Stories with Tag interactive theorem proving

• Show HN: Formalizing Principia Mathematica using Lean

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  Posted: 2025-04-25 18:49:30

  Verbose tl;dr

  Andrew N. Aguib has launched a project to formalize Alfred North Whitehead and Bertrand Russell's Principia Mathematica within the Lean theorem prover. This ambitious undertaking aims to translate the foundational work of mathematical logic, known for its dense symbolism and intricate proofs, into a computer-verifiable format. The project leverages Lean's powerful type theory and automated proof assistance to rigorously check the Principia's theorems and definitions, offering a modern perspective on this historical text and potentially revealing new insights. The project is ongoing and currently covers a portion of the first volume. The code and progress are available on GitHub.

  This GitHub repository, titled "Formalizing Principia Mathematica using Lean," documents an ambitious project undertaken by Andrew N. Aguib to translate and verify Alfred North Whitehead and Bertrand Russell's monumental work, Principia Mathematica, within the Lean proof assistant. Principia Mathematica, a three-volume work published between 1910 and 1913, represents a landmark attempt to ground mathematics in symbolic logic, aiming to demonstrate that all mathematical truths could be derived from a small set of logical axioms and inference rules. Aguib's project leverages the power of Lean, a modern, interactive theorem prover, to rigorously formalize the definitions, theorems, and proofs presented in Principia. This involves translating the often-opaque symbolic language and intricate arguments of the original text into Lean's precise and computationally verifiable format. The repository contains Lean code corresponding to various sections of Principia, effectively creating a digital, interactive version of the foundational text. This allows for a level of scrutiny and analysis not possible with the original printed work. The formalization process not only helps verify the correctness of the original proofs but also provides a clearer and more accessible representation of the underlying logic. The project is ongoing, with the current state reflecting progress made in formalizing different parts of Principia. The repository serves as a dynamic record of this effort, allowing others to follow, contribute to, and benefit from the ongoing work of translating this historically significant mathematical text into a modern, computationally verifiable form. This endeavor offers valuable insights into the foundations of mathematics, the evolution of logical systems, and the capabilities of modern proof assistants in tackling complex mathematical formalizations. By making Principia Mathematica accessible and verifiable within a powerful computational framework like Lean, the project aims to facilitate a deeper understanding and appreciation of this seminal work in mathematical logic.

  • lean
  • Theorem Prover
  • Principia Mathematica
  • Formalization
  • mathematics
  • Logic
  • Automated Reasoning
  • formal systems
  • Russell
  • Whitehead
  • proof assistant
  • type theory
  • dependent types
  • interactive theorem proving

  Summary of Comments ( 32 )
  https://news.ycombinator.com/item?id=43797256

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  Hacker News users discussed the impressive feat of formalizing parts of Principia Mathematica in Lean, praising the project for its ambition and clarity. Several commenters highlighted the accessibility of the formalized proofs compared to the original text, making the dense mathematical reasoning easier to follow. Some discussed the potential educational benefits, while others pointed out the limitations of formalization, particularly regarding the philosophical foundations of mathematics addressed in Principia. The project's use of Lean 4 also sparked a brief discussion on the theorem prover itself, with some commenters noting its relative novelty and expressing interest in learning more. A few users referenced similar formalization efforts, emphasizing the growing trend of using proof assistants to verify complex mathematical work.

  The Hacker News post titled "Show HN: Formalizing Principia Mathematica using Lean" (https://news.ycombinator.com/item?id=43797256) has generated a modest number of comments, primarily focusing on the complexities and nuances of formalizing mathematical proofs, particularly within the context of Principia Mathematica.

  One commenter highlights the historical context of Principia, emphasizing its significance as a pre-computer-era attempt to formalize mathematics. They point out the inherent challenges in such an endeavor, particularly the book's verbose and intricate symbolic notation. This comment also touches on the evolution of logical systems and proof assistants, contrasting Principia's methods with more modern approaches.

  Another commenter questions the practical applications of formalizing Principia. They acknowledge the intellectual value of the project but wonder about its relevance to modern mathematics, particularly given the availability of more efficient and powerful proof assistants. This sparks a discussion about the differences between simply verifying existing proofs and actually creating new ones within a formal system.

  A subsequent comment clarifies the distinction between verifying and formalizing a proof. They explain that formalization involves encoding the entire logical structure of a proof within a computer system, allowing for automated checking of each individual step. This is contrasted with mere verification, which might involve a human checking the overall logic of a proof without necessarily breaking down every minute detail.

  Another thread delves into the specifics of Lean, the proof assistant used in this project. Commenters discuss its strengths and weaknesses, comparing it to other systems like Coq and Isabelle. The discussion touches upon the tradeoffs between the accessibility and expressiveness of different proof assistants. One commenter mentions the potential of using Lean for educational purposes, given its relatively user-friendly interface.

  Finally, a comment praises the project for its ambition and potential to contribute to the field of automated theorem proving. They acknowledge the limitations of current technology but express optimism about the future of formal mathematics and the role projects like this can play in advancing the field.

