# What exactly does a scientific law describe?

## Theories, models, experiments 2/3: Laws of nature [Updated]

This is the second part of a three-part short series on basic concepts of science - with a special focus on physics. In the first part of this I had described the connection between phenomena and their simplified images, the models. One path leads from models to more general statements about nature: to natural laws and scientific theories.

**Laws of Nature and Theories**

Models are first of all situation-specific and reflect the properties of a very specific phenomenon. But from there it is only a small step towards generalization to entire classes of phenomena, just as when we talk about phenomena we almost automatically switch to talking about classes of phenomena: We do not think of, let's say: every rainbow has its own word but talk about rainbows in general.

If I construct a simple physical model for a rainbow, then the chances are good that other rainbows can also be described with it. The two circumstances that the model does not reproduce every detail of the rainbow in its simplicity, but that it is on the other hand the details in which rainbows differ from one another, can interlock here.

Can, but do not have to - if each model describes only partial aspects of a phenomenon, one could also devise models that depict precisely those partial aspects that cannot simply be generalized to entire classes of phenomena; in the case of a rainbow, for example, the position in the sky and the course of brightness along the arc. But physical models are typically designed in such a way that *that* they describe a whole class of phenomena at once - and that is intentional.

Successful physical models can be adapted to many different situations. One example is the harmonic oscillator introduced by Einstein Understanding in Part V - the spring, in which the restoring force is directly proportional to the deflection (Hooke's law). This model concept not only describes a very specific spring, but - at least in the approximation of small deflections - almost all springs, as well as related systems such as torsion pendulums, certain systems in electrodynamics and quantum mechanics, and so on.

If models are equally suitable for whole classes of systems - if we can describe all phenomena with certain properties by one and the same type of model - then it is reasonable to assume that there could be a statement in these models that for *all* Phenomena with the properties mentioned apply because they are directly linked to these properties, not to the specific individual cases.

From the abstract to the more concrete: If you find that stones, iron balls, logs and an immense variety of other bodies, if you lift them off the ground and then let go, fall to the ground, then falling to the ground is evidently a general property.

The generality claim is also expressed in the choice of words: The general relationships that apply to a variety of situations and objects are also called *physical laws*, as *physical laws* or *Laws of nature* designated. This is misleading in some ways, because the human-made laws in the legal books of lawyers have the property that a person can also break the law (both consciously and in ignorance of the law in question). Laws of nature, on the other hand, state that a certain behavior *always* occurs. This is a description of the world around us, not an imposed norm of behavior to which nature would have to be educated with appropriate sanctions.

**Inset on verification and falsification**

The* always* this statement is of course not without its problems. Karl Popper stumbled upon this fact with his nose to anyone who thinks about science: We can never fully verify scientific theories, i.e. prove them to be true. We can only try as comprehensively as possible with our experiments to falsify them, i.e. to show that the theory is wrong. If a theory survives even the most sophisticated attempts to refute it experimentally, then we can as* proven* consider.

**Thrift and consistency**

[This section new on 1/30/2014, 10:30 am]

We arrive at scientific theories through the detour to models, and thus through simplifications. Simplification in this context means that the model does not reflect all the properties of the modeled phenomenon, but only a part of the properties. Conversely, models themselves can have properties that the modeled phenomenon lacks. An example is the following orrery, i.e. a mechanical model of the orbit of the planets around the sun (Image: Sage Ross under CC BY-SA 3.0 via Wikimedia Commons):

This model has many properties that the real solar system does not have: The metal connecting rods, the materials used for the bodies, the metal base housing and the gears hidden in it, which ensure the correct transmission ratio and thus the correct relative orbital speeds of the model planets, and so on. Arbitrariness goes hand in hand with each of these properties: the metal rods could also have a completely different shape, have a round or angular cross-section, they and the other model components could consist of a completely different material, the gears all have a common multiple of teeth, and so on.

At the beginning, if a phenomenon is not yet properly understood, one may use such models in science as well, with additional properties that are not necessary to explain the phenomenon. But one objective in science is to introduce as few additional properties, auxiliary constructions, things, connections, and events as possible into the description of nature. For what is enumerated in the long list of the last sentence (“Properties, auxiliary constructions, ...”), the common general term is that of *entity*. And the Target, aka *Thrift* or (probably a bit ahistorical) *Occam's razor*, in relation to scientific models and theories, reads: Entities should not be multiplied without necessity. In other words: Models and theories should not contain more entities (concepts, relationships between these concepts, regularities ...) than necessary.

