Complexity Wrapped Uncertainty in the Bazaar

economic_growth_atlas

One could conceive a financial market as a set of N agents each of them taking a binary decision every time step. This is an extremely crude representation, but capture the essential feature that decision could be coded by binary symbols (buy = 0, sell = 1, for example). Although the extreme simplification, the above setup allow a “stylized” definition of price.

Let Nt0, Nt1 be the number of agents taking the decision 0, 1 respectively at the time t. Obviously, N = Nt0 + Nt1 for every t . Then, with the above definition of the binary code the price can be defined as:

pt = f(Nt0/Nt1)

where f is an increasing and convex function which also hold that:

a) f(0)=0

b) limx→∞ f(x) = ∞

c) limx→∞ f'(x) = 0

The above definition perfectly agree with the common believe about how offer and demand work. If Nt0 is small and Nt1 large, then there are few agents willing to buy and a lot of agents willing to sale, hence the price should be low. If on the contrary, Nt0 is large and Nt1 is small, then there are a lot of agents willing to buy and just few agents willing to sale, hence the price should be high. Notice that the winning choice is related with the minority choice. We exploit the above analogy to construct a binary time-series associated to each real time-series of financial markets. Let {pt}t∈N be the original real time-series. Then we construct a binary time-series {at}t∈N by the rule:

at = {1 pt > pt-1

at = {0 pt < pt-1

Physical complexity is defined as the number of binary digits that are explainable (or meaningful) with respect to the environment in a string η. In reference to our problem the only physical record one gets is the binary string built up from the original real time series and we consider it as the environment ε . We study the physical complexity of substrings of ε . The comprehension of their complex features has high practical importance. The amount of data agents take into account in order to elaborate their choice is finite and of short range. For every time step t, the binary digits at-l, at-l+1,…, at-1 carry some information about the behavior of agents. Hence, the complexity of these finite strings is a measure of how complex information agents face. The Kolmogorov – Chaitin complexity is defined as the length of the shortest program π producing the sequence η when run on universal Turing machine T:

K(η) = min {|π|: η = T(π)}

where π represent the length of π in bits, T(π) the result of running π on Turing machine T and K(η) the Kolmogorov-Chaitin complexity of sequence π. In the framework of this theory, a string is said to be regular if K(η) < η . It means that η can be described by a program π with length smaller than η length. The interpretation of a string should be done in the framework of an environment. Hence, let imagine a Turing machine that takes the string ε as input. We can define the conditional complexity K(η / ε) as the length of the smallest program that computes η in a Turing machine having ε as input:

K(η / ε) = min {|π|: η = CT(π, ε)}

We want to stress that K(η / ε) represents those bits in η that are random with respect to ε. Finally, the physical complexity can be defined as the number of bits that are meaningful in η with respect to ε :

K(η : ε) = |η| – K(η / ε)

η also represent the unconditional complexity of string η i.e., the value of complexity if the input would be ε = ∅ . Of course, the measure K (η : ε ) as defined in the above equation has few practical applications, mainly because it is impossible to know the way in which information about ε is encoded in η . However, if a statistical ensemble of strings is available to us, then the determination of complexity becomes an exercise in information theory. It can be proved that the average values C(η) of the physical complexity K(η : ε) taken over an ensemble Σ of strings of length η can be approximated by:

C|(η)| = 〈K(η : ε) ≅  |η| – K(η : ε), where

K(η : ε) = -∑η∈∑p(η / ε) log2p(η / ε)

and the sum is taking over all the strings η in the ensemble Σ. In a population of N strings in environment ε, the quantity n(η)/N, where n(s) denotes the number of strings equal to η in ∑, approximates p(η / ε) as N → ∞.

Let ε = {at}t∈N and l be a positive integer l ≥ 2. Let Σl be the ensemble of sequences of length l built up by a moving window of length l i.e., if η ∈ Σl then η = aiai+1ai+l−1 for some value of i. The selection of strings ε is related to periods before crashes and in contrast, period with low uncertainty in the market…..

