John Baez's post "Surprises in Logic" explores counterintuitive results within mathematical logic. It highlights the unexpected power of first-order logic, capable of expressing sophisticated concepts like finiteness and the natural numbers despite its seemingly simple structure. Conversely, it demonstrates limitations, such as the inability of first-order theories of the natural numbers to capture all true statements about them (Gödel's incompleteness theorem). The post emphasizes the surprising disconnect between a theory's ability to define a concept and its ability to characterize it completely, using examples like Peano arithmetic. This leads to the exploration of second-order logic and its increased expressive power, though at the cost of losing the completeness and compactness theorems enjoyed by first-order logic. The overall message is that even seemingly basic logical systems can harbor deep and often unintuitive complexities.
A blog post challenges readers to solve a math puzzle involving predicting the output of a hypothetical AI model trained on specific numerical sequences. The AI, named "Predictor," is trained on sequences like 1,2,3,4,5 -> 6 and 2,4,6,8,10 -> 12, seemingly learning to extrapolate the next number in simple arithmetic progressions. However, when given the sequence 1,3,5,7,9, the AI outputs 10 instead of the expected 11. The puzzle asks readers to determine the underlying logic of the AI and predict its output for the sequence 1,2,3,5,8. A symbolic prize (bragging rights) is offered to anyone who can crack the code.
HN users generally found the AI/Math puzzle unimpressive and easily solvable. Several commenters quickly pointed out the solution involves recognizing the pattern as powers of 2, leading to the answer 2^32. Some criticized the framing as an "AI" puzzle, arguing it's a straightforward math problem solvable with basic pattern recognition. Others debated the value of the $100 prize and whether it justified the effort. A few users noted potential ambiguity in the problem's wording, but these concerns were largely dismissed by others who found the intended pattern clear. There was some discussion about the puzzle's suitability for testing AI, with skepticism expressed about its ability to distinguish genuine intelligence.
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.
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 Peirce Edition Project (PEP) is dedicated to creating a comprehensive, scholarly edition of the writings of American philosopher Charles Sanders Peirce. The project, based at Indiana University–Purdue University Indianapolis (IUPUI), makes Peirce's vast and complex body of work accessible through various print and digital publications, including the 30-volume Writings of Charles S. Peirce, selected shorter works, and the digital archive Arisbe, which contains transcribed and encoded manuscripts. PEP's goal is to facilitate scholarship and understanding of Peirce's significant contributions to pragmatism, semiotics, logic, and the philosophy of science. The project provides essential resources for researchers, students, and anyone interested in exploring Peirce's multifaceted thought.
Hacker News users discuss the Peirce Edition Project, praising its comprehensive approach to digitizing Charles Sanders Peirce's works. Several commenters highlight the immense scope and complexity of Peirce's philosophical system, noting its influence on fields like semiotics and pragmatism. The project's importance for researchers is emphasized, particularly its robust search functionality and the inclusion of manuscripts. Some express excitement for exploring Peirce's lesser-known writings, while others recommend specific introductory texts for those unfamiliar with his work. The technical aspects of the digital edition also receive attention, with users commending the site's navigation and performance.
Jürgen Schmidhuber's "Matters Computational" provides a comprehensive overview of computer science, spanning its theoretical foundations and practical applications. It delves into topics like algorithmic information theory, computability, complexity theory, and the history of computation, including discussions of Turing machines and the Church-Turing thesis. The book also explores the nature of intelligence and the possibilities of artificial intelligence, covering areas such as machine learning, neural networks, and evolutionary computation. It emphasizes the importance of self-referential systems and universal problem solvers, reflecting Schmidhuber's own research interests in artificial general intelligence. Ultimately, the book aims to provide a unifying perspective on computation, bridging the gap between theoretical computer science and the practical pursuit of artificial intelligence.
