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What is a physical law, a theory, and what is the relationship among them?

I know this is a very basic question, so any reference to epistemology will be greatly appreciated :)

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dmckee is right (in the comments), the distinction between "theory" and "law" is quite subjective and varies a lot depending on who you ask and the context in which they are used. Sometimes they can be nearly synonymous. I would advise you to take any information you get about the difference between these two terms (including the remainder of this post) with a grain of salt, and never be afraid to ask someone to clarify what they mean by "theory" or "law" if it matters to your conversation.

The one difference that I think most people will agree upon is that in order for something to be called a "law," there must (or at least should) be experimental evidence supporting it. There is no such requirement to be called a "theory." So it is possible for a theory to be "upgraded" to a law, once there is enough experimental evidence to make it seem true. However, even when that happens, it doesn't mean people are going to stop calling it a theory; for example, many people still use the terms "theory of gravity" and "theory of relativity" (and many others) even though both those theories have been confirmed by many, many experiments and have unquestionably achieved "law" status.

One other difference that I think is common is that "law" often (but not necessarily always) refers to a single principle, typically something that can be expressed as a single equation or a set of closely related equations. A "theory" can be more broad. For example, when someone says "law of gravity," they're probably talking about the equation

$$\mathbf{F} = -\frac{G m_1 m_2}{r^2}\hat{\mathbf{r}}$$

or a related equation. But I generally don't hear people using the term "law of relativity," possibly because special relativity involves several equations and a set of related concepts.

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    $\begingroup$ The relationship between laws and theories is quite fuzzy, and seems that their relationship is only semantical rather than conceptual. Am I getting it? $\endgroup$
    – deps_stats
    Commented Mar 2, 2011 at 23:52
  • $\begingroup$ @deps_stats: I think so. Basically what I'm saying is that the choice of the word "theory" or "law" to describe something doesn't mean that much. $\endgroup$
    – David Z
    Commented Mar 3, 2011 at 1:16
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In my understanding, laws are the "what" and theories are the "why". Ohm's law tells you what the relationship between voltage and current in a resistor is. The theory of electrons in solids tells you why.

So, the laws tell you how physical systems behave, and the theories tell you why they do so.

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  • $\begingroup$ But if a theory is wrong does it mean that the laws are also wrong? I don't refer to a limiting case like Newton's law. Consider we have two theories make the same predictions about a phenomenon and different predictions for another. Is this possible? $\endgroup$
    – Antonios Sarikas
    Commented May 26, 2020 at 22:23
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A law is a precise statement about a physical relationship, ie Newton's laws. A law is a subset of atleast one theory, which provides a context and framework for the law, ie Classical Mechanics, and can also provide relations between laws.

A consistent theory normally has a set of axioms, or assumptions, and restrictions on the situation where they hold, and any statement derived from these can be considered laws.

You can ask why of a law, and it can be backtracked inside the theory and back and forth even, between these axioms. Some will call this process an explanation.

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Laws are the building blocks of a theory. A theory is a mathematical model which is a collection of one or more laws.

Edit: I don't think, the answer of @Lagerbaer is right. Here is a confusion. "The theory of electrons in solids" are nothing but a set of more general laws. Take chemistry for example. All of the laws of chemistry are derived in principle from more general and fundamental laws of quantum mechanics. Theories are no more than a set of rules relating different quantities of a model or in other words a set of laws. These laws can be explained in terms of deeper and more fundamental laws. It's not that theory explains any "why" of its own laws.

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  • $\begingroup$ true, although one could then say the laws of quantum mechanics degrade Ohm's law to a corollary. I wouldn't dare stating that about chemistry, though $\endgroup$
    – Tobias Kienzler
    Commented Mar 3, 2011 at 15:22
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I recently stumbled upon this thread after reading a duplicate question. I agree entirely that the context in which the words are used is important, but in the context of science, the words have drastically different meanings.

I will frame this response in the context of science and I will provide references instead of conjecture.

tl;dr - the words mean different things and people need to understand why knowing the difference is important.

Laws are explanations for what is occurring and typically described observable relationships in the natural world. If an equation is involved, typically the relationship is a "scientific law".

Theories provide explanations for why or how something is occurring.

Theories do not become laws when they are "proven".

This is detailed in the work "What is Science?" by Norman Campbell (1953).
Further evidence can be found written by P.W. Bridgman.

(I know more research exists on this topic, unfortunately my sources are all printed papers so I don't have links to post at this time).

