Electoral model of special relativity.

Proportions of representation within and across constituencies.

The basic concepts of physics and elections were compared on one page of this site. Another page presents a case for a statistical basis to special relativity. This page compares special relativity to an electoral system with statistical distributions of constituencies and votes. Statistics suggests that observers are given a chance to choose how they observe a given event.

The electoral system chosen is largely that described on the page about the Ross quota. That involves an electoral system in which proportional representation is considered not only within constituencies but across a whole system of constituencies. ( A further possibility is a PR between constituency systems or changing the number of seats per constituency. )

Proportional representation is by definition within multi-member constituencies. A proportion is an equality between ratios, as of votes per seat. That means at least two ratios and therefore at least two seats in the constituency.

Next is the issue of how many votes per seat. What is the proportion or quota of votes in the constituency that any candidate needs to win to become a representative? Simply dividing votes per seat ( V/R ) is called the Hare quota. But in a single member constituency the Hare quota would require the winning candidate to win all the votes.

We know that the elected candidate wins on just over half the votes. In fact, just half the votes will do. Tho, just two candidates, both with half the votes, would have to resort to a tie-break. So, for practical elections, the Hare quota must be modified from V/R to V/(R+1). This ratio is the Droop quota ( as Droop originally conceived it ).

The more seats in the constituency, the closer the Hare quota becomes to the Droop quota as an elective hurdle for the candidates. But the Hare quota's higher quota is still a marginally higher obstacle to electing the candidates, who are most prefered anyway. This assumes that the voters are given a preference vote, allowing ranked choice of candidates.

Moreover, the Droop quota's lower elective hurdle means that winning candidates have more surplus votes, above the Droop quota than they would have with the Hare quota. These votes are transferable to the next most prefered candidates, helping them, in turn, to be elected. Thus the Droop quota facilitates competitive elections more in accord with the voters' wishes. This system of the proportional counting of preference voting is called the single transferable vote.

The Droop quota only gives a proportional representation, in a constituency, of the number of seats or representatives times the ( Droop ) quota needed to elect each of them. That is R/(R+1) of the votes V, or, VR/(R+1). With the Hare quota, it is possible that in some elections all the voters prefer a single candidate for a seat, or half the voters each prefer two candidates for two seats. As a rule, the Hare quota obliges the voters to express more preference than they may have, for certain candidates to elect enough candidates for all the vacancies.

The Droop quota offers a proportional representation of R/(R+1), which means a loss of representation of 1/(R+1). To offset this loss, there would have to be a counter-balancing PR of total representation plus that loss. That is 1 + 1/(R+1) or (R+2)/(R+1). This doesnt make sense until a constituency is considered in relation to other constituencies in a system of constituencies.

If the system is a uniform-member system of, say, five seats per constituency then they must all have the same number of voters, for the constituencies to be equally represented. For this purpose of allocating equal representation, the Hare quota is used. This equality was compromised by electoral preference, facilitated by the Droop quota, within constituencies.

Also, equality may be compromised across constituencies. That is because communities differ in size. Rather than break them up to make exactly equal constituencies of all of them, some latitude may be allowed to preserve their integrity. I reasoned how this might be done on my page about the 'Ross quota.' This quota V/(R+2) equalled the Droop quota in the smallest permitted constituency and also equalled the margin of difference in votes between that smallest constituency and the average size constituency.

If some constituency is only (R+1)/(R+2) of the size of the original constituency, then it will be over-represented by (R+2)/(R+1) compared to it. The smaller constituency will have the standard PR within constituencies, R/(R+1). But its smallness will give it a comparitively larger across-constituencies proportional representation of the within-constituencies PR. Namely, R(R+2)/(R+1)˛.

