2. The structure of Newtonian mechanics

In the tradition of the semantic view of theories, I will present Newtonian gravitation (NG) as a class of models. The kinematically possible models (KPMs) of a theory are models that contain the correct sort of mathematical objects to formulate the theory’s dynamics. The dynamically possible models (DPMs) of a theory are those KPMs that, in addition, satisfy the theory’s equations of motion.

Here is a first proposal for the structure of NG, which will require significant revision later on. The KPMs of NG are of the form:

$${\rm{NG }}1:\langle D,\mathbb{E},\mathbb{T},{x_i}(t),{m_i},{\mathbb{R}^ + }\rangle, $$

where $D$ is a bare set of particles, $\mathbb{E}$ is a three-dimensional Euclidean affine space, $\mathbb{T}$ is a one-dimensional Euclidean affine space, $x:D \times \mathbb{T} \to \mathbb{E}$ is a smooth function that represents the particles’ trajectories, and $m:D \to {\mathbb{R}^ + }$ is a function into the positive real numbers that represents the particles’ masses (in some unit of mass). Because $\mathbb{E}$ is an affine space, for any pair of points $x,y \in \mathbb{E}$ , there is a unique vector from $x$ to $y$ denoted $y - x$ , and likewise for $\mathbb{T}$ . I will denote the associated vector spaces ${\mathbb{V}_E}$ and ${\mathbb{V}_T}$ , respectively. Moreover, both vector spaces come attached with a positive real-valued norm $|.|:\mathbb{V} \to {\mathbb{R}^ + }$ that represents the magnitude of each vector (in some unit of length or time).

Notice that it is perhaps more common to set NG on a Newtonian spacetime $\langle M,{t_{ab}},{h^{ab}},\nabla ,{\sigma ^a}\rangle $ , where $M$ is a differentiable manifold and $t$ , $h$ , $\nabla $ , and $\sigma $ are certain geometric objects. But the idea that NG is more appropriately set on a Euclidean affine space is familiar from Stachel (Reference Stachel, Earman, Janis and Gerald1993); see also Saunders (Reference Saunders2013) and Dewar (Reference Dewar2015). I will not discuss this issue further.

The DPMs of NG are those KPMs that satisfy the following equation of motion:

(1) $${\ddot x_i}(t) = \sum\limits_{j \ne i} G {{{m_j}} \over {|{{\bf{r}}_{ij}}(t){|^3}}}{{\bf{r}}_{ij}}(t),$$

where ${{\bf{r}}_{ij}}(t)$ is the unique vector between ${x_i}(t)$ and ${x_j}(t)$ , $|{{\bf{r}}_{ij}}(t)|$ is the real-valued norm of that vector, and ${m_j}$ is the mass value of $j$ . The value of $G$ is an experimentally determined real number. ^{Footnote 6}

The left-hand side of this equation deserves some comment. For a given particle $i$ , ${x_i}(t)$ is a function $\mathbb{T} \to \mathbb{E}$ . The velocity of particle $i$ at time $t$ is a measure of how much distance $i$ covers (and in which direction) over an infinitesimal period of time ${\bf{t}}$ from $t$ . In other words, velocity is the directional derivative of ${x_i}(t)$ :

(2) $${\nabla _{\bf{t}}}{x_i}(t):=\mathop {\lim }\limits_{\varepsilon \to 0} {{{x_i}(t + \varepsilon {\bf{t}}) - {x_i}(t)} \over \varepsilon },$$

where $\varepsilon $ is a real number. ^{Footnote 7} This means that the velocity of $i$ at $t$ is represented by a function from ${\mathbb{V}_T}$ into ${\mathbb{V}_E}$ .

We can then define acceleration as the directional derivative of velocity:

(3) $${\nabla _{\bf{s}}}{\nabla _{\bf{t}}}{x_i}(t):=\mathop {\lim }\limits_{\varepsilon \to 0} {{{\nabla _{\bf{t}}}{x_i}(t + \varepsilon {\bf{s}}) - {\nabla _{\bf{t}}}{x_i}(t)} \over \varepsilon }.$$

From this definition, it follows that acceleration is a function from $\mathbb{V}_T^2 \to {\mathbb{V}_E}$ . We are normally only interested in the case in which ${\bf{t}} = {\bf{s}}$ , so acceleration really is another function from ${\mathbb{V}_T} \to {\mathbb{V}_E}$ . We can turn this into a vector quantity by choosing the unique unit vector ${\bf{\hat t}}$ in the positive time direction as our unit of time. $\mathbb{T}$ has no privileged orientation, so the choice of direction is conventional. Once this choice is made, let ${\ddot x_i}(t):=\nabla _{{\bf{\hat t}}}^2{x_i}(t)$ . This quantity takes value in ${\mathbb{V}_E}$ .

It hardly seems necessary to point out that equation (1) is a well-defined equation. On the left-hand side is a displacement vector. The quantities $G$ , $m$ , and $|{\bf{r}}|$ are all real numbers, and ${\bf{r}}$ is a vector; scalar multiplication of vectors is well defined, so on the right-hand side of equation (1) is also a displacement vector. The equation of motion compares like with like. However, I will show later in the article that this is an artifact of the redundant real number structures encoded into the theory’s kinematics. When this redundant structure is removed, equation (1) is no longer well-defined. In order to resolve this puzzle, an account of the nature $G$ is required. I will give one that does so in section 5.

3. Redundant real numbers

The theory of NG has three dimensionful quantities: length, time, and mass. Each of these quantities has an associated value space. Consider, for example, mass value space. This space represents the determinable quantity mass; elements of it represent determinate mass magnitudes. The structure of mass value space encodes the relations between mass magnitudes. For example, the magnitude “being 10 kg” stands in the relation of “being twice as much” to the magnitude “being 5 kg.” Similarly, lengths and durations stand in certain relations to each other.

