TL;DR – The laws of physics are not the laws of nature. They can only describe what is experimentally accessible.

I am often cautious about sharing my interest in fundamental physics to strangers. “Ah, fundamental physics!” They may get excited and step too close to me. “What can you tell me about the nature of the universe? What can we tell about nature itself? Does God play dice?” To which I reply: “Well, one thing I can definitely tell you: you are standing on my foot.”

The notion that the basic laws of physics can, by themselves, provide explanations for an incredible variety of physical systems is mind boggling. Atomic energy, material properties, light, electricity, fluid dynamics, heat engines, planetary motion: all these phenomena can be reduced to applications of the basic laws. Naturally, some pieces of the puzzles are still missing. For example: when and how do socks disappear? Is it during the laundry? Science can’t say. Yet, it’s still all pretty impressive.

It is natural, then, to think that we are somehow reaching some fundamental truth about the universe, that we will understand how nature works and that we are probing her innermost secrets. Which implies that either nature is really poor at keeping them or they are not secrets at all. On the other hand, we may dislike that the very description of fundamental particles is statistical on the grounds that “God does not play dice.” I tend to agree as it would be very boring for Him.

But is that really what’s going on? Are the laws of physics the actual laws of nature or are they something else? Are they the rules that nature has to follow or are they the rules of those systems upon which we can perform scientific investigation? Let’s explore the difference with another allegorical and preposterous thought experiment. They are so much fun! At least to some people… This actually happened to a friend of mine.

The box of salvation

Imagine you are a scientist. Maybe I ask this too often… But bear with me. You are minding your own business while suddenly loud trumpets sound out of nowhere and a bright light illuminates you. So bright you can barely keep your eyes open. “<Insert your name here>! Rise!” You were already standing. “I have chosen thee to be the bearer of the box of salvation!” A black box floats in front of you. Yeah, I like black boxes. They fascinate me: so black, so boxy.

“Thou shalt study the box!” The only thing you see is a one digit display that keeps changing every second. “Thou will find the rule to predict the numbers! If thou succeedest, the world will be saved! If thou failest, the world will be damned!” and with a puff of smoke everything goes back to normal. You start looking at the box when suddenly, “Oh, I forgot! You have 24 hours! No pressure…” and then another puff of smoke.

You wonder why He would do such a thing. Then you remember that He also created an apple tree with the specific intent to see whether humans would disobey him. I personally hope that it was a Granny Smith or a Pink Lady: it would make no sense to lose paradise for, say, a Red Delicious. Plus, I never quite understood who created the snake. Anyway, you realize that it may not be unlike Him.

You start methodically writing down the numbers, and they do not seem to fit any pattern. They just seem to be randomly distributed. The hours are passing by so you shout, “Could I have a hint? This does not seem to be a reasonable task. There are far too many possibilities to explore and 24 hours is not much time!” Trumpets and bright light reappear “Knowest thou this! The digits are the decimal expansion of an irrational number!” and puff of smoke. This means, for example, that the box may be calculating pi=3.1415926… and showing one digit every second, in the correct sequence.

At first you are relieved: you just need to guess the right number. Then you realize that there are an infinite number of… numbers. And that the decimal expansion of irrational numbers is infinite and never repeats and that any sequence of digits will fit in any number. For example, the first few digits of Euler’s number, 2.7182, appear also in pi at position 28,024. You can add more digits, say 2.7182818, and this also appear in pi as well at position 73,154,827. So you ask “Could I have another hint? It is not possible for me to fully identify a rational number with only a finite sequence of its decimal expansion.” “Knowest thou this! The number is pi!” and there is a solemn echo “Pi! Pi! Pi! Pi!”

You feel relieved. Now you are sure it’s pi, you only have to figure out what position the box is at in the decimal expansion! But you realize that any finite series of digits will be present in the decimal expansion of pi an infinite number of times. For example, the sequence 27182818 appears 3 times in the first 200 million digits. So, knowing the past numbers is not enough to figure out what position of pi the box is at. This is a lost cause: you cannot do it. You cannot predict the next digit even if you know exactly how the black box works. You can ask for the position itself, but then there is nothing left for you to figure out. You fail either way.

You realize that, most likely, He would know that. In fact, you realize that there is kind of no point in telling Him anything, since He would have already known what you were going to say. He’s probably just messing with you. You simply wait for him. He appears right away. “Thou hast learned the lesson. Thou cannot find the rule to predict the numbers. It is beyond the ability of a human being. Thou cannot save the world.” And He disappears with the box.

You wonder what the point of this whole exercise was. You turn on the radio. The news, as usual, if full with the different opinions of politicians, economists, pundits, … and then it hits you.

What science can study

From outside, the box of salvation seems just to be spewing random digits. The inner mechanism, though, is absolutely deterministic: there is a well-defined law. Yet, we cannot reconstruct the law just by looking at the digits. We would only be able to characterize the system with a statistical distribution. Just like with those scratch-and-didn’t-win things. Maybe God plays those?

Suppose, though, that you figure out how to open the box and see its inner workings. Or that you had special glasses that allowed you to read under the scratchy thingies. Then you would know exactly what was coming: you would be able to write the law that predicts each and every digit. You see: your new ability changed your description, the law of physics, while the functioning of the box, the law of nature, remained the same. That is: the laws of physics are influenced by what processes are at our disposal for scientific investigation.

If you studied a bit of control theory, you may have heard of “observability” (the ability to learn the state of a system by looking at its output) and of “controllability” (the ability to put a system in any state given its inputs). The box of salvation is neither observable nor controllable. That’s the problem. If we have a system that is not observable, we cannot know its internal state, how can we deduce its laws? If we have a system that is not controllable, we can’t put it in an arbitrary state, how can we test those laws experimentally? Most likely, though, you haven’t studied any control theory… don’t feel bad: most scientists haven’t either. Plus, it allows me to show off my more sophisticated and holistic perspective. So, that’s great!

To put it more equivocally: the correct answer for “Could I have a glass of water?” is not, like many may think, “Yes, you could”. It is not a question of whether there exists such a thing as a glass of water and whether, in principle, I could drink it. It is rather a question of whether you happen to have one and you are willing to give it to me. So, the correct answer is, “Here it is!” or “I am sorry, I just have the one.” Similarly, the answer to “Could I predict the future state of this object?” would be “By all means, here is the law!” or “I am sorry, I have only this law and it applies to cats.”

If you studied a bit of computational complexity theory, you may know that some problems are “intractable.” In theory they have a solution, but in practice the time required to calculate it is too long to be useful. For example, it may require so long that you reach the heat death of the universe first. Personally, I am not planning to be there at that time: it’s going to be a dreadful place. If we insist that our laws of physics are actually experimentally verifiable, it means we have to restrict ourselves to laws that are tractable. Again, you may have not studied computational complexity… that’s ok. You can be impressed on how I weave arguments from the entirety of human knowledge. I think there is a technical term for that. Pulling things out of your…. what was it… hat! Pulling things out of your hat!

Anyway, the laws of physics are at least restricted to be tractable and about systems that can be accessible experimentally. They describe what we can learn about the universe with the instruments at our disposal, as parts of the universe itself. The laws of nature do not have that restriction. What reason do we have to believe that they, coincidentally, follow it? Therefore one should always be careful in extrapolating from the laws of physics to things like “the universe” or “nature” or “God.” God does not play dice. We just have no clue what’s really going on.