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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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I suspect that many people will be surprised to learn how very subjective this question is, but it really comes down to a matter of opinion (and in some case the ego of the person choosing the name): they get applied willy-nilly and have no well settled meanings. – dmckee Mar 2 '11 at 23:08
I'm trying to clarify some matters regarding this very topic, that the names may be subjective. A colleague said that theory was some sort of previous step to get into a law. I thought that laws where similar to mathematical axioms, they are the rules of the game. On the other hand, theories are the "why" and "how" as Lagerbaer said. I thought that theories cannot become laws, but I'm unsure. – deps_stats Mar 2 '11 at 23:19
This is also something you could consider asking at english.SE, although they would probably give you a different perspective because the way in which physicists use these terms is (probably) different from the way in which the general English-speaking population uses them. – David Z Mar 2 '11 at 23:45
@David Zaslavsky That's why I didn't asked there :) Maybe I should give it a try at Philosophy.SE, maybe this is concerned with philosophy of science. – deps_stats Mar 3 '11 at 0:55
not a bad idea, when the philosophy proposal launches. – David Z Mar 3 '11 at 1:07
up vote 6 down vote accepted

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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The relationship between laws and theories is quite fuzzy, and seems that their relationship is only semantical rather than conceptual. Am I getting it? – deps_stats Mar 2 '11 at 23:52
@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. – David Z Mar 3 '11 at 1:16

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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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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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 – Tobias Kienzler Mar 3 '11 at 15:22

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 hear 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 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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But why not using the word "law"? Because it is polysemic and confusing? – deps_stats Mar 3 '11 at 15:28
@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... – Tobias Kienzler Mar 3 '11 at 15:34

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