This raises the question of which particular relations these quantities stand in to each other. What is the structure of the theory’s value spaces? In the foregoing, I assumed that these value spaces are isomorphic to the positive real numbers ( ${\mathbb{R}^ + }$ ). For instance, the mass function $m$ simply assigned each particle a positive real number. This is often tacitly assumed: in foundational treatments of NG considered as a field theory, for instance, the mass density field $\rho $ is defined as a scalar field; that is, a function from spacetime points into (positive) real numbers (Friedman Reference Friedman1983; Malament Reference Malament2012). But it turns out that this real number structure is too rich for a dimensionful quantity. This will lead to a second, more realistic proposal, namely, that value spaces have an additive extensive structure.

The problem with ${\mathbb{R}^ + }$ is that it, in effect, endows quantities with a preferred unit. Consider the case of mass, and suppose that, for some particle $i$ and some positive real number $x$ , $m(i) = x$ . Because $x \in {\mathbb{R}^ + }$ , the mass of $i$ is associated to a particular real number. And because mass value space is meant to represent objective physical structure, this association does not depend on any particular choice of unit. But this association between numbers and masses is spurious. In Martens’s (Reference Martens and Read2021) words: there is nothing “five-ish” about a 5-kg mass. So, mass value space cannot have the structure of the real numbers. Wolff (Reference Wolff2020, § 8.2.1) presents a similar reductio: her conclusion is that when mass value space has no nontrivial automorphisms, then there is a unique homomorphism from that value space into the positive real numbers—contrary to our freedom to choose a mass scale.

Martens (Reference Martens2021, 2518) correctly claims that the received view amounts to a “fail[ure] to distinguish between physical magnitudes and the numerical quantities used to represent them.” Instead, both Martens and Wolff endorse a type of structure that is studied under the guise of “measurement theory.” ^{Footnote 8} These structures are concisely characterized as so-called principal homogeneous spaces (PHSs) for the group $\langle {{\rm{R}}^ + }, \times \rangle $ of positive real numbers under multiplication. The PHS of ${\mathbb{R}^ + }$ is a set ${{\rm{R}}^ + }$ over which one has defined a regular action of ${\mathbb{R}^ + }$ . The result is a structure that has “forgotten” the multiplicative identity of ${\mathbb{R}^ + }$ but retains the latter’s order and composition structure.

There is a different way of defining these structures that is physically more perspicuous. Known as additive extensive structures, they are of the form $\langle M,$ ⩽, $ \circ \rangle $ , where $M$ is a set of cardinality ${2^{{\aleph _0}}}$ , ⩽ imposes a total order on $M$ , and $ \circ $ is an associative binary function. When these relations satisfy certain axioms, the resulting structure is equivalent to the PHS defined previously. ^{Footnote 9} In physical terms, ⩽ is interpreted as the binary relation of one mass being less than or equal to another, and $ \circ $ as the tertiary relation of the sum of two masses equaling a third. The intuitive thought behind the fact that $ \circ $ admits of no inverses or identity is that there are neither negative nor zero masses.

Hölder (Reference Hölder1901) proves that one can represent $\langle M,$ ⩽, $ \circ \rangle $ on ${\mathbb{R}^ + }$ in the following sense: there exists a function ${f_r}:M \to {\mathbb{R}^ + }$ such that (i) x ⩽ y if and only if (iff) $f_r(x) \le f_r(y)$ , and (ii) $x \circ y = z$ iff $f_r(x) + f_r(y) = f_r(z)$ . Furthermore, this representation is unique up to multiplication by a positive constant $\alpha $ , so that ${f_r}$ and ${f_{r'}}$ both represent the same structure $\langle M,$ ⩽, $ \circ \rangle $ iff there is some $\alpha \gt 0$ such that ${f_{r'}} = \alpha {f_r}$ . This nonuniqueness provides a precise sense in which the former has less structure than the latter. Unlike the real number structure, an additive extensive structure does not define a privileged system of units. The fact that ${f_r}$ is defined up to multiplication by a positive constant means that any system of units related by such a transformation represents the structure of mass value space equally well.

Therefore, additive extensive structures are apt to represent the value spaces of dimensionful quantities such as mass, length, and time. This means that the norm over ${\mathbb{V}_E}$ is not real-valued; it takes value in a value space isomorphic to the PHS for ${\mathbb{R}^ + }$ . The same is true for the norm over ${\mathbb{V}_T}$ . I will let ${{\cal V}_M}$ , ${{\cal V}_L}$ , and ${{\cal V}_T}$ denote the value spaces for mass, length, and time, respectively. Note that these value spaces are distinct, albeit isomorphic. There is no canonical isomorphism between them.

4. A puzzle for intrinsic NG

With these revisions to the theory in mind, I will rewrite the models of NG as follows:

$${\rm{NG }}2:\langle D,\mathbb{E},\mathbb{T},{{\cal V}_M},{{\cal V}_L},{{\cal V}_T},{x_i}(t),{m_i}\rangle. $$

Although the value spaces for length and time are part of the vector space structure over $\mathbb{E}$ and $\mathbb{T}$ , I write them out explicitly in order to emphasize that the norm over these vector spaces is not real-valued. Call this formulation intrinsic NG.

Let’s return to NG’s equation of motion:

(1) $${\ddot x_i}(t) = \sum\limits_{j \ne i} G {{{m_j}} \over {|{{\bf{r}}_{ij}}(t){|^3}}}{{\bf{r}}_{ij}}(t).$$

When ${m_j}$ and $|{{\bf{r}}_{ij}}|$ are interpreted as real numbers, the right-hand side of this equation is a well-defined vector quantity. But in the previous section, I argued that these quantities are not real numbers but elements of the PHS for ${\mathbb{R}^ + }$ . The product of a vector in ${\mathbb{V}_E}$ by these elements is ill-defined. In particular, there is no canonical way to associate such elements to a unique real number.

