Welfare Economics, or Social Psychic Wellbeing. Note Quote.

The economic system is a social system in which commodities are exchanged. Sets of these commodities can be represented by vectors x within a metric space X contained within the non-negative orthant of an Euclidean space RNx+ of dimensionality N equal to the number of such commodities.

An allocation {xi}i∈N ⊂ X ⊂ RNx+ of commodities in society is a set of vectors xi representing the commodities allocated within the economic system to each individual i ∈ N.

In questions of welfare economics at least in all practical policy matters, the state of society is equated with this allocation, that is, s = {x_i}_i∈N, and the set of all possible information concerning the economic state of society is S = X. It is typically taken to be the case that the individual’s preference-information is simply their allocation x_i, s_i = x_i. The concept of Pareto efficiency is thus narrowed to “neoclassical Pareto efficiency” for the school of economic thought in which originates, and to distinguish it from the weaker criterion.

An allocation {x_i}_i∈N is said to be neoclassical Pareto efficient iff ∄{x_i}_i∈N ⊂ X & i ∈ N : x′_i ≻ x_i & x′_j ≽ x_j ∀ j ≠ i ∈ N.

A movement between two allocations, {x_i}_i∈N → {x′_i}_i∈N is called a neoclassical Pareto improvement iff ∃i∈N : x′_i ≻ x_i & x′_j ≽ x_j ∀ j ≠ i ∈ N.

For technical reasons it is almost always in practice assumed for simplicity that individual preference relations are monotonically increasing across the space of commodities.

If individual preferences are monotonically increasing then x′_i ≽_i x_i ⇐⇒ x′_i ≥ x_i, and x′ ≻ x_i ⇐⇒ x_i > x′_i².

This is problematic, because a normative economics guided by the principle of implementing a decision if it yields a neoclassical Pareto improvement where individuals have such preference relations above leads to the following situation.

Suppose that individual’s preference-information is their own allocation of commodities, and that their preferences are monotonically increasing. Take one individual j ∈ N and an initial allocation {x_i}_i∈N.

– A series of movements between allocations {{x_i}^t_i∈N → {x′_i}^t_i∈N}^T_t=1 such that x_i≠j = x′_i≠j ∀ t and x′_j > x_j ∀ t and therefore that x_j − x_i → ∞∀i≠j ∈ N, are neoclassical Pareto improvements. Furthermore, if these movements are made possible only by the discovery of new commodities, each individual state in the movement is neoclassical Pareto efficient prior to the next discovery if the first allocation was neoclassical Pareto efficient.

Admittedly perhaps not to the economic theorist, but to most this seems a rather dubious out- come. It means that if we are guided by neoclassical Pareto efficiency it is acceptable, indeed de- sirable, that one individual within society be made increasingly “richer” without end and without increasing the wealth of others. Provided only the wealth of others does not decrease. The same result would hold if instead of an individual, we made a whole group, or indeed the whole of society “better off”, without making anyone else “worse off”.

Even the most devoted disciple of Ayn Rand would find this situation dubious, for there is no requirement that the individual in question be in some sense “deserving” of their riches. But it is perfectly logically consistent with Pareto optimality if individual preferences concern only to their allocation and are monotonically increasing. So what is it that is strange here? What generates this odd condonation? It is the narrowing of that which the polity care about to each individual allocation, alone, independent of others. The fact that neoclassical Pareto improvements are distribution-invariant because the polity is supposed to care only about their own individual allocation x_i ∈ {x_i}^t_i∈N alone rather than broader states of society s_i ⊂ s as they see it.

To avoid such awkward results, the economist may move from the preference-axiomatic concept of Pareto efficiency to embrace utilitarianism. The policy criterion (actually not immediately representative of Bentham’s surprisingly subtle statement) being the maximisation of some combination W(x) = W {u_i(x_i)}_i∈N of individual utilities u_i(x_i) over allocations. The “social psychic wellbeing” metric known as the Social Welfare Function.

In theory, the maximisation of W(x) would, given the “right” assumptions on the combination method W (·) (sum, multiplication, maximin etc.) and utilities (concavity, montonocity, independence etc.) fail to condone a distribution of commodities x extreme as that discussed above. By dint of its failure to maximise social welfare W(x). But to obtain this egalitarian sensitivity to the distribution of income, three properties of Social Welfare Functions are introduced. Which prove fatal to the a-politicality of the economist’s policy advice, and introduce presuppositions which must lay naked upon the political passions of the economist, so much more indecently for their hazy concealment under the technicalistic canopy of functional mathematics.