Relationist and Substantivalist meet by the Isometric Cut in the Hole Argument

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To begin, the models of relativity theory are relativistic spacetimes, which are pairs (M,gab) consisting of a 4-manifold M and a smooth, Lorentz-signature metric gab. The metric represents geometrical facts about spacetime, such as the spatiotemporal distance along a curve, the volume of regions of spacetime, and the angles between vectors at a point. It also characterizes the motion of matter: the metric gab determines a unique torsion-free derivative operator ∇, which provides the standard of constancy in the equations of motion for matter. Meanwhile, geodesics of this derivative operator whose tangent vectors ξa satisfy gabξaξb > 0 are the possible trajectories for free massive test particles, in the absence of external forces. The distribution of matter in space and time determines the geometry of spacetime via Einstein’s equation, Rab − 1/2Rgab = 8πTab, where Tab is the energy-momentum tensor associated with any matter present, Rab is the Ricci tensor, and R = Raa. Thus, as in Yang-Mills theory, matter propagates through a curved space, the curvature of which depends on the distribution of matter in spacetime.

The most widely discussed topic in the philosophy of general relativity over the last thirty years has been the hole argument, which goes as follows. Fix some spacetime (M,gab), and consider some open set O ⊆ M with compact closure. For convenience, assume Tab = 0 everywhere. Now pick some diffeomorphism ψ : M → M such that ψ|M−O acts as the identity, but ψ|O is not the identity. This is sufficient to guarantee that ψ is a non-trivial automorphism of M. In general, ψ will not be an isometry, but one can always define a new spacetime (M, ψ(gab)) that is guaranteed to be isometric to (M,gab), with the isometry realized by ψ. This yields two relativistic spacetimes, both representing possible physical configurations, that agree on the value of the metric at every point outside of O, but in general disagree at points within O. This means that the metric outside of O, including at all points in the past of O, cannot determine the metric at a point p ∈ O. General relativity, as standardly presented, faces a pernicious form of indeterminism. To avoid this indeterminism, one must become a relationist and accept that “Leibniz equivalent”, i.e., isometric, spacetimes represent the same physical situations. The person who denies this latter view – and thus faces the indeterminism – is dubbed a manifold substantivalist.

One way of understanding the dialectical context of the hole argument is as a dispute concerning the correct notion of equivalence between relativistic spacetimes. The manifold substantivalist claims that isometric spacetimes are not equivalent, whereas the relationist claims that they are. In the present context, these views correspond to different choices of arrows for the categories of models of general relativity. The relationist would say that general relativity should be associated with the category GR1, whose objects are relativistic spacetimes and whose arrows are isometries. The manifold substantivalist, meanwhile, would claim that the right category is GR2, whose objects are again relativistic spacetimes, but which has only identity arrows. Clearly there is a functor F : GR2 → GR1 that acts as the identity on both objects and arrows and forgets only structure. Thus the manifold substantivalist posits more structure than the relationist.

Manifold substantivalism might seem puzzling—after all, we have said that a relativistic spacetime is a Lorentzian manifold (M,gab), and the theory of pseudo-Riemannian manifolds provides a perfectly good standard of equivalence for Lorentzian manifolds qua mathematical objects: namely, isometry. Indeed, while one may stipulate that the objects of GR2 are relativistic spacetimes, the arrows of the category do not reflect that choice. One way of charitably interpreting the manifold substantivalist is to say that in order to provide an adequate representation of all the physical facts, one actually needs more than a Lorentzian manifold. This extra structure might be something like a fixed collection of labels for the points of the manifold, encoding which point in physical spacetime is represented by a given point in the manifold. Isomorphisms would then need to preserve these labels, so spacetimes would have no non-trivial automorphisms. On this view, one might use Lorentzian manifolds, without the extra labels, for various purposes, but when one does so, one does not represent all of the facts one might (sometimes) care about.

In the context of the hole argument, isometries are sometimes described as the “gauge transformations” of relativity theory; they are then taken as evidence that general relativity has excess structure. One can expect to have excess structure in a formalism only if there are models of the theory that have the same representational capacities, but which are not isomorphic as mathematical objects. If we take models of GR to be Lorentzian manifolds, then that criterion is not met: isometries are precisely the isomorphisms of these mathematical objects, and so general relativity does not have excess structure.

This point may be made in another way. Motivated in part by the idea that the standard formalism has excess structure, a proposal to move to the alternative formalism of so-called Einstein algebras for general relativity is sought, arguing that Einstein algebras have less structure than relativistic spacetimes. In what follows, a smooth n−algebra A is an algebra isomorphic (as algebras) to the algebra C(M) of smooth real-valued functions on some smooth n−manifold, M. A derivation on A is an R-linear map ξ : A → A satisfying the Leibniz rule, ξ(ab) = aξ(b) + bξ(a). The space of derivations on A forms an A-module, Γ(A), elements of which are analogous to smooth vector fields on M. Likewise, one may define a dual module, Γ(A), of linear functionals on Γ(A). A metric, then, is a module isomorphism g : Γ(A) → Γ(A) that is symmetric in the sense that for any ξ,η ∈ Γ(A), g(ξ)(η) = g(η)(ξ). With some further work, one can capture a notion of signature of such metrics, exactly analogously to metrics on a manifold. An Einstein algebra, then, is a pair (A, g), where A is a smooth 4−algebra and g is a Lorentz signature metric.

