How is a Scientific law/theory's accuracy tested?

[LittleRookie]

How is a Scientific law or theory's accuracy tested? I've searched online about verification of Newton's law but the materials are unhelpful. Many show an experiment that verifies a particular instance of the law and then concludes the law is verified. We all know that there are infinite combinations of values for the variables involved in the law, and thus there are infinite instances of the law to be checked and verified by the law. I have a few burning questions.

-How does Scientist cover as many instances as possible? What are some good methods? Is it possible or not possible to cover all?

-Does the Scientist really always get exactly that same number as predicted by the law? (I highly doubt so we can achieve this for just a single experiment)

-If not (to question 2), i) Is the Scientific law considered not able to exactly describe the nature, but serves as currently the best approximate model of nature? OR ii) Is the Scientific law regarded as an exact description of nature, but the results are either interfered by our own measurement error or other aspects of nature? OR iii) Is the Scientific law abandoned immediately and the search for better one continues? (I feel this is too extreme since we end up with nothing).

-If not (to question 2), does the Scientist then keep track of the difference between the expected (by the Scientific law) value and the obtained (from experiment) value? How does the Scientist determine whether the difference is acceptable or unacceptable, and thus the Scientific law is deemed to be accurate to a certain degree/inaccurate?

Sorry if I post in the wrong section.

 

[anorlunda]

Sometimes, we can disprove the negative. If we didn't have Newtons 2nd law, energy would not be conserved. If we didn't have Newtons 3rd law, momentum would not be conserved.

If we didn't have those conservation laws, the laws of physics might change one minute to the next or from one side of the room to the other. There is the principle of least action underlying those laws.

Principles, like causality, we don't prove. We just observe that they are always true.

 

[phinds]

-How does Scientist cover as many instances as possible? What are some good methods? Is it possible or not possible to cover all?

Depends on the law / theory but NOBODY covers "all" possibilities unless the part of the theory being tested is something like "such and such a collision will produce exactly this many particles".

-Does the Scientist really always get exactly that same number as predicted by the law? (I highly doubt so we can achieve this for just a single experiment)

Depends on the law / theory. Most confirmations have an experimental error range. For example, Quantum Mechanics, Richard Feynman was fond of saying, has been tested to one part in 10E12.

-If not (to question 2), i) Is the Scientific law considered not able to exactly describe the nature, but serves as currently the best approximate model of nature? OR ii) Is the Scientific law regarded as an exact description of nature, but the results are either interfered by our own measurement error or other aspects of nature? OR iii) Is the Scientific law abandoned immediately and the search for better one continues? (I feel this is too extreme since we end up with nothing).

NO law or theory is considered an exact description of nature since a counter example could always come up. Consider Newton's law of gravity. Worked great for a very long time and is still used in many applications today but it turned out to be only an approximation.

 

[Dale]

How is a Scientific law or theory's accuracy tested?

This is a good question, and is not so obvious.

Typically, in order to test a theory you first need to create a new theory called a test theory. The test theory usually is very general and has a bunch of free parameters. It contains multiple theories of interest as specific free parameter settings of the test theory.

Then, you use the test theory to analyze a bunch of possible experiments. You look for experiments that are technically feasible and are highly sensitive to one or more of the test theory parameters that distinguish the theories of interest.

Once you have identified such an experiment then you perform it. After the experiment you have a range of values for the test theory parameter. If your experiment is good then it should be a very narrow range. That will then allow you to exclude one or both of the theories of interest.

We all know that there are infinite combinations of values for the variables involved in the law

Yes. But there are a finite number of free parameters. So although there are an infinite number of values there are finite ranges of the parameters.

 

[Vanadium 50]

Let's go with a seasonal example, Consider the hypothesis "reindeer can't fly". I go on the roof with a herd of reindeer, and start pushing them off one by one.

1. How many reindeer do I have to go through to prove reindeer can't fly?
2. How many reindeer do I have to go through so that no reasonable person would believe reindeer really can fly?
3. How many reindeer do I have to go through before I am arrested for cruelty to animals?