Firstly, it is so famous a result as to be called the “third theorem of welfare economics” that any such function W(·) as has certain “uncontroversially” desirable technical properties will impose upon the polity N the preferences of a dictator i ∈ N within it. The preference of one individual i ∈ N will serve to determine the preference indicated between by society between different states by W(x). In practice, the preferences of the economist, who decides upon the form of W(·) and thus imposes their particular political passions (be they egalitarian or otherwise) upon policy, deeming what is “socially optimal” by the different weightings assigned to individual utilities u_i(·) within the polity. But the political presuppositions imported by the economist go deeper in fact than this. Utilitari-anism which allows for inter-personal comparisons of utility in the construction of W(x) requires utility functions be “cardinal” – representing “how much” utility one derives from commodities over and above the bare preference between different sets thereof. Utility is an extremely vague concept, because it was constructed to represent a common hedonistic experiential metric where the very existence of such is uncertain in the first place. In practice, the economist decides upon, extrapolates, assigns to i ∈ N a particular utility function which imports yet further assumptions about how any one individual values their commodity allocation, and thus contributes to social psychic wellbeing.

And finally, utilitarianism not only makes political statements about who in the polity is to be assigned a disimproved situation. It makes statements so outlandish and outrageous to the common sensibility as to have provided the impetus for two of the great systems of philosophy of justice in modernity – those of John Rawls and Amartya Sen. Under almost any combination method W(·), the maximization of W(·) demands allocation to those most able to realize utility from their allocation. It would demand, for instance, redistribution of commodities from sick children to the hedonistic libertine, for the latter can obtain greater “utility” there from. A problem so severe in its political implications it provided the basic impetus for Rawls’ and Sen’s systems. A Theory of Justice is, of course, a direct response to the problematic political content of utilitarianism.

So Pareto optimality stands as the best hope for the economist to make a-political statements about policy, refraining from making statements therein concerning the assignation of dis-improvements in the situation of any individual. Yet if applied to preferences over individual allocations alone it condones some extreme situations of dubious political desirability across the spectrum of political theory and philosophy. But how robust a guide is it when we allow the polity to be concerned with states of society in general? Not only their own individual allocation of commodities. As they must be in the process of public reasoning in every political philosophy from Plato to Popper and beyond.

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Weyl and Automorphism of Nature. Drunken Risibility.

In classical geometry and physics, physical automorphisms could be based on the material operations used for defining the elementary equivalence concept of congruence (“equality and similitude”). But Weyl started even more generally, with Leibniz’ explanation of the similarity of two objects, two things are similar if they are indiscernible when each is considered by itself. Here, like at other places, Weyl endorsed this Leibnzian argument from the point of view of “modern physics”, while adding that for Leibniz this spoke in favour of the unsubstantiality and phenomenality of space and time. On the other hand, for “real substances” the Leibnizian monads, indiscernability implied identity. In this way Weyl indicated, prior to any more technical consideration, that similarity in the Leibnizian sense was the same as objective equality. He did not enter deeper into the metaphysical discussion but insisted that the issue “is of philosophical significance far beyond its purely geometric aspect”.

Weyl did not claim that this idea solves the epistemological problem of objectivity once and for all, but at least it offers an adequate mathematical instrument for the formulation of it. He illustrated the idea in a first step by explaining the automorphisms of Euclidean geometry as the structure preserving bijective mappings of the point set underlying a structure satisfying the axioms of “Hilbert’s classical book on the Foundations of Geometry”. He concluded that for Euclidean geometry these are the similarities, not the congruences as one might expect at a first glance. In the mathematical sense, we then “come to interpret objectivity as the invariance under the group of automorphisms”. But Weyl warned to identify mathematical objectivity with that of natural science, because once we deal with real space “neither the axioms nor the basic relations are given”. As the latter are extremely difficult to discern, Weyl proposed to turn the tables and to take the group Γ of automorphisms, rather than the ‘basic relations’ and the corresponding relata, as the epistemic starting point.