  In summary, the comments on this Hacker News post reflect a nuanced understanding of the challenges and opportunities associated with formalizing mathematical proofs. They range from discussions about the historical significance of Principia Mathematica to the practicalities of using modern proof assistants like Lean. The overall tone is one of cautious optimism, acknowledging the limitations of current technology while recognizing the potential for future advancements.

• A proof checker meant for education

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  Posted: 2025-03-21 11:47:37

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  Deduce is a proof checker designed specifically for educational settings. It aims to bridge the gap between informal mathematical reasoning and formal proof construction by providing a simple, accessible interface and a focused set of logical connectives. Its primary goal is to teach the core concepts of formal logic and proof techniques without overwhelming users with complex syntax or advanced features. The system supports natural deduction style proofs and offers immediate feedback, guiding students through the process of building valid arguments step-by-step. Deduce prioritizes clarity and ease of use to make learning formal logic more engaging and less daunting.

  The webpage introduces Deduce, a proof checker specifically designed for educational purposes, aiming to bridge the gap between informal mathematical reasoning and the rigor demanded by formal proof assistants. It emphasizes practicality and ease of use over comprehensive theorem proving capabilities. Deduce operates within the confines of a web browser, eliminating the need for local installations or complex setup procedures, thus offering a frictionless entry point for students.

  The system utilizes a syntax intentionally crafted to resemble conventional mathematical notation as closely as possible, enhancing readability and reducing the cognitive overhead associated with learning specialized syntax commonly found in other proof assistants. This design choice prioritizes pedagogical clarity, making the transition from textbook mathematics to formal verification smoother and more intuitive. Furthermore, Deduce incorporates features designed to assist users in constructing proofs, providing helpful feedback and guidance throughout the process. It offers support for common mathematical objects like sets, functions, and natural numbers, providing a foundational framework within which students can explore fundamental mathematical concepts.

  While acknowledging its current limitations in terms of advanced features and extensibility compared to more mature proof assistants, the webpage highlights Deduce's focus on pedagogical value. It positions itself as a tool particularly suited for introductory logic and mathematics courses, enabling students to engage with formal proof construction in a more accessible and less daunting manner. The project explicitly welcomes contributions and feedback, indicating its ongoing development and commitment to improvement. In essence, Deduce presents itself as a pragmatic and user-friendly educational tool, specifically tailored to introduce students to the principles of formal proof without overwhelming them with the complexities of full-fledged proof assistant software.

  • proof checker
  • automated theorem proving
  • Formal Verification
  • Education
  • teaching
  • Logic
  • Computer Science
  • mathematics
  • deductive reasoning
  • interactive theorem proving
  • proof assistant
  • formal methods

  Summary of Comments ( 22 )
  https://news.ycombinator.com/item?id=43434503

  Verbose tl;dr

  Hacker News users discussed the educational value of the Deduce proof checker. Several commenters appreciated its simplicity and accessibility compared to other systems like Coq, finding its focus on propositional and first-order logic suitable for introductory logic courses. Some suggested potential improvements, such as adding support for natural deduction and incorporating a more interactive tutorial. Others debated the pedagogical merits of different proof styles and the balance between automated assistance and requiring students to fill in proof steps themselves. The overall sentiment was positive, with many seeing Deduce as a promising tool for teaching logic.

  The Hacker News post titled "A proof checker meant for education" (https://news.ycombinator.com/item?id=43434503) discussing the Deduce proof checker (https://jsiek.github.io/deduce/index.html) has a modest number of comments, focusing primarily on comparisons to other proof assistants and the potential role of Deduce in education.

  Several commenters compare Deduce to Lean, a popular interactive theorem prover. One commenter points out that Lean's steeper learning curve might make it less suitable for introductory logic courses, while Deduce's simplicity could be beneficial for beginners. This comment highlights the potential niche Deduce fills by prioritizing ease of use over advanced features. Another echoes this sentiment, suggesting Deduce's focus on natural deduction could be a pedagogical advantage compared to Lean's more complex tactics. The user praises Deduce's accessibility, particularly for those unfamiliar with the intricacies of dependent type theory.

  Another discussion thread centers around the practical applications of proof assistants in education. One commenter questions the overall value proposition of teaching formal proofs, arguing that it might not be the most efficient use of limited class time. They express skepticism about whether the rigor of formal proofs translates to improved "informal reasoning" skills valuable in other mathematical contexts. A counter-argument suggests that, while the direct benefits might not be immediately apparent, the process of constructing formal proofs can enhance a student's understanding of logical structure and the importance of precise definitions.

  Another comment focuses on the target audience for Deduce. The commenter speculates that it seems most appropriate for students already comfortable with mathematical reasoning, rather than complete beginners. This implies Deduce serves as a bridge to more advanced tools like Lean, rather than a replacement for introductory logic texts.

  Finally, one commenter expresses interest in the technical details of Deduce's implementation, specifically how it handles quantifier instantiation and substitution. This suggests a desire for more documentation or transparency about the internal workings of the system. However, this thread does not receive any further replies.

  In summary, the comments generally appreciate Deduce's simplicity and potential for educational use, particularly in introductory logic courses. The discussion revolves around comparisons with other tools like Lean, the pedagogical benefits of formal proofs, and the specific target audience for Deduce. There's also a brief, unanswered question about the technical details of its implementation.