Is there a law in our universe that says explanations with fewer entities are more likely?
Is there a law in our universe that says explanations that have fewer entity types are more likely?
Is there a law in our universe that says explanations that have fewer explanations are more likely?
Is there a law in our universe that says explanations with fewer entities are more likely?
Explanations are produced by minds--or, if you prefer, brains.
The term "entities" here seems to connote some philosophical paradigm, but explanations typically do not presuppose any specific number of entities. For example, quantum physics does not postulate any specific number of electrons. Instead, theories usually postulate species of things, such as for example types of elementary particles, or types of force fields etc.
Explanations presupposing fewer species of things also do not appear to be necessarily more likely to be true than explanations presupposing more species of things. Reality is what it is, and the only explanation which is likely to be true is the one which presupposes the exact number of species of things which there is in reality.
That being said, it makes sense to presuppose as many species of things as empirical evidence suggests that there are. However, a naïve view of the world will suggest one species of things for each species of things that we can observe. Thus, we may want to say that cows is one species of things and pigs are another. This is what zoologists decided that they had to do, but biologists eventually didn't follow them. The number of animal species is very large, but this dizzying variety can be explained by just one sort of thing, namely DNA, which is two strands of polynucleotides which themselves only require six species of things, namely, four species of nucleobases (cytosine, guanine, adenine and thymine), a sugar (deoxyribose) and a phosphate group. And for now at least, this theory seems to work remarkably well.
There are no such laws. Occam's razor is merely a maxim, a heuristic that has been found to be useful through experience. It isn't a law.
Here is an overly simplistic intuition as to why it tends to be useful. Say a randomly chosen component of an explanation has a probability of being wrong given by P. If an explanation is comprised of N such components, then the probability of the explanation being free of errors is (1 - P)^N, so the more components, the greater the probability of the explanation being wrong because it contains at least one faulty component.
This is just common sense, but unfortunately the model is way to simple to be of practical use, mainly because different components have different probabilities of being wrong and you may be better off having a model comprised of many reliable components than a simple model composed of a few unreliable components.
It is a useful guide, nothing more.
I suggest you read Solomnoff's mathematical treatment of Occam's razor.
A simple "proof" of why fewer entities in a hypothesis imply better chances of being true follows:
Hypothsis A
Assumption X
Hypothesis B
Assumption X
Assumption Y
Given initially P(X true) and P(Y true) values, rules of probability show that P(X true) > P(X true AND Y true). Every new assumption you add to your hypothesis (multiplying entities) brings down the probability of your hypothesis being true.
Hence, pluralitas non est ponenda sine necessitate (do not multiply entities without necessity).
We humans have an intuitive preference for "simpler". We have not confirmed this is true of the universe. Instead, the universe seems to be immensely complex. For instance, just physics is now so hard to understand, that only a very few of us are even able to grasp theoretical physics. Add in things like crystal beavior, electrical conduction, etc, and physics has multiple specialty areas that even a PhD physics generalist will not be fully competent in.
Add in our other sciences, and engineering, and other areas of practical skills, and the complexity of our world is astonishing. The functions and behaviors within even a "simple" creature like an earthworm require years of study to understand, were not even fully understandable until the last several decades when we developed far better micro-resolution non-destructive imaging techniques. And there are areas of ongoing study because we still don't understand an issue fully, such as atmospheric dynamics and climate behavior.
There is a however. Despite the intrinsic complexity of our world, we have found the "assume no more complexity than is needed to match the data" is an excellent heuristic. As is "look for more global simplifications that can compress the complexity of our model". These work well in improving our understanding.
Are there laws in our universe that say simpler explanations are more likely?
I’m not aware of any such laws. If we think of the universe as encompassing everything that exists, along with all the factual statements that are true, then an "explanation" is either true or false. For the universe, a statement is binary: it’s either true (1) or false (0). Therefore, discussing the "likelihood" of an explanation doesn’t really make much sense, except pragmatically from the perspective of an agent with limited, imperfect knowledge who is performing statistical analysis. In contrast, an omniscient being, possessing perfect knowledge of all true facts in the universe, would instantly know whether an explanation is true (1) or false (0), rendering the entire task of estimating likelihoods as continuous values between 0 and 1 nonsensical.
You can propose a very simple explanation that turns out to be false, just as you can propose a complex explanation that is also false. On the other hand, the universe itself can be extraordinarily complex, and a truly accurate explanation of an event X might very well be equally complex. In the end, complexity is irrelevant—the only thing that truly matters is whether the explanation is true, whether complex or otherwise.
If you’re looking for a physical law expressed mathematically that quantifies the statistical likelihood of an "explanation," to the best of my knowledge, no such law is known to exist. At least there’s nothing in the equations of relativity, quantum mechanics, string theory, or other physical theories that fulfills this role. The closest we have to a "law" involving likelihood is in quantum mechanics, where probabilities are described by wave functions. However, there is a significant gap between that and a "law" governing the likelihood of explanations. The concept of an "explanation" itself is quite vague and has not been rigorously defined in a way that would allow the laws of physics to meaningfully apply to it.
This is objective; you may be able to express it as a law.
Suppose we are doing machine learning on soybean plants. One is in my yard, is green, and is alive. One is in Nakamura's farm, brown, dead. We also have Hernandez, black, dead; Smith, green, alive; Owe, green, alive, and 100 others.
We can make this rule to predict life: it's alive if it's belongs to me, Smith, Owe, and some 50 other names.
Or this one: it's alive if it's green.
Both perfectly match the data we have.
So here's the question. We get a new soybean plant. Which rule will better predict if it's alive? It's theoretically possible it's alive if it belongs to owners with that set of last names, but it's not the way to bet. The simple rule is way more likely to be predictive.
It's even stronger than that. Suppose that in a few cases, the green one is dead. "Green implies alive" is still way more likely to be right, even if there are a few exceptions.
This isn't about consciousness or human preference, but just about predictive ability. It's not absolute, but I suppose not everything has to be.