Hence we come much nearer to the actual state of affairs if we start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations. Once the group is known, we know what it means to say of a relation that it is objective, namely invariant with respect to Γ.

By such a well chosen constitutive stipulation it becomes clear what objective statements are, although this can be achieved only at the price that “…we start, as Dante starts in his Divina Comedia, in mezzo del camin”. A phrase characteristic for Weyl’s later view follows:

It is the common fate of man and his science that we do not begin at the beginning; we find ourselves somewhere on a road the origin and end of which are shrouded in fog.

Weyl’s juxtaposition of the mathematical and the physical concept of objectivity is worthwhile to reflect upon. The mathematical objectivity considered by him is relatively easy to obtain by combining the axiomatic characterization of a mathematical theory with the epistemic postulate of invariance under a group of automorphisms. Both are constituted in a series of acts characterized by Weyl as symbolic construction, which is free in several regards. For example, the group of automorphisms of Euclidean geometry may be expanded by “the mathematician” in rather wide ways (affine, projective, or even “any group of transformations”). In each case a specific realm of mathematical objectivity is constituted. With the example of the automorphism group Γ of (plane) Euclidean geometry in mind Weyl explained how, through the use of Cartesian coordinates, the automorphisms of Euclidean geometry can be represented by linear transformations “in terms of reproducible numerical symbols”.

For natural science the situation is quite different; here the freedom of the constitutive act is severely restricted. Weyl described the constraint for the choice of Γ at the outset in very general terms: The physicist will question Nature to reveal him her true group of automorphisms. Different to what a philosopher might expect, Weyl did not mention, the subtle influences induced by theoretical evaluations of empirical insights on the constitutive choice of the group of automorphisms for a physical theory. He even did not restrict the consideration to the range of a physical theory but aimed at Nature as a whole. Still basing on his his own views and radical changes in the fundamental views of theoretical physics, Weyl hoped for an insight into the true group of automorphisms of Nature without any further specifications.

Discontinuous Reality. Thought of the Day 61.0

Convention is an invention that plays a distinctive role in Poincaré’s philosophy of science. In terms of how they contribute to the framework of science, conventions are not empirical. They are presupposed in certain empirical tests, so they are (relatively) isolated from doubt. Yet they are not pure stipulations, or analytic, since conventional choices are guided by, and modified in the light of, experience. Finally they have a different character from genuine mathematical intuitions, which provide a fixed, a priori synthetic foundation for mathematics. Conventions are thus distinct from the synthetic a posteriori (empirical), the synthetic a priori and the analytic a priori.

The importance of Poincaré’s invention lies in the recognition of a new category of proposition and its centrality in scientific judgment. This is more important than the special place Poincaré gives Euclidean geometry. Nevertheless, it’s possible to accommodate some of what he says about the priority of Euclidean geometry with the use of non-Euclidean geometry in science, including the inapplicability of any geometry of constant curvature in physical theories of global space. Poincaré’s insistence on Euclidean geometry is based on criteria of simplicity and convenience. But these criteria surely entail that if giving up Euclidean geometry somehow results in an overall gain in simplicity then that would be condoned by conventionalism.

The a priori conditions on geometry – in particular the group concept, and the hypothesis of rigid body motion it encourages – might seem a lingering obstacle to a more flexible attitude towards applied geometry, or an empirical approach to physical space. However, just as the apriority of the intuitive continuum does not restrict physical theories to the continuous; so the apriority of the group concept does not mean that all possible theories of space must allow free mobility. This, too, can be “corrected”, or overruled, by new theories and new data, just as, Poincaré comes to admit, the new quantum theory might overrule our intuitive assumption that nature is continuous. That is, he acknowledges that reality might actually be discontinuous – despite the apriority of the intuitive continuum.

Ricci-flow as an “intrinsic-Ricci-flat” Space-time.