In other words, models and theories should be as simple as possible. Of two competing theories that have the same explanatory power, the simpler one is preferable.

In general, the principle of economy is not that easy to grasp - it is not possible to define what is meant by “simpler”. In most practical cases, there is consensus about which entities are redundant. The material properties and mechanical implementation of the orrery, the details of which do not influence the descriptiveness of this material model, are examples of this; compare them with Newtonian mechanics and the theory of gravity, from which one cannot simply omit a term (force, mass, ...) without severely limiting the explanatory power of the theory. This does not mean that Newtonian mechanics and the theory of gravity do not have properties that should not be found in reality - the infinitely fine divisibility of distances and time intervals is a counterexample. But without this infinitely fine divisibility and the mathematics possible on this basis (infinitesimal calculus), the mathematical model on which the theory is based could by no means be formulated so easily.

**Theory: everyday meaning vs. technical term**

[01/25/2014, 18:10: In this section some changes - thanks to M. Holzherr for the comments]

The word *theory* more than one meaning. On the one hand, the word describes a coherent collection of physical quantities and more general concepts and general statements about how these are related to one another. In this sense, a law of nature would be something like a little theory. In practice, the word theory is usually reserved for larger and more complex structures.

In some cases, there is a disproportion between scientific and everyday language use. If someone expresses in everyday life that something is “just a theory” or “pure theory”, then that means that a certain claim is pure speculation, a mere assumption, possibly without any real reference to reality (for the latter see “gray theory”).

In science, however, a theory can be a fundamental collection of natural laws, concepts, ideas and statements about the world that conclusively describe a large number of phenomena and also prove themselves, i.e. a considerable number of tests of their validity through experiment and / or Has survived observations - in other words, the best we have in terms of a description of the world.

Much of what you mean in everyday life when you say that “you have such a theory” would be called a hypothesis in science (or a guess - it doesn't always have to be a technical term).

Important theories in physics are those *classic mechanics* (see just Understanding Einstein Part IV and Part V) that *Special theory of relativity* as a fundamental theory of space and time in the absence of strong gravity (this is what Einstein is working towards), the *general theory of relativity* as the theory of space, time, gravity and the relationship between these three, as well as the *Theory of Classical Electrodynamics *based on the equations found by James Clerk Maxwell, which describe how electrical charges lead to interactions and how they react to such interactions. These theories are well tested and, within their (well-known) limits of validity, are considered reliable descriptions of what is happening in the world. An example of a theory that has not yet passed direct experimental tests and is therefore not yet confirmed is string theory.

**Mathematical models and generalization**

The fact that mathematical formulations and especially mathematical models are used in physics - i.e. mapping of phenomena to mathematical structures - inevitably entails certain possibilities for generalization.

Whenever we describe the development of a certain quantity through a mathematical function, this description has two aspects: a functional relationship and parameter values.

A simple example: we consider the phenomenon of radioactive decay. Atoms of one type A decay radioactively into atoms of another type B. If we find that the law of decay

describes in good approximation how many atoms N (t) with the number N originally present at time t = 0_{0} are still present at time t, with λ a constant characteristic of the atomic type (the correct technical term for “type” in this context is “isotope”), then this form alone provides a direction for the generalization. Wherever concrete numerical values such as N in the function_{0} and λ, you can certainly use other values. Such *parameter* In a description using mathematical functions, if mathematics alone has its way, an infinite number of different values can be assumed.

A special choice of values is necessary to model a specific situation - but the mere fact that there are many other possible values suggests that the functional relationship itself, i.e. the fact that we are dealing with an exponential function of time , is more general - or at least can be.

And indeed: the above formula can, depending on the choice of parameter values, be used to describe very different quantities of very different types of atoms.

**Laws and initial conditions**

The variety of situations has mathematically formulated laws of nature built into physics in another way. The fundamental laws of nature, for example Newton's axioms of mechanics, do not specify in detail how things work, but they typically make statements about *Rates of change*.

However, rates of change do not, in and of themselves, provide a clear rule of what happens. A primitive example: Suppose I head south on my bike at 20 kilometers per hour. That defines my speed (in physics, the direction of movement is part of the definition of speed), but not my path (when I am where).