Einstein algebras arguably provide a “relationist” formalism for general relativity, since one specifies a model by characterizing (algebraic) relations between possible states of matter, represented by scalar fields. It turns out that one may then reconstruct a unique relativistic spacetime, up to isometry, from these relations by representing an Einstein algebra as the algebra of functions on a smooth manifold. The question, though, is whether this formalism really eliminates structure. Let GR1 be as above, and define EA to be the category whose objects are Einstein algebras and whose arrows are algebra homomorphisms that preserve the metric g (in a way made precise by Rosenstock). Define a contravariant functor F : GR1 → EA that takes relativistic spacetimes (M,gab) to Einstein algebras (C(M),g), where g is determined by the action of gab on smooth vector fields on M, and takes isometries ψ : (M, gab) → (M′, g′ab) to algebra isomorphisms ψˆ : C(M′) → C(M), defined by ψˆ(a) = a ◦ ψ. Rosenstock et al. (2015) prove the following.

Proposition: F : GR1 → EA forgets nothing.

Moishe Postone: Capitalism, Temporality, and the Crisis of Labor. Note Quote.

Moishe Postone: Capitalism, Temporality, and the Crisis of Labor from The American Academy in Berlin on Vimeo.

Moishe Postone’s work establishes a crucial distinction between the critique of capitalism from the standpoint of labour and the critique of labor in capitalism.The former implies a transhistorical account of work, while the latter situates labor as a consistent category – capable of “social synthesis” – within the capitalist mode of production. But, does this distinction require us to abandon any form of ontological account of labour? As Postone would say,

It depends what you mean by an ontological account of labour. It does force us to abandon the idea that transhistorically there is an on-going development of humanity which is effected by labour, that human interaction with nature as mediated by labour is a continuous process which is led to continuous change. And that labour is in that sense a central historical category. That position is closer actually to Adam Smith than it is to Marx. I think that the centrality of labour to something called historical development can be posited only for capitalism and not for any other form of human social life. On the other hand, I think one can retain the idea that humanity’s interaction with nature is a process of self-constitution.

One of the most important contributions of Time, Labour and Social Domination is a novel theory of impersonal domination in capitalist society. To him “traditional Marxism” is a criticism of capitalism from the standpoint of labor. Postone’s Marxism, by contrast, is a critique of labor in capitalism. Since Marx’s theory refers to capitalism, not society in general, labor cannot be a transhistorical category. Instead, it must be understood as an integrated part of capitalism. This means that labor cannot provide a standpoint from which to criticize capitalism, and neither can the proletariat: “the working class is integral to capitalism, rather than the embodiment of its negation”. The struggle, then, should not be a struggle of labor against capital, as traditional Marxists thought, but a struggle against labor seen as an integral part of the valorization of capital.  This conclusion has implications for Postone’s understanding of domination in capitalism. Rather than being a matter of class relations, it takes the form of domination by impersonal and quasi-objective mechanisms such as fetishism, in the construction of which labor is deeply implicated. The benefit of this reinterpretation, according to Postone, is that it shows the usefulness of Marx’s theory not only in a criticism of liberal nineteenth-century capitalism but also in a criticism of contemporary welfare-state capitalism or Soviet-style state-capitalism. The latter forms of capitalism are just as capitalist as the former since they all build on the valorization of capital built on labor. Abolishing private ownership or rearranging the distribution of goods is not enough to escape capitalism. Postone both builds on and criticizes the approaches of Lukács and the Frankfurt School. There is much in his book that shows his affinities especially to the latter – such as the criticism of welfare state capitalism or the stress on fetishism – but he nevertheless criticizes these earlier thinkers for being bound to a transhistorical conception of labor. Lukács in particular is singled out for heavy criticism since he saw the proletariat as the Subject of history, as capable of grasping totality and hence offering the standpoint of critique. Engaging with the Hegelian legacy, or should I quip lunacy in Lukács, Postone arrives at one of his most important and provocative arguments. “Marx suggests that a historical Subject in the Hegelian sense does indeed exist in capitalism, yet he does not identify it with… the proletariat”. Instead it is capital that is portrayed as a Hegelian Geist – as a subject and self-moving substance, following its own immanent historical logic. Hegelian dialectics, then, is specific to capitalism and is not a tool for grasping history in general. Thus, to Marx, the “totality” was not the whole in general, and certainly not a standpoint which he affirmed. Instead, he identified totality with the capitalist system and made it the object of his critique: “the historical negation of capitalism would not involve the realization, but the abolition, of the totality”, Postone argues. The working class cannot lead history towards this negation. In fact, it is only by breaking with the logic consitutive of this totality, in which the working class forms part, that a different, post-capitalist society can be born.