From the comparitive size of constituencies, of (R+1)/(R+2) we can give a size for the smallest and average constituencies. Simply multiply numerator and denominator by the same constant, to turn a ratio of representatives into a ratio of the voters they represent. It can be any constant but perhaps the simplest is to multiply above and below by R, for R(R+1)/R(R+2). This gives minimal numbers of voters per constituency. ( The numbers approximate candidates per constituency, as there cannot be less voters than candidates. ) The numbers are not realistic, they are purely for comparing sizes of constituency across a constituency system.

We started with any constituency, having a Droop quota's PR. This constituency may be identified with the average size constituency of R(R+2). The smaller constituency size is R(R+1). The latter may be identified with the minimum permitted size in the constituency system, because it came from considering a counter-balancing PR to the Droop quota's PR.

What is the maximum permitted constituency? Well, the minimum constituency enjoys over the average constituency a greater proportion of representation. The numerator R(R+2), of the amount, equals the average constituency size. The denominator corresponds to some greater constituency size, (R+1)˛. This is a ready-made maximum constituency. Compared to the average constituency, it is under-represented in proportion to its greater number of voters for the same number of seats.

Thus the minimum constituency's cross-constituency PR of the within-constituency PR, compared to the average constituency, equals the across-constituency PR of the maximum constituency compared to the average constituency. The page on the Ross quota described this PR across constituencies.

Suppose there are enough seats on a council to represent the public at 60 voters per representative. This is the Hare quota's measure of the rate of representation. Much larger numbers, like sixty thousand, would be realistic for considering variations in constituency size as a result of the differing geographies of communities. To keep it simple this example is in tens rather than tens of thousands.

Again for simplicity, suppose a system of only two members per constituency. The required average size constituency is 2 times 60 equals 120 voters. The ratio of maximum to average constituencies is (R+1)˛/R(R+2). Multiplied by 120, this works out at 135 voters as the maximum constituency.

The minimum constituency is 120 times R(R+1)/R(R+2) = 120.(R+1)/(R+2) = 90.

Averages and dispersions of proportional representation.

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The idea of taking a counter-balancing PR of (R+2)/(R+1) compared to R/(R+1) was essentially taking an arithmetic mean. If you add the two ratios and divide by two, the result is one, for whole or complete representation. There are other possible averages for these two ratios as end-limits to a range of representation.

In fact, the ratio R(R+2)/(R+1)˛ is the harmonic mean of this range. It is calculated by adding the inverses of the two ratios, dividing by two, and inverting the result.

The other average is the geometric mean, which comes from taking the square root of the product of the two ratios. That is square root{R(R+2)}/(R+1). The geometric mean is smaller than the harmonic mean, so it may signify a smaller range. In our example, the harmonic mean happens to equal the product of the range ratios. This suggests treating the geometric mean, here, as a product of a range from square root{R/(R+1)} to square root{(R+2)/(R+1)}.

A narrower range of proportional representation means less permitted variation in the sizes of the constituencies in a uniform system. The wider range was from a maximum size of (R+1)˛ compared to an average of R(R+2) and a minimum of R(R+1). The terms have changed with respect to the example above, where the basic rate of representation is given as 120 voters per two-member constituency.

Then the maximum constituency was found as the product of 120 by (R+1)˛/R(R+2), which equals 135 voters. The narrower dispersion uses a contracted product of (R+1)/square rootR(R+2), which works out at just over 127 to the nearest whole number of voters.

( The page on the Ross quota suggested a permitted variation in constituency sizes, that was, in effect, the wide dispersion. On the page about the statistics of special relativity, the wide and narrow dispersion products correspond, respectively, to t' and t. )

Finding the minimum vote for the narrow dispersion uses the contracted product square root{(R+1)/(R+2)} of 120, which works out at 104 votes, to the nearest whole number. Both contracted products are square roots of the original products, so that the essential structure of the wide dispersion is preserved in the narrow dispersion.

The ratio, of the maximum constituency's variation from average, over the minimum constituency's variation from average, is essentially the same in the wide and the narrow dispersion. In the wide dispersion the ratio is 1/R. For example, (135-120)/(120-90) = 15/30 =1/2. Compare the narrow dispersion of approximately ( 127-120)/(120-104) ~ 7/16 < 1/2. But as one increases the number of seats per constituency, the ratio of the narrow dispersion converges to 1/R.