The left-hand side also requires a different interpretation. In section 2, I defined ${\ddot x_i}(t)$ as the second directional derivative of $i$ ’s trajectory with respect to the unit vector ${\bf{\hat t}}$ in the positive time direction. But when vectors in ${\mathbb{V}_T}$ take value not in ${\mathbb{R}^ + }$ but in a PHS over ${\mathbb{R}^ + }$ , there is no unique unit vector even after a temporal orientation is fixed by convention: there is no unique vector ${\bf{t}}$ such that $|{\bf{t}}|$ is canonically mapped onto the multiplicative identity. The best option is therefore to redefine acceleration such that ${\ddot x_i}(t):=\nabla _{\bf{t}}^2{x_i}(t)$ , where ${\bf{t}}$ is a variable rather than a fixed input. This means that acceleration becomes a function from ${\mathbb{V}_T} \to {\mathbb{V}_E}$ .

This leads to a puzzle for the intrinsic formulation of NG. Although the left-hand side of equation (1) is a function ${\mathbb{V}_T} \to {\mathbb{V}_E}$ , the right-hand side is a product of (a) a vector in ${\mathbb{V}_E}$ , (b) the norm of that vector, (c) a mass value, and (d) the constant $G$ . Even if we can make sense of this product, it is unclear what it means to say that these quantities are equal. It seems that equation (1), interpreted in this way, does not compare like with like. This is a fatal problem for intrinsic NG: it leaves the theory’s dynamics undefined.

Of course, once we attach numbers to these quantities, we can equate them with ease. Using Hölder’s representation theorem, it is possible to assign positive real numbers to mass values in a way that is faithful to mass value space’s internal structure, and similar for length and time. Based on this procedure, one could formulate NG’s dynamics as follows: a KPM of NG is a DPM iff for any faithful maps from ${{\cal V}_M}$ , ${{\cal V}_L}$ , and ${{\cal V}_T}$ onto ${\mathbb{R}^ + }$ , equation (1) is satisfied for that model. This avoids the puzzle raised earlier.

In this interpretation, the fundamental equation of NG quantifies over assignments of numbers to physical quantities, or systems of units. Field has criticized such quantification as illegitimate. Numbers are physically inert: the laws are not true by virtue of facts about them. As Field wrote about $G$ :

Although I will not defend Field’s program of intrinsic physics here, I find these concerns persuasive. ^{Footnote 10} The excision of redundant structure from the theory’s kinematics is a virtue; this virtue is undone if the same redundancies recur in the dynamics. I will therefore develop an alternative interpretation of equation (1) in which no quantification over numbers is required. This interpretation requires an account of the role of $G$ , to which I now turn.

5. The nature of dimensionful constants

I have purposefully remained vague about the role of the gravitational constant, $G$ , in Newtonian mechanics. In the first approximation of NG, the gravitational constant was another real number, but from an intrinsic point of view, this is clearly unsatisfactory. The theory has reached a breaking point that can only be overcome by considering the dynamical role of $G$ in more detail. That is the aim of this section.

5.1 Example: Hooke’s law

Consider first a simpler example. Hooke’s law relates the restoring force of a spring to the displacement of the load away from equilibrium:

(4) $${F_s} = - kx,$$

where $k$ is a “stiffness” constant that depends on the material constitution of the spring in question. This constant is quite unlike $G$ in that it is specific to a particular (kind of) object and not part of any fundamental law. But like $G$ , it is dimensionful, which will allow us to draw a comparison between them.

Hooke’s law relates one quantity, force, to another, displacement. We cannot equate these quantities directly. It makes no sense to say that the force on a spring is equal to a certain amount of displacement, although this is obscured when we express both quantities vectorially. This is in fact the same puzzle as the one discussed in the previous section: when quantities are interpreted “intrinsically,” equations of motion such as equations (1) and (4) seem ill-defined.

I propose that the role of the constant $k$ is to facilitate comparisons of forces and displacements. The spring constant determines an “exchange rate” between these quantities: such-and-such displacement is “worth” so much force. This is the role of dimensionful constants: to convert the value of one type of quantity into the value of another type of quantity. In this way, constants can restore the balance between both sides of the equation.

In more detail, recall that displacement vectors live in a vector space ${\mathbb{V}_E}$ over $\mathbb{E}$ . In my formulation of NG, forces had no existence of their own: by combining Newton’s second law with the law of universal gravitation, forces dropped out of the equation. But suppose for the sake of example that forces are real. Then their values would live in a distinct value space $\mathbb{F}$ , which is a vector space isomorphic to ${\mathbb{V}_E}$ . Despite the isomorphism, there is no canonical map between ${\mathbb{V}_E}$ and $\mathbb{F}$ . Although one can compare the directions of vectors in either vector space, one cannot compare their magnitudes. There simply is no answer to the question: How much force is equivalent to a certain displacement in the same direction?

But the spring constant $k$ provides a link between these value spaces. From the many maps between $\mathbb{F}$ and ${\mathbb{V}_E}$ , it picks out one as privileged. By $k$ ’s standards, we can say whether a vector in $\mathbb{F}$ is equivalent to a vector in ${\mathbb{V}_E}$ . We can construe the spring constant as a function $k:{\mathbb{V}_E} \to \mathbb{F}$ . This means that equation (4) should really be read as

(5) $${F_s} = - k(x),$$

where $x$ now functions as input for $k$ . On both the left-hand side and the right-hand side of equation (5) are force-valued quantities: like is compared with like. The puzzle for intrinsic dynamics is solved without recourse to the real numbers.

Because $k$ should ultimately be derived $k$ from more fundamental physics, there is no reason to believe that $k$ represents any fundamental relations between displacements and forces. But the same account also applies to more fundamental constants, such as $G$ .

