Philosophy of Science 101: Laws of Nature

Foreword

The Philosophy of Science series explores both general questions about the nature of science and specific foundational issues related to the individual sciences. When applied to such subject areas, philosophy is particularly good at illuminating our general understanding of the sciences. This 101 series will investigate what kinds of serious—often unanswered—questions a philosophical approach to science exposes through its heuristic lens. This series, more specifically, will look at the ‘Scientific Realism’ debate throughout, which questions the very content of our best scientific theories and models.

Philosophy of Science 101 will be divided into the following chapters of content:

1. Philosophy of Science 101: The Relationship Between Philosophy and Science

2. Philosophy of Science 101: Scientific Realism

3. Philosophy of Science 101: Anti-Realism

4. Philosophy of Science 101: Realism and Anti-Realism ‘Compromise’

5. Philosophy of Science 101: Causation

6. Philosophy of Science 101: Scientific Models

7. Philosophy of Science 101: Models of Explanation

8. Philosophy of Science 101: Laws of Nature

9. Philosophy of Science 101: Science and Social Context

Philosophy of Science 101: Laws of Nature

Previously, the Philosophy of Science 101 series explored models of explanation, in particular Carl Hempel and Paul Oppenheim’s (1948) Deductive Nomological Model (i.e., the DN model) where the modern philosophical investigation into this area begins. The former article considered the DN model’s various objections, extensions, and proposed alternative models in response. Indeed, both the DN model and alternative models offered to rely on accounting for (scientific) explanations via laws. Laws make up the core of a valid deductive argument, for instance, in the DN model or statistical laws may be used to make up the structure of an explanation—or prediction—in science.

As discussed last time, competing models of scientific explanation encounter interrelated issues. Oftentimes, the said issues relate to a model’s use of laws to comprise the explanation. A major problem, Nancy Cartwright (1979) argues, arises when models of explanation cannot explain the structure of laws themselves (or, put differently, when laws do not possess any explanatory power). Do laws have their own structure? Are they merely regularly occurring patterns of events? Can models of explanation appeal to more general laws and can they explain laws? The series now turns to these very questions, investigating the philosophy of science concerning the laws of nature. More specifically, this article will introduce two major schools of thought, namely a regularity theory of laws and a necessitarian theory of laws. The article will then also examine the so-called Best Systems Approach to Laws, also known as the MRL account since it is credited to John Stuart Mill, Frank Ramsey, and David Lewis (Pfeifer, 2012) (hence MRL—Mill-Ramsey-Lewis). Finally, this addition to the series evaluates instrumentalism about the laws of nature.

Figure 1: Kepler’s law of areas states that a line connecting a planet to the sun covers an equal area over equal periods of time (Silverman, 2021).

Two Schools of Thought: Regularity versus Necessitarian

Laws are undeniably essential to all sciences, whether that be a natural science or a social science. Consider some basic examples to illustrate the significance of laws:

1. Isaac Newton’s Law of Universal Gravitation in Astrophysics, that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centres (Chang & Hsiang, 2008).

2. The Cayley-Hamilton theorem in Linear algebra credited to Arthur Cayley and William Hamilton states that every square matrix (i.e., a matrix with the same number of rows and columns) over a commutative ring (i.e., a ring in which the multiplication operation is commutative) satisfies its own characteristic equation (Straubing, 1983).

3. Stanley Dermott’s (1968) Law in Celestial mechanics, an empirical formula for the orbital period of major satellites orbiting planets in the Solar System.

4. Gauss’s Law for magnetism in Mathematics and Physics, one of the four Maxwell’s equations that underlie classical electrodynamics equivalent to the statement that magnetic monopoles (i.e., hypothetical elementary particles that are an isolated magnet with only one magnetic pole) do not exist (Poole, 2018).

5. Law of Supply and Demand in Economics, the theory that prices are determined by the relationship between supply and demand (i.e., if the supply of a good or service outstrips the demand for it, prices will fall, whereas if demand exceeds supply, then prices will rise) (Lowe, 1942).

The list of scientific laws—often named after their creators—that are crucial in their respective fields may be extended ad nauseam. They are of particular importance for a multitude of reasons. First and foremost, expectations about the behaviour of systems are based on the laws that govern them (Bunge, 1961). Interventions in a system are also based on such laws, and scientific predictions are indeed trustworthy because they are based on laws (Hanzel, 1999). Further, as discussed at length in the previous article of the series, laws figure prominently in explanations on multiple accounts, too.