A Ricci flow solution {(M^m, g(t)), t ∈ I ⊂ R} is a smooth family of metrics satisfying the evolution equation

∂/∂t g = −2Rc —– (1)

where M^m is a complete manifold of dimension m. We assume that sup_M |Rm|_g(t) < ∞ for each time t ∈ I. This condition holds automatically if M is a closed manifold. It is very often to put an extra term on the right hand side of (1) to obtain the following rescaled Ricci flow

∂/∂t g = −2 {Rc + λ(t)g} —– (2)

where λ(t) is a function depending only on time. Typically, λ(t) is chosen as the average of the scalar curvature, i.e. , 1/m ∱Rdv or some fixed constant independent of time. In the case that M is closed and λ(t) = 1/m ∱Rdv, the flow is called the normalized Ricci flow. Starting from a positive Ricci curvature metric on a 3-manifold, Richard Hamilton showed that the normalized Ricci flow exists forever and converges to a space form metric. Hamilton developed the maximum principle for tensors to study the Ricci flow initiated from some metric with positive curvature conditions. For metrics without positive curvature condition, the study of Ricci flow was profoundly affected by the celebrated work of Grisha Perelman. He introduced new tools, i.e., the entropy functionals μ, ν, the reduced distance and the reduced volume, to investigate the behavior of the Ricci flow. Perelman’s new input enabled him to revive Hamilton’s program of Ricci flow with surgery, leading to solutions of the Poincaré conjecture and Thurston’s geometrization conjecture.

In the general theory of the Ricci flow developed by Perelman in, the entropy functionals μ and ν are of essential importance. Perelman discovered the monotonicity of such functionals and applied them to prove the no-local-collapsing theorem, which removes the stumbling block for Hamilton’s program of Ricci flow with surgery. By delicately using such monotonicity, he further proved the pseudo-locality theorem, which claims that the Ricci flow can not quickly turn an almost Euclidean region into a very curved one, no matter what happens far away. Besides the functionals, Perelman also introduced the reduced distance and reduced volume. In terms of them, the Ricci flow space-time admits a remarkable comparison geometry picture, which is the foundation of his “local”-version of the no-local-collapsing theorem. Each of the tools has its own advantages and shortcomings. The functionals μ and ν have the advantage that their definitions only require the information for each time slice (M, g(t)) of the flow. However, they are global invariants of the underlying manifold (M, g(t)). It is not convenient to apply them to study the local behavior around a given point x. Correspondingly, the reduced volume and the reduced distance reflect the natural comparison geometry picture of the space-time. Around a base point (x, t), the reduced volume and the reduced distance are closely related to the “local” geometry of (x, t). Unfortunately, it is the space-time “local”, rather than the Riemannian geometry “local” that is concerned by the reduced volume and reduced geodesic. In order to apply them, some extra conditions of the space-time neighborhood of (x, t) are usually required. However, such strong requirement of space-time is hard to fulfill. Therefore, it is desirable to have some new tools to balance the advantages of the reduced volume, the reduced distance and the entropy functionals.

Let (M^m, g) be a complete Ricci-flat manifold, x₀ is a point on M such that d(x₀, x) < A. Suppose the ball B(x₀, r₀) is A⁻¹−non-collapsed, i.e., r^−m₀|B(x₀, r₀)| ≥ A⁻¹, can we obtain uniform non-collapsing for the ball B(x, r), whenever 0 < r < r₀ and d(x, x₀) < Ar₀? This question can be answered easily by applying triangle inequalities and Bishop-Gromov volume comparison theorems. In particular, there exists a κ = κ(m, A) ≥ 3^−mA^−m−1 such that B(x, r) is κ-non-collapsed, i.e., r^−m|B(x, r)| ≥ κ. Consequently, there is an estimate of propagation speed of non-collapsing constant on the manifold M. This is illustrated by Figure

We now regard (M, g) as a trivial space-time {(M, g(t)), −∞ < t < ∞} such that g(t) ≡ g. Clearly, g(t) is a static Ricci flow solution by the Ricci-flatness of g. Then the above estimate can be explained as the propagation of volume non-collapsing constant on the space-time.

However, in a more intrinsic way, it can also be interpreted as the propagation of non-collapsing constant of Perelman’s reduced volume. On the Ricci flat space-time, Perelman’s reduced volume has a special formula

V((x, t)r²) = (4π)^-m/2 r^-m ∫_M e^{-d2(y, x)/4r2} dv_y —– (3)

which is almost the volume ratio of B_g(t)(x, r). On a general Ricci flow solution, the reduced volume is also well-defined and has monotonicity with respect to the parameter r², if one replace d²(y, x)/4r² in the above formula by the reduced distance l((x, t), (y, t − r²)). Therefore, via the comparison geometry of Bishop-Gromov type, one can regard a Ricci-flow as an “intrinsic-Ricci-flat” space-time. However, the disadvantage of the reduced volume explanation is also clear: it requires the curvature estimate in a whole space-time neighborhood around the point (x, t), rather than the scalar curvature estimate of a single time slice t.