The abolition of the totality would, then, allow for the possible constitution of very different, non-totalizing, forms of the political coordination and regulation of society.

In a question asked about if the capitalist form of domination not better defined as the appearance of truly abstract relations as if they were concrete, personal relations? Furthermore, does this inversion, or at least the recognition of the crucial role of abstraction in capitalism, render a definition of class struggle untenable, or are we rather in need of a concept of class that takes this distance from the concrete into consideration? Postone says,

I am not sure that I would fully agree with the attempted reformulation. First of all, with regard to the quote “relations between people appearing as relations between things” what is left out of this version of what Marx said is that he adds that relations among people appear as they are, as social relations between things and thingly relations between people. Marx only explicitly elaborated the notion of fetishism with the fetishism of commodity. All three volumes of Capital, are [our change] in many respects, however, a study on fetishism even when he doesn’t use that word. And fetishism means that because of the peculiar, double character of the structuring social forms of capitalism, social relations disappears from view. What we get are thingly relations: we also get abstractions. However, one dimension of the fetish is, as you put it, that abstract relations appear concrete.They appear in the form of the concrete. So, for example, the process of creating surplus value appears to be a material process, the labour process. It appears to be material-technical, rather than moulded by social forms. And yet there are also abstract dimensions and regularities that don’t appear in the form of the concrete. I am emphasising this is because certain reactionary forms of thought only view capitalism in terms of those abstract regularities and refuse to see that the concrete itself is moulded by, and is really drenched with, the abstract. I think a lot of forms of populism and anti-Semitism can be characterised that way. Now I am not sure that this appropriation of the categories of Marx’s critique of political economy renders a definition of class struggle untenable, but it does indicate that class struggle occurs within and is moulded by the structuring social forms.This position rejects the ontological centrality or the primacy of class struggle, as that which is truly social and real behind the veil of capitalist forms. Class struggle rather is moulded by the capitalist relations expressed by the categories of value, commodity, surplus value, and capital.

Postone’s approach only seems far-fetched if we continue to equate capitalism with the economy. Not if we think of it as a form of life. For example, after Darwin wrote, natural processes, such as adaptation or sexual selection, came to be seen as operating within history. This gave us the naturalist novel of Zola or Norris. “Nature” was seen to structure history (the Rougon-Marcquet saga, the strike in Germinal etc.) as well compel individuals from within. For Postone, it is not “natural” Darwinian processes that do this but an historic process, capitalism. Another example of the same idea is Max Weber’s “spirit of capitalism.” Weber can be read, and wanted to be read, as saying that there are forces outside capitalism on which capitalism depends, such as religious ethics. However, Postone is suggesting that such “spiritual” Weberian forces as asceticism, compulsivity and hypocrisy (Weber’s famous triad) are internal to capitalism, structuring its motion. But, there are ambivalences to his theory, and especially ones concerning science and technology. Postone rejects the view, associated with traditional Marxism, that sees industrial production as a neutral, purely technical process that could be salvaged from capitalism and carried on in similar form in socialism. To criticize capitalism, he argues that we also need to criticize industrial production, or at least the form it has assumed in capitalism. The problem is that he simultaneously argues – based primarily on a famous passage in Grundrisse – that science and technology creates the preconditions for an overcoming of capitalism, since they enable human beings to create unprecedented “material wealth” in a way that relies less and less on human labor. Since in capitalism “value” can only be created by labor, capitalism increasingly comes to be characterized by a contradiction between the processes generating “wealth” and “value”. Unlike “value”, Postone appears to think that “wealth” is a category that it is fine to apply transhistorically. “Wealth” existed in precapitalist societies and must also be imagined as something that can exist in post-capitalist, socialist societies. What happens with capitalism is that the creation of “wealth” can only take place through the production of “value”, i.e. through the exploitation of labor and valorization of capital. However, by showing that “wealth” can be produced in abundance without relying on labor, science and technology open up possibilities of overcoming capitalism. Here Postone portrays science and technology, not as irremediably implicated in capitalism, but as potentially liberating forces that point beyond capitalism. That is of course fine, but the question then becomes how to distinguish the good and bad moments of science and technology. Postone calls for a transformation of not only of “relations of production” but also of the “mode of production”, but without giving us much in the way of explaining how much or how radically the latter needs to be changed.