HN users discuss the density and breadth of "Matters Computational," praising its unique approach to connecting diverse computational topics. Several commenters highlight the book's treatment of randomness, floating-point arithmetic, and the FFT as particularly insightful. The author's background in physics is noted, contributing to the book's distinct perspective. Some find the book challenging, requiring multiple readings to fully grasp the concepts. The free availability of the PDF is appreciated, and its enduring relevance a decade after publication is also remarked upon. A few commenters express interest in a physical copy, while others suggest potential updates or expansions on certain topics.
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.
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.
This paper explores using first-order logic (FOL) to detect logical fallacies in natural language arguments. The authors propose a novel approach that translates natural language arguments into FOL representations, leveraging semantic role labeling and a defined set of predicates to capture argument structure. This structured representation allows for the application of automated theorem provers to evaluate the validity of the arguments, thus identifying potential fallacies. The research demonstrates improved performance compared to existing methods, particularly in identifying fallacies related to invalid argument structure, while acknowledging limitations in handling complex linguistic phenomena and the need for further refinement in the translation process. The proposed system provides a promising foundation for automated fallacy detection and contributes to the broader field of argument mining.
Hacker News users discussed the potential and limitations of using first-order logic (FOL) for fallacy detection as described in the linked paper. Some praised the approach for its rigor and potential to improve reasoning in AI, while also acknowledging the inherent difficulty of translating natural language to FOL perfectly. Others questioned the practical applicability, citing the complexity and ambiguity of natural language as major obstacles, and suggesting that statistical/probabilistic methods might be more robust. The difficulty of scoping the domain knowledge necessary for FOL translation was also brought up, with some pointing out the need for extensive, context-specific knowledge bases. Finally, several commenters highlighted the limitations of focusing solely on logical fallacies for detecting flawed reasoning, suggesting that other rhetorical tactics and nuances should also be considered.
The paper "Efficient Reasoning with Hidden Thinking" introduces Hidden Thinking Networks (HTNs), a novel architecture designed to enhance the efficiency of large language models (LLMs) in complex reasoning tasks. HTNs augment LLMs with a differentiable "scratchpad" that allows them to perform intermediate computations and logical steps, mimicking human thought processes during problem-solving. This hidden thinking process is learned through backpropagation, enabling the model to dynamically adapt its reasoning strategies. By externalizing and making the reasoning steps differentiable, HTNs aim to improve transparency, controllability, and efficiency compared to standard LLMs, which often struggle with multi-step reasoning or rely on computationally expensive prompting techniques like chain-of-thought. The authors demonstrate the effectiveness of HTNs on various reasoning tasks, showcasing their potential for more efficient and interpretable problem-solving with LLMs.
Hacker News users discussed the practicality and implications of the "Hidden Thinking" paper. Several commenters expressed skepticism about the real-world applicability of the proposed method, citing concerns about computational cost and the difficulty of accurately representing complex real-world problems within the framework. Some questioned the novelty of the approach, comparing it to existing techniques like MCTS (Monte Carlo Tree Search) and pointing out potential limitations in scaling and handling uncertainty. Others were more optimistic, seeing potential applications in areas like game playing and automated theorem proving, while acknowledging the need for further research and development. A few commenters also discussed the philosophical implications of machines engaging in "hidden thinking," raising questions about transparency and interpretability.
This article dissects the structure of a formal mathematical proof, illustrating it with a simple example about even and odd numbers. It emphasizes the distinction between informal proofs aimed at human understanding and formal proofs designed for automated verification. Formal proofs meticulously lay out every logical step, referencing specific axioms and inference rules within a chosen formal system. This detailed approach, while tedious for humans, enables computer-assisted verification and eliminates ambiguity, ensuring absolute rigor. The article highlights the importance of choosing appropriate axioms and the role of proof assistants in constructing and checking these complex formal structures, ultimately increasing confidence in mathematical results.
HN commenters discuss the accessibility of formal proof systems, particularly referencing Lean. Some express excitement about the potential of formal proofs to revolutionize mathematics, while others are more skeptical, citing the steep learning curve and questioning the practical benefits for most mathematicians. Several commenters debate the role of intuition versus rigor in mathematical practice, with some arguing that formalization can enhance understanding and others suggesting it might stifle creativity. The feasibility of formalizing existing mathematical knowledge is also discussed, with varying opinions on the timescale and resources required for such a project. Some users highlight the potential of AI in assisting with formalization efforts, while others remain cautious about its current capabilities. The overall tone is one of cautious optimism, acknowledging the challenges but also recognizing the potential transformative power of formal proof systems.