 

[LittleRookie]

Let's go with a seasonal example, Consider the hypothesis "reindeer can't fly". I go on the roof with a herd of reindeer, and start pushing them off one by one.

1. How many reindeer do I have to go through to prove reindeer can't fly?
2. How many reindeer do I have to go through so that no reasonable person would believe reindeer really can fly?
3. How many reindeer do I have to go through before I am arrested for cruelty to animals?

1. No matter how many reindeer we have checked, we can't prove that reindeer can't fly.
2. I guess a couple of reindeer? But I will think that reindeer can't fly based on current findings, but perhaps there is one that flies and we don't know of it.
3. One

 

[LittleRookie]

This is a good question, and is not so obvious.

Typically, in order to test a theory you first need to create a new theory called a test theory. The test theory usually is very general and has a bunch of free parameters. It contains multiple theories of interest as specific free parameter settings of the test theory.

Then, you use the test theory to analyze a bunch of possible experiments. You look for experiments that are technically feasible and are highly sensitive to one or more of the test theory parameters that distinguish the theories of interest.

Once you have identified such an experiment then you perform it. After the experiment you have a range of values for the test theory parameter. If your experiment is good then it should be a very narrow range. That will then allow you to exclude one or both of the theories of interest.

Yes. But there are a finite number of free parameters. So although there are an infinite number of values there are finite ranges of the parameters.

I don't really understand. Can you explain in simpler terms? Like I'm five.

 

[Ibix]

1. No matter how many reindeer we have checked, we can't prove that reindeer can't fly.
2. I guess a couple of reindeer? But I will think that reindeer can't fly based on current findings, but perhaps there is one that flies and we don't know of it.
3. One

Good answers. And it's the same with any scientific hypothesis. For two and a half centuries we believed Newtonian physics was exactly correct, because no matter how many reindeer we (metaphorically) pushed off the roof, they never flew. Maxwell, and ultimately Einstein, eventually found a way to push reindeer off the roof so they do fly. It turns out that Newton predicts approximately the same as Einstein at low speeds, and we weren't able to do tests at high enough speeds to spot the difference.

It will happen again (we strongly suspect quantum physics and relativity aren't quite complete, and they definitely don't play well together). That's why we stopped calling new stuff "laws" in the 19th century. Our discoveries are conditional on what we've been able to test.

 

[Ibix]

I don't really understand. Can you explain in simpler terms? Like I'm five.

Newton says $F=ma$. A test theory for this might say $F=m^\alpha a^\beta$ - not for any physical reason, just because. Then we try to measure $\alpha$ and $\beta$ and show that they are both equal to 1.

Sometimes you can just have competing theories and compare their predictions. For example, Newton makes a prediction about the orbit of Mercury. Einstein's general relativity makes a slightly different one. The data matches Einstein, so we reject Newton and accept Einstein as our current best theory, pending future discoveries that may show errors in that.

 

[BWV]

Popper may be passe for some, but the falsifiability criteria is a good yardstick-

a theory must make predictions that can be tested, if it cannot then its not science

as in the example of GR, often what happens is some new phenomena is discovered that the old theory cannot explain, which motivates the generation of new science.

does the Scientist really always get exactly that same number as predicted by the law?

predictions of Quantum Electrodynamics match experimental results to 10 parts in a billion

 

[Vanadium 50]

Then we try to measure alpha and beta and show that they are both equal to 1.

And, as a practical matter, we can never measure exactly one. What we end up saying is that alpha is, for example, between 0.99 and 1.01. As a realistic example, the exponent in Coulomb's Law is -2 to within a few parts per billion.

 

[LittleRookie]

Depends on the law / theory. Most confirmations have an experimental error range. For example, Quantum Mechanics, Richard Feynman was fond of saying, has been tested to one part in 10E12.