Then there is the problem of dialectics. As mentioned, Postone confines Geist and totality to capitalism. This claim has some antecedents in earlier critical theory. Adorno, for instance, claims that the role of Spirit in capitalism is taken by “value”: “The objective and ultimately absolute Hegelian spirit [is] the Marxist law of value that comes into force without men being conscious of it” (Adorno). The posture of taking up arms against “totality” itself is of course also familiar from older critical theory. Adorno, however, never confined dialectics in toto to capitalism. Although Postone does allow for some forms of dialectical interaction (e.g. people changing their own nature reflexively through acting on nature or the reciprocal constitution of social practice and social structure), he argues that such interaction only becomes “directionally dynamic” in capitalism. In other words, dialectics in the sense of a historical logic or necessity only exists in capitalism. This raises the question of how capitalism can be overcome. If there is no Geist but capital, then dialectics cannot point the way out of capitalism. Liberation can only mean liberating oneself from dialectics, by creating a world in which it is no longer dominant.

The indication of the historicity of the object, the essential social forms of capitalism, implies the historicity of the critical consciousness that grasps it; the historical overcoming of capitalism would also entail the negation of its dialectical critique.

However, sometimes Postone himself seems to grasp the relation between capitalism and its outside dialectically, as when he uses the term “determinate negation” for the movement whereby capitalism is transcended. But if the overcoming of capitalism is a determinate negation, doesn’t that require the premise of a totality transcending the capitalist system, as Lukács thought?  Sometimes Postone writes as if the totality of capitalism were driven towards its own abolition by its inner contradictions. However, apart from the discussion of technology and wealth referred to above, it is hard to see that he specifies anywhere what kind of contraditions might bring about this self-abolition.

Classical Theory of Fields

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Galilean spacetime consists in a quadruple (M, ta, hab, ∇), where M is the manifold R4; ta is a one form on M; hab is a smooth, symmetric tensor field of signature (0, 1, 1, 1), and ∇ is a flat covariant derivative operator. We require that ta and hab be compatible in the sense that tahab = 0 at every point, and that ∇ be compatible with both tensor fields, in the sense that ∇atb = 0 and ∇ahbc = 0.

The points of M represent events in space and time. The field ta is a “temporal metric”, assigning a “temporal length” |taξa| to vectors ξa at a point p ∈ M. Since R4 is simply connected, ∇atb = 0 implies that there exists a smooth function t : M → R such that ta = ∇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 hab restricted these surfaces is (the inverse of) a flat, Euclidean, and complete metric. In this sense, hab 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 ξa, called the 4-velocity of the particle, satisfies ξata = 1 along the curve. The derivative operator ∇ then provides a standard of acceleration for particles, which is given by ξnnξa. 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, ta, hab, ∇, ηa). The first four elements are precisely as in Galilean spacetime, with the same assumptions. The final element, ηa, is a smooth vector field satisfying ηata = 1 and ∇aηb = 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 ξa, the (absolute, spatial) velocity of the particle at p is ξa − ηa.

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 ta, hab, ∇, and ηa. 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, ∇aaφ = 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 ξnnξa = −∇aφ, where ξa is the 4-velocity of the particle. We write models as (M, ta, hab, ∇, φ).

The models of GNG, meanwhile, may be written as quadruples (M,ta,hab,∇ ̃), where we assume for simplicity that M, ta, and hab are all as described above, and where ∇ ̃ is a covariant derivative operator compatible with ta and hab. Now, however, we allow ∇ ̃ to be curved, with Ricci curvature satisfying the geometrized Poisson equation, Rab = 4πρtatb, 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 ta, hab, ∇, and φ; and a category GNG whose objects are models of GNG and whose arrows are automorphisms of M that preserve ta, hab, 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.