When I drive off in front of the Federal Chancellery fence, after three quarters of a minute I am at the crossing Scheidemannstrasse / Yitzak-Rabin-Strasse / Heinrich-von-Gagern-Strasse. When I start cycling at the Klosterstern in Hamburg, I am in front of the Museum of Ethnology in just under four and a half minutes. And if I drive off right in front of the Bunsengymnasium here in Heidelberg, I will reach one of the Neckar bridges in just over a minute.

How exactly my path runs depends not only on my speed - the rate of change in my location - but also on *Initial conditions*, in this case from where I started. My specified speed only fully describes the situation when we have specified initial conditions - and vice versa that means: Specifying a speed allows many different paths, depending on the initial condition.

When physical theories specify rates of change - typically not speeds, but *Accelerations* of various kinds - then it is not yet clearly defined how the given system will develop over time. Only when one additionally stipulates suitable initial conditions (for given accelerations these are: initial speeds and locations), one arrives at a clear model for the specific situation in question. The freedom to specify initial conditions automatically means that the laws of nature govern a large number of possible situations at the same time. The generalizability of models is due to the fact that we use mathematical models, so to a large extent already created.

*That* mathematical models are proving so successful in our world is, if you think harder, quite remarkable. The mathematical physicist Eugene Wigner has dedicated an essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, to this fact. The physicist Max Tegmark even asked whether the successes in mathematical modeling ultimately mean that the physical world itself is a mathematical structure.

**Theories: Consequences and Review**

I talked about testing theories earlier. This is a central part of science - theories should not just be beautiful stories, but should pass all the tests we can think of for them.

Such tests start with considering certain consequences of the theory. All the consequences that can be logically deduced from the theory are important here. It often makes sense to first look at extreme cases - which are often particularly simple - and otherwise run through as many possible scenarios as possible and see what the theory says about them. This “what if” is a theoretical exploration of the consequences of theory in a variety of conceivable situations.

Such a theoretical exploration with the help of hypothetical situations is also called *Thought experiment* designated. Good thought experiments make valuable contributions to understanding the structure and statements of a theory. Even if a theory contains logical contradictions, this fact can be worked out with creative thought experiments. Then the theory is either refuted as a whole, or at least the limits of the scope of a theory have been demonstrated.

Meeting theories with a persistent scrutiny of what-ifs is something that scientists are so accustomed to, that they sometimes also do this in everyday life. If one is outside of the sciences, this can definitely lead to cultural conflicts and upset. I know that from discussions about religion, which usually do not take place in the context of theological science, but on a more everyday level. If the paradox "Can God create a stone that is so heavy that he cannot lift it?" comes to mind, in many cases my counterpart gets the impression that I am making a laugh out of something and not taking his view of the world very seriously - while in reality it is just the other way around: in science, checking consequences is an expression of this, *that* one takes a theory or, more generally, an assertion seriously. But in everyday conversation mode, many statements are meant much more vaguely than in science - when my counterpart speaks of the omnipotence and omniscience of the God in whom he or she believes, then these are not scientific statements, but attempts, an idea of a god To give form that is far above all human abilities.

**Verification, experiments and models again**

But back to scientific statements. Physics, as a field of science in which theories are largely mathematical in nature and formulated in the form of equations, benefits from the fact that mathematics is downright the science of what can be clearly deduced from the consequences of simple basic assumptions. In particular, it offers reliable rules for how equations can be combined and transformed and other equations can be derived from them. This enables a highly effective and stringent what-if.

Once you have deduced certain consequences of the theory, you can test the theory by whether these consequences also show in reality. However, this does not work directly, but only indirectly via models - models of the situation that we want to use to check the theory as well as any measurement arrangements that we use in the course of the check.

**Falling bodies as an example**

Let us consider Newtonian gravity as a simple example in the context of classical mechanics. In this case, the theory describes the connection between movement and forces, specifically Newton's gravitational force with its characteristic dependence on the masses that attract each other and their mutual distance.

One consequence of the theory is that things fall down on earth. The statement itself can be confirmed by many examples - for example, if I hold a steel ball in my hand and let go, the ball does indeed fall down. But of course that does not exhaust the predictive power of the combination of classic mechanics and Newton's gravitational force by far.