Additionally, Richard Feynman indirectly discusses the demarcation of laws and theories in his conversation regarding the ball in the wagon. He explains how people say "the ball moves to the back of the wagon because of inertia" (explanation for what is happening i.e. the "law of inertia") however, they do not know why this happens. He clearly lays out the framework that laws play a specific role in science.

In the context of science laws do not become theories. I repeat this because it is a very common misconception. There is not a single historical example of a "theory" becoming a "law" in science, because they do different things.

To relegate the difference to "semantics", which by definition is literally the "meaning of the word or phrase" does a disservice to those individuals trying to promote an accurate understanding of the Nature and Philosophy of Science to students, peers, and other citizens.

To further support this claim, look at some examples of scientific laws and theories.

The theory of gravity is still undergoing current research. Wikipedia Why would scientists be conducting research if the "theory" of gravity had already been proven to be Newton's "law" of gravity? Answer: Because laws and theories do different things in the context of science.

Kinetic molecular theory is a very popular theory used to explain the ideal gas law(s). The laws tell you "what is expected", but they offer no explanation for "why".

Diseases are spread via "germ theory", but you don't hear people say "Oh, germs are just a theory, I don't need to practice hygiene." Humans have seen the microscopic organisms that spread disease, but this hasn't become "germ law".

Perhaps the pinnacle of the answer should explain why knowing the difference between a law and a theory is important.

One of the most common rebuttals to "controversial" science is, "well that's just a theory". If people don't understand that established theories of science are supported by a multitude of evidence, then they will be more inclined to disregard the knowledge. These people grow up to vote on scientific issues or advocate for or against science research. The misinformation and personal bias could become the difference between an entity receiving the funding necessary for a scientific breakthrough that would benefit all of society.

The Nature and Philosophy of Science is a topic I love to discuss and research. Rather than down vote or disregard this as "another opinion", leave a comment explaining where you would like to see more evidence.

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As the answers of Lagerbaer and sb1 show, the answer can seem quite contradictory:

On the one hand, you can say a law is like a mathematical axiom, i.e. the starting point for bootstrapping your theories. I guess the example here are Newton's laws of motion. There is no explanation why motion should always stem from second order in time ODEs, and yet this assumption has (in the context of classical mechanics) proven to be correct.

On the other hand, you can say a law is a set of equations (like Ohms law) that can be explained by a theory. But since you can use Ohms law as an axiom as well, I think this latter definition is (IMHO) less correct.

But let's have a look at wikipedia, specifically http://en.wikipedia.org/wiki/Theorem#Terminology links law to the article Laws of science, starting with

The laws of science are various established scientific laws, or physical laws as they are sometimes called, that are considered universal and invariable facts of the physical universe. Laws of science may, however, be disproved if new facts or evidence contradicts them. A "law" differs from hypotheses, theories, postulates, principles, etc., in that a law is an analytic statement, usually with an empirically determined constant. A theory may contain a set of laws, or a theory may be implied from an empirically determined law.

not mentioning the word axiom anywhere. However, there's the article Physical Law, which states:

A physical law or scientific law is a scientific generalization based on empirical observations of physical behaviour (i.e. the law of nature). Laws of nature are observable. Scientific laws are empirical, describing observable patterns. Empirical laws are typically conclusions based on repeated scientific experiments and simple observations, over many years, and which have become accepted universally within the scientific community. The production of a summary description of our environment in the form of such laws is a fundamental aim of science. These terms are not used the same way by all authors. Some philosophers e.g. Norman Swartz use "physical law" to mean what others mean by "natural law"/"law of nature".

And further down:

Some mathematical theorems and axioms are referred to as laws because they provide logical foundation to empirical laws.

Personally, I prefer not using the word law at all (I mean in physics, don't get the wrong idea there ;)

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  • $\begingroup$ But why not using the word "law"? Because it is polysemic and confusing? $\endgroup$
    – deps_stats
    Commented Mar 3, 2011 at 15:28
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    $\begingroup$ @deps_stats: exactly. I mean, in well-established expressions, sure. But what one considers a law and what not is too subjective, as you can see from the example of Ohm's law - electrical engineers consider that a law, but a particle physicist could consider it a deduction from quantum mechanics... $\endgroup$
    – Tobias Kienzler
    Commented Mar 3, 2011 at 15:34
  • $\begingroup$ @TobiasKienzler So the best we can do is somehow always referring to the theory that a "law" has a meaning. For example using Lagrangian mechanics we can derive "Newton's laws" but in the context of Newtonian mechanics these equations are considered as laws. $\endgroup$
    – Antonios Sarikas
    Commented May 26, 2020 at 22:18

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