The range has been 'contracted', with a 'contraction factor' multiplied by the harmonic mean to equal the geometric mean. Here, then, is an electoral analogy to special relativity. On the statistics of special relativity page, the arithmetic mean was taken of the range (c-v)/c to (c+v)/c. There c was the constant speed of light and v the relative velocity between observers. Suppose two observers are moving away from each other with a certain velocity v and a light beam is going in the direction of one observer and away from the other observer.

Common sense says that the observer moving with the light will see the beam moving at its natural speed less his own velocity, or (c-v). Conversely, the observer moving away from the beam will register a speed of (c+v). ( Dividing both terms by c is simply a way of expressing the magnitude or size of the differences from light speed. ) Common sense is based on classical physics, which says this is so, and gives it the name of Galilean relativity. This relating one observer's velocity to another's is called a Galilean transformation. This works for ordinary experience, which does not have to deal with motion anywhere near the speed of light.

The slightly more complicated Lorentz transformations are more general formulas that also match the findings of experiment for speeds significantly approaching light speed. The page on Lorentz transformations gives an arithmetic example of how observers, in high relative velocity, cannot measure more than a constant velocity for light.

Simply taking the arithmetic mean of the range (c-v)/c to (c+v)/c has the like effect of canceling relative velocity to leave the constant, one, ( which is the constant c expressed in terms of itself as c/c ).

We can compare the physical range (c-v)/c to (c+v)/c with the electoral range R/(R+1) to (R+2)/(R+1). In the electoral case, the arithmetic mean also does a cancelation to arrive at a constant. The marginal under-representation given by R/(R+1) is canceled out by a corresponding over-representation (R+2)/(R+1), derived thru an arithmetic mean of whole or complete representation, at unity or (R+1)/(R+1).

The speed of light is a physical limit, just as complete representation is an electoral limit. The act of choosing some candidates over others means that some voters are going to be more or less satisfied than others, and the limit of complete representation approached rather than attained.
But electoraly possible representation may be measured by other averages than the arithmetic mean. The harmonic mean and the geometric mean average more or less slower velocities than light speed in the Michelson-Morley experiment, if one considers greater or lesser 'ether drags' in terms of greater or lesser statistical dispersions, related thru a contraction factor of the wider to the narrower dispersion.

Constituency distributions in uniform and non-uniform systems.

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On the Ross quota page, it was pointed out that the maximum permitted constituency size varied, from the required average size constituency, less than the minimum permitted constituency varied from the average constituency. The difference was in the ratio of one to the number of seats, R. That implies there has to be R maximum-sized constituencies for one minimum sized constituency. This is needed to balance the over-representation of the smallest constituency, which is greater than the under-representation of the largest constituency.

Consider a binomial distribution of such constituencies, essentialy Pascal's triangle. Take the row: 1, 4, 6, 4, 1, representing a distribution of 16 constituencies. Then weight each term in succession from one to five ( where R = 5 ). That gives a total of 48 constituencies.

Further statistical interpretation may improve the analogy. So far, R, the number of representatives per constituency has been considered for a uniform member constituency system. Every constituency has been regarded as having a given number of seats, three seats per constituency or whatever.

In that case, the constant, c, would be analgous to R+1, and the velocity analgous to one seat, which was either added or taken away from c. ( The constant could be adjusted to unity, as it is in physics. Then, the values for velocity, v, would have corresponding adjustments. ) In special relativity, the velocity can be anything up to light speed. Suppose a non-uniform member system, in which the constant number of seats, to a uniform member system, becomes the average number of seats per constituency.