If I want to take a closer look, I have to model the situation, which means: simplify it in a targeted and sensible way. The fact that the steel ball ultimately consists of atoms with electrons and atomic nuclei, which are arranged in a crystal lattice, has almost no influence on their trap properties in the earth's gravitational field. Nor do I need to know every detail of the globe; From the orders of magnitude of the earth's radius and mass and the degree of homogeneity of the earth, it follows that a fall over a few meters near the earth's surface is a good approximation using the formula

is described, with z as the coordinate for the vertical direction, h_{0} the height from which the body is dropped and at which it is at time t_{0} was at rest at the beginning of its fall, and g is the (local) gravitational acceleration, which can vary slightly from place to place, but always around 9.81 m / s^{2} lies. Here, too, by the way, a description with two aspects: With a functional form (quadratic dependence on time) and situation-specific parameter values.

The simplifications that lead us to this model are many. The first simplification steps are often not explicitly named, because they concern all those influences that we leave aside in this case: I already mentioned the atomic structure and details of the earth, but all other possible interactions must also be taken into account with the greatest possible care pull and check that they can be neglected: electrical and magnetic forces, the gravitational influences of all other masses in the universe, the effects of the earth's rotation (inertial accelerations such as Coriolis or centrifugal force) as well as the buoyancy and friction forces due to the air surrounding the body.

In this sense, every time we want to test a specific part of one of our physical theories, all of the rest of the physics stands quietly in the background. And physicists can count themselves lucky that they live in a world, the partial aspects of which can be largely isolated from one another and examined separately - undoubtedly an important prerequisite for us to be able to understand physics to the extent that it is in fact the case.

**Why models?**

Part of the need for simplified models arises from the theories themselves. Their statements about what happens in a given situation are often indirect in a very specific way, as already mentioned above: Instead of specifying which values certain physical quantities such as the To assume the position of a certain particle at any given time is often only given by the laws of physics *Rates of change* for such sizes. From the axioms of classical mechanics (as described in Understanding Einstein Part IV), if all interactions (read: forces) are taken into account, first of all the acceleration of the observed particles, i.e. the rate of change of speed over time. This results in so-called *Equations of motion* for a system that is *Differential equations* acts - this is the mathematical expression for "equations for rate of change of a quantity of interest".

Above we were concerned with the fact that in this way the initial conditions automatically come into play with the freedom to describe a large number of situations with a single law - each concrete situation corresponds to a possible choice for the initial conditions.

The fact that only rates of change are given has another consequence. It means that you still have to do some work to determine the orbits of the observed particles from the equations of motion and given initial conditions. This step is called “finding a solution to the equation of motion”. A similar situation - given generalized accelerations, sought solutions of the corresponding equation of motion - results quite generally from the physical laws of nature for the development of interacting systems. Finding solutions is anything but easy in general. Often enough, you can only write down solutions for very simple cases in the form of mathematical functions and you also have to switch to computer simulations, for example. And so models already play a role on the theoretical level - because constructing a description of a given situation from such simple cases or from computer simulations is also a form of physical model construction.

Back to more general statements: The example of the falling body illustrates the important role that models play as a link between the general statements of theories and those concrete situations in which we can test such theories through observations or experiments. Without this link, neither the theories nor the technical applications in which we put our physical knowledge into practice would be possible.

In the philosophy of science there is a lively discussion of how this and some other possible roles of models relate to theories - see the section Models and Theory in the aforementioned Stanford Encyclopedia article. If I start from the use of language among physicists, i.e. among my colleagues, in the lectures I attended as a student and in the specialist physical literature, then the way in which I have used the term here is quite widespread, at least among physicists .

After these preparations, in which models came into play again in a different role than before, we have arrived at the targeted experiments with which scientific theories, claims and hypotheses can be checked. More on this in Part 3.

Continue with part 3

[The original version of this article was published on the SciLogs on January 25, 2014.]

Markus Pössel had already noticed during his physics studies at the University of Hamburg: The challenge of working through and presenting physical topics in such a way that they can also be understood by non-physicists was at least as interesting for him as the actual research work. After completing his doctorate at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) in Potsdam, he stayed with the institute as an "outreach scientist", was involved in various exhibition projects during the Einstein year 2005 and created the Einstein Online web portal. At the end of 2007 he moved to the World Science Festival in New York for a year. Since the beginning of 2009 he has been a research assistant at the Max Planck Institute for Astronomy in Heidelberg, where he heads the House of Astronomy, a center for astronomical public and educational work. Pössel blogs, is the author / co-author of several books, and writes regularly for the magazine Sterne und Weltraum.

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