2. Coordinates and structure

We begin with North’s method of uncovering the structure that a theory posits. Her basic idea is that the coordinate systems that a theory privileges provide us with a window into the underlying structure of the theory: “[a] preference for certain coordinates, in the sense that the laws take a simple or natural form in them, is indicative of, it is evidence for, underlying structure” (112). This method is interesting in its own right, but it is also used throughout the book to draw surprising conclusions—for example, that Newtonian and Lagrangian mechanics are inequivalent theories. It is therefore worth discussing in detail.

North’s core idea is that a piece of structure can be indirectly characterized by singling out a class of privileged coordinates. We will illustrate this idea with her example of the two-dimensional Euclidean plane (17–26). One is familiar with the usual way of directly characterizing the metric structure of the Euclidean plane: we simply define the standard metric tensor on ${{\mathbb {R}}^2}$ . North’s idea is that although “the metric tensor more directly encapsulates the geometry of the plane” (23), we can also indirectly characterize this structure by pointing to a particular collection of “privileged coordinates.” North reminds us that “[f]rom among all the coordinate systems we can use for the plane, there is a particularly nice kind, the Cartesian coordinate systems, which have straight, mutually orthogonal coordinate axes, and whose numerical values reflect the relative locations of the points in a particularly clear manner” (18). These Cartesian coordinates are the privileged ones for the Euclidean plane. They are the ones in which the Euclidean metric takes a particularly “simple form” (25). North points out that if we know that these are the privileged coordinates, we can use this fact to uncover the structure of the space:

The fact that all of the privileged coordinates agree on the distance between points tells us that the plane comes equipped with metric structure. As North puts it, metric structure is “part of the intrinsic, objective nature of this space” (19). This example demonstrates that in some cases singling out a collection of privileged coordinates on a space suffices to “characterize” or “determine” or “implicitly define” some structures on it—those structures that are agreed upon by all of the privileged coordinates. North puts this basic idea as follows: “There are […] two ways of characterizing a given structure, and two corresponding routes to learning about it. A structure can be characterized more directly, as in the case of the Euclidean plane and the metric tensor. Or it can be characterized less directly, by means of the coordinate systems we can use for the space and the features that are invariant under transformations of them” (23).

Many parts of this idea merit further attention, but I will examine here the sense in which singling out a collection of coordinates suffices to characterize a structure. In particular, I will consider the following “conjecture schema.” In what follows, we let $(M,{g_{ab}})$ and $(M,{{g}^{\prime} _{ab}})$ be manifolds with Riemannian metric.^{Footnote 1}

If true, this conjecture would capture a sense in which one can, at least in the case of a metric, use collections of coordinates to “implicitly define” a piece of structure on a manifold. By singling out a collection of coordinates as privileged, one would be pointing to a unique metric. If the conjecture were true for arbitrary structures, rather than merely metrics on the manifold, then this would allow North to move seamlessly from claims about theories “privileging coordinates” [by their laws taking a particularly “simple or natural form” (112) in those coordinates] to claims about the structures that those theories posit. If the conjecture is false, then coordinates are not a perfect guide to underlying structure. Providing the collection of privileged coordinates would not uniquely pick out a structure; the privileged coordinates would not tell us everything about that structure. The truth of the conjecture would, at least in the case of a metric, rule out this possibility. North’s discussion provides another way of putting this idea. She asks: “What is the nature of [the two-dimensional Euclidean plane]? One way of getting at an answer to this question, though it is somewhat indirect, is to consider the different kinds of coordinate systems we can use” (17). If the above conjecture is false, then the privileged coordinates would not in general provide a complete answer to that question. The conjecture is therefore an important plank in the project of using coordinates to learn about structure.

Before assessing whether or not it is true, however, we need to make the conjecture precise. In particular, we need to say exactly what the “privileged coordinates” are for a manifold with metric. North’s idea is that these are the coordinates in which the structure looks “nice” or “simple” or “natural.” We begin with the following attempt, which North explicitly suggests (22). Let $(U,\varphi )$ be a coordinate chart on $(M,{g_{ab}})$ ; that is, U is a subset of M and $\varphi :U \to {{\mathbb R}^n}$ is a smooth injective map that assigns “coordinate values” to each point in U. We will say that $(U,\varphi )$ is orthonormal on $(M,{g_{ab}})$ if ${g_{ab}} = \sum\nolimits_{i = 1}^n {d_a}{u^i}{d_b}{u^i}$ in the region U, where the scalar functions ${u^i}$ are the coordinate functions of $\phi $ on U. This is one natural way to make precise the idea that ${g_{ab}}$ looks nice in the coordinate patch $(U,\varphi )$ . One can easily verify that the orthonormal coordinates on Euclidean space suffice to characterize the Euclidean metric in the sense of our conjecture: if any metric on ${{\mathbb R}^n}$ has the same orthonormal coordinates as the Euclidean metric does, then that metric is just the Euclidean metric.