There are two major schools of thought as to what constitutes a law. Firstly, the regularity theory, whereby laws describe the way things "actually do" behave so to speak. Secondly, the necessitarian theory argues that laws describe how things must behave (Swartz, 1985). There are interdisciplinary debates as to whether David Hume’s work is necessitarian or not—and this deserves attention in another paper—but what is clear is that the regularity view of laws results from Hume’s famous concept of causation. This is indeed where modern study into the laws of nature begins. According to Hume, causation may be constituted by (1) succession (i.e., a cause comes before an effect), (2) contiguity (i.e., a cause is spatiotemporally contiguous to an effect, (3) constant conjunction (i.e., all causes are invariably followed by effects), and (4) no necessity, since the concept holds that this is a psychological illusion (Baumgartner, 2008). There are therefore no necessities involved since the theory is not saying what must happen but what does happen. On Hume’s regularity view of laws that results from his concept of causation, laws thus state what is universally the case (Baumgartner, 2008). Laws are universal generalisations, making up the Humean theory of laws and the basis for the first school of thought here.

Figure 2: David Hume, oil on canvas. Modern study into the laws of nature begins with Hume’s work on causation (Cranston & Jessop, 2023).

Following Hume, the late philosopher Alfred Jules Ayer is a proponent of the regularity view of laws (Earman, 1984). Ayer argues that all that is required for there to be laws in nature is the existence of de facto constancies (Curd & Cover, 1998). Constancy consists in the fact that events or properties of different types are conjoined with one another (Ayer, 1956). Ayer therefore defends regularity theory, and further to the basic Humean theory, holds that laws are universally quantified conditional statements:

(∀x) (Fx → Gx)

Yet again, there is no necessity involved. Rather in Ayer’s view, the laws of nature are (a) contingent (i.e., not a truth of logic), (b) true at all times and at all places, and (c) contain only non-local or natural predicates, that is, predicates that do not only apply in special circumstances (Curd & Cover, 1998). The containment of only non-local or natural predicates is notably important, ruling out, for example, a statement like "all flowers in my garden are lilies" as a law-like statement.

Delving a little deeper into the regularity view, now consider the truth table for material conditionals where "T" is true and "F" is false on Ayer’s account (figure 3):

Figure 3: Truth Table for material conditionals (Ikenaga, 2019).

As exhibited by the truth table, whenever a conditional has a false antecedent, the conditional itself is true. Importantly, the regularity theory faces problems as such. First, there is the problem of vacuous laws. The fact that a material conditional is always true when an antecedent is false has implications here, for if F is a predicate with an empty extension (i.e., one that does not apply to anything in the world), then Fx is always false and "(∀x) (Fx → Gx)" is always true (Curd & Cover, 1998). The claim that "all pink dragons are vegetarians" is a law of nature, for example, simply because there are no pink dragons in existence. Since it is a true universal generalisation that there are no pink dragons, "all pink dragons are vegetarians" is a law of nature according to regularity theory which proves to be a serious problem.

One can perhaps address the problem of vacuous laws by adding a further condition:

(∃x) (Fx) & (∀x) (Fx → Gx)

In this view, "all pink dragons are vegetarians" is not a law because the first clause, namely that there are pink dragons, is false. This, however, is too restrictive. The problem remains, for there are laws of nature that do in fact have an empty antecedent (Mellor, 1980). To give an example, Newton’s Law of Inertia says that if there is no force acting on a body, then the body either remains at rest or moves with constant velocity (Earman & Friedman, 1973).

Figure 4: English philosopher A.J. Ayer, proponent of the regularity view of laws (Rogers, 2023).

One therefore must distinguish between ultimate laws and derivative laws, whereby a law like the law of inertia is the latter (i.e., derived from other laws, often ultimate laws) and strictly only ultimate laws (such as "F = ma") have to be instantiated. For example, the force acting on an object is always equal to the mass of an object times its acceleration (Earman & Friedman, 1973), as illustrated by what actually happens everyday and often resulting in various other derivative laws. Distinguishing between the two is indeed only a potential response, and one must decide whether the problem of vacuous laws proves too serious and inescapable for the regularity theory.

Another problem that the regularity theory faces results from continuous values since many laws notably express functional dependencies between two or more magnitudes (Urbach, 1988). Consider a brief example:

Gas Law: pV = nRT

The variables in this equation can assume any real number as values. No one, however, has ever observed all continuous values the variables can assume; actual observation only provides one with a finite set of values (Urbach, 1988). Continuous values are thus a problem for the regularity view of laws since they conflict with the regularity theory’s idea that laws are summaries of actually observed regularities (recall that regularity theory says that laws describe the way things "actually do" behave).

Figure 5: Laws like the Ideal Gas Law create a problem for regularity theory (Encyclopædia Britannica, 2023).