If the scientist obtain a number from experiment that is different from the calculated number according to some Scientific law, is the Scientific law disproved?

-If not (to question 2), does the Scientist then keep track of the difference between the expected (by the Scientific law) value and the obtained (from experiment) value? How does the Scientist determine whether the difference is acceptable or unacceptable, and thus the Scientific law is deemed to be accurate to a certain degree/inaccurate?

I need help on these questions.

predictions of Quantum Electrodynamics match experimental results to 10 parts in a billion

And, as a practical matter, we can never measure exactly one. What we end up saying is that alpha is, for example, between 0.99 and 1.01. As a realistic example, the exponent in Coulomb's Law is -2 to within a few parts per billion.

I'm very curious in knowing how does Scientist test the accuracy of a Scientific law/theory. I see many statements such as "Experiments have shown that theory/law X has been shown accurate to <some very small number y>". But I don't know exactly what such statement means.
If I were to test the theory/law X, then my experiment will definitely give a result, whose difference from the number calculated using theory/law X will be no more than y?

 

[Dale]

I don't really understand. Can you explain in simpler terms? Like I'm five.

No, I don’t think that this subtle of a point can be taught to a five year old. However, I can provide an example.

Robertson developed a test theory of special relativity, as described here:
https://en.m.wikipedia.org/wiki/Test_theories_of_special_relativity

This test theory can be written:
$t’=A t + E x$
$x’=B(x-vt)$
$y’=Dy$
$z’=Dz$

For $A=B=D=1$ and $E=0$ we get the Galilean transform of classical Newtonian mechanics. For $A=1/B=\sqrt{1-v^2/c^2}$ and $D=1$ and $E=-v/c^2$ we get the Lorentz transform of special relativity.

So by doing experiments to measure $A$ we can distinguish between relativity and Newtonian mechanics, but doing experiments to measure $D$ won’t distinguish between the two theories. Also since $A=\sqrt{1-v^2/c^2}$ in relativity and since $A=1$ in Newtonian mechanics we cannot do experiments in the limit as $v$ approaches zero.

 

[phinds]

If the scientist obtain a number from experiment that is different from the calculated number according to some Scientific law, is the Scientific law disproved?

If there is ONE well verified experiment that differs from the theory then the theory is wrong. It could be that the theory just needs to be tweaked to account for the anomaly or it may be that a whole new theory is needed (as happened when it was finally observed that Newton's Law of Gravitation didn't actually work for all cases).

Here is the most famous, brief, statement of that :

 

[PeroK]

I'm very curious in knowing how does Scientist test the accuracy of a Scientific law/theory. I see many statements such as "Experiments have shown that theory/law X has been shown accurate to <some very small number y>". But I don't know exactly what such statement means.
If I were to test the theory/law X, then my experiment will definitely give a result, whose difference from the number calculated using theory/law X will be no more than y?

It probably doesn't quite work like that. Let's take Newton's Law of Gravitation. From his law Newton was able to show that planets should move in ellipses around the Sun - and, in fact, explain Kepler's three laws of planetary motion. This is a good start. But, it doesn't prove the law of gravitation. And, in fact, some people at the time were sceptical. For example, a scientist of the time wrote:

"That gravity should be innate inherent & essential to matter so that one body may act upon another at a distance through a vacuum without the mediation of any thing else by & through which their action or force may be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it."

Can you guess who said that?

Spoiler

That was Isaac Newton himself!

Then, some scientists working with Newton's law found they could explain the detailed motion of the other planets if there was an unknown planet. They worked out where it must be - and when a telescope was pointed at the sky, the planet Neptune was discovered exactly where it was predicted to be. You can read about this here:

https://en.wikipedia.org/wiki/Discovery_of_Neptune

Using your law or theory to predict something that no one knows about is generally more powerful than explaining known things.