A non-uniform member system typicly has a binomial distribution of constituencies, with the maximum number of constituencies having an average number of seats. For instance, there might be six constituencies with an average three seats per constituency. One constituency, covering a large, sparsely populated area, might only have one seat. On the other hand, one small, densely populated constituency might be entitled proportionly to five seats. The average, of these two extremes of population dispersion, still works out at three seats per constituency. Likewise, for the inter-mediate constituencies: For those familiar with the binomial distribution ( or Pascal's triangle ), one would expect, also, four two-member constituencies and four four-member constituencies.

With the above example, statistical usage may also cast the distribution of constituencies not for one to five seats but for zero to four seats. A boundary commission would not draw up any area that was too sparsely populated to qualify equitably for any seat. But in terms of a country's geography, there may be such a region, in truth. Such an aberration may be over-looked, when one more seat does not make much difference to a country which returns a large number of members to parliament.

The maximum variation of seats, from the average number of seats, is just equal to that average. For instance, with a distribution range of from zero to four seats, the average is two seats per constituency. The minimum of zero seats and the maximum of four seats are minus or plus two seats per constituency from the average of two seats.

In this way, an average number of seats per constituency fits the analogy with light speed, c, and the relative velocity, v, which may vary up to light speed, just as the variation of seats per constituency may vary up to the average seats.

Conceivably, the distribution range starting at one, rather than zero, might also have value in the electoral analogy with relativity. With a range of, say, one to five, the mid-point three is greater than the limits of variation, plus or minus two. For a much greater range, say 1 to 99, the average, 50, still only exceeds the maximum variation, 49, by 1.
This pattern is reminiscent of the physics of objects that can never quite reach the speed of light. In short, starting one's distribution with zero or one may suit different circumstances, as it generally does in statistics.

Time distributions and representation distributions.

The page about the diffusion equation as a difference equation showed the 'diffusion' of constituency distributions. The diffusion essentialy follows the pattern from adding wider and wider rows to Pascal's triangle. The distribution of constituencies 'diffuses' as the maximum number of seats per constituency increases. The usual equation for diffusion, or the similar process of heat conduction, depends on diffusion in space over time.

The Michelson-Morley calculations are of times slowed in relation to ranges of velocity, on average, less than light speed, unaffected by more or less 'ether drag', or, as later considered, by the relative velocities of observers.
In terms of the just-mentioned diffusion equation, light speed, c, is analgous to the average seats per constituency. The relative velocity, v, is analgous to the variations about the average. The time is analgous to the distribution or diffusion. This distribution can consist of constituencies. It can further consist of the number of seats there are for each class of constituencies in the distribution.
If there are six constituencies with the average of three seats per constituency, then the total of seats in the average size constituencies is 3 times 6 equals 18 seats.

The physical concept of time may be compared to the electoral distribution of diffusion in this way. Time is not just measured by how long it takes to go a certain distance in a straight line. Time can also be measured in terms of circular motion, as round a clock face. How long it takes a wheel to turn on its circumference is called its period. This basic unit of time is then multiplied, by how many times the wheel turns, to arrive at a given length of time. Time, t is defined as n times period P, or t = nP.

Similarly, to the time, the number of seats to a given class of constituencies is its basic 'period', multiplied by the number of this class of constituencies. How many times this class of constituencies is repeated is analgous to how many times circular motion is repeated.

Mass distribution and vote distribution.

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The value of the analogy doesnt end there. My page about the Lorentz transformations ended with a comparison of the kinematic and the dynamic versions of the Lorentz transformations. Dynamics introduces mass into the equations. The point is that the transformations of times are of the same form as the transformations of mass. Special relativity is of observations of local times and proportionately similar local masses.

This similarity can be carried over into an electoral analogy of times with representatives and of masses with voters. The number of representatives should be proportional to the number of voters. We suggested a diffusion equation with velocity as an independent variable. Velocity multiplied by mass gives momentum.

