Is there is town/gown gap between pure mathematics and automated proof checking?

Meta AI, Perplexity AI and Microsoft Copilot say Yes. Google Gemini gave a kind of clueless middle school response.

Meta AI

Yes, there is a town/gown gap between pure mathematics and automated proof checking. Pure mathematicians often focus on abstract mathematical structures and proofs, while automated proof checking is a more applied field that deals with the implementation of algorithms and software to verify proofs.

Town/Gown Gap:

• Town: Automated proof checking is often seen as a practical tool for verifying proofs, and is frequently used in industry and computer science applications. It is often associated with the "town" of applied mathematics and computer science.
• Gown: Pure mathematics, on the other hand, is often seen as a more theoretical and abstract field, and is frequently associated with the "gown" of academia and theoretical mathematics.

Gap:

• There is a gap between the two fields, as pure mathematicians may not always be familiar with the latest developments in automated proof checking, and vice versa.
• The gap is also reflected in the different languages and tools used by each community. Pure mathematicians often use mathematical notation and language, while automated proof checking relies on programming languages and software.
• Additionally, the gap can lead to a lack of communication and collaboration between the two fields, which can hinder progress in both areas.

However, there are efforts to bridge this gap, such as:

• The use of proof assistants like Coq, Agda, and Mizar, which allow mathematicians to formalize and verify their proofs using automated tools.
• The development of formal proof libraries, such as the Mizar Mathematical Library, which provide a common language and framework for mathematicians and automated proof checking tools.
• Collaborations between mathematicians and computer scientists, such as the Formal Proof community, which aims to bring together researchers from both fields to advance the state of the art in formal proof.

By bridging this gap, we can facilitate the development of new mathematical theories and proofs, and increase the reliability and efficiency of automated proof checking tools.

Perplexity AI

There appears to be a divide between pure mathematicians and those working on automated theorem proving and proof assistants like Lean. Here are a few key points:

• Many pure mathematicians are skeptical or resistant to the idea of fully automating mathematical reasoning and proof writing. They view proving theorems as a fundamentally human endeavor requiring creativity and intuition that machines cannot replicate.4
• However, there is growing interest and adoption of proof assistants like Lean, Coq, Isabelle etc. among some mathematicians and logicians. These tools are useful for verifying proofs and formalizing mathematics, but still require significant human guidance.5
• Researchers in automated reasoning and interactive theorem proving believe that increasing automation of "mundane" proof steps and case analyses is possible and desirable. This could make proofs more readable by offloading tedious details to machines.5
• There are efforts to use machine learning over large corpora of formal proofs (e.g. in Lean) to develop systems that can suggest proof steps or even generate full proofs automatically with some human guidance.35 But fully automated mathematical reasoning remains an immense challenge.
• Proponents argue proof assistants could transform math education by making formal proofs more accessible and allowing students to experiment.5 But many mathematicians remain skeptical about replacing traditional proof writing.

In summary, while some believe automated reasoning will eventually automate most mathematical proofs, many mathematicians are doubtful this is possible or desirable, creating a divide between the two camps.45 But both agree proof assistants have value for verifying proofs and perhaps offloading routine proof steps.12

Microsoft Copilot

Yes, there is indeed a gap between pure mathematics and automated proof checking, often referred to as the “town/gown” gap. This gap arises from the different objectives and methodologies of the two fields.

Pure mathematics is primarily concerned with abstract concepts and theoretical results. It values rigorous proofs and deep understanding of mathematical structures. On the other hand, automated proof checking is a field within computer science that focuses on the development of algorithms and software to verify the correctness of mathematical proofs automatically1.

Vladimir Voevodsky, a renowned mathematician, identified this gap and aimed to bridge it. He proposed two strategic goals for mathematics: developing a computerized library of mathematical knowledge that supports automated proof verification, and bridging the gap between pure and applied mathematics2. His research towards the first goal led to the development of the Univalent foundations of mathematics, which is a new foundation of mathematics that supports automated proof-checking2.

The Lean Theorem Prover is another effort to bridge this gap. It aims to combine interactive and automated theorem proving by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs3.

However, despite these efforts, the gap between pure mathematics and automated proof checking still exists. Bridging this gap is a challenging task that requires further research and collaboration between mathematicians and computer scientists. It involves not only technical issues but also philosophical and methodological considerations about the nature of mathematical knowledge and proof.

Google Gemini

There's a connection between the "town/gown gap" concept and the relationship between pure mathematics and automated proof checking, but it's not a perfect fit. Here's a breakdown:

Town/Gown Gap:

This term refers to a divide between a university (gown) and the surrounding community (town). There can be a lack of understanding or communication between academics focused on theoretical pursuits and the general public with more practical concerns.

Pure Math vs. Automated Proof Checking:

• Pure Math: Focuses on abstract concepts, exploring the foundations of mathematics and its logical structures.
• Automated Proof Checking: Emphasizes applying computer programs to verify the correctness of mathematical proofs.

There's a difference in goals:

• Pure mathematicians are interested in developing new ideas and understanding the nature of mathematics itself.
• Automated proof checking is a tool to ensure the accuracy of existing proofs, often for complex theorems.

The Connection:

• Some mathematicians might see automated proof checking as a less creative or "pure" pursuit compared to developing new theorems. This aligns somewhat with the town/gown gap idea, where theoretical work might be seen as less practical by those focused on applications.

The Difference:

• However, automated proof checking is crucial for mathematicians. Long and intricate proofs can be error-prone, and automation helps to ensure their validity.
• Additionally, many pure mathematicians are actively involved in developing new automated proof checking techniques.

In Conclusion:

There's a difference in emphasis between pure mathematics and automated proof checking, but it's more of a collaboration than a true "town/gown gap." Both areas are essential for advancing mathematics.