excess of light.
The Lorentz transformations of velocities are usually derived from combining the transformations for space and for time. First, the time transformations, numbering the first equation, (1):
t = t'( 1 - u'.v/c² )/F
where F is the so-called contraction factor: ( 1 - v²/c² ).
Conversely, equation ( 2):
t' = t( 1 + u.v/c² )/F.
The times, t' and t, can be shown as geometric means, that transform into each other, by combining their converse equations. An average implies a range of items or values that it is the average of. Therefore, the trick is to express the Lorentz forms as such ranges.
Expressing the times, t and t', as a ratio means that the contraction factor, F, in both their equations, can be cancelled. Hence (3):
t/t' = t'( 1 - u'.v/c² )/t( 1 + u.v/c² ).
Therefore (4),
t²/t'² = ( 1 - u'.v/c² )/( 1 + u.v/c² )
= { 1 - (u'.v)/c}{ 1 + (u'.v)/c}/{ 1 - i(u.v)/c}{ 1 + i(u.v)/c}.
The numerator and the denominator of the equation have both been
factorised in terms of one plus and one minus the ratio of (u'.v)/c and i(u.v)/c, respectively.
In the latter case, i means the square root of minus one, to make possible
the factorisation of the denominator.
These four terms ( each to the left hand side of one, in the above equation ) may be given a name, which I'll call "transfer values". The arithmetic of transferable voting uses essentially these arithmetic forms. One plus the transfer value equals the keep value, in a standard count of elective keep values. I have extended this form to a novel transferable voting system that would also make use of unelective keep values, given by: one minus the transfer value.
If we were taking an arithmetic mean of these two ranges plus or minus
one, then their average would be estimated in both cases to be just one. This
assumes a constant ( "arithmetic" ) increase in values from the given lowest
to highest terms in the range. The arithmetic mean is calculated by adding
the highest and lowest terms and dividing by two, resulting in an answer of
one.
This taking the arithmetic mean also works for the denominator even tho the
square root of minus one is involved, because the two imaginary terms cancel
each other out. They also cancel to allow taking the geometric mean.
( The weighted arithmetic mean and weighted geometric mean take into account all the distribution values, between the highest and lowest values, if available. )
In this instance, the arithmetic mean is not a suitable average to take, because we are not dealing with constant increases in velocity. Instead, the Lorentz transformations are adapted to also take into account geometric distributions, one way or another. The cumulative decrease in the rate of velocity increase of an object, corresponds to its cumulative increase in mass, as light speed is approached. A geometric mean is more suited to measure the average of the distribution of high velocities as they approach light speed.
Before showing the working of Lorentz transformations as of geometric
means, it is worth mentioning ( as foreshadowed, in my early paper, "The laws
of motion and election" ) a similarity between the mathematical structure of
proportional representation and special relativity.
Elections that require representatives to have more than their own vote never
achieve completely proportional representation. In fact, the ratio of
representation, using the Droop quota, is VR/( R + 1 ), where V is the number
of voters and R the number of representatives. Obviously, the more
representatives and voters, the closer the ratio of representation, R/( R + 1
), to unity. But it will never quite get there.
Much the same is true in special relativity. As an object increases velocity to a significant fraction of light speed, its ever greater increase in mass will drag ever more heavily against achieving light speed. Only at infinite mass, could the body achieve light speed. And the ratio of representation, R/( R + 1 ), would need an infinite number of representatives to be considered absolutely complete representation.
Self-representation means everyone only needs one's own vote for oneself, for everyone to be fully represented. Self-representation is an ultimate proportional representation, which yet requires no increasing "mass" of voters to bring it closer to full representation: it is already there without any mass of voters. This is mathematicly akin to a light beam, which, having no mass, moves at an ultimate constant velocity.
Thus, with ( proportionly ) representative democracy, there are
diminishing returns to putting more energy of representation, or more and
more representatives, into an election, for fuller representation. And the
pattern of this diminishing returns distribution is most representatively
measured by an average such as the geometric mean.
Whereas, the arithmetic mean averages for a distribution of values that show
a constant increase: this plainly does not fit the facts of proportional
representation -- except in the instance of self-representation, where each
extra self-representative measures or "represents" a constant increase in
fully proportional representation.
In the above section on the deficit transfer value and unelective keep value, it was suggested how the latter could be multiplied by the conventional elective keep value. This is a form of averaging. ( Taking the square root of that multiple would be akin a geometric mean keep value. ) As explained above, this novel procedure would be more proportional than conventional PR using the Droop quota, with just elective keep values, because less transfer value of votes is left over, at the end of an over-all keep value count, untransfered to representatives of choice.