Noneism. Part 2.

Noneism is a very rigourous and original philosophical doctrine, by and large superior to the classical mathematical philosophies. But there are some problems concerning the different ways of characterizing a universe of objects. It is very easy to understand the way a writer characterizes the protagonists of the novels he writes. But what about the characterization of the universe of natural numbers? Since in most kinds of civilizations the natural numbers are characterized the same way, we have the impression that the subject does not intervene in the forging of the characteristics of natural numbers. These numbers appear to be what they are, with total independence of the creative activity of the cognitive subject. There is, of course, the creation of theorems, but the potentially infinite sequence of natural numbers resists any effort to subjectivize its characteristics. It cannot be changed. A noneist might reply that natural numbers are non-existent, that they have no being, and that, in this respect, they are identical with mythological Objects. Moreover, the formal system of natural numbers can be interpreted in many ways: for instance, with respect to a universe of Skolem numbers. This is correct, but it does not explain why the properties of some universes are independent from subjective creation. It is an undeniable fact that there are two kinds of objectual characteristics. On the one hand, we have the characteristics created by subjective imagination or speculative thought; on the other hand, we find some characteristics that are not created by anybody; their corresponding Objects are, in most cases, non-existent but, at the same time, they are not invented. They are just found. The origin of the former characteristics is very easy to understand; the origin of the last ones is, a mystery.

Now, the subject-independence of a universe, suggests that it belongs to a Platonic realm. And as far as transafinite set theory is concerned, the subject-independence of its characteristics is much less evident than the characteristic subject-independence of the natural numbers. In the realm of the finite, both characteristics are subject-independent and can be reduced to combinatorics. The only difference between both is that, according to the classical Platonistic interpretation of mathematics, there can only be a single mathematical universe and that, to deductively study its properties, one can only employ classical logic. But this position is not at all unobjectionable. Once the subject-independence of the natural numbers system’s characteristics is posited, it becomes easy to overstep the classical phobia concerning the possibility of characterizing non-classical objective worlds. Euclidean geometry is incompatible with elliptical and hyperbolic geometries and, nevertheless, the validity of the first one does not invalidate the other ones. And vice versa, the fact that hyperbolic and other kinds of geometry are consistently characterized, does not invalidate the good old Euclidean geometry. And the fact that we have now several kinds of non-Cantorian set theories, does not invalidate the classical Cantorian set theory.

Of course, an universally non-Platonic point of view that includes classical set theory can also be assumed. But concerning natural numbers it would be quite artificial. It is very difficult not to surrender to the famous Kronecker’s dictum: God created natural numbers, men created all the rest. Anyhow, it is not at all absurd to adopt a whole platonistic conception of mathematics. And it is quite licit to adopt a noneist position. But if we do this, the origin of the natural numbers’ characteristics becomes misty. However, forgetting this cloudiness, the leap from noneist universes to the platonistic ones, and vice versa, becomes like a flip-flop connecting objectological with ontological (ideal) universes, like a kind of rabbit-duck Gestalt or a Sherrington staircase. So, the fundamental question with respect to the subject-dependent or subject-independent mathematical theories, is: are they created, or are they found? Regarding some theories, subject-dependency is far more understandable; and concerning other ones, subject-independency is very difficult, if not impossible, to negate.

From an epistemological point of view, the fact of non-subject dependent characteristic traits of a universe would mean that there is something like intellectual intuition. The properties of natural numbers, the finite properties of sets (or combinatorics), some geometric axioms, for instance, in Euclidean geometry, the axioms of betweenness, etc., would be apprehended in a manner, that pretty well coincides with the (nowadays rather discredited) concept of synthetical a priori knowledge. This aspect of mathematical knowledge shows that the old problem concerning the analytic and the a priori synthetical knowledge, in spite of the prevailing Quinean pragmatic conception, must be radically reset.