SudokuVariants.com lets you play and create a wide variety of Sudoku puzzles beyond the classic 9x9 grid. The website offers different grid sizes, shapes, and rule sets, including variations like Killer Sudoku, Irregular Sudoku, and even custom rule combinations. Users can experiment with existing variants or design their own unique Sudoku challenges using a visual editor, and then share their creations with others via a generated link. The site aims to provide a comprehensive platform for both playing and exploring the vast possibilities within the Sudoku puzzle format.
Hacker News users generally expressed interest in the SudokuVariants website. Several praised its clean design and the variety of puzzles offered. Some found the "construct your own variant" feature particularly appealing, and one user suggested adding a difficulty rating system for user-created puzzles. A few commenters mentioned specific variant recommendations, including "Killer Sudoku" and a variant with prime number constraints. There was also a brief discussion about the underlying logic and algorithms involved in generating and solving these puzzles. One user pointed out that some extreme variants might be NP-complete, implying significant computational challenges for larger grids or complex rules.
Summary of Comments ( 6 )
https://news.ycombinator.com/item?id=43763291
Hacker News users discuss various aspects of the surprises in mathematical logic presented in the linked article. Several commenters delve into the implications of Gödel's incompleteness theorems, with some highlighting the distinction between truth and provability. The concept of "surprising" itself is debated, with some arguing that the listed examples are well-known within the field and therefore not surprising to experts. Others point out the connection between logic and computation, referencing Turing machines and the halting problem. The role of axioms in shaping mathematical systems is also mentioned, alongside the challenge of finding "natural" axioms that accurately reflect our intuitive understanding of mathematics. A few commenters express appreciation for the article's clear explanations of complex topics.
The Hacker News post titled "Surprises in Logic (2016)" linking to John Baez's article has generated several comments discussing various aspects of logic, set theory, and their implications.
One commenter highlights the significance of Löwenheim-Skolem theorem, which states that if a first-order theory has an infinite model, then it has a model of every infinite cardinality. They explain how this theorem can be counterintuitive, especially when applied to theories intended to describe a unique structure, like the real numbers. They suggest it implies there are "countable models of set theory that think they contain uncountable sets," a concept they find fascinating and paradoxical.
Another comment dives into the implications of Gödel's incompleteness theorems, specifically focusing on their impact on Hilbert's program. They mention how Gödel's work demonstrated the inherent limitations of formal systems in proving all true statements within a given set of axioms. This commenter further connects this to the concept of truth being "larger" than provability, emphasizing that there will always be true statements that are unprovable within a given formal system.
Further discussion revolves around the nature of infinity and its various interpretations within set theory. One comment clarifies the distinction between countable and uncountable infinities, using the analogy of integers versus real numbers. They point out that while both sets are infinite, the real numbers are "denser" than the integers, leading to the concept of uncountability.
The conversation also touches upon the implications of the Axiom of Choice, a fundamental principle in set theory that allows for making infinitely many arbitrary choices. Some comments express how counterintuitive this axiom can be, even though it's necessary for many important mathematical theorems. They mention how it can lead to seemingly paradoxical results, like the Banach-Tarski paradox, which demonstrates how a sphere can be decomposed and reassembled into two spheres identical to the original.
A few commenters also delve into the philosophical implications of these mathematical concepts, questioning the nature of mathematical truth and its relationship to reality. They discuss whether mathematical structures are discovered or invented, and whether the limitations of formal systems reflect limitations on our ability to understand the universe.
Finally, some comments offer additional resources for exploring these topics further, including links to relevant Wikipedia pages, books, and online lectures. These recommendations provide avenues for those interested in gaining a deeper understanding of the discussed concepts.