Picture an equilateral triangle—a triangle with three sides of equal length. Now imagine a line extending from one vertex of the triangle to the opposite side, such that the line bisects the angle of the vertex and is perpendicular to the opposite side. Then, mentally flip the triangle along this line, so that the side to the left of the line is now to the right and vice versa. If you have done everything right, the image you arrive at is indistinguishable from the one you started with. Equilateral triangles have a bilateral symmetry along this axis: transforming them by flipping them along this axis leaves all of their properties unchanged. Their structure is invariant under this reflection.
Symmetries are easiest to picture in geometric shapes. But the notion is much broader: a transformation of a mathematical object is a symmetry transformation just in case it leaves the structure of that object unchanged. Physical theories too have symmetries: transformations between solutions of the central equations of the theory that leave the form of the equations of motion unchanged. So, for example, Newtonian gravitational dynamics is symmetric under spatial and temporal shifts, kinematic boosts, and rotations. Just as the bilateral symmetry of an equilateral triangle can be expressed in terms of a transformation that leaves the structure of the triangle unchanged, so these symmetries can be expressed in terms of transformations that leave the system’s equations of motion unchanged.
Symmetry principles are statements of the symmetries of a physical theory: they tell us under which transformations the dynamics of the theory is invariant. But their presence raises questions in the metaphysics of laws of nature: Are symmetry principles themselves laws of nature? Do the symmetry principles constrain the laws? Or are they just a result of the dynamics we contingently have in the actual world? Here, I put forward a Humean view of the metaphysical status of symmetries. On this view, the symmetries, like the laws, give us useful information about the behavior of physical systems. This information tells us which differences matter to that behavior: two systems that differ only by a property that varies under symmetry transformations (rather than one that is invariant) behave the same way. Humeans hold that laws are true, informative generalizations; here, I argue that symmetry principles are laws in this sense.
The article is structured as follows. In section 1 I provide some background on the philosophical significance of symmetries. In section 2 I thoroughly review Lange’s account and argue that its central claim, that symmetry principles are law-governing laws, is a mischaracterization. Then, in section 3, I provide a Humean view that puts the symmetries in their proper place. Finally, I conclude by drawing conclusions for the metaphysics of laws.
1. Physical Symmetries
Recall that the symmetries of a physical theory are a set of transformations on the space of solutions to that theory. They are a function from solutions to solutions, such that some relevant mathematical structure is the same in any two symmetry-related models. An account of the physical significance of symmetries is an account of what two symmetry-related models have in common.
If two physical situations are internally identical but merely at different places or times, then they can be described by the same equations of motion, and their internal dynamics are identical. Similarly, if two physical situations are internally identical but one is rotated relative to the other or moving at a different velocity to another, they will behave identically.
This is the first thing to note about symmetry-related physical states. They are empirically indistinguishable. No experiment can distinguish a world from one related to it by a symmetry; no experiment within an isolated subsystem of the world can distinguish it from a symmetry-related subsystem (although external observers can tell the difference between, say, a system and its rotated counterpart). The relationship between different sorts of symmetries, taken as global transformations of the state of the world, and the behavior of subsystems is discussed in, among others, Brading and Brown (Reference Brading and Brown2004), Greaves and Wallace (Reference Greaves and Wallace2014), and Dasgupta (Reference Dasgupta2016).
Many philosophers of physics take symmetries to be a guide to connecting the structure of a physical theory to the structure of the world. On this view, two symmetry-related solutions to physical equations are simply different descriptions of the same situation. There are not two worlds in one of which I am here and in the other I am three feet to the left, with everything else similarly shifted. Instead, there is just this world and two mathematical descriptions of it. The fact that those descriptions put the origin at different places does not indicate any difference between the worlds, as the origin in our mathematical description did not correspond to anything in the world anyway. The symmetries tell us what structure the world does not have. See Greaves and Wallace (Reference Greaves and Wallace2014) for a discussion of this view.