Unfortunately, this is not the case for arbitrary metrics. The conjecture is false if we take orthonormal coordinates to be the privileged ones.^{Footnote 2}

This result immediately follows from the lemma above. If ${g_{ab}}$ and ${{g}^{\prime} _{ab}}$ are both flat on no region of M, then they both have no orthonormal coordinates, but they are not necessarily equal. If we consider orthonormal coordinates to be the privileged ones for a manifold with metric, then distinct metrics can have the same (empty) collection of privileged coordinates. We might therefore try to recover the conjecture by considering more coordinates to be privileged. We will say that a coordinate chart $(U,\varphi )$ on $(M,{g_{ab}})$ is diagonal if ${g_{ab}} = \sum\nolimits_{i = 1}^n {\alpha ^i}{d_a}{u^i}{d_b}{u^i}$ in the region U for some smooth scalar functions ${\alpha ^i}$ on U. This is another way to make precise the idea that the metric looks “nice” or “simple” in the coordinate chart. Unfortunately, this makes the conjecture false again.

Of course, this does not show that the conjecture is false for all ways of characterizing privileged coordinates for a manifold with metric. It only shows that the conjecture is false for two of the most natural such ways. Further work is required to assess the more general status of the conjecture.^{Footnote 3} But there is another degree of freedom we have when considering the conjecture. One can formulate corresponding conjectures for geometric structures other than a metric. We conclude with one case where a structure is perfectly characterized by a natural class of privileged coordinates. A symplectic form ${\Omega _{ab}}$ on a $2n$ -dimensional smooth manifold M is a smooth tensor field that is closed, nondegenerate, and antisymmetric. We will say that the coordinate chart $(U,\varphi )$ is symplectic if

$${\Omega _{ab}} = {1 \over 2}\sum\limits_{i = 1}^n {{d_a}} {u^i}{d_b}{u^{i + n}} - {d_b}{u^i}{d_a}{u^{i + n}}$$

in the region U. The symplectic coordinates are those in which the symplectic form takes a particularly natural and simple form; they are analogous to the orthonormal coordinates for a metric. Darboux’s theorem states that for every point p in the symplectic manifold $(M,{\Omega _{ab}})$ there exists a symplectic chart $(U,\varphi )$ with p in U (Abraham and Marsden Reference Abraham and Marsden1978, 3.2.2). This immediately yields the following result.

One wonders whether versions of the conjecture are true for other structures as well, in addition to symplectic forms. The structures for which a version of the conjecture holds are those structures that can be characterized in North’s “indirect” manner, by singling out the privileged coordinates. The more such structures there are, the better the prospects for using coordinates as a window into the underlying structure that a theory posits. Ensuing work on these questions about coordinates and structure will be fruitful, and that is a testament to the clarity and creativity that North’s discussion brings to the topic.^{Footnote 4}

3. Classical mechanics

We turn to the second topic of this review: the case study of classical mechanics that North presents in chapter 4. North previously argued that the Hamiltonian and Lagrangian formulations of classical mechanics are, contrary to the standard view, inequivalent theories (North Reference North2009). A centerpiece of her new book is her argument for the claim that the standard formulation of Newtonian mechanics is inequivalent to Lagrangian mechanics. This dissents from “the usual view [that] Lagrangian and Newtonian mechanics are wholly equivalent theories, mere notational variants, differing at most in calculational ease” (107). North argues for the following claim.

C. Standard Newtonian mechanics posits more structure than Lagrangian mechanics.

This claim then implies that the two theories do not “say the same thing” about the world. They disagree in the same way that, for example, the Newtonian and Galilean theories of spacetime disagree (61). And moreover, insofar as we should prefer theories that posit less structure, C implies that we have reason to prefer the Lagrangian formulation.

North’s main argument for C relies on the following two premises.^{Footnote 5}

• P1. Newton’s law $F = ma$ privileges Cartesian coordinates.

• P2. The Euler–Lagrange equations do not privilege any kind of coordinates.

In brief, North argues that “the Lagrangian equations are invariant under a wider range of coordinate transformations [than Newton’s law], which indicates that they require less structure” (116). The dynamical laws of Lagrangian mechanics—the Euler–Lagrange equations—privilege fewer coordinates than the dynamical law $F = ma$ of Newtonian mechanics does. P1 and P2 support C by way of North’s method of using coordinates to learn about structure. If a theory privileges fewer coordinate systems, that will indicate to North that the theory posits more structure, since there will be more features that are agreed upon by all of the coordinate systems in this smaller class. For the purposes of this discussion we will simply grant that North’s method of using coordinates to learn about structure is successful. At the very least, the basic idea behind the inference of C from P1 and P2 is clear. P2 indicates that the Euler–Lagrange equations “do not distinguish or recognize differences among different coordinate systems [which] means that they do not require or presuppose the mathematical structure that would underlie a distinguished or preferred type of coordinate system” (109). P1 indicates that Newton’s law does presuppose the mathematical structure that underlies these privileged coordinates. And so C follows.