5.2 The gravitational constant

The dynamical role of $G$ is to determine the contribution of the gravitational attraction of a massive particle to the acceleration of another particle some distance away from the first. Recall that acceleration is a function from ${\mathbb{V}_T} \to {\mathbb{V}_E}$ . So, $G$ is a function from mass values $m \in {{\cal V}_M}$ and displacement vectors ${\bf{v}} \in {\mathbb{V}_E}$ to functions from ${\mathbb{V}_T}$ to ${\mathbb{V}_E}$ :

(6) $$G:{\cal V} \times {\mathbb{V}_E} \to ({\mathbb{V}_T} \to {\mathbb{V}_E}).$$

For a given mass and displacement, $G$ yields an acceleration. The function is surjective but not injective: for every acceleration, there is some mass–duration pair that yields that acceleration, but this pair is not unique. ^{Footnote 11}

But $G$ is not just any such function. It satisfies these four requirements, where $\lambda $ is a real number:

1. (i) $G(\lambda m,{\bf{v}})({\bf{t}}) = \lambda G(m,{\bf{v}})({\bf{t}})$ ;

2. (ii) $G(m,\lambda {\bf{v}})({\bf{t}}) = {\lambda ^{ - 2}}G(m,{\bf{v}})({\bf{t}})$ ;

3. (iii) $G(m,{\bf{v}})({\bf{t}}) \propto {\bf{v}}$ ; and

4. (iv) $G(m,{\rm{R}}{\bf{v}})({\bf{t}}) = {\bf{R}}G(m,{\bf{v}})({\bf{t}})$ .

These requirements say that (i) $G$ scales proportionally with mass, (ii) $G$ scales inversely proportional to the square of distance, (iii) the acceleration is in the same direction as the displacement, and (iv) rotations have no effect on the magnitude of the acceleration. These requirements are not a priori principles; they are determined experimentally. It is through careful observation that we know that a test particle accelerates twice as fast toward a body that is twice as massive, and so on.

These constraints jointly determine $G$ up to a scale factor. We can think of this scale factor as a “choice of unit”—but note that we are not concerned here with numerical scales. This residual freedom in the definition of $G$ will lead to the possibility of different “values” for $G$ considered in the second half of this article.

The aforementioned only concerns the mathematical representation of $G$ . I do not intend to suggest that the world itself contains a function that causally contributes to the motions of particles. In my view, $G$ is a complex piece of real cross-value space structure between ${\mathbb{V}_E}$ , ${\mathbb{V}_T}$ , and ${{\cal V}_M}$ . Unlike $k$ , it does not connect just two value spaces, but three. $G$ represents a complex relation that holds between masses, displacements, and durations. It is similar to the relations between values of the same quantity, such as mass ratios, in that it is a physical relation between quantity values; the difference is that it holds between values of different quantities. We can visualize these relations as threads between the elements of these various value spaces. One can follow the thread from one space and arrive at the value of a quantity in a different one. For example, one can follow the thread from a certain mass: this thread has many branches, one for each displacement. Follow one such branch, and one arrives at an acceleration; follow another, and one arrives at a different acceleration. Or start from a displacement, and one is faced with many branches that correspond to different masses. Choose one, and arrive at an acceleration, which may or may not be distinct from before. Thus value spaces are not isolated islands; they are interconnected networks. ^{Footnote 12}

6. The structure of Newtonian mechanics revisited

I have argued that the gravitational constant $G$ represents the real kinematical structure of NG. This kinematical structure connects the theory’s various value spaces.

Because $G$ represents physical structure, it deserves a place in the theory’s models. This yields the third and final version of NG’s kinematics:

$${\rm{NG }}3:\langle D,\mathbb{E},\mathbb{T},{{\cal V}_M},{{\cal V}_L},{{\cal V}_T},{x_i}(t),{m_i},G\rangle .$$

It is not always necessary to explicitly list value space structure among the theory’s posits. I have not explicitly written down the action of ${\mathbb{V}_E}$ on $\mathbb{E}$ , for instance, because this is just part of the structure of $\mathbb{E}$ . Yet this is not the case for $G$ because $G$ links the various value spaces the theory posits; it is not part of any one value space. Therefore, it really is necessary to include $G$ in the theory’s models for a full account of the kinematical structure of NG.

This account of the gravitational constant also provides a solution to the puzzle raised in section 2. Recall the puzzle: the left-hand side of equation (1) is an acceleration—a function from ${\mathbb{V}_T}$ to ${\mathbb{V}_E}$ —whereas the right-hand side of equation (1) is a complicated product. It seems that these are distinct quantities that cannot be compared. But $G$ was the missing piece of the puzzle. Given a mass and a displacement, $G$ outputs an acceleration that we can compare directly to the left-hand side of equation (1). In other words, equation (1) should read as follows:

(7) $${\ddot x_i}(t) = \sum\limits_{j \ne i} G({m_j},{{\bf{r}}_{ij}}(t)),$$

where, as before, ${{\bf{r}}_{ij}}(t):={x_j}(t) - {x_i}(t)$ .

It may seem odd that this novel equation only involves the displacement vector between ${x_i}$ and ${x_j}$ but not the norm of that vector. But the norm is itself a function from vectors of ${\mathbb{V}_E}$ into ${{\cal V}_L}$ , so $G$ may involve the norm indirectly. This is indeed the case. The occurrence of $|{\bf{r}}{|^{ - 2}}$ in the law of universal gravitation tells us that acceleration is inversely proportional to the square of distance; it famously is an inverse-square law. Here, that fact is encoded in condition (ii) on $G$ . Likewise, the occurrence of the unit vector $|{\hat v_{ij}}|$ in the law of universal gravitation tells us that acceleration is in the same direction as displacement. That fact is here encoded in conditions (iii) and (iv). Despite the “disappearance” of distance, then, equation (7) has the same empirical content as equation (1).

This proves that an account of $G$ is not just an optional element of an interpretation of NG. Insofar as we are driven by a desire for intrinsic theories, an account of $G$ is crucial to the enterprise. Once we have an account of $G$ , we can consistently interpret the equation of motion of NG intrinsically.