But this view is not universally held; others think that we should take the mathematical structure of theories seriously and hold that a distinct possible world corresponds to each model of our theory. Nonetheless, on this view symmetry-related models correspond to empirically indistinguishable worlds. See Belot (Reference Belot, Brading and Castlellani2003) for an expression of this view.
2. Langian Metalaws
How should metaphysicians of law regard symmetries? Symmetries constrain our physical theories and the structure of our state spaces. Laws of nature, at least in physics, are typically differential equations. Rather than a list of generalizations in a first-order extensional logic, we get a single differential equation and constraints on how its free parameters can be set. So in Lagrangian mechanics we get the Euler-Lagrange equation and a set of constraints on the Lagrangian. Some of these constraints are requirements that the Lagrangian be invariant under certain transformations (these are the symmetry principles), while others specify the kinetic and potential energy of the system (these are typically understood as force laws and the boundary specification of, say, the number and masses of any particles in the system). Recognizing that some of these constraints are correctly regarded as law-like while others seem to be boundary or initial conditions requires metaphysicians of law to get into the weeds of philosophy of physics. Which parameters are which? And where do we put the symmetry constraints?
2.1. Lange on Symmetries
Lange (Reference Lange2007, Reference Lange2009, Reference Lange2011a, Reference Lange2011b) follows Morrison (Reference Morrison and Weinert1995) in arguing that symmetry principles are laws of nature and that, even more interestingly, they are laws that govern not first-order facts but the laws themselves. Symmetries are metalaws: laws of the laws.
Lange draws inspiration for this view from the writing of some physicists. Here is Einstein describing the meaning of the Lorentz transformation, which gives us the central symmetry of special relativity: “The content of the [special] relativity theory can … be summarized in one sentence: all natural laws must be so conditioned that they are covariant with respect to Lorentz transformations” (Einstein Reference Einstein1954, 329, quoted in Lange Reference Lange2011b, 216). Similarly, when providing a Lagrangian (or Hamiltonian) formulation of classical mechanics, we specify the theory’s symmetry group—those symmetry transformations under which the Lagrangian (or Hamiltonian) must be invariant. Lange takes this ‘must’ seriously. For Lange, the symmetries constrain the laws with a form of natural necessity—one stronger than that of the first-order laws.
Here is Lange: “A law is ‘symmetric’ in a certain respect exactly when it remains unchanged under a certain transformation. … Generalizing from one symmetry exhibited by one law, a ‘symmetry principle’ ascribes some symmetry to the laws as a whole” (Reference Lange2009, 105–6). Lange points out that some “regularities” of the laws, like some regularities of fact, can be accidents rather than necessary. He thereby distinguishes between consequences of the laws that merely happen to hold and those that must hold. Lange calls the first ‘by-products’ of the laws and the second ‘metalaws’. For example, it is a consequence of the laws that there is no inverse cubed force law: there could have been, but there is not. This regularity is a mere by-product. He then argues that the symmetry principles are not mere by-products. It is not that there could have been a law violating Lorentz invariance, or violating the isometry of space, but there is not: there must not be such a law. Had we ended up with different laws, they would have been Lorentz invariant, and they would have been symmetric under rotations, shifts, and boosts.
Having established that symmetries are regularities of the laws but are not accidental regularities, Lange infers that they are metalaws—higher-order principles governing the laws.
2.2. Lange on Laws
Having argued that symmetry principles are higher-order laws, Lange now faces the burden of showing that his view of laws delivers the correct result. This, I think, he does exceptionally well. Lange’s view of first-order laws is based on their counterfactual resiliency; he is easily able to extend this view to higher-order laws. According to Lange’s view, L is a law if and only if L would be true if p were true, for any p logically compatible with L, where L and p are both subnomic—not nomically modal statements or explicitly about laws.Footnote 1
What, then, are metalaws? According to Lange, metalaws are those statements that are nomically stable: they would have held even if any facts, subnomic or not, had been different. That is, they are truths that would have held even had the laws been different. According to Lange: “A closed set of truths that are nomic or sub-nomic qualifies as ‘nomically stable’ exactly when (whatever the conversational context) the set’s members would all still have held (indeed, none of their negations might have held) under every nomic or sub-nomic supposition logically consistent with the set—however many such suppositions are nested” (Reference Lange2009, 114). Lange argues that the symmetry principles would have held even had the other laws not held, including the conservation laws.