We begin with North’s argument for P1: “Newton’s law […] prefers Cartesian coordinates, a preference that’s revealed by the change in form of the equation in non-Cartesian coordinates, and by how the theory treats systems naturally characterized in terms of such coordinates” (104–5). North points out that the Newtonian equation of motion $F = ma$ “when expressed in a different coordinate system, does not always have the same mathematical form it did in the original coordinate system: the equation needn’t have the same form when expressed in terms of the new coordinate system as it did in the old” (110). She carefully works through the special case of Cartesian and polar coordinates (96) and shows that Newton’s law does indeed take a simpler form in Cartesian coordinates than it does in polar coordinates. North concludes that standard Newtonian mechanics privileges Cartesian coordinates.

North’s idea behind P2—the claim that “Lagrangian mechanics eliminates the favoritism for any type of coordinate system” (102)—is that the Euler–Lagrange equations “take the same form” in any generalized coordinates. The mathematical fact that underlies this claim requires a bit of background. Suppose that we are modeling a system of n particles in Lagrangian mechanics. The statespace for this system is the 6n- dimensional tangent bundle $TQ$ of configuration space Q. A point in $TQ$ represents the positions and velocities of all of the particles in the system. The Lagrangian $L:TQ \to {\mathbb R}$ of the system encodes the system’s “activity” or “liveliness.” Insofar as the Lagrangian is sufficiently well behaved, it gives rise to a vector field $X_L^a$ on $TQ$ . The field $X_L^a$ tells us how the system will evolve. Given a point in $TQ$ representing the initial condition of the system, there is a unique integral curve of $X_L^a$ through that point. The state of the system evolves along that integral curve; one can picture the state of the system “flowing” along the vector field $X_L^a$ .

We can now state the following mathematical fact. Let $(U,\varphi )$ be a coordinate chart on the $3n$ -dimensional configuration space Q with coordinate functions ${q^1}, \ldots ,{q^{3n}}$ . These coordinates on M naturally induce a coordinate chart $(TU,T\varphi )$ on the tangent bundle $TQ$ .^{Footnote 6} We use the notation ${q^1}, \ldots ,{q^{3n}},{\dot q^1}, \ldots ,{\dot q^{3n}}$ to denote the coordinate functions for this induced chart. Now if we are given an integral curve $(u(t),v(t))$ of $X_L^a$ , then the Euler–Lagrange equations

$${d \over {dt}}\left( {{{\partial L} \over {\partial {{\dot q}^i}}}} \right) = {{\partial L} \over {\partial {q^i}}}$$

hold in this induced chart $(TU,T\varphi )$ for each $i = 1, \ldots 3n$ (Abraham and Marsden Reference Abraham and Marsden1978, 3.5.17). This captures the sense in which the Euler–Lagrange equations do not privilege any coordinate system. They take the same form in any coordinate chart $(TU,T\varphi )$ on $TQ$ .

I have two main concerns with this argument for C. In brief, it seems that Newtonian mechanics posits less structure than North suggests, while Lagrangian mechanics posits more. My first concern is with the argument for P1. North is correct that Newton’s law privileges particular coordinate systems, but they are not the Cartesian ones. North works through the special case of polar coordinates and Cartesian coordinates, but it is worth working through the general case here. Consider standard Newtonian mechanics formulated on Galilean spacetime $({{\mathbb R}^4},{h^{ab}},{t_a},\nabla )$ , where the derivative operator $\nabla $ is the “standard” derivative operator on ${{\mathbb R}^4}$ .^{Footnote 7} This theory says that if a particle has mass m, then it will traverse a smooth timelike curve whose tangent field ${\xi ^a}$ satisfies ${t_a}{\xi ^a} = 1$ and

(1) $${F^a} = m{\xi ^n}{\nabla _n}{\xi ^a}$$

where ${F^a}$ is the vector field representing the net force acting on the particle. This equation is just $F = ma$ , expressed using the derivative operator $\nabla $ ; the vector field ${\xi ^n}{\nabla _n}{\xi ^a}$ is just the acceleration field of the particle. North points out that this equation takes a different form in different coordinates. Indeed, one can compute that in a coordinate patch $(U,\varphi )$ on Galilean spacetime, equation (1) takes the form

$${F^a} = m\left[\sum\limits_{i = 1}^4 {\sum\limits_{j = 1}^4 {\mathop \xi \limits^j } } {{\partial \mathop \xi \limits^i } \over {\partial {u^j}}}{({\partial \over {\partial {u^i}}})^a} + \sum\limits_{i = 1}^4 {\sum\limits_{j = 1}^4 {\mathop \xi \limits^i } } \mathop \xi \limits^j {({\partial \over {\partial {u^j}}})^n}{\nabla _n}{({\partial \over {\partial {u^i}}})^a}\right]$$