The geometric mean can be found by multiplying the end values of a distribution range and taking their square root. In equation (4), of t²/t'², the numerator and denominator are both multiplications of end values: the two factors in curly brackets are the lowest and the highest velocities. Hence, the square root of the numerator gives a geometric mean, as does the square root of the denominator. But these respectively, are equal to t and t', which are, therefore, geometric mean times.
That is (5):
t/t' = ({ 1 - (u'.v)/c} { 1 + (u'.v)/c})/( { 1 - i(u.v)/c}{ 1 + i(u.v)/c}).
Similarly, for Lorentz transformation of geometric mean distances.
(6):
x = t'( u' - v )/F and x' = t( u + v)/F.
(7):
x/x' = t'(u' - v )/t( u + v ) = ut/u't' =
(8):
( u' - v ){ ( 1 + i{uv}/c)( 1 - i{ uv}/c)}/( u + v ){( 1 - {u'v}/c) ( 1 + {u'v}/c)}.
Notice that the geometric means of the times, t and t', are inverted for the distances, x and x'. For example, the geometric mean that appeared in the numerator, corresponding to t, in the ratio t/t', now appears in the denominator corresponding to x', in the ratio x/x'. Tho, the geometric means of distances, x and x' ( as times multiplied by velocities ) also have the coefficent factors ( u' - v ) and ( u + v ).
It is possible to get rid of these coefficient factors in the ratio x/x' for an equivalent form, as follows.
From the ratios of the Lorentz time transformations and velocity transformations, (9):
t²/t'² = {( u' - v )/u}/{( u + v )/u'}.
Therefore (10):
t/t' = ( {( u' - v )/u}/{( u
+ v )/u'}).
And (11):
x/x' = ut/u't' = ( {( u' - v
)u}/{( u + v )u'})
= {( u' - v )/u'}/ {( u + v )/u}
(12):
= {( 1 - { v/u'})( 1 + {v/u'})}/ {( 1 + i{ v/u} )( 1 - i{v/u} )}.
The above expression for x/x' is a simple ratio of a geometric mean in the
numerator and in the denominator.
Thus, a ratio of velocities, u/u', could be expressed as the x/x' ratio of
distance geometric means, divided by the ratio of time geometric means, t/t'.
That is, u/u' equals equation (12) divided by equation (5). Or equivalently,
equation (8) divided by equation (5).
In terms of velocity, time is the inverse of distance, or v = x/t. And if
an electoral analogy is to consider distance and time in terms of keep
values, then there must be inverse keep values, for the analogy to hold good.
Indeed, it is possible to calculate a transferable vote not only with
election keep values but with exclusion keep values. The latter may be
inverted to give an elective effect, which is then multiplied by the election
keep values to give an over-all election result.
This simple principle but complicated arithmetic is exemplified on my page
about Binomial STV, which gives an example of a second order count. Another
page gives the simplest version, a first order binomial STV count, with just
two unqualified counts: one election count and one exclusion count. There-by,
the election keep values of the candidates are multiplied by their inverted
exclusion keep values for their resulting keep-values.
When the Lorentz transformations were treated as ratios of one to another observer's values of time ( or distance ) this cancelled the contraction factor found in both observers' versions of the Lorentz transformation for time ( or distance ). But the contraction factor itself has the form of a geometric mean. This first alerted me to the possibility of "A statistical basis for special relativity".
Frankly, I was puzzled on re-reading that page, written a year or so previously. It gives a different statistical interpretation to special relativity, in that observers of the relatively wider range of observations take a harmonic mean and observers of the narrower range, take a geometric mean, the two averages being related by the contraction factor. This rigmarole puzzled me at the time. Perhaps, it is another way of giving this page's procedure, which "simply" gives the Lorentz transformations as of geometric means determined by their observers' related but different ranges of observation of the same event. As these observers' ranges may both be considered as geometric distributions, this latter formulation may be more intelligible.
The contraction factor, F, comes into our present reckoning, when a special case for the Lorentz transformations of time is considered. That is when u' =v and therefore u = 0.
From equation (1), t = t'( 1 - u'.v/c² )/F. When u' = v, t = t'( 1 - v²/c² )/F. Therefore, t = t'F, because F is the square root of ( 1 - v²/c² ). Taking the ratio t/t' = F/1, if t is taken to be F as geometric mean, then the value of t' = 1. There is no geometric mean for t' because a mean or average must be an average of a range. And the velocity, u being zero or at rest, does not furnish a range at all -- there is no question of a velocity ranging diminishingly towards light speed.
In other words, equation (2), t' = t( 1 + u.v/c² )/F, reduces to t' = t( 1 + 0 )/F when u = 0. Then the ratio of equations (1) to (2) for t/t' can no longer be expressed as the ratios of two geometric means but instead as only one geometric mean, in the familiar guise of the contraction factor.