7. Active mass symmetries

I have thus established the first aim of this article: providing an account of the gravitational constant, $G$ . I will now use this account to illuminate the recent debate about mass symmetries. I start by defining active mass scalings in this section; in the next section, I consider scalings of $G$ .

Unlike passive transformations, which only alter which numbers we use to represent particular masses, an active transformation changes the actual mass values assigned to particles themselves. The effect of an active mass scaling is that each particle’s mass (as represented on ${\mathbb{R}^ + }$ ) is multiplied by a constant $\alpha $ .

Take any arbitrary representation $f$ of mass values on ${\mathbb{R}^ + }$ (where $f$ is the “internal space” analogue of a coordinate chart). ^{Footnote 13} We can carry out a passive transformation on this representation that transforms $f(m)$ into $f'(m) = \alpha f(m)$ . The effect of this transformation is to assign each mass value $m$ a different real number. In order to turn this passive transformation into an active one, we keep fixed the representation relation $f$ and consider an automorphism ${\phi _\alpha }$ on ${{\cal V}_M}$ defined such that ${\phi _\alpha }(m) = ({f^{ - 1}} \circ \alpha f)(m)$ . If a mass $m$ is associated in some system of units to some number $x$ , then ${\phi _\alpha }$ maps $m$ onto the unique mass that is associated to the number $\alpha x$ in the same system of units.

The automorphism ${\phi _\alpha }$ in turn acts on $m(x)$ —the function from particles into ${{\cal V}_M}$ —as ${\phi _\alpha } \circ m(i) \equiv ({\phi _\alpha } \circ m)(i)$ . In this way, an active mass scaling induces a transformation on the models of NG:

(8) $$\langle D,\mathbb{E},\mathbb{T},{{\cal V}_M},x_i(t),m(i)\rangle \mathop \to \limits^{{\rm{Scaling}}} \langle D,\mathbb{E},\mathbb{T},{{\cal V}_M},x_i(t),({\phi _\alpha } \circ m)(i)\rangle .$$

This transformation is a symmetry of NG only if the model before the transformation is a solution to NG’s equations of motion whenever the same model after the transformation is also a solution. ^{Footnote 14}

Are active mass scalings symmetries of NG? The obvious answer is no. This follows immediately from NG’s equations of motion: the right-hand side of equation (7) contains a mass quantity and so changes value with mass—in accordance with condition (i) on $G$ —whereas the left-hand side is independent of mass and so remains the same. Unless $\alpha = 1$ , this transformation affects the satisfaction of NG’s equation of motion.

Martens (Reference Martens2021) illustrates this with the “comparativist’s bucket,” an analogue of Newton’s famous bucket experiment based on a thought experiment attributed to Baker (Reference Baker, Bennett and Dean2020). The escape velocity of a projectile is given by

(9) $${v_{{\rm{esc}}}} = \sqrt {{{2G{m_2}} \over r}} ,$$

where $G$ is the gravitational constant, ${m_2}$ is the mass of the body the projectile is escaping from (say, Earth), and $r$ is the distance between them. Suppose that in one model, the actual velocity of the projectile $v$ is just over ${v_{{\rm{esc}}}}$ —so the projectile escapes. But under an active mass scaling, Earth multiplies in mass by a factor $\alpha $ , and hence ${v_{{\rm{esc}}}}$ is multiplied by a factor $\sqrt \alpha $ . Consequently, $v$ may no longer exceed ${v_{{\rm{esc}}}}$ : the projectile crashes back to Earth. Because the actual trajectory ${x_i}(t)$ of the projectile is left the same by an active mass scaling, the transformed model is inconsistent with the equations of motion. Therefore, active mass scalings are not symmetries of NG.

8. Scaling G

We have not yet considered the gravitational constant, $G$ . $G$ is a constant with dimensions proportional to ${M^{ - 1}}$ . It is commonly thought that if one were to scale $G$ with all particle masses, active mass scalings would become symmetry-like transformations after all (Roberts Reference Roberts2016; Wolff Reference Wolff2020). Wolff, for instance, argues that because we fix the reference of the term “ $G$ ” by ostension, it will have a different referent in a mass-scaled world. Although that may be true, it is irrelevant to the question of whether $G$ itself has a different value in such a world. The inhabitants of a mass-scaled world use the same symbol “ $G$ ” to denote a different constant, but this does not mean that the constant we denote with “ $G$ ” has a different value in that world. I therefore concur with Martens (Reference Martens2022) that there is no reason to believe that one must scale $G$ when we scale all particle masses.

It remains possible that one could change the value of $G$ in addition to scaling all particle masses. This allows us to define another transformation—call it an inclusive active mass scaling—which is a symmetry-like transformation of NG insofar as it seems to relate empirically equivalent physical possibilities. But Martens goes on to make the stronger claim that we cannot change the value of $G$ without moving to a different theory: “It is not at all surprising that if one were allowed to change (the strength of) the laws at will for each possible world one could get any (or at least many) of the evolutions one may have wanted. That is simply not an option within the rules of the game we are playing” (Martens Reference Martens2022, 12). Martens claims that changing $G$ is akin to changing the laws, which is against the “rules of the game.” But who set out these rules? It seems to me that there are different conceptions of what a constant of nature is, which in turn have different consequences for whether a change in $G$ is “allowed.” In particular, once we reject real number structures as redundant, it becomes untenable to hold that $G$ is a parameter with a fixed numerical value. The cross-value space structure of $G$ is determined by conditions (i)–(iv) listed earlier, but these conditions only determine $G$ up to a “choice of unit.” To demand that $G$ is fully fixed is to introduce an “unnecessary global assumption,” in the words of Earman and Norton (Reference Earman and Norton1987), akin to hard-coding the particular Minkowski metric $diag( - 1,1,1,1)$ into the fabric of relativistic spacetime. It may turn out that different $G$ -functions ultimately represent the same cross-value space structure, as I will argue later in the article. This follows not from any predetermined rule book but from a philosophical analysis of the role of $G$ in NG.

