Mathematics and physics cannot prove the whole truth

Mathematics and physics cannot prove the whole truth

Manon Bischoff

All mathematical truths can never be proven. For me, this incompleteness theorem discovered by Kurt Gödel is one of the most surprising results in mathematics. This may not be surprising to everyone, since everything in everyday life is unprovable, but to mathematicians, this idea was shocking. In the end, they can build their own world through a few basic building blocks, so-called axioms. Only the rules they create apply there, and all truth is made up of these building blocks and those rules. Experts have long believed that if we find the right framework, we should be able to prove the whole truth in some way.

But in 1931 Gödel proved otherwise. There are always truths that fall outside the basic mathematical framework and cannot be proven. And this is not a purely abstract discovery that has no real-world implications. Soon after Gödel’s groundbreaking work, the first unprovable problems arose. For example, it is never possible to clarify how many real numbers exist within the mathematical framework currently in use. And unsolvable problems aren’t limited to math. For example, in certain card and computer games (such as Magic: The Gathering), situations may arise where it is impossible to determine which player will win. And in physics, it is not always possible to predict whether a crystal system will conduct electricity.

Now experts, including physicist Toby Cubitt of University College London, We found another way in which the incompleteness theorem is reflected in physics.. They described a system of particles that undergo a phase transition – a change similar to the change that occurs when water freezes below 0 degrees Celsius. However, the important parameter for phase transitions to occur in this particle system is can’t do it Unlike water, it is calculated. “Our results … demonstrate how incalculable numbers can appear in physical systems,” the physicists wrote in a preprint paper posted last month on the server arXiv.org.

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Uncertain phase transition

This isn’t the first time experts have experienced unpredictable phase transitions. Cubit again in 2021 Two of his colleagues described another physical system in which transitions are unpredictable. However, in this case an infinite number of phase transitions were possible. Such situations do not occur in nature. Researchers therefore asked themselves whether unpredictability could arise in realistic systems.

In the new study, Cubitt and his colleagues investigated a very simple system called a finite square lattice, which contains an array of multiple particles interacting with their nearest neighbors. These models are commonly used to describe solids. This is because the atoms are arranged in a regular structure and the electrons can interact with the electrons of the atoms immediately surrounding them. In Cubitt’s model, the strength of interaction between electrons depends on parameters. ∅—the bigger the better ∅ That is, the force that pushes the particles in the atomic shell against each other becomes stronger.

In case of resistance ∅ The small outer electrons are mobile. It can move back and forth between atomic nuclei. The stronger ∅ That means more electrons are frozen in place. This different behavior is also reflected in the energy of the system. You can see the ground state (lowest total energy) and the next highest energy state. if ∅ Because it is so small, the total energy of the system can continuously increase. As a result, the system conducts electricity without any problems. For large values ∅, But the situation is different. With these values, the energy increases gradually. There is a gap between the ground state and the first excited state. In this case, depending on the size of the gap, the system becomes either a semiconductor or an insulator.

To date, physicists have created thousands of similar models to describe all kinds of solids and crystals. However, because the system presented by Cubitt and his colleagues exhibits two different behaviors, there must be a transition between the conducting and insulating phases. That is, it has the following values: ∅ Beyond that, a gap suddenly appears in the energy spectrum of the system.

an immeasurable number

Cubitt and his team determined the following values: ∅ This is where the gap arises. And this corresponds to the so-called Chaitin constant Ω. This number may sound familiar to math nerds because it’s one of the few examples of numbers that can’t be calculated. An irrational number whose decimal places last forever and do not repeat regularly. Unlike computable irrational numbers such as π or π, E, However, the values ​​of uncomputable numbers cannot be approximated with arbitrary precision. There is no algorithm that outputs Ω if run for an infinitely long time. Without being able to calculate Ω, it is also impossible to specify when phase transitions occur in the systems studied by Cubitt and his colleagues.

The Argentine-American mathematician Gregory Chaitin defined Ω precisely to find uncomputable numbers. For this he used the famous halting problem in computer science. According to this, for any possible algorithm, no machine can determine at what point the computer running it will stop. If you give a computer an algorithm, it may be possible to determine whether that algorithm can be executed in a finite amount of time. However, there is obviously no way to do this for every conceivable program code. Therefore, the halting problem is also a direct application of Gödel’s incompleteness theorem.

The Chaitin constant Ω corresponds to the probability that a theoretical model of a computer (Turing machine) will stop for a given input.

In this equation blood Represents any program that stops after a finite runtime, |blood| Describes the length of the program in bits. To calculate Chaitin’s constant accurately, we need to know which programs are maintained and which are not. This is not possible due to retention issues. In 2000, mathematician Cristian Calude and his colleagues I succeeded in calculating the first few digits of the Chaitin constant 0.0157499939956247687… It is impossible to find all decimal places.

Cubitt’s team was thus able to mathematically prove that his physical model undergoes a phase transition for the following values: ∅ = Ω: Changes from a conductor to an insulator. However, since Ω cannot be calculated precisely, the state diagram of the physical system is also undefined. To be clear, this has nothing to do with the fact that your current computer doesn’t have enough power or you don’t have enough time to fix the problem. The fact is that the task is clearly unsolvable. “Our results show that the incalculable number become known Even when all the underlying microscopic data are fully computable, they are used as phase transition points in physics-like models,” the physicists wrote in the paper.

Technically, the precision with which the Chaitin constant can be specified is sufficient for real-world applications. But the work of Cubitt and his colleagues shows once again how incredibly broad Gödel’s insights were. Even after 90 years, there are still new cases of unprovable statements. There is a high possibility of serious physical problems, Like finding a theory about everything.It is influenced by Gödel’s incompleteness theorem.

This article originally appeared on: Spectrum der Wissenschaft Reproduced with permission.