November 29, 2024
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Mathematics and physics cannot prove all truths
Physicists have described systems that require incomputable numbers to fully understand. This is another example of an unprovable mathematical puzzle.
Mathematicians have known for decades that some problems are unsolvable. Now, physicists have identified a new example.
You can never prove every mathematical truth. For me, this incompleteness theorem discovered by Kurt Gödel is one of the most amazing achievements in mathematics. It may not come as a surprise to anyone – all sorts of unprovable things exist in everyday life – but for mathematicians, the idea was shocking. After all, they are able to build their own world from a few basic building blocks, the so-called axioms. There only the rules they created apply, and all truth consists of these basic building blocks and corresponding rules. Experts have long believed that if you find the right framework, you should somehow be able to prove any truth.
But in 1931 Gödel proved otherwise. There are always truths that circumvent the basic mathematical framework and are impossible to prove. And this is not a purely abstract finding and has no impact on real situations. Shortly after Gödel’s groundbreaking work, the first unprovable problems appeared. For example, it is never possible to determine how many real numbers exist within the currently used mathematical framework. And unsolvable problems are not limited to mathematics. 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. Also, physics cannot always predict whether a crystalline system will conduct electricity.
Currently, experts including physicist Toby Cubitt of University College London, Discovered another way to apply the incompleteness theorem to physics. They described a system of particles that undergoes a phase transition, a change similar to that which occurs when water freezes below zero degrees Celsius. However, the important parameters for the phase transition in this particle system are can’t do it Unlike water, it is calculated. “Our results… show how uncomputable numbers appear in physical systems,” the physicists wrote in a preprint paper posted last month on the server arXiv.org.
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undecidable phase transition
This is not the first time that experts have encountered unpredictable phase transitions. Cubitt Back in 2021 And two of his colleagues described another physical system whose transitions are unpredictable. However, in that case an infinite number of phase transitions are possible. Such a situation does not occur in nature. So the researchers asked themselves whether unpredictable events could occur in a realistic system.
In the new study, Cubitt and his colleagues studied a very simple system: a finite square lattice containing an array of several particles, each interacting with its nearest neighbor. Such models are typically used to describe solids. This is because their atoms are arranged in a regular structure and their electrons can interact with the electrons of the atoms immediately surrounding them. In Cubitt’s model, the strength of the interaction between electrons depends on a parameter. φ— greater than φ That is, the particles within the atomic shell repel each other more strongly.
If there is repulsion φ When is small, the outer electrons are mobile and can move back and forth between the nuclei. The stronger the φ That means more electrons freeze 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 φ is so small that the total energy of the system can increase continuously. As a result, electricity flows through the system without any problems. For large values, φ, However, the situation is different. With such values, the energy increases only 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 a semiconductor or an insulator.
To date, physicists have created thousands of similar models to describe all kinds of solids and crystals. However, since the system presented by Cubitt et al. exhibits two different behaviors, there should be a transition between the conducting and insulating phases. That is, it has the following values: φ Beyond this, a sudden gap appears in the energy spectrum of the system.
immeasurable numbers
Cubitt and his team determined the following values: φ Where this gap occurs. And this corresponds to the so-called Chaitin constant Ω. This number may be familiar to math geeks, as it is one of the few known examples where it cannot be calculated. These are irrational numbers where the decimal places last forever and do not repeat regularly. In contrast to computable irrational numbers such as π and picture, However, the value of an uncomputable number cannot be approximated to any precision. No algorithm outputs Ω when run infinitely long. If we cannot calculate Ω, we also cannot determine when the phase transition occurs in the system studied by Qubit et al.
Argentinian-American mathematician Gregory Chaitin defined Ω precisely for the purpose of finding incalculable numbers. To do this, he used computer science’s famous halting problem. According to it, for every possible algorithm, no machine exists that can determine whether the computer running them will stop at some point. If you give a computer some algorithm, it may be able to determine whether the algorithm can be executed within a finite amount of time. But there is clearly no way to do this for all possible 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 formula p | indicates all programs that stop after a finite execution time.p|Represents the length of the program in bits. To calculate the Chaitin constant accurately, we need to know which programs are retained and which are not, which is impossible according to the retention problem. However, in 2000 mathematician Cristian Calud and colleagues I’ve successfully calculated the first few digits of the Chaitin constant 0.0157499939956247687…, but it’s never possible to find all the digits to the right of the decimal point.
Qubit’s team was therefore able to prove mathematically that his physical model undergoes a phase transition at the following values: φ = Ω: From a conductor to an insulator. However, since Ω cannot be calculated accurately, the phase diagram of the physical system is also not defined. To be clear, this has nothing to do with the fact that current computers are not powerful enough or that there is not enough time to solve the problem, i.e. the task is clearly unsolvable. There is no. “Our results show that the incalculable numbers appear “It serves as a phase transition point in physics-like models even when all the underlying microscopic data are fully computable,” the physicists wrote in the paper.
Technically, the precision with which Chaitin constants can be specified is sufficient for practical applications. But the work of Cubitt and his colleagues shows once again how incredibly far-reaching Gödel’s insights were. More than 90 years later, there are still new examples of statements that cannot be proven. Possibly widespread physical problems; Exploring the theory of everything, etc.is affected by Gödel’s incompleteness theorem.
This article was first published Wissenschaft spectrum Reprinted with permission.