It remains possible that Martens simply has a different conception of what counts as “the” theory of NG: perhaps in Martens’s conception, the value of $G$ is hard-coded into that theory. In that case, the disagreement is over the precise definition of NG’s space of models rather than over the behavior of $G$ in particular. Nevertheless, if one adopts a more liberal account of NG, as I have urged earlier, this has consequences for the possible values of $G$ . In what follows, I will assume such a more liberal account.

I will thus formulate an inclusive active mass scaling as follows. For any automorphism ${\phi _\alpha }$ of ${\cal V}$ :

(10) $$\langle D,\mathbb{E},\mathbb{T},{\cal V},x_i(t),m(i)\rangle $$

(11) $$\mathop \to \limits^{{\rm{Inc}}{\rm{. Scaling}}} \langle D,\mathbb{E},\mathbb{T},{\cal V},x_i(t),({\phi _\alpha } \circ m)(i),\phi _\alpha ^* G\rangle ,$$

where $\phi _\alpha ^*$ is the pullback of ${\phi _\alpha }$ , defined such that $\phi _\alpha ^* G(m,{\bf{v}}) = G(\phi _\alpha ^{ - 1}(m),{\bf{v}})$ . The effect of this transformation is to change the cross-value space relations between ${{\cal V}_M}$ and ${\mathbb{V}_E}$ . If $m$ and ${\bf{v}}$ were previously connected to some acceleration $\ddot x$ , for instance, then after this transformation, they are connected to some different acceleration $\alpha \ddot x$ .

It is easy to see that such a transformation preserves the dynamics: when $\phi $ acts on both $G$ and $m$ , ${\phi ^*}G(\phi (m(i)),{\bf{v}}) = G(m(i),{\bf{v}})$ . Given a particle’s mass value and a displacement vector, $G$ yields the same acceleration before and after an inclusive mass scaling. This transformation therefore also preserves escape velocities. Suppose, for instance, that ${\phi _\alpha }(m) = \alpha m$ ; then $\phi _\alpha ^* G = G/\alpha $ . In the formula for escape velocity, these factors of $\alpha $ precisely cancel each other out.

Perhaps it would be a misnomer to call this transformation a symmetry. ^{Footnote 15} The best definition of symmetries is a matter of current debate, but Earman’s (1989) definition of dynamical symmetries is a useful first approximation. ^{Footnote 16} As the name suggests, dynamical symmetries act on a theory’s dynamical objects. In the case of NG, these are the position and mass quantities. ^{Footnote 17} Importantly, a dynamical symmetry does not act on the theory’s kinematical structure, which here includes $G$ . So, an inclusive mass scaling is not a dynamical symmetry because it does act on $G$ . Earman contrasts dynamical symmetries with spacetime symmetries, which leave the dynamical objects alone and instead affect the theory’s (fixed) spacetime structure. We can generalize this definition to include nonspatiotemporal kinematic structure. ^{Footnote 18} In this definition, kinematic symmetries are transformations only of the theory’s kinematic structure—potentially including $G$ . Again, inclusive mass scalings are not kinematic symmetries because they transform both kinematical and dynamical structure.

Elsewhere in the literature, the kind of transformation we are interested in is called a similarity. ^{Footnote 19} This term originates in a passage from Poincaré ([1908] Reference Poincaré2003):

The case in which all mass-valued quantities are scaled follows the same pattern as set out by Poincaré. Following Poincaré, I will say that an inclusive mass scaling is a similarity of NG.

Despite the fact that this transformation is a similarity rather than a symmetry, it poses similar metaphysical and epistemological questions. In particular, the models related by an inclusive mass scaling describe empirically equivalent states of affairs. But in certain views, those states are physically distinct. This would lead to a worrisome form of underdetermination: particle masses become undetectable even in principle because mass measurements have the same outcome in similar worlds despite the fact that particles have different masses in them. ^{Footnote 20} The presence of such undetectable quantities is a hallmark of redundant theoretical structure.

The physical distinctness of mass-scaled solutions follows from Martens’s (2021) belief that mass values possess primitive identities, or quiddities: nonqualitative properties that are uniquely possessed by individual mass magnitudes. ^{Footnote 21} The reason Martens postulates quiddities is that the elements of ${{\cal V}_M}$ are absolutely indiscernible: for no $m \in M$ does there exist a formula with one free variable that is true of $m$ only. ^{Footnote 22} However, Martens claims, the laws treat different masses differently—the more massive an object is, the slower it accelerates in response to gravitational attraction. Therefore, Martens continues, we must introduce quiddities that tell forces how to “latch on to” masses: they are “required for the forces to be well-defined, in the sense of uniquely matching up instances of initial conditions, including masses, with, say, accelerations” (Martens Reference Martens2021, 2519).

When masses are scaled, each particle is assigned a different mass value. When each mass comes attached with a quiddity, this represents a real physical difference. The presence of qualitatively identical yet physically distinct possibilities is just the sort of problematic underdetermination that is familiar from debates around the structure of spacetime. When a theory’s spacetime symmetries match the dynamical symmetries, symmetry-related models seem to represent worlds that differ merely over which spacetime point plays which qualitative role. This is the case, for example, for boost-related models of NG set on Galilean spacetime. There, a renouncement of primitive identities as redundant solves the issue: the difference between boost-related models becomes a distinction without a difference. I claim that in the present case, the quiddities of mass values are also redundant. Without quiddities, the specter of underdetermination fades away.