So Lange’s view is as follows: (a) the symmetry principles are nomically stable. (b) The symmetry principles explain the conservation laws by governing them; that is, the explanation of the conservation laws from the symmetry principles is an example of a covering law explanation. (c) The modal force, or counterfactual robustness, of these generalizations tells us which is the metalaw.
2.3. Contra Lange
Here I aim to present arguments against b: I will argue that symmetry principles are not properly thought of as higher order. Instead, they are generalizations about the first-order facts.
Lange claims that symmetry principles are laws governing other laws. This requires them to have nomic content—they must be about laws rather than first-order facts. Discussions of symmetry principles by physicists (like the Einstein quote in sec. 2.1) and textbook presentations of symmetry principles as constraints on the mathematical formulation of Lagrangian theories can give us the impression that the symmetry principles are primarily constraints on the laws or their formulation. But this is misleading.
Recall from section 1 that explaining the significance of symmetries involves saying what symmetry-related models have in common. There are two principal views here: on one view, the two symmetry-related models are distinct, but empirically or qualitatively indistinguishable, possibilities. On this view, the symmetries tell us the limits of our empirical discernment. On the second view, symmetry-related states are qualitatively identical. They represent the same situation or possibility. On this view, the symmetries tell us what structure—present in our mathematically formulated theory—is lacking in the world.
On either view, symmetry principles provide a constraint on first-order facts. They tell us either what systems are empirically indistinguishable or what systems are qualitatively identical. These first-order facts about what structure the world has leads to constraints on the laws: our dynamics cannot make distinctions without a difference, either in what occurs or in what can be observed. By focusing on these constraints, we can be lead to believe that the symmetries are primarily about the laws. This is a mistake.
The view that symmetry principles are primarily about first-order facts, and thereby give rise to constraints on the laws, also finds support among physicists. Here is physicist Eugene Wigner agreeing with this sentiment: “The geometric principles of invariance, though they give a structure to the laws of nature, are formulated in terms of the events themselves” (Reference Wigner1967, 17). So, rather than being higher-order principles to which the laws must adhere, the symmetry principles are generalizations about the first-order facts of the world. They are justified because they encode information about the structure of space-time (in the case of global external symmetries, those we have been focusing on thus far) or the property structure of the world (as, e.g., the charge conjugation symmetry of classical electrodynamics does).
This information provides us with some important empirical information: in addition to giving us quite general information about what properties and relations are (and are not) instantiated in the world, symmetry principles tell us when two isolated subsystems will behave in the same way, despite having different connections to the rest of the world. (There is some debate on the connection between different sorts of symmetries and empirically indistinguishable worlds and subsystems [see Brading and Brown Reference Brading and Brown2004; Greaves and Wallace Reference Greaves and Wallace2014; Friederich Reference Friederich2015].)
If symmetry principles are first-order generalizations, then they are not about the laws at all. Rather, physical theories are required to obey symmetry constraints because symmetry constraints are first-order facts that are known to be true. When we look beyond our current physics and consider possible new theories, we expect them to obey the symmetries of our old theory (and, plausibly, new symmetries we have discovered experimentally, as was the case with Lorentz invariance and special relativity) not because we have inductively discovered these symmetry principles by enumerating old, false theories and generalizing but because these symmetries encode some of the most general empirical information we have about the world, which in many cases can be retained through theory change.
If the symmetry principles are first-order generalizations, then they cannot explain the conservation laws as part of a covering-law explanation. So Lange’s view about the relationship between conservation laws and symmetry principles fails. Note that this does not mean that the symmetry principles cannot explain the conservation laws in some other way. I have not here argued against Lange’s claim that the symmetry principles have a “greater modal force” than other laws; I have only shown that that force does not come from their governing those laws from a higher level.