Before the article goes on to look at the necessitarian theory in contrast, a third—and final—problem for the regularity view arises due to accidental generalisations. Think about the claim "all US presidents are men". As mentioned, such a claim can be dismissed (as a law-like statement) on the grounds that the regularity theorist uses non-local or natural predicates (Suchting, 1974). Consider, however, another example. The claim "all dogs born at sea are spaniels", or the claim that "all mountain peaks are less than 10km above sea level". Both claims satisfy the criteria of a law according to regularity theory. Yet both claims are (clearly) accidental generalisations. How might one distinguish real laws from accidental generalisations? An intuition might be to say that accidental generalisations, unlike laws, merely say how things happen to be. As such, perhaps one can move from:

"For all x: if x is F, then x is G"

To

"For all x: if x were an F, then x would be a G".

This, however, simply contradicts the Humean spirit (i.e., laws describing how things "actually do" behave). Yet again, one must decide whether regularity theory can overcome a problem like that of accidental generalisations.

Moving on, the necessitarian view proves quite different. In this approach, laws do not only say what universally is the case, but they also say what must happen (Bigelow, Ellis, & Lierse, 1992). Laws, therefore, express necessities. Hence: necessarily, all Fs are Gs. American philosopher Fred Dretske’s (1977) famous necessitarian idea follows, namely, that lawfulness is a relation between universals. On this account, law statements are not statements about the extensions of the predicates F and G. Rather, they are statements about universal properties: F-ness and G-ness (Dretske, 1977). As such, laws are necessitation relations between universals. Instead of "for all x: if x is F then x is G" one has "F-ness necessitates G-ness":

This x is F.

____________________________________________________________________________________________

This x must be a G.

It is questionable whether this inference is valid, and a further problem for the necessitarian is known as the identification problem. Broadly, it is unclear what exactly has to be the case in the world for something to qualify as a law (Armstrong, 1993). Philosopher David Armstrong (1993) offers a response, arguing that properties are universals and necessity is a second-order relation. The necessitarian school of thought is not uncontroversial, however, and is perhaps unconvincing as an alternative to the regularity theory. A newer and more popular account of the laws of nature is the MRL account, discussed next.

Figure 6: British philosopher John Stuart Mill, associated with the ‘MRL’ account of laws (Anschutz, 2023).

Best Systems Approach to Laws

Perhaps the most significant contemporary theory of lawhood is the Best System (/MRL) view on which laws are true generalisations that best systematise knowledge (Cohen & Callender, 2009). Associated with Mill, Ramsey, and Lewis, the MRL theory holds that the laws of nature unify (seemingly) diverse phenomena (Beebee, 2000). Such phenomena are unified in a frame of mechanics, which Mill claims is the defining feature of the laws of nature (Beebee, 2000). Lewis then adds that a generalisation is a law of nature if it appears as a theorem (or axiom) in each of the true deductive systems that achieve the best combination of simplicity and strength (Mumford, 2005). This is in relation to the Humean Mosaic, a collection of everything that actually happens; that is, all occurrent facts at all times (‘occurrent’ means that irreducible modalities, powers, propensities, necessary connections and so forth are not part of the mosaic) (Mumford, 2005). Indeed, the deductive systems that Lewis talks of must strike the best balance of simplicity and strength with respect to the Humean mosaic. The MRL account is therefore no longer an epistemic notion of law since it takes known as well as unknown facts into account via the Humean mosaic (Beebee, 2000). Many uniformities may not be known now, but there are true sentences corresponding to them and it is an objective matter of fact that these sentences enter into certain deductive relationships (Mumford, 2005). Hence, the notion of law is independent of what one knows now (or even of what one ever comes to know).

Strength and Simplicity on MRL

Simplicity and strength—two virtues of a deductive system—are in tension with one another according to the MRL theory of lawhood. This article shall first investigate strength, which is ultimately a matter of how many things are covered by the laws of a system (Pfeifer, 2012). "How many things" can refer both to tokens of events and to the types under which the tokens are subsumed. So, adding a law to a system that applies to, say, 109 distinct events adds more strength than adding a law that applies to only 100 events. Adding a rule that covers five different types of systems adds more strength than adding a law that covers just one new kind of system (Pfeifer, 2012).

Figure 7: Physicists propose that violations of a fundamental law of physics in early stages of the universe are responsible for the mysterious dark energy (Mann, 2023).

As for simplicity, there are various different types of this at play in the theory. Numerical simplicity, for example, which refers to the number of laws, may constitute one dimension of simplicity (Pfeifer, 2012) since a system is less simple with the more laws it has. Derivational simplicity, on the other hand, refers to how easily predictions can be derived from the laws (Mumford, 2005). How easily, for example, one can predict that magnetic monopoles do or do not exist according to Gauss’s Law for magnetism (Poole, 2018). Finally, the MRL then points to the simplicity of formulation which explores just how complicated the formulation of laws is (Beebee, 2000). Assuming a given language, for instance, then a law that takes many pages to write down is less simple than a law that can be compactly expressed on a single line.