Time, as the dependent variable, may be replaced by mass. ( In this case, mass is usually treated as a multiple of the light speed squared, which defines mass in terms of energy. That is from the celebrated equivalence of energy to mass, in E = mc˛. ) After all, mass is considered in terms of its 'massive' distributions. This is analgous to replacing the distribution of representatives with a ( 'massive' ) distribution of voters. Vote distribution must be with respect to its own independent variable or variables. Take an arithmetic series other than the number of seats or representatives per constituency. One that will do is the number of preferences per voter.

This assumes that the voters are given a preference vote to rank as many candidates, as they please, in order of choice. Some voters may be pleased with less candidates than others. Some voters may not prefer any of the candidates. In that case they come in the first category of the number of voters with no preferences. The next category of voters prefer only one candidate, the next category prefer just two candidates, and so on.

By the way, the voting system, we have in mind here, the single transferable vote, does not count later preferences against former preferences. There is no constraint to skew the distribution of voters towards registering few preferences rather than many. ( Tho, it is possible that other factors might sway the distribution from a regular binomial or multinomial distribution. )

As with the case of seats per constituency, there is the question of whether to start with zero preferences or one preference. An election will count the total number of voters. Subtracting this from the total number of registered voters gives an estimate of the number of constituents who didnt vote: the zero-preference vote. If one just wishes to compare those who do vote, then one starts the distribution at one preference per voter. The number of preferences per voter can go up to the number of candidates.

In general, the distribution for preferences per voter should be multinomial. The binomial distribution really applies only to an all-or-nothing vote, an X-vote, which means a one-preference yes for one candidate and no to the rest. If there are more than two candidates, the X-vote gives a minimum of information about the relative choice between the candidates. As explained at the end of pages on the diffusion equation and on measurement of language and logic, preference isnt merely a binary choice with a binomial distribution but a 'multinary' choice with a multinomial distribution.

The two cases of uniform or non-uniform member constituency distributions have an analog with voting distributions. Non-uniform voting distributions, binomial or multinomial, have been mentioned. A non-uniform constituency system may be transformed into a uniform member system. Take the non-uniform system from one to five members per constituency. If there is one five member constituency, two of its members might be transfered to the one-member constituency, producing two three-member constituencies, as well as the probable six three-member constituencies, previously mentioned for this case of the binomial distribution.

Likewise, for four four-member constituencies, which might each transfer a seat to four two-member constituencies. There are many more voters than constituencies, but the distributions are similar and votes may be transfered from a random to a uniform distribution.

This transfering of the vote resembles the process of an electoral count. Candidates become representatives on achieving a quota or proportion of the votes in a multi-member constituency. Some candidates will have more votes than the quota they need, but this surplus of votes may be transfered, according to their voters' next preferences, to help other candidates achieve the quota.
Thus, the non-uniform distribution of votes is transfered into a uniform distribution of votes, analgous to the non-uniform and uniform constituency cases. Using the constituency example as a mini-election, a spread of preferences, for up to five candidates, is reduced to a spread for up to three candidates, suggestive of electing three out of five candidates.

This description doesnt explain how votes are transfered by the election called the single transferable vote ( STV ). But there is a simple version of the system that transfers random ( therefore probably representative ) samples of a winning candidate's voters, equal to the size of his surplus vote over the quota.

Postscript: An electoral analogy to the Interval.

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In the above page, I compared time and mass, which are directly proportional, to representation and votes, respectively. But in special relativity, distance and mass are inversely proportional. This is shown for distance considered as the radius of a particle, which has greater mass, the more crowded into a smaller radius. For the purposes of an electoral analogy, some term like the "mass density" of votes per candidate may be needed as the inverse of the number of candidates. This suggests the loss of a straight analogy between mass, space and time with voters, candidates and representatives.

On the other hand, energy is more readily given an electoral equivalent from mass as votes per candidate. Very roughly speaking, suppose the speed of light as a maximum is analgous to maximum candidates per representative, which might be treated as per one representative. Then the analogy of energy, in terms of the energy equation, may be something like voters per candidate times the square of the candidates. That is voters times candidates, which may be interpreted as the number of voters times the number of candidates prefered per voter. That is the total number of preferences could be an electoral analog of energy.