But I am suspicious that Lange can provide a satisfying explanation of this “greater modal force” if symmetry principles are not taken to be higher-order laws. While Lange is able to show that any set of statements that is “nomically robust” (i.e., a candidate for metalawhood) will be a subset of the (subnomically robust) laws, he does not provide any reason for believing the symmetry principles occupy this space.
3. Humean Maxilaws
In what follows, I argue that Humeans have the resources to countenance some laws as more counterfactually robust than others. I do this by discussing the traditional Humean view of what laws are and their relationship to counterfactuals in section 3.1. I then argue that Humeans can identify an especially informative subset of the laws and call these the ‘maxilaws’, in section 3.2. I argue that this view of counterfactuals leads neatly to the view that the maxilaws are more modally robust than other laws.
3.1. Humean Laws and Counterfactuals
The modern received Humean view of laws is the Best System Account (BSA), which has its roots in the writing of David Lewis (see Lewis Reference Lewis1983, 367). According to the BSA, the laws of nature are those true generalizations that best balance simplicity and informative strength.
To understand the place of counterfactuals for a proponent of the BSA, let us briefly look at the advantages of the view. The BSA comes equipped with a ready explanation of our interest in laws. We are interested in believing truths; the laws are statements that (a) we can learn (because they are simple) and (b) are such that, if we use them as axioms of our reasoning, we will infer lots of truths.
The motivation for taking the axioms of the best system to be laws is that adopting them as axioms of our reasoning will lead us to believe a large number of interesting and useful truths. If we do this—take these laws as axioms—we will have quite a number of beliefs purely in virtue of reasoning on the basis of the laws. Some of these beliefs will be consciously endorsed; others (possibly infinitely many others) will be purely dispositional beliefs—things we believe not because we are actively thinking about them but because we would endorse them if we were asked or if they were relevant to a decision situation.
Now consider the Ramsey test for counterfactuals (Ramsey Reference Ramsey and Mellor1929/1990): according to the Ramsey test, we should endorse a counterfactual just in case, were we to add the antecedent to our stock of beliefs and change our belief set as little as possible, we would come to believe the consequent. I do not think that the Ramsey test is an adequate philosophical account of counterfactuals or even all there is to say concerning our knowledge of counterfactual truth. But I do think that any successful account of counterfactuals needs to show that the Ramsey test is truth conducive: that is, it is at least a reliable way of evaluating counterfactuals, and by employing it we can come to know which counterfactuals are true (at least in most situations and provided we start with mostly true beliefs).
If the Ramsey test is truth conducive, then the best system laws are counterfactually robust. For suppose we took some generalization to be a law. Then—because the laws are those generalizations that are particularly useful for inferring truths—we would take it as an axiom in our reasoning and have many beliefs on its basis. So removing it from our belief set would result in a large change in our beliefs, as we would also lose those beliefs (including dispositional beliefs, possibly infinitely many) that we have on its basis. Hence, removing the laws from our belief set would result in a less-than-minimal change. Recall that the Ramsey test asks us to add the antecedent of a counterfactual to our beliefs and change our beliefs as little as possible and then see whether the result includes the consequent. Since removing the laws will never be a minimal change, we will only do so if the antecedent of the counterfactual contradicts them. So, if the Ramsey test is truth conducive, the members of the best system will remain true under any counterfactual supposition with which they are logically compatible. And while the Ramsey test does not supply the semantics of counterfactuals, it is a good barometer of their truth—not because of anything specific to Humeanism but because of the general principle that we are epistemically conservative, and change our beliefs as little as possible in the face of new evidence.
3.2. The Maximally Simple System
It is a frequent complaint of BSA that we are given little guidance as to how simplicity and informativeness are to be balanced. But I think this criticism is misguided. Humeans need not hold that there is just one way of weighting these that applies to every context in which we evaluate counterfactuals. One can think of them as two dials. Dial up the simplicity, and fewer axioms will qualify as laws, but the range of counterfactual suppositions one can evaluate will be broader, as more counterfactual antecedents will be compatible with these laws. Raise the value of informativeness, and more generalizations will count as laws: one will be able to more precisely evaluate a smaller set of counterfactual suppositions.