Recall that Martens claims that quiddities are necessary for forces to “latch on to” the right mass magnitudes. This ignores the gravitational constant. The real way in which forces latch on to masses is via the cross-value space structure represented by $G$ . When we include $G$ within the theory’s kinematical structure, quiddities become unnecessary. For example, consider once more the escape velocity of a projectile. Which velocity the projectile requires to escape from a massive body with mass ${m_2}$ depends on the acceleration the projectile acquires as a result of the gravitational pull of the massive body. For a displacement ${\bf{r}}$ , this acceleration is equivalent to $G({m_2},{\bf{v}})$ . We can thus find out the acceleration of the projectile without the quiddity of ${m_2}$ . The gravitational attraction can simply latch on to masses via $G$ to produce the observed accelerations.

There is a helpful analogy here with Aristotelian spacetime. Aristotelian spacetime results from adding a privileged worldline to a Newtonian spacetime. This additional structure is surplus in NG, but it would be necessary to express the laws if forces were to depend on absolute positions. It is exactly because translations of all material bodies are not symmetries of such an Aristotelian theory that we would need to introduce a worldline to represent the “center” of the universe. Similarly, it is because of the fact that mass scalings are not dynamical symmetries of NG that there must exist some structure that distinguishes the points of mass value space.

Furthermore, translations of both all material bodies and the center-of-the-universe worldline are symmetries of the Aristotelian theory; models related by such translations are isomorphic. Likewise, a scaling of all particle masses and $G$ is a similarity of NG; models related by such scalings are isomorphic, too. Once one rejects primitive identities, however, these models therefore represent the same possibility. Consider the privileged Aristotelian worldline. If we think of it as a property of a certain special collection of spacetime points, it seems that a translation of the center of the universe assigns this special property to a different collection of points. The result is an empirically equivalent yet physically distinct state of affairs. It seems more natural to consider the Aristotelian worldline as a property that individuates spacetime points: to be this spacetime point is just to be the point that is located in a certain way relative to the center of the universe. This view entails that a shift of all particles and spacetime structure has no physical effect: matter remains in the same location with respect to the Aristotelian worldline. Compare this with Stachel’s (Reference Stachel, Earman, Janis and Gerald1993) view of the metric in general relativity (GR) as a dynamical individuating field. Both views follow from anti-haecceitism: the view that spacetime points have no primitive identities but are qualitatively individuated.

The analogue of anti-haecceitism for quantities is anti-quidditism: physical magnitudes have no primitive identities but are qualitatively individuated. I propose anti-quidditism about mass. ^{Footnote 23} The idea is that mass magnitudes are identified by their pattern of instantiation. Teller (Reference Teller1991, 393) puts the idea as follows:

Because the structural relations that the particles’ masses stand in to each other and to the elements of ${\mathbb{V}_E}$ and ${\mathbb{V}_T}$ (as determined by $G$ ) are the same across models related by inclusive scalings, anti-quidditism implies that the same masses are instantiated in these models. The differences between isomorphic models are representationally irrelevant because they concern quidditistic facts about which particular mass values are instantiated. This is also a distinction without a difference.

This conclusion allows us to avoid the specter of underdetermination that normally accompanies symmetry-like transformations. If inclusive mass scalings merely change the way a possibility is represented, rather than that possibility itself, then there are no empirically equivalent yet physically distinct possibilities to speak of. Therefore, scaling $G$ with all particle masses does not lead to a distinct physical possibility. This conclusion does not follow from any preconceived notion of $G$ as a fixed parameter but from an account of the metaphysical nature of $G$ in conjunction with anti-quidditism. The position I have presented is therefore structurally similar to sophisticated substantivalism about spacetime (cf. Wolff’s (Reference Wolff2020) “sophisticated quantity substantivalism”).

9. Constants and symmetries

I have presented an account of what dimensionful constants represent: cross-value space relations. I have then used this account to argue in favor of anti-quidditism about mass values. But a further question could be raised: Why are there constants in the first place? It is not an a priori fact that distinct value spaces should be connected in this way. This is not the type of question that always has an answer. But in this case, I believe that there is an answer, namely, that constants are necessary in order to coordinate the dynamical symmetries of different quantities. I will now elaborate on this idea.

Recall Earman’s distinction between dynamical and kinematical symmetries, where I will include the symmetries of nonspatiotemporal kinematical structure in the latter. Earman (Reference Earman1989) famously posited a pair of symmetry principles:

The motivation for SP1 is to avoid redundant structure. If SP1 fails, then the theory posits kinematical structure that is dynamically inefficacious. The classical example here is the standard of absolute rest of Newtonian spacetime: because the dynamics of NG are boost-invariant, such a standard is superfluous. The motivation for SP2, on the other hand, is to make sure that the theory has enough structure to sustain the dynamics. If SP2 fails, then the dynamics distinguish between elements that are qualitatively the same. An example is Leibnizian spacetime, which lacks a standard of absolute acceleration. Because NG is sensitive to absolute accelerations, Leibnizian spacetime does not have enough structure to sustain the theory’s dynamics.

Consider now the structure of mass value space, ${{\cal V}_M}$ . On the one hand, the mass scalings ${\phi _\alpha }$ —applied just to particles—are not dynamical symmetries of NG. On the other, they are symmetries of the kinematical structure of ${{\cal V}_M}$ because the relations ⩽ and $ \circ $ defined over $M$ are invariant under the ${\phi _\alpha }$ transformation. Consider arbitrary ${m_1},{m_2}$ such that ${m_1}$ ⩽ ${m_2}$ . By Hölder’s representation theorem, $f({m_1}) \le f({m_2})$ for any representation $f:{\cal V} \to {\mathbb{R}^ + }$ . Because $\alpha $ is positive, $\alpha f({m_1}) \le \alpha {f_r}({m_2})$ . By another application of the representation theorem, $({f^{ - 1}} \circ \alpha f)({m_1})$ ⩽ $({f^{ - 1}} \circ \alpha f)({m_2})$ . Because ${\phi _\alpha }(m) = ({f^{ - 1}} \circ \alpha f)(m)$ , this shows that ${m_1}$ ⩽ ${m_2}$ iff $\phi ({m_1})$ ⩽ $\phi ({m_2})$ ; hence, ⩽ is invariant under the action of $\phi $ . The proof for $ \circ $ is analogous.