The above two paragraphs are just a token recognition of one of the loose ends to the above page. The real reason for this post-script is that Ive come across another electoral analogy with special relativity, or that later treatment of it, known as the Interval ( or Minkowski's Interval ). This happened thru developing a reform of voting method. The single transferable vote has a residual defect in that its count elects, but does not exclude, candidates by proportional representation. To amend this, I proposed a Reversible STV, that counted exclusion quotas of the least prefered candidates, as well as election quotas of the most prefered candidates.

Confusingly, there appeared to be two different ways to conduct a reverse count. Later, I realised that these were indications of four logicly possible ways to conduct a transferable vote. Conventional STV methods calculate a "keep value" for a candidate, which is the proportion of a superfluous vote that the candidate retains, to ensure he keeps just enough votes to be elected on a quota of votes in a proportionly represented multi-member constituency.

The votes a candidate reaches in surplus of a quota are transferable to that candidate's voters' next preferences, in proportion to the size of the surplus. This rate is called the "transfer value". Each voter is given an equal power of election in one vote, so that the sum of the keep value and transfer value of votes for a candidate, greater than the quota, must equal one.

Retransferable voting ( RV ) has to extend the range of keep values and transfer values to cover the case not only of surplus candidates but also deficit candidates. The reason for this is that RV is a balanced-out count between four different ways to transfer the vote, all with keep values and transfer values assigned to every candidate. Some candidates will not be surplus candidates. But that may not matter if a candidate who is a deficit candidate, on one of the four count runs, is a relatively strong surplus candidate on the other runs.

For a deficit candidate, as for a surplus candidate, the keep value plus the transfer value still must equal one, adhering to the democratic rule of voter equality. But a candidate in deficit of a quota, in effect has a negative surplus. This results in a negative transfer value. So, the keep value for a deficit candidate is calculated as one minus a negative transfer value, which results in a keep value of greater than one.

A candidate can only be elected on a keep value of no more than one. A keep value of one signifies that the candidate has no more and no less than the quota of votes required to be elected. However, with retransferable voting, there remains the hope that on another of the four count runs the candidate may have a keep value sufficiently below one, which may make-up for any keep value above one in another count run.

Consider a graph in which the horizontal axis is a scale related to keep values and the vertical axis is a scale related to transfer values. The axes may be in terms of polar co-ordinates, which are size of angle of turn and length of radius of turn. That is on a circle, like a compass with a needle. The compass also has rectangular co-ordinates ( y and x ) axes, that point north-south and east-west. The radius forms the hypotenuse of a right-angled triangle, with horizontal side ( the x-axis ) and vertical side ( the y-axis ).

By Pythagoras' theorem, the square of the hypotenuse equals the sum of the squares of the other two sides. Let the radius length represent one vote. As one vote squared, it still equals one. For this to equal keep value plus transfer value, the x-axis must be set at the square root of the keep value, and the y axis must be set at the square root of the transfer value.

This brings us to the situation of Minkowski's Interval in special relativity. It has been written about thousands of times. I have described it on other web pages. Essentially, the Interval consists of Euclid's geometry of three dimensions of space, with time treated as the fourth dimension. Time is space-like except it has an "imaginary" number, i = √-1, for a coefficient. This means that when time is squared, the result is a negative number.

Thus, Pythogoras' theorem applies to Minkowski's Interval, as it does in Euclid's geometry, but with this peculiar reverse, that the square of time, on the fourth dimension, involves a subtraction rather than the addition of the squares of the spatial sides, in order to obtain the value of the hypotenuse squared. The Interval is this hypotenuse squared. ( Physicists dont usually bother to take its square root. ) The point or purpose of the Interval is that, as a combined measure of space and time or "space-time", it is the same for every observer ( given special relativity's restriction to "uniform translatory motion" ) of an event.