where ${u^1}, \ldots ,{u^4}$ are the coordinate functions for $(U,\varphi )$ and we write ${\xi ^n} = \sum\nolimits_{i = 1}^4 {{\xi ^i}} {({\partial \over {\partial {u^i}}})^n}$ in these coordinates. Notice that the right-hand term of the acceleration field will vanish if ${\nabla _n}{({\partial \over {\partial {u^i}}})^a} = {\bf{0}}$ ; that is, if the coordinate curves in the patch $(U,\varphi )$ are straight according to $\nabla $ . This means that, as North clearly points out, Newton’s equation of motion will take “a different form” in polar coordinates, where the coordinate curves are not straight according to $\nabla $ , than it does in standard Cartesian coordinates, where the coordinate curves are straight according to $\nabla $ . In the first case, the right-most term of the acceleration field will not vanish, while in the second case it will. In Cartesian coordinates, $F = ma$ looks simpler, nicer, and more natural than it does in coordinates whose coordinate curves “bend” according to $\nabla $ . North argues that this “reveals that [Newton’s law] does distinguish or recognize differences among different coordinate systems—it does not say the same thing regardless. This, in turn, means that the law requires or presupposes the mathematical structure that underlies the preferred type of coordinate system” (110). North claims that the preferred type of coordinate system is the Cartesian coordinates, and the mathematical structure that underlies it is a Euclidean metric.

This last claim, however, does not follow. The right-hand term in the acceleration field will vanish for any coordinate system whose coordinate curves are straight according to $\nabla $ , not just for the Cartesian ones. Newton’s equation (1) will take the same nice form in any “straight” coordinates. The class of privileged coordinate systems is therefore broader than merely the Cartesian ones, since Cartesian coordinates have straight coordinate axes that are also mutually orthogonal. Note that there is a sense in which this is not surprising. Newton’s law explicitly appeals to “straightness structure” in the form of the covariant derivative operator that appears in its acceleration term. So it is natural that the straight coordinates are the privileged ones. Indeed, it is a mark in favor of North’s method of using coordinates to investigate structure that these end up being the privileged ones. But as far as the above argument is concerned, this means that P1 is false. Crucially, it means that Newton’s law privileges more coordinates than what North suggests, which means that it gives rise to less structure than what North suggests, since a broader class of privileged coordinates will agree on fewer features. In particular, the coordinates with “straight” coordinate curves will not all have the same coordinate metric.^{Footnote 8} And that is a problem for North’s argument. It means that Newtonian mechanics posits less structure than P1 suggests it does.

My second concern has to do with P2. Recall that the Euler–Lagrange equations hold in the coordinates $(TU,T\varphi )$ on $TQ$ that are induced by coordinates $(U,\varphi )$ on Q. Not all coordinates on $TQ$ are of this form, and indeed, one can show that the Euler–Lagrange equations do not hold in all coordinate systems on $TQ$ . One way to see this is simply by examining the equations themselves. They do not make sense in an arbitrary coordinate system on $TQ$ , since they explicitly appeal to the coordinate functions ${q^i}$ and ${\dot q^i}$ , which are only defined when we are working in coordinates $(TU,T\varphi )$ . So the Euler–Lagrange equations do privilege a particular kind of coordinate system, contrary to what P2 asserts.

In sum, my first concern with the argument was that Newton’s equations privilege more coordinates than P1 suggests, and my second concern was that the Euler–Lagrange equations privilege fewer coordinates than P2 suggests. Even so, North may argue that Newton’s equations still privilege fewer coordinate systems than the Euler–Lagrange equations do. But one then wonders what sense of “fewer” is meant. The two most natural explications will not work. First, it seems unlikely that the set of coordinates privileged by Newton’s equations has a smaller cardinality than the set privileged by the Euler–Lagrange equations. (One would guess that both sets have the same cardinality as the real numbers.) And second, it cannot be that the former set is a proper subset of the latter, since the former are coordinates on a spacetime, while the latter are coordinates on a statespace; elements of the one set of coordinates simply are not elements of the other. We are therefore left without a compelling argument for C.

It should be clear, however, that North’s book opens up entirely new and promising lines of inquiry. It covers a remarkable amount of material. (I have not even touched on North’s general discussion of equivalence in chapter 7 nor her contribution to the debate on substantivalism and relationalism in chapter 5, both of which will receive much attention.) And it does so in an original and engaging manner. The ideas put forward in the book will have an impact in metaphysics, general philosophy of science, philosophy of mathematics, and philosophy of physics. They will be discussed and debated for years to come.