There is thus a mismatch between the kinematical and dynamical symmetries of ${{\cal V}_M}$ in violation of SP2. Earman’s (1989) argument in favor of SP2 (applied to masses) is that were the principle to fail, “the theory would have to contain names, regarded as rigid designators, of [elements] of [mass value space]” (47). This is exactly what Martens believes: that the laws of NG “latch on to” masses via their quiddities. The laws must pick out masses independently from their qualitative features, so in this sense, the theory “names” particular mass magnitudes. Earman (Reference Earman1989), however, sees this as a defect: “But such a difference in lawlike behavior is reason to suppose that [ ${m_1}$ ] and [ ${m_2}$ ] differ in some structural property that grounds the difference in behavior” (47). In other words, the fact that gravity acts in a certain way on a certain mass should follow from facts about what that mass is like—not just from the irreducible fact that it is that mass! This suggests that mass value space has more than an additive extensive structure.

Let us suppose not only that mass scalings are not dynamical symmetries but also that no transformation that acts solely on ${m_i}$ is a symmetry of NG. Put differently, no nontrivial automorphisms of ${{\cal V}_M}$ are dynamical symmetries. It is then tempting to conclude, by SP2, that mass value space itself has no nontrivial automorphisms. This would mean that ${{\cal V}_M}$ has the sort of rigid real number structure discussed in section 3. The real number structure is richer than an additive extensive structure exactly because it has fewer symmetries. ^{Footnote 24} Unfortunately, this does not solve the inconsistency with SP1 and SP2. We have already seen that this structure endows mass value space with a privileged unit, contrary to the conventional (and correct) wisdom that the choice of unit is arbitrary. But there is a more general argument against this sort of structure, which turns on the scale independence of NG.

Suppose one were to double all particle masses, and all distances between particles, and all durations between events. It turns out that this is a symmetry of NG. Generally, consider a joint scaling of mass by a factor $\mu $ , of length by a factor $\lambda $ , and of time by a factor $\tau $ : this transformation is a dynamical symmetry of NG iff ${\lambda ^3} = \mu {\tau ^2}$ . One can easily see from dimensional analysis that under such a joint scaling, the left-hand side of equation (7) is multiplied by a factor of $\lambda {\tau ^{ - 2}}$ , whereas the right-hand side is multiplied by a factor of $\mu {\lambda ^{ - 2}}$ . These factors are equal iff ${\lambda ^3} = \mu {\tau ^2}$ . The special case of $\lambda = \tau = \mu $ shows that the theory is invariant under a joint rescaling of all quantities by the same factor; that is, it is scale-independent.

These transformations are dynamical symmetries: they act on the dynamical objects ${m_i}$ and ${x_i}(t)$ (and their derivatives). From SP1, it follows that there are corresponding kinematical symmetries. But these kinematical symmetries are special in that they act simultaneously on quantities with different value spaces. We cannot account for these symmetries with changes to the structure of the theory’s value spaces in isolation. If we were to let ${{\cal V}_M}$ , ${{\cal V}_L}$ , and ${{\cal V}_T}$ individually remain invariant under scale transformations, for instance, then this would entail that scale transformations of mass, length, and time are dynamical symmetries of NG even when ${\lambda ^3} \ne \mu {\tau ^2}$ , contrary to the conclusion of section 7 that mass scalings are not dynamical symmetries.

In order to find the kinematical counterpart of NG’s scale independence, a piece of structure that connects the theory’s value spaces is required: the gravitational constant. On the one hand, $G$ is not invariant under scale transformations of mass, length, or time by themselves. This correctly entails that such transformations are not dynamical symmetries of NG. On the other hand, $G$ is invariant under any joint scale transformation for which ${\lambda ^3} = \mu {\tau ^2}$ . From (i) and (ii), it follows that $G(\mu m,\lambda {\bf{v}}) = \mu {\lambda ^{ - 2}}G(m,{\bf{v}})$ . But the resultant value of $G(m,{\bf{v}})$ is an acceleration, which scales with $\lambda {\tau ^{ - 2}}$ . The function $G$ is therefore preserved whenever $\mu {\lambda ^{ - 2}} = \lambda {\tau ^{ - 2}}$ , or equivalently, whenever ${\lambda ^3} = \mu {\tau ^2}$ . Although each individual value space is invariant under scale transformations, the connections between these value spaces are only invariant when these transformations are related in this particular way.

Is the occurrence of constants a consequence of the dynamics’ scale independence? Or are the dynamics scale independent because of the occurrence of $G$ in the equations of motion? This question is reminiscent of the debate between the geometrical approach and the dynamical approach to spacetime structure (Brown and Read Reference Brown, Read, Knox and Wilson2021). In the former view, the dynamical symmetries of NG are a consequence of the theory’s spacetime structure. In the case under discussion, this translates into the view that the scale independence of NG is a consequence of the occurrence of $G$ in the equation of motion. Suppose that the equations of motion did not contain $G$ ; then they would not remain invariant under scale transformations of the sort discussed earlier. The latter view reverses the order of explanation: spacetime structure is only a reflection of the theory’s dynamical symmetries. In such a view, the gravitational constant merely codifies the details of NG’s scale independence. The particular transformation properties of $G$ reflect the fact that these dynamics are invariant whenever ${\lambda ^3} = \mu {\tau ^2}$ . $G$ becomes a “glorious nonentity,” to borrow a phrase from Brown and Pooley (Reference Brown, Pooley and Dieks2004). Whether the parallel between the gravitational constant and spacetime structure is sufficiently strong to warrant a similar treatment for both remains to be seen. The benefit of a clear account of constants is that it affords us the means to ask such questions in the first place.