Consequentialism -X- (Pareto Efficiency) -X- Deontology

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Let us check the Polity to begin with:

1. N is the set of all individuals in society.

And that which their politics concerns – the state of society.

2. S is the set of all possible information contained within society, so that a set s ∈ 2S (2S being the set of all possible subsets of S) contains all extant information about a particular iteration of society and will be called the state of society. S is an arbitrary topological space.

And the means by which individuals make judgements about that which their politics concerns. Their preferences over the information contained within the state of society.

3. Each individual i ∈ N has a complete and transitive preference relation ≽i defined over a set of preference-information Si ⊂ S such that si ≽ s′i can be read “individual i prefers preference information si at least as much as preference-information s′i”.

Any particular set of preference-information si ⊂ Si can be thought of as the state of society as viewed by individual i. The set of preference-information for individual i is a subset of the information contained within a particular iteration of society, so si ⊂ s ⊂ S.

A particular state of society s is a Pareto efficient if there is no other state of society s′ for which one individual strictly prefers their preference-information s′i ⊂ s′ to that particular state si ⊂ s, and the preference-information s′j ⊂ s′ in the other state s′ is at least as preferred by every other individual j ≠ i.

4. A state s ∈ S is said to be Pareto efficient iff ∄ s′ ∈ 2S & i ∈ N : s′i ≻ si & s′j ≽ sj ∀ j ≠ i ∈ N.

To put it crudely, a particular state of society is Pareto efficient if no individual can be made “better off” without making another individual “worse off”. A dynamic concept which mirrors this is the concept of a Pareto improvement – whereby a change in the state of society leaves everyone at least indifferent, and at least one individual in a preferable situation.

5. A movement between two states of society, s → s′ is called a Pareto improvement iff ∃ i ∈ N : s′i ≻ si & s′j ≽ sj ∀ j ≠ i ∈ N .

Note that this does not imply that s′ is a Pareto efficient state, because the same could potentially be said of a movement s′ → s′′. The state s′ is only a Pareto efficient state if we cannot find yet another state for which the movement to that state is a Pareto improvement. The following Theorem, demonstrates this distinction and gives an alternative definition of Pareto efficiency.

Theorem: A state s ∈ 2S is Pareto efficient iff there is no other state s′ for which the movement s → s′ is a Pareto improvement.

If one adheres to a consequentialist political doctrine (such as classical utilitarianism) rather than a deontological doctrine (such as liberalism) in which action is guided by some categorical imperative other than consequentialism, the guide offered by Pareto improvement is the least controversial, and least politically committal criterion to decision-making one can find. Indeed if we restrict political statements to those which concern the assignation of losses, it is a-political. It makes a value judgement only about who ought gain (whosoever stands to).

Unless one holds a strict deontological doctrine in the style, say, of Robert Nozick’s Anarchy state and Utopia (in which the maintenance of individual freedom is the categorical imperative), or John Rawls’ A Theory of Justice (in which again individual freedom is the primary categorical imperative and the betterment of the “poorest” the second categorical imperative), it is more difficult to argue against implementing some decision which will cause a change of society which all individuals in society will be at worst indifferent to. Than arguing for some decision rule which will induce a change of society which some individual will find less preferable. To the rationalisitic economist it seems almost petty, certainly irrational to argue against this criterion, like those individuals who demand “fairness” in the famous “dictator” experiment rather than accept someone else becoming “better off”, and themselves no “worse off”.

Finsler Space as a Locally Minkowskian Space: Caught Between Curvature and Torsion Tensors.

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The extension of Riemannian “point”-space {xi} into a “line-space” {xi, dxi} make things clearer but not easier: how do you explain to a physicist a geometry supporting at least 3 curvature tensors and five torsion tensors? Not to speak of its usefulness for physics! Fortunately, the “impenetrable forest” by now has become a real, enjoyable park: through the application of the concepts of fibre bundle and non-linear connection. The different curvatures and torsion tensors result from vertical and horizontal parts of geometric objects in the tangent bundle, or in the Finsler bundle of the underlying manifold.

In essence, Finsler geometry is analogous to Riemannian geometry: there, the tangent space in a point p is euclidean space; here, the tangent space is just a normed space, i.e., Minkowski Space. Put differently: A Finsler metric for a differentiable manifold M is a map that assigns to each point x ∈ M a norm on the tangent space TxM. When referred to the almost exclusive use of methods from Riemannian geometry it means that this norm is demanded to derive from the length of a smooth path γ : [a, b] → M defined by ∫ab ∥ dγ(t)/dt ∥ dt. Then Finsler space becomes an example for the class of length spaces.

Starting from the length of the curve,

dγ(p, q):= ∫pq Lx(t), dx(t)/dt dt

the variational principle δdγ(p, q) = 0 leads to the Euler-Lagrange equation

d/dt(∂L/∂x ̇i) – ∂L/∂xi = 0

which may be rewritten into

d2xi/dt2 + 2Gi(xl, x ̇m) = 0

with Gi(xl, x ̇m) = 1/4gkl(-∂L/∂xl + ∂2L/∂xl∂x ̇m), and 2gik = ∂2L/∂x ̇l∂x ̇m, gilgjl = δij. The theory then is developed from the Lagrangian defined in this way. This involves an important object Nil := ∂Gi/∂yl, the geometrical meaning of which is a non-linear connection.

In general, a Finsler structure L(x, y) with y := dx(t))/dt = x ̇ and homogeneous degree 1 in y is introduced, from which the Finsler metric follows as:

fij = fji = ∂(1/2L2)/∂yi∂yj, fijyiyj = L2, yl∂L/∂yl = L, fijyj = L∂L/∂yi

A further totally symmetric tensor Cijk ensues:

Cijk := ∂(1/2L2)/∂yi∂yj∂yk

which will be interpreted as a torsion tensor. As an example of a Finsler metric is the Randers metric.

L(x.y) = bi(x)yi + √(aij(x)yiyj)

The Finsler metric following is

fik = bibk + aik + 2b(iak)lyˆl − aillakmm(bnn)

with yˆk := yk(alm(x)ylym)−1/2. Setting aij = ηij, yk = x ̇k, and identifying bi with the electromagnetic 4-potential eAi leads back to the Lagrangian for the motion of a charged particle.

In this context, a Finsler space thus is called a locally Minkowskian space if there exists a coordinate system, in which the Finsler structure is a function of yi alone. The use of the “element of support” (xi, dxk ≡ yk) essentially amounts to a step towards working in the tangent bundle TM of the manifold M.