Unlocking the secrets of a universe governed by undecidability, researchers explore the frontiers of knowledge and the limits of human understanding. Can we truly predict the behavior of complex systems, or are there inherent limitations to our comprehension?
The article discusses the concept of undecidability in physics, which refers to the idea that certain problems or systems are inherently impossible to solve or predict with complete accuracy. This concept was first introduced by Alan Turing, who showed that some problems were undecidable, meaning that there was no algorithm that could solve them exactly.
Undecidability is a fundamental concept in computability theory, describing the limits of computation.
A problem is undecidable if there is no algorithm that can determine its solution with certainty.
This means that for some problems, it's impossible to create an effective procedure to solve them within a finite amount of time and resources.
The halting problem, proposed by Alan Turing in 1936, is a classic example of an undecidable problem.
It questions whether there exists an algorithm that can determine if any given program will eventually halt or run indefinitely.
The article highlights several examples of undecidability in physics, including:
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The spectral gap problem: this refers to the energy required to excite a quantum system from its lowest energy state. If it takes some amount of energy, the system is ‘gapped.’ If it can become excited at any moment without infusing energy, it is ‘gapless.’
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The halting problem: this is a fundamental problem in computer science that asks whether there exists an algorithm that can determine, given an arbitrary program and input, whether the program will run forever or eventually halt.
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Fluids as computers: researchers have found that certain fluids can exhibit properties similar to those of computers, such as the ability to slosh around and respond to inputs.
The halting problem is a fundamental result in computability theory, proving that there cannot exist an algorithm to determine whether any given program will run indefinitely or eventually halt.
In 1936, Alan Turing demonstrated this by showing that no such algorithm can exist.
This has significant implications for programming and computer science, as it means that some problems may be inherently unsolvable.
The halting problem is often cited as a key example of the limits of computation.
The article also discusses the implications of undecidability in physics, including:
- The limitation of computational power: even with the most advanced supercomputers, there are some problems that are inherently unsolvable.
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The importance of approximations: many physical systems can be modeled using infinite theories, but these theories are often approximations of reality.
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The role of infinity in physics: while infinity may not exist in reality, it is a useful tool for modeling and understanding physical systems.
The article concludes that undecidability is an inherent aspect of physics, and that physicists must accept this limitation when trying to understand the behavior of complex systems. While it may be impossible to solve certain problems exactly, researchers can still make progress by identifying patterns and approximations that are useful for prediction and control.
Some key quotes from the article include:
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‘We live in a universe where you can build computers… Computation is everywhere.’ – Alan Moore
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‘There is no such thing as perfect knowledge, because you cannot touch it.‘ – Karl Svozil
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‘These are very important results. They are very, very profound.‘ – Toby Cubitt
Toby Cubitt is a British philosopher known for his work on the foundations of mathematics and its relationship to physics.
He has written extensively on the subject, including books such as 'To Be or Not To Be: A Guide to Existentialism' and 'The Philosophy of Mathematics'.
Cubitt's research focuses on the nature of mathematical truth and its implications for our understanding of reality.
His quotes often highlight the complexities of human knowledge and the importance of critical thinking.
Overall, the article highlights the importance of understanding undecidability in physics and its implications for our understanding of the universe.