Geometric mean Lorentz transformations imply a radical re-interpretation of special relativity, which assumed that the Lorentz transformations are just a minor formula adjustment of classical mechanics, in which space and time are calculated unambiguously, tho there may be statistical errors in their experimental determination.
Instead, the Lorentz transformations are of geometric means of space, time and velocity. ( This "kinematics" has a dynamic version, whose Lorentz transformations, in terms of mass, energy and momentum, follow closely proportional forms. ) So, special relativity may be rendered as a statistical theory. ( An above-mentioned page gave my first inklings of this, which shows how averages, arithmetic mean, harmonic mean and geometric mean seem to be implicit and perhaps more basic to relativity than classical determinism. )
Electoral science of proportional representation by transferable voting may help to understand the meaning of special relativity interpreted in statistical terms. With prototype transferable voting system, an election count can be considered as an averaging of controled re-counts. For instance, my "Binomial STV" derives its name from using a logical or non-commutative expansion of the binomial theorem to determine the sequence and composition of each re-count in terms of preference or unpreference. ( These are the two factors that put the bi- in binomial. )
These re-counts give an average of results, for the candidates, which are expressed in keep values. These over-all keep values use more information than the current practise, which only expresses elective keep values. I also used unelective keep values, by extending the simple arithmetic to consider the case of deficit transfer values as well as surplus transfer values. This could be done with my prototype STV because I was taking averages of results, such that a candidate might get an elective keep value in one count and an unelective keep value in another run of the count.
What matters is whether, on average, the over-all keep value, for each candidate, comes out elective or not. Essentially the same applies to the re-runs that are exclusion counts of unpreference rather than election counts of preference. Except that the exclusion keep values are inverted to give them an effective elective value. Thus, a candidate may "fail" to get an exclusion keep value of less than unity. But an exclusion value of more than one, when inverted for its elective value, is less than one. And election keep values of less than one are elective.
After this fashion, I multiplied the counts of election keep values and
exclusion keep value inverses for the respective candidates to determine
whether their over-all keep values, for the final count, were more or less
than unity, unelective or elective, respectively.
In the case of Binomial STV, I didnt ( till 2 december 2005 ) take the square
root of a candidate's two keep values, which I had multiplied together, thus
treating it as a sort of child of two parents. Such a way of taking an
average, of the range between a candidate's keep values, is called the
geometric mean.
I came to the conclusion that different logical analyses of how to conduct count re-runs, other than Binomial STV for instance, could make a difference, if usually marginal. Also, the logical principle underlying the system of re-runs could make a difference whether, or not, taking square roots, of multiplied keep values, can make a difference, however slight, to the final keep values of the candidates, and the apparent level and order of popularity they show.
In that case, and where there was a range of more than two keep values per candidate to average, then that average might be more correctly the weighted geometric mean. In short, re-count analysis soon complicates complications and becomes strictly for computers.
Asking why are there different logical principles of a count is like asking why are there different distributions to the binomial distribution. As explained at the end of my page on Binomial STV, a count is like a theory to explain the test of a vote and a theory must sometimes be modified to fit the facts. Theoretical physics may only be so comprehensive as it takes into account all the choice possibilities of experimentation. Relativity and Quantum mechanics are notably theories to cover the observational choices that experimenters can make.
Thus, the theoretical calculations of physical results, as classicly supposed of Relativity, do not so much arrive at some determinate value. The classical emphasis on a unique determination of values misses a more important point, the finding of representative values of one's range of choices with respect to other points of view. Statistical measurement of ranges of choice is not the inferior law or measure to classical determinism, which precludes choice. Experimental physics may be likened better to representative elections or tests of popular points of view. The representative is not some logicly or mathematicly necessary conclusion to an electoral system, which is determinate only with the help of pre-conceptions. Rather, a best estimate is made, in the sense of taking the most appropriate average of the spread of votes.
Special relativity, then, may be considered as transformations between observers' results, which average relative ranges of observations. The physical values, that such observers calculate, are their respective representative values, depending on how they choose to observe the same event in relation to each other, ranging from sooner or later, nearer or further, faster or slower.
Special relativity uses the Lorentz transformations and in particular the Fitzgerald-Lorentz contraction as a sort of mathematical scaffolding that uniquely configures or determines the structure of the "house" or architecture of high-energy physics. But the statistical interpretation of special relativity with geometric mean Lorentz transformations and contraction factor may be compared rather to a more versatile and convenient mobile elevator that may reach other inaccessible high points.
The statistical interpretation is not an ad hoc mathematical scaffolding of no meaning beyond its application to special relativity, rather it is a general purpose mathematical structure - the statistics of ranges of choice - applicable, for instance, to the method of proportional representation by transferable voting, given an averaged count of systematicly controled counts of the preference vote ( including its reverse ), like "Binomial STV".
Richard Lung.
21 october 2005;
minor amendment, 2 december 2005.