But, there was a little thing called the precession of Mercury, which Newton's law was not able to explain. That might have been a hint that his law was not the final word on gravitation. And, of course, it was Einstein's theory of General Relativity (GR) that explained the precession of Mercury. Einstein's theory also predicted new phenomena, such as the bending of light by the Sun:

https://www.smithsonianmag.com/scie...ry-relativity-baffled-press-public-180973427/

Gravitational time dilation, which was experimentally verified in 1959:

https://en.wikipedia.org/wiki/Pound–Rebka_experiment

Gravitational lensing:

https://en.wikipedia.org/wiki/Gravitational_lens

And, of course, the existence of Black Holes.

And, it predicted an expanding universe.

But, Einstein's theory does not explain how gravity arises from elementary particles. This is why most scientists believe that a theory of Quantum Gravity is required; if not to supersede GR, then at least to show how GR arises from elementary particles.

That's more the way the various laws of gravitation have arisen, been accepted and eventually been superseded or modified.

 

[Ibix]

If the scientist obtain a number from experiment that is different from the calculated number according to some Scientific law, is the Scientific law disproved?

Yes and no. We'd need to convince ourselves that the theory was applied correctly. First that the maths was right. Second we might see if we can correctly predict the experimental result by proposing things we weren't aware of (e.g. Neptune, as mentioned by @PeroK), or just failed to account for (google for the "Pioneer anomaly"). And sometimes we aren't sure what the answer is for a while (dark matter vs. MOND, for a current example).

But sometimes, yes, we do replace theories with newer ones (the precession of Mercury that PeroK mentioned, explained by general relativity) because there is no other explanation for the difference between theory and experiment.

I'm very curious in knowing how does Scientist test the accuracy of a Scientific law/theory. I see many statements such as "Experiments have shown that theory/law X has been shown accurate to <some very small number y>". But I don't know exactly what such statement means.

Our experiments have a certain precision. Our calculations have a certain precision. This ends up giving us a certain range of experimental results that are reasonable if the theory is right.

For example - if you launch a ball straight upwards with speed $u$, according to Newton's theory of gravity the ball ought to land at a time $t=2u/g$ after launch, where $g$ is the acceleration due to gravity. So I launch a ball upwards at 5m/s, I know that g is 10m/s², and I calculare that the ball ought to land after 2×5/10=1s. But quoting $g$=10m/s doesn't mean exactly 10.0000000... It means somewhere between 9.5m/s and 10.5m/s. Which means that any time measurement between 10/10.5$\simeq$0.95s and 10/9.5$\simeq$1.05s is consistent with theory. And I haven't even started to worry about the experimental side of it - the reliability of my 5m/s launch speed and accuracy of my time measurement.

So I'm predicting 1s, but my rough-and-ready numbers mean that quite a wide range of results are consistent with theory. Better research on the value of $g$ (it's about 9.81m/s², and the next decimal place depends noticeably on your height above sea level) would make for a more precise prediction. And if I don't time the flight with a stopwatch, but use electronic timers (and maybe Doppler radar to measure the launch speed of the ball), I can make my experiments more precise as well.

So when you see someone claiming a theory matches experiment to ten decimal places it means that they know all their numbers accurately enough to say that the theory predicts an answer between 0.999999999s and 1.0000000001s (ten decimal places, if I counted right) and the experiments are controlled to a similar precision and they produced a number in that range.

If the experiment said 1.005, and that result could be repeated, and we couldn't find anything we'd forgotten about (for example, I never mentioned air resistance, which would slow the ball), we'd have to consider that the theory was wrong.

The theory of errors, of how uncertainty and inaccuracy affects results, is extremely dull, but very important.

 

[mfb]

You can never prove a law - or any other general statement. There might always be a counterexample that gets discovered later. You can disprove a law with a counterexample (limiting the range of it, showing it is just an approximation, or (rarely) getting rid of it completely).

And, of course, the existence of Black Holes.

And the existence of gravitational waves. Predicted 1915, first measured 2015.
100 years and we are still testing more and more predictions of general relativity and improving previous tests to repeat them with more accuracy.