The Interval supplies a democratic equality and freedom of observation of an event: whatever the difference ( which may be freely chosen ) of each observer's spatial and temporal measurements, all observers equally find the same measurement, the Interval, when their measurements are space-time combined. Whereas the Interval is a result for physics, its equivalent in electoral science is a first principle. Thus the vote is given in equal measure to all possible observers of candidature. The one vote is the "Interval" measurement that all voters observe. All voters split up their vote into different fractions of keep values and transfer values for different candidates, akin to the different space and time measures between differently situated physical observers.

And just as the square of a time measurement is negative, so deficit candidates produce a negative transfer value. Both special relativity and its electoral analogy are about limits, whether the limiting speed of light or limiting proportions of votes that candidates may find themselves in deficit of, but which no candidate may be in surplus of, without the surplus being transfered away.

It should be possible to refine the analogy between the Interval and transferable voting in its above-mentioned reform. There seems to be no reason why the keep value for a candidate need be confined to one dimension, usually the dimension of a political election. For example, candidates could stand for election to either the first or the second chamber of government. Suppose the first chamber was the dimension of political representation and the second chamber the dimension of economic or occupational representation. As well as expressing their order of preference for candidates, the voters could put those preferences on either the political or economic dimension.

The total value of each vote would still be one. But there would be an economic, as well as a political, keep value for each candidate, together with the transfer value, to make up the one vote.

One can extend this argument to a third dimension of parliamentary representation, say an education parliament, which might include training, retraining and research responsibilities of representation. Or, we might quite naturally consider parliamentary representation in three dimensions of science. There could be a parliament or discussion of theory, a parliament of experiment, and an operational parliament of translation or interpretation of the language of theory into its experimental meaning.
Notice that "translation" literally means to carry over, in which spatial dimensions are implicit.

I am not advocating this system. ( The system I presently advocate is the political first chamber and an economic or vocational second chamber both elected by transferable voting, the latter perhaps while unions and professions and other occupations hold the elections to their representative bodies. ) I have not even fully spelled out the voting mechanics of three dimensions of keep values and one transfer value dimension. But this electoral analogy with the Interval, as four-dimensional Euclid, should be possible.

Having advocated two successive reforms to STV, namely, Reversible STV and the Retransferable Vote, I suspected that these systems might be the first few of a series of approximations, to a more accurate count of the transferable vote, if such were required. And I was able to show how this might be the case, on my page entitled Binomial STV. In principle, this count is simple but the calculations are much more involved than ordinary STV and would need to be programmed to be of any practical use.

My use of the binomial theorem is not the usual expansion by the rules of arithmetic, as is used to derive the famous approximation E = mc², but an expansion by ( non-commutative ) rules of logic.

I believe there is a moral for physics in general, including relativity, to be derived from Binomial STV ( following from my discussion in the last section of that web page ). The point about the binomial series is that it is just one of any number of possible distributions, to base the count on, where they are applicable to any possible number of patterns of voting.
Thus, the count can be considered as a mathematical structure or theory of an election result as a test or experiment of the vote as the people's wishes.

In turn, science may be thought an "election" where democratic considerations of free and equal choice in experimental observations all count towards standard theoretical laws. ( In election studies, there is, for example, a law of majority rule, or rather its rational generalisation by the Droop quota into multi-member majority rule. )

As C S Lewis pointed out, the concept of natural law for science is taken from jurisprudence. Seeing science in terms of election law makes explicit an ethical relationship that believers in natural determinism seemed to borrow despite themselves. This is perhaps seen in the law of natural selection and in the bracketing of relativity theory with "classical", meaning deterministic, physics.
But the commonplace fact is that knowledge and freedom are inseparable. Perhaps, if scientists recognised this more fully, their science would be truer, and people's right to free speech would be more assured against a dictatorial determinism.

Richard Lung
11 october 2003; postscript 30 november 2004.

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