November 29, 2024
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Mathematics and Physics cannot prove all truths
Physicists have described a system that requires an incomputable number to fully understand, another example of the unprovable puzzles of mathematics.
Mathematicians have known for decades that some problems are unsolvable. Now physicists have identified a new example.
You will never be able to prove every mathematical truth. For me, this incompleteness theorem discovered by Kurt Gödel is one of the most incredible results in mathematics. It may not surprise everyone – there are all sorts of unprovable things in everyday life – but for mathematicians this idea was surprising. Ultimately, they can build their world from a few basic building blocks, the so-called axioms. Only the rules they have created apply there, and all truth is made up of these basic elements and their respective rules. If you find the right frame, experts have long believed, so you can somehow prove all the truths.
But in 1931 Gödel proved the opposite. There will always be truths that escape the basic mathematical framework and are impossible to prove. And this is not a purely abstract finding, without implications for practical situations. Shortly after Gödel’s pioneering work, the first unprovable problems appeared. For example, it will never be possible to figure out how many real numbers there are in the mathematical framework in use today. And unsolvable problems are not limited to mathematics. For example, in some card and computer games (such as Magic: The Gathering), situations may arise in which it is impossible to determine whether the 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, They found another way to reflect the incompleteness theorem in physics. They described a system of particles that undergoes a phase transition, similar to the change that occurs when water freezes below zero degrees Celsius. But the critical parameter in which the phase transition of this particle system occurs can’t it has to be calculated, unlike water. “Our result … shows how computable numbers can appear in physical systems,” the physicists wrote in a preprint paper published last month on the arXiv.org server.
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An indeterminate phase transition
This is not the first time that experts have come across an unexpected phase transition. Cubitt returned 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 were possible. Such situations do not occur in nature. Therefore, the researchers asked themselves whether unexpected things can happen in realistic systems.
In the new work, Cubitt and his colleagues investigated a relatively simple system: a finite square lattice with an arrangement of several particles, each interacting with its nearest neighbor. Such models are often used to describe solids. This is because their atoms are arranged in a regular structure, and their electrons can interact with immediately surrounding atoms. In Cubitt’s model, the strength of the interaction between electrons depends on a parameter f– the bigger f that is, the more strongly the particles in the atomic shells repel each other.
If rejected f is small, the outer electrons are mobile: they can jump back and forth between atomic nuclei. The stronger f that is, the more the electrons are frozen in place. This different behavior is also reflected in the energy of the system. You can see the base state (lowest total energy) and the next highest energy state. If there is f is very small, the total energy of the system can grow continuously. As a result, the system carries electricity without any problems. For large values f however, the situation is different. With such 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 would be a semiconductor or an insulator.
So far, physicists have created thousands of similar models to describe all kinds of solids and crystals. But since the system presented by Cubitt and his colleagues shows two different behaviors, there must be a transition between the conducting and insulating phases. In other words, there is a value f above that the energy spectrum of the system suddenly has a gap.
Impossible number
Cubitt and his team have determined the value of f that’s where this gap occurs. And that Chaitin’s constant corresponds to the so-called Ω— A number popular with math nerds because it’s one of the few examples of numbers that can’t be calculated. They are irrational numbers whose decimal digits last forever and never repeat. Compared to computable irrational numbers π or and, however, the value of a non-computable number cannot be approximated with arbitrary precision. There is no algorithm that yields Ω if run infinitely long. If Ω cannot be calculated, then it is also not possible to determine when the phase transition occurs in the system studied by Cubitt and his colleagues.
Argentinian-American mathematician Gregory Chaitin precisely defined Ω with the aim of finding a computable number. To do this, he used the famous halting problem in computing: according to it, there is no machine that can judge, for all possible algorithms, whether or not the computer running them will ever stop. If you give any algorithm to a computer, it may judge whether that algorithm can run in finite time. But any method that can do this for all program code is provable. The halting problem is therefore a direct application of Gödel’s incompleteness theorem.
Chaitin’s constant Ω corresponds to the probability that the theoretical model of a computer (Turing machine) stops for a given input:
In this equation p It represents all programs that stop after a finite execution time, and |p| describes the length of the program in bits. To calculate the Chaitin constant accurately, you should know which programs are maintainable and which are not, which is not possible, depending on the maintainability problem. Even in 2000 the mathematician Cristian Calude and his colleagues Chaitin managed to calculate the first digit of the constant, 0.0157499939956247687…, it will never be possible to find all the decimal numbers.
Cubitt’s team was therefore able to mathematically demonstrate that their physical model undergoes a phase transition for a value. f = Ω: it changes from a conductor to an insulator. Because Ω cannot be calculated exactly, however, the phase diagram of the physical system is also undetermined. To be clear, this has nothing to do with current computers not being powerful enough or not having enough time to solve the problem – that task is clearly insurmountable. “Our results show incalculable numbers emerge as phase transition points in similar models of physics, even when all the underlying microscopic data are fully computable,” the physicists wrote in their paper.
Technically the accuracy with which the Chaitin constant can be determined makes it sufficient for real-world applications. But the work of Cubitt and his colleagues still shows how incredibly broad Gödel’s vision is. Even after more than 90 years, there are still new examples of unprovable statements. You are likely to have major physical problems, such as the search for a theory of everythingIt is influenced by Gödel’s incompleteness theorems.
This article originally appeared the spectrum of science and reproduced with permission.
