MHB Are there practical applications for Axiomatized Formal Theories?

AI Thread Summary
The discussion highlights the practical applicability of logic theories, particularly in fields like mathematics, engineering, and computer science. It emphasizes that logic, once viewed as an abstract area of mathematics, has evolved into a crucial tool for modeling and analyzing software and hardware, forming the foundation of formal methods. Key applications include proof theory for formal verification of hardware and software, as well as solving complex combinatorial problems using satisfiability solvers. The compactness theorem illustrates further mathematical applications, such as extending partial orders and finding roots of polynomials. Additionally, the conversation notes the presence of numerous logic-related conferences worldwide, indicating a vibrant community focused on these topics. Quantum logic is mentioned as a potential connection to physics, although the specifics are less explored. Overall, the discussion underscores the relevance of logic beyond theoretical frameworks, showcasing its significant role in practical problem-solving across various disciplines.
agapito
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The study of these theories is interesting in itself,if only to understand Godel's work. My question concerns their practical applicability to the solution of problems in different disciplines like mathematics, engineering, physics and the like.

Are there reasons for studying them beyond the strictly abstract realm?

All contributions appreciated.
 
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Look at the list of logic-related conferences. If you include smaller events that are related to logic, but not necessarily have mathematical logic as the main focus, they happen around the world almost every week.

Most of them are related to computer science. Logic turned out to be a wonderful tool for building models and investigating the properties of software and hardware. It's one of the main foundations of formal methods. So even though in the beginning of the 20th century logic was one of the most abstract areas of mathematics, now it is one of the most applied ones.

One of the most direct applications of proof theory is formal verification of hardware and software as well as of regular mathematical results. A great deal of research is devoted to making proving properties of programs convenient and, hopefully, not significantly harder than writing programs themselves.

Logic can be applied to other areas of mathematics. For example, using compactness theorem, it is easy to show that every partial order can be extended to a linear one, and that every field $F$ has an extension $F'$ where every polynomial with coefficients from $F$ has a root. Satisfiability solvers can be used to solving computationally complex combinatorial problems, such as finding shortest routes that visit all cities. I am not a specialist about connection with physics, but there are some, such as quantum logic.
 
Evgeny.Makarov said:
Look at the list of logic-related conferences. If you include smaller events that are related to logic, but not necessarily have mathematical logic as the main focus, they happen around the world almost every week.

Most of them are related to computer science. Logic turned out to be a wonderful tool for building models and investigating the properties of software and hardware. It's one of the main foundations of formal methods. So even though in the beginning of the 20th century logic was one of the most abstract areas of mathematics, now it is one of the most applied ones.

One of the most direct applications of proof theory is formal verification of hardware and software as well as of regular mathematical results. A great deal of research is devoted to making proving properties of programs convenient and, hopefully, not significantly harder than writing programs themselves.

Logic can be applied to other areas of mathematics. For example, using compactness theorem, it is easy to show that every partial order can be extended to a linear one, and that every field $F$ has an extension $F'$ where every polynomial with coefficients from $F$ has a root. Satisfiability solvers can be used to solving computationally complex combinatorial problems, such as finding shortest routes that visit all cities. I am not a specialist about connection with physics, but there are some, such as quantum logic.

Many thanks for your response, am
 
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