Mathematics is very rigorous, and they show great strength in physics. Physicists create laws of physics based on mathematics, which can describe various phenomena in the universe, so the application of mathematics in physics is very successful.
But even so, many mathematical theorems prove that the power of mathematics is limited. So how much do these theorems have to do with science, especially physics?
Godel's incomplete theorem
Mathematician Godel found that in all formal systems containing Piano axioms (five axioms about natural numbers, namely elementary number theory), propositions that can neither be proved nor falsified can be constructed, and their compatibility can not be proved. This discovery broke the cognition that mathematicians had formed two thousand years ago, and mathematics was actually incomplete.
However, Godel's incomplete theorem has nothing to do with scientific practice. This is because mathematicians can always use another axiom to expand the original axiom system, but this axiom does not indicate whether the previously undecided proposition is true or not.
Uncertainty: how to deal with the incompleteness of mathematics in physics?
In physics, a theory is a set of mathematical axioms, just like those involved in Godel's incomplete theorem. However, the theory of physics also provides a method for how to determine the mathematical structure with measurable quantities. After all, physics is a science, not pure logical mathematics.
Therefore, if there is any undecidable proposition in physics, physicists will judge it through experimental measurement, and then introduce an axiom consistent with the result. Or, if the undecidable proposition has no observable result, physicists can ignore it.
Uncomputability
The mathematical theorem about computability is also irrelevant in physics, but for different reasons. The uncountable problem is that it always comes from infinite things, but in fact nothing is really infinite. Therefore, these theorems do not actually apply to anything we can find in nature.
Turing downtime problem
Turing downtime can illustrate this problem. Turing, the father of computer science, put forward an idea to try to find a meta-algorithm, which can tell us whether another algorithm will end in a limited time. Turing proved that such meta-algorithm does not exist. This is an infinite class. In reality, we never need an algorithm to answer an infinite number of questions.
In mathematics, most real numbers are uncountable, because no algorithm can approximate to a limited precision in a limited time. But in physics, physicists never deal with real numbers. Physicists deal with numbers with finite decimal places and error lines.
unpredictability
Quantum mechanics has an unpredictable factor, but this unpredictability is quite boring because it is obtained through assumptions. More interesting is the unpredictability of chaotic systems.
For some chaotic systems, they have a special unpredictability. Even if we know any exact initial conditions, we can only make predictions in a limited time. In the real world, it is not clear whether this will really happen.
An example of this unpredictable equation is the Naville-Stokes equation (N-S equation), which is often used in weather forecasting. It is unclear whether the N-S equation will lead to unpredictable situations in some cases, which is one of the most difficult mathematical problems today.
Assuming that this problem has been solved, sometimes it is really impossible for the N-S equation to make a prediction in a limited time. So, what can we learn from it? Not many, because we already know that the N-S equation is only an approximation.
In fact, gas and liquid are composed of microscopic particles, which should be described by quantum mechanics, but quantum mechanics is not unpredictable. But quantum mechanics may not be the correct theory after all. Therefore, we really can't say whether nature is predictable or unpredictable.
This is a common problem in applying the impossibility theorem to nature. We never know whether the mathematical assumptions made in physics are really correct or whether they will be replaced by something better one day. Physics is science, not mathematics. Physicists use mathematics because it is useful, not because physicists think that nature is mathematics.
Maybe the N-S equation is not the correct equation to predict the weather at all, but we are using it at present. Because of this, it is very important to know when unpredictable things will happen, so as to avoid mistakes. This is not feasible for the weather, but it is feasible for some chaotic systems, such as plasma in nuclear fusion.
In the process of nuclear fusion, plasma sometimes produces instability and destroys the protective shell. Therefore, in the case of instability, nuclear fusion must be stopped quickly, which will make nuclear fusion very inefficient. If we can know when unpredictable situations will happen, we can prevent them from happening in the first place.
All these problems, which sound uncertain, uncountable and unpredictable, belong to mathematics and have nothing to do with science. But in a sense, the impossibility theorems in mathematics are relevant in science, not because they tell us something about nature, but because we use mathematics to understand the observed natural phenomena in practice, and the theorems can tell us what predictions can be made by physical theory.