Quantum Geometrodynamics and Emergence of Time in Quantum Gravity

It is clear that, like quantum geometrodynamics, the functional integral approach makes fundamental use of a manifold. This means not just that it uses mathematical continua, such as the real numbers (to represent the values of coordinates, or physical quantities); it also postulates a 4-dimensional manifold M as an ‘arena for physical events’. However, its treatment of this manifold is very different from the treatment of spacetime in general relativity in so far as it has a Euclidean, not Lorentzian metric (which, apart from anything else, makes the use of the word ‘event’ distinctly problematic). Also, we may wish to make a summation over different such manifolds, it is in general necessary to consider complex metrics in the functional integral (so that the ‘distance squared’ between two spacetime points can be a complex number), whereas classical general relativity uses only real metrics.

Thus one might think that the manifold (or manifolds!) does not (do not) deserve the name ‘spacetime’. But what is in a name?! Let us in any case now ask how spacetime as understood in present-day physics could emerge from the above use of Riemannian manifolds M, perhaps taken together with other theoretical structures.

In particular: if we choose to specify the boundary conditions using the no-boundary proposal, this means that we take only those saddle-points of the action as contributors (to the semi-classical approximation of the wave function) that correspond to solutions of the Einstein field equations on a compact manifold M with a single boundary Σ and that induce the given values h and φ₀ on Σ.

In this way, the question of whether the wave function defined by the functional integral is well approximated by this semi-classical approximation (and thus whether it predicts classical spacetime) turns out to be a question of choosing a contour of integration C in the space of complex spacetime metrics. For the approximation to be valid, we must be able to distort the contour C into a steepest-descents contour that passes through one or more of these stationary points and elsewhere follows a contour along which |e^−I| decreases as rapidly as possible away from these stationary points. The wave function is then given by:

Ψ[h, φ₀, Σ] ≈ ∑_p e^{−Ip/ ̄h}

where I_p are the stationary points of the action through which the contour passes, corresponding to classical solutions of the field equations satisfying the given boundary conditions. Although in general the integral defining the wave function will have many saddle-points, typically there is only a small number of saddle-points making the dominant contribution to the path integral.

For generic boundary conditions, no real Euclidean solutions to the classical Einstein field equations exist. Instead we have complex classical solutions, with a complex action. This accords with the account of the emergence of time via the semiclassical limit in quantum geometrodynamics.

On the Emergence of Time in Quantum Gravity

Classical Theory of Fields

Galilean spacetime consists in a quadruple (M, t_a, h^ab, ∇), where M is the manifold R4; t_a is a one form on M; h^ab is a smooth, symmetric tensor field of signature (0, 1, 1, 1), and ∇ is a flat covariant derivative operator. We require that t_a and h^ab be compatible in the sense that t_ah^ab = 0 at every point, and that ∇ be compatible with both tensor fields, in the sense that ∇_at_b = 0 and ∇_ah^bc = 0.

The points of M represent events in space and time. The field t_a is a “temporal metric”, assigning a “temporal length” |t_aξ^a| to vectors ξ^a at a point p ∈ M. Since R4 is simply connected, ∇_at_b = 0 implies that there exists a smooth function t : M → R such that t_a = ∇_at. We may thus define a foliation of M into constant – t hypersurfaces representing collections of simultaneous events – i.e., space at a time. We assume that each of these surfaces is diffeomorphic to R3 and that h^ab restricted these surfaces is (the inverse of) a flat, Euclidean, and complete metric. In this sense, h^ab may be thought of as a spatial metric, assigning lengths to spacelike vectors, all of which are tangent to some spatial hypersurface. We represent particles propagating through space over time by smooth curves whose tangent vector ξ^a, called the 4-velocity of the particle, satisfies ξ^at_a = 1 along the curve. The derivative operator ∇ then provides a standard of acceleration for particles, which is given by ξⁿ∇_nξ^a. Thus, in Galilean spacetime we have notions of objective duration between events; objective spatial distance between simultaneous events; and objective acceleration of particles moving through space over time.

