The blog post explores the limitations of formal systems, particularly in discerning truth. It uses the analogy of two goblins, one always truthful and one always lying, to demonstrate how relying solely on a system's rules, without external context or verification, can lead to accepting falsehoods as truths. Even with additional rules added to account for the goblins' lying, clever manipulation can still exploit the system. The post concludes that formal systems, while valuable for structuring thought, are ultimately insufficient for determining truth without external validation or a connection to reality. This highlights the need for critical thinking and skepticism even when dealing with seemingly rigorous systems.
Mathematicians are exploring the boundaries of provability using large language models (LLMs) and other automated theorem provers. While these tools can generate novel and valid proofs, they often rely on techniques too complex for human comprehension. This raises questions about the nature of mathematical truth and understanding. If a proof is too long or intricate for any human to verify, does it truly constitute "knowledge"? Researchers are investigating ways to make these computer-generated proofs more accessible and understandable, potentially revealing new insights into mathematical structures along the way. The ultimate goal is to find a balance between the power of automated proving and the human need for comprehensible explanations.
HN commenters discuss the implications of Gödel's incompleteness theorems and the article's exploration of concrete examples in Ramsey theory and Diophantine equations. Some debate the philosophical significance of undecidable statements, questioning whether they represent "true" mathematical facts or merely artifacts of formal systems. Others highlight the practical limitations of proof assistants and the ongoing search for more powerful automated theorem provers. The connection between computability and the physical universe is also raised, with some suggesting that undecidable statements could have physical implications, while others argue for a separation between abstract mathematics and the concrete world. Several commenters express appreciation for the article's clarity in explaining complex mathematical concepts to a lay audience.
Summary of Comments ( 0 )
https://news.ycombinator.com/item?id=43285485
The Hacker News comments generally praise the clarity and engaging presentation of the article's topic (formal systems and the halting problem, illustrated by a lying goblin puzzle). Several commenters discuss the philosophical implications of the piece, particularly regarding the nature of truth and provability within defined systems. Some draw parallels to Gödel's incompleteness theorems, while others offer alternate goblin scenarios or slight modifications to the puzzle's rules. A few commenters suggest related resources, such as Raymond Smullyan's work, which explores similar logical puzzles. There's also a short thread discussing the potential applicability of these concepts to legal systems and contract interpretation.
The Hacker News post "A brief meditation on formal systems and lying goblins" has generated several comments discussing the article's premise and exploring related concepts.
Several commenters engage with the core idea of the article, which uses the analogy of lying goblins to illustrate how seemingly sound logical systems can lead to incorrect conclusions if based on false premises. One commenter points out the parallel to real-world scenarios where misinformation or flawed assumptions can corrupt a system, regardless of its internal consistency. They mention how this applies to areas like political discourse and conspiracy theories.
Another commenter delves deeper into the concept of formal systems, emphasizing the distinction between validity and soundness. They clarify that a valid argument can still be untrue if its premises are false, echoing the goblin analogy. They also introduce the idea of Gödel's incompleteness theorems, suggesting that even consistent formal systems can contain unprovable truths.
The discussion extends to the practical implications of these ideas. One commenter reflects on the challenge of identifying false premises in real-world situations, highlighting the importance of critical thinking and questioning assumptions. Another commenter draws a connection to Bayesian reasoning, suggesting that incorporating prior probabilities can help mitigate the risk of being misled by false information.
Further comments explore related philosophical themes, touching on the nature of truth and the limits of knowledge. One commenter mentions the concept of "unknown unknowns," emphasizing the difficulty of accounting for information that we are not even aware of.
Some commenters also offer alternative analogies to illustrate the same principles. One suggests the image of a perfectly functioning calculator that produces incorrect results due to a user inputting the wrong numbers.
Overall, the comments on the Hacker News post provide a thoughtful and engaging discussion of the article's core ideas, exploring their implications in various contexts and connecting them to broader philosophical and mathematical concepts. They highlight the importance of critical thinking, the limitations of formal systems, and the challenges of navigating a world filled with potentially misleading information.