Rather importantly, strength and simplicity compete. One can achieve maximal strength, for example, simply by making every sentence an axiom—and thereby sacrificing simplicity. One can then make a system simple by advocating just some straightforward axioms whilst excluding everything that is not deducible from them – thereby sacrificing strength. The challenge, however, is to find the system that strikes the best balance. Hence, the best systems approach to laws systematises the Humean mosaic on a strength and simplicity balance. This is not, however, without encountering various issues. On the one hand, there are notable virtues of the MRL account—namely that MRL (a) explains why laws have law-like features such as generality and (b) connects scientific practice (Torza, 2022). Problems, however, result since it is unclear how exactly strength and simplicity are measured (Torza, 2022). As discussed earlier, one can reach intuitive characterisations of both strength and simplicity. Yet the MRL account does not go further than this, arguably making the theory quite ambiguous. Another problem is that it is also not obvious how to handle accidental generalisations, especially considering strength—one might indeed add accidental generalisations mistaking these as an increase in the system’s strength (for example, accidentally assuming that a regularly occurring event is a law even though it simply just regularly occurs and is not law-like). Further, some realists object, there remains an element of subjectivity in the notion of "law" with strength and simplicity (Woodward, 1992). Then, in the same vein, one could object on the grounds that "systematising" is too vague. Systematising what? It is unclear as to what "systematising the Humean mosaic" really means. It is perhaps unclear as an account because too much focus—as with multiple theories of lawhood—is placed on laws as generalisations. Many, accordingly, favour Instrumentalism instead (as the essay will consider hereafter).

Figure 8: The simplicity of the Humean Mosaic (Sartenaer, 2019).

Instrumentalism

Unlike regularity or necessitarian (realist) theories of laws, instrumentalism holds that laws are not supposed to represent what happens in the world (Cartwright, 1983). Rather, laws are instruments that help to predict the outcomes of observations. If these predictions are in good agreement with observations, one may accept the law. Cartwright (1983) is indeed a prominent figure of this view, arguing that the laws of physics do not describe true facts about reality (hence not realists). In her famous book, How The Laws of Physics Lie, Cartwright (1983) claims that there is a trade-off between the factual content and explanatory power of laws: either the laws are false but have explanatory power and wide scope, or they are true but fail to explain and are only highly restricted in scope. Whereas the former refers to fundamental laws, the latter refers to phenomenological laws, Cartwright (1983) asserts. Cartwright’s argument, broadly speaking, is that if the laws were true then they would describe how things behave. This is not the case, however, and laws simply cannot explain things and be exact at the same time. The law of gravitation, for example, describes the force between two bodies varying in their masses (Cartwright, 1979). A possible solution is to limit the scope of the law to those bodies who interact only gravitationally, yet Cartwright argues that there is still a problem since this limits the scope of the law so drastically that it becomes practically useless for any explanation. Therefore, Cartwright (1983) concludes that one cannot be a realist about explanatory fundamental laws (i.e., one cannot hold that laws must describe certain processes/entities literally and truthfully).

Conclusion

The laws of nature usually fall under one of two camps: regularity theory or necessitarian theory. Laws are indeed very important in many sciences, and so deciding what lawhood truly consists of has become a significant philosophical endeavour. More often than not, the modern discussion starts with Hume’s work. According to Hume, laws describe the way things actually do behave (Baumgartner, 2008). Various proponents have then gone on to extend and develop this theory, such as Ayer (1956) who argues that the need for the laws of nature to occur is the existence of de facto constancies (Curd and Cover, 1998). Both Hume’s simple account and Ayer’s (1956) more developed regularity theory—this article went on to discuss—encounter issues, however, such as vacuous laws, accidental generalisations, and continuous values. Hence "the alternative view" was considered, namely the necessitarian approach: laws state what must happen. Dretske’s (1977) prominent necessitarian theory was explored specifically, before turning to the contrasting MRL account. This "best systems approach" to laws is perhaps the most well-known contemporary theory of lawhood and is credited to Mill, Ramsey, and Lewis. It does, however, face serious issues due to its enigma and lack of clarity (concerning strength and simplicity in particular). Thence a final approach to laws was briskly considered, adding yet another potential way to think of laws. Unlike any realist point of view, instrumentalism regards laws as instruments that help to predict the outcomes of observations. According to Cartwright (1984), for example, the laws of nature do not describe how things behave. As ever, the philosophy of science thus questions what humans do, do not, can, or cannot, know (in this case regarding the laws of nature). The series, therefore, pans out entirely and delves into the social dimensions of scientific knowledge next time. In this upcoming (and final) chapter, the series turns to the sort of study that encompasses the effects of scientific research on human life and social relations, the effects of social relations and values on scientific research, and the social aspects of inquiry itself.

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