One can then imagine dialing simplicity to a maximum while retaining an interest in informativeness. We would then arrive at a system of laws that is as simple as it could be while retaining some informational content. Call this the Maximally Simple System (MSS). What would the MSS contain? I think it is very plausible that it would include (and be limited to) the symmetry principles. For the symmetry principles tell us the broadest information we have about the property and relational structure of our world. The space-time symmetries principally do two things: they describe the isometries of space, and they provide us with information about which subsystems we should expect to behave in the same way.Footnote 2 Other symmetries, for example, charge conjugation symmetry and tell us about the property structure underlying our physical laws. But these symmetry principles are considerably simpler than the laws: they omit the dynamic principles linking these properties to trajectories through space-time, and they lack the physical constants that encode the strength of the forces.
The argument given in section 3.1 naturally extends to show that the maxilaws—elements of the MSS—are robust under counterfactual variations in the dynamic laws. Just as we have many true beliefs in virtue of taking the dynamic laws as axioms of our reasoning, so too we have many true beliefs on the basis of symmetry principles. If we are asked to evaluate a counternomic counterfactual—one whose antecedent violates the laws—that does not directly contradict the symmetry principles, we will retain our beliefs in the symmetries because to do otherwise would result in a nonminimal change in our belief set. So (provided the Ramsey test leads to truth) the symmetry principles would hold even if dynamic laws were different. Our reasons for employing laws in evaluating counterfactuals are equally good reasons to employ maxilaws—when they are available—to evaluate counternomic counterfactuals.
4. Conclusion
I conclude that the Humean view has an advantage in explaining the explanatory structure within the laws of physics. To recap: symmetry principles describe quite abstract features of the spatial, temporal, and quantitative structure of our world. These abstract features provide us with information about which parts of the world are qualitatively alike and which are different. Because our laws only respond to qualitative features, symmetry principles provide quite strong constraints on the structure that the laws can take. But they provide these constraints because they are true, informative generalizations that we know when we are formulating our physical theories, not because they have a mysterious power to make the theories or laws develop one way rather than another.Footnote 3
The Humean view, if it succeeds for first-order laws, provides a link between the informativeness and simplicity of a set of generalizations and their modal resilience and explanatory power. This link arises because of the pragmatic utility of holding such generalizations fixed when reasoning and acting, and it naturally extends to symmetry principles, which are counterfactually robust and explanatory even among the laws. So the Humean view is quite powerful: it not only gets right the nature of symmetry principles and their explanatory status within the laws, but it connects naturally to a philosophical explanation of why we, as agents mostly interested in finding out about the actual world, would be interested in this counterfactual stability. It arises from their power as axioms of reasoning and our epistemic conservatism.
Can other views of laws reap this success? I am not sure. But I do think they will face challenges. As Lange (Reference Lange2009) and French (Reference French2014, 251) note, dispositional essentialists like Bird (Reference Bird2007) are in quite a pickle: whether or not the symmetry principles are metalaws, they do seem to be robust under counterfactual changes in the actual laws. But for the dispositional essentialist, the laws are metaphysically necessary. So it is not clear how these counterfactuals should be evaluated.
Proponents of sui generis laws, like Maudlin (Reference Maudlin2007), are similarly out on a limb. If the laws are part of the ontological structure of the world, what should we make of their internal explanatory structure? Plausibly, such a structure can be built in by hand. But it is more satisfying if we are able to explain it from the scruples of our view.
Similarly, I think there is hope for Lange. For while I have argued that he is wrong to take the symmetries to be metalaws, he still has the resources to make them more counterfactually robust, and so (on his view) more explanatory, than conservation principles and other dynamic laws. My chief complaint with this strategy is that it seems to build the explanatory structure in by hand: by stipulating that these counterfactuals are true and fundamental, we lose any ability to explain why these facts, rather than others, are the counterfactually robust ones. We thereby have a less satisfying philosophical explanation of the explanatory power of symmetries relative to the other laws. Perhaps, at the end of inquiry, the Humean view will not succeed, and we will have to accept this shallower explanation. But I am not yet ready to give up.