However, Galilean spacetime does not support an objective notion of the (spatial) velocity of a particle. To get this, we move to Newtonian spacetime, which is a quintuple (M, t_a, h^ab, ∇, η_a). The first four elements are precisely as in Galilean spacetime, with the same assumptions. The final element, η_a, is a smooth vector field satisfying η^at_a = 1 and ∇_aη^b = 0. This field represents a state of absolute rest at every point—i.e., it represents “absolute space”. This field allows one to define absolute velocity: given a particle passing through a point p with 4-velocity ξ^a, the (absolute, spatial) velocity of the particle at p is ξ^a − η^a.

There is a natural sense in which Newtonian spacetime has strictly more structure than Galilean spacetime: after all, it consists of Galilean spacetime plus an additional element. This judgment may be made precise by observing that the automorphisms of Newtonian spacetime – that is, its spacetime symmetries – form a proper subgroup of the automorphisms of Galilean spacetime. The intuition here is that if a structure has more symmetries, then there must be less structure that is preserved by the maps. In the case of Newtonian spacetime, these automorphisms are diffeomorphisms θ : M → M that preserve t_a, h^ab, ∇, and η^a. These will consist in rigid spatial rotations, spatial translations, and temporal translations (and combinations of these). Automorphisms of Galilean spacetime, meanwhile, will be diffeomorphisms that preserve only the metrics and derivative operator. These include all of the automorphisms of Newtonian spacetime, plus Galilean boosts.

It is this notion of “more structure” that is captured by the forgetful functor approach. We define two categories, Gal and New, which have Galilean and Newtonian spacetime as their (essentially unique) objects, respectively, and have automorphisms of these spacetimes as their arrows. Then there is a functor F : New → Gal that takes arrows of New to arrows of Gal generated by the same automorphism of M. This functor is clearly essentially surjective and faithful, but it is not full, and so it forgets only structure. Thus the criterion of structural comparison may be seen as a generalization of the latter to cases where one is comparing collections of models of a theory, rather than individual spacetimes.

To see this last point more clearly, let us move to another well-trodden example. There are two approaches to classical gravitational theory: (ordinary) Newtonian gravitation (NG) and geometrized Newtonian gravitation (GNG), sometimes known as Newton-Cartan theory. Models of NG consist of Galilean spacetime as described above, plus a scalar field φ, representing a gravitational potential. This field is required to satisfy Poisson’s equation, ∇^a∇_aφ = 4πρ, where ρ is a smooth scalar field representing the mass density on spacetime. In the presence of a gravitational potential, massive test point particles will accelerate according to ξⁿ∇_nξ^a = −∇^aφ, where ξ^a is the 4-velocity of the particle. We write models as (M, t_a, h^ab, ∇, φ).

The models of GNG, meanwhile, may be written as quadruples (M,t_a,h^ab,∇ ^̃), where we assume for simplicity that M, t_a, and h^ab are all as described above, and where ∇ ^̃ is a covariant derivative operator compatible with t_a and h^ab. Now, however, we allow ∇ ^̃ to be curved, with Ricci curvature satisfying the geometrized Poisson equation, R_ab = 4πρt_at_b, again for some smooth scalar field ρ representing the mass density. In this theory, gravitation is not conceived as a force: even in the presence of matter, massive test point particles traverse geodesics of ∇ ^̃ — where now these geodesics depend on the distribution of matter, via the geometrized Poisson equation.

There is a sense in which NG and GNG are empirically equivalent: a pair of results due to Trautman guarantee that (1) given a model of NG, there always exists a model of GNG with the same mass distribution and the same allowed trajectories for massive test point particles, and (2), with some further assumptions, vice versa. But in an, Clark Glymour has argued that these are nonetheless inequivalent theories, because of an asymmetry in the relationship just described. Given a model of NG, there is a unique corresponding model of GNG. But given a model of GNG, there are typically many corresponding models of NG. Thus, it appears that NG makes distinctions that GNG does not make (despite the empirical equivalence), which in turn suggests that NG has more structure than GNG.

This intuition, too, may be captured using a forget functor. Define a category NG whose objects are models of NG (for various mass densities) and whose arrows are automorphisms of M that preserve t_a, h^ab, ∇, and φ; and a category GNG whose objects are models of GNG and whose arrows are automorphisms of M that preserve t_a, h^ab, and ∇ ^̃. Then there is a functor F : NG → GNG that takes each model of NG to the corresponding model, and takes each arrow to an arrow generated by the same diffeomorphism. This results in implying

F : NG → GNG forgets only structure.