Are Scientific Laws Invented or Discovered?
Don't mix up reality with our model of reality. It is a really important ontological distinction.
A scientific law is a discovered regularity in nature that we have given a name to so we can talk about the world. Calling them "laws" is an unfortunate choice.
What is a Law?
A scientific law is a DESCRIPTION of regularities in nature. It is an unfortunate term calling it a law because it sound like something that someone commanded to happen, and perhaps the first scientists to use this term were thinking that these were things that God commanded.
The scientific law is actually only an approximation. It often simplifies things, and assumes that nothing else effects the result. The law gives a result based on the idea that we can simplify the situation, and that side effects are minimal. What reality does is different from this equation.
Laws will be upgraded from time to time. Newton wrote the law of gravity, and then Einstein wrote a better one. The behavior of gravity certainly did not change between the times these two scientists lived. Clearly the new law is a better approximation, but the facts laws can be made better gives us insight into what they are.
What is Discovery? What is Invention?
Note: We are using the term "discovery" in a special way here. There is a standard meaning of discovery used when we talk about scientific discoveries. For this discussion however we want to ask if those discoveries are really discoveries in the ontological sense, or whether they are ontic inventions. A discovery would refer to something that was there before; it had an existence before the scientist wrote the law down. An invention is something that did not exist before, and therefor came into existence when the scientist wrote it down. These are traditional terms used in this kind of discussion, but please understand that an important invention is something called a discovery in a different sense.
Discovery requires that the thing exist before someone thinks of it, while invented means it was created the first time it was thought of. To be discovered, the law would have to exist in some form, before it occurred to the first person who thought of it. The question is: did the scientific law exist before the first person thought to write it down?
I am not trying to quibble over the word "discovery" but I am trying to highlight the difference. We don't actually see nature as it is, but we invent models of nature and see that.
What we know of as "laws" are actually simplified models of the natural phenomena. They are approximate and for that reason NOT exactly what nature does, so we could not have discovered them in nature.
Once we have a law, we can describe nature in terms of that law. We can talk about the strength of gravity on the surface of the earth, and know what that means.
Are there no Discoveries in Science?
Sure there are. Mix hydrogen and oxygen, provide a spark, and a nice explosion results. This is a real thing that actually happens. Observations can be discoveries. You get exactly the explosion you get. These are regularities in nature. They occur, we don't always know why they do, but we can observe them.
Then we abstract the law out of the observations. We simplify. We treat masses like point sources. We assume there is no air resistance. We assume that nothing else (like the spinning earth, or the magnetic field) have a significant effect. Then we write a simple equation for the simplified situation.
It is important to note that nature never exactly matches the simplified situation, and therefor never does exactly what the law says. The prediction though will be close enough to be useful for our purposes.
To find the law, there is a search. Take a number of measurements, plot the results on a chart, and then find a formula that fits the closest. The scientist might propose several possible equations to see which one fits best. It may take many iterations to find the right equation to represent the law. I am sympathetic to those who would describe this as a method of discovery, because it is a systematic way to work out the best formula, but the formula did not actually exist before.
If we had access to what nature is actually doing, then we might look at that, write it down, and call it a discovery. There is certainly a way that nature works. The problem is that we can't really discover that directly.
Example: Gravity
Take for instance the law of gravity: f = Gmm/rr
This formula describes the behavior of objects in the universe in many certain circumstances. It assumes that matter is point-like objects, and ignores relativistic effects. This law is good enough to predict eclipses quite accurately.
Did this law exist out there to be discovered? I think not, because this law precisely as stated does not actually exist in nature. What nature does is different than this law, however the law is a good approximation in certain circumstances. Because nature does not really "do" what the law says, it can't have come from nature. Instead, it came from the mind of people and it was found to be a good enough model of what is happening. The model is a fiction made up, but the model is good enough for our purpose.
Newton's law of gravity was a distinct expression in math. Later Einstein invented his law of gravity, and was found to be better. Gravity itself did not change behavior between the two. What Newton came up with was not what nature exactly does. Einstein's version is probably not exact either.
The exact behavior of nature is not the same thing as the "laws" we write down to describe nature. The laws are invented. They are an attempt to describe nature, but nature is distinct from the law.
But there was Gravity before Newton
Newton might have invented the law of gravity, that is not to say that Newton invented gravity itself. This is obvious, and I don't really need to go into it, but it is surprising how many comments have gotten from people along this line.
We need to grasp how the formula is different from nature. Nature unfolds as it unfolds. We may never know the real reason that it unfolds in a certain way. Nature has regularities. What we are doing is simplifying the patterns we see in nature, and then matching a formula to it. This formula is useful as it allows us to calculate the trajectories of objects in space and elsewhere.
Nature does not sit down and perform this calculation. It does not add up all the atoms in the sun and the earth, and calculate the force, and then curve the orbit of Earth around the sun. Nature has no need for this formula. Instead, the way it works is just the way it works. If you start at a point, and emit a ray, then chance of it hitting something is proportional to the surface of a sphere around that point, and that surface increases as the square of the distance, diluting the force by the square of the distance. The dropping of gravity over distance that way might be related to this but we don't know.
What we do know is that we can model things this way. A planet is a bunch of atoms, but we can treat it as a single thing to a certain approximation. We simplify the planet and the sun to a point, and then use a simple equation, and the result is good enough.
But this is the key point: we never ACTUALLY observe what nature does. We observe the models working. We think about planets as point sources of mass, and we ignore the details that don't matter. We understand the world through our models and in terms of our models, but nature is not actually those models.
It is a mistake, I think, to assume that nature is "doing" what the models say. You can never know what nature actually does, you can only make a model.
Questions and Answers
You roll marbles down an incline. You notice they go farther in the third second than they did in the second second, which was farther than it went in its first second. You have discovered that the relationship that describes this motion is a second power relationship.
The motion is described by x = at^2. This particular equation with those symbols: did they somehow appear floating over the rolling ball? Did the balls scratch this out into the surface.
There are two things to note. (1) the equation was invented, and (2) the equation is only approximate. In reality you need to worry about air resistance, Coriolis effects, and relativistic effects -- which are all small but cause a deviation from the ideal. The equation describes some ideal situation that does not exist in nature.
You are correct in that you keep trying possible equations until you get one that fits better than any other, so in that sense it is a discovery, but it seems there are two senses of discovery. In the philosophical sense, for something to be discovered, it would have to have been there BEFORE it was discovered. That equation (which is only approximate) was not there before Newton wrote it down. The equation was in this sense invented -- while at the same time the process of inventing and testing different equations is a kind of seeking the best fit.