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.
Lehmer's continued fraction factorization algorithm offers a way to find factors of a composite integer n. It leverages the convergents of the continued fraction expansion of √n to generate pairs of integers x and y such that x² ≡ y² (mod n). If x is not congruent to ±y (mod n), then gcd(x-y, n) and gcd(x+y, n) will yield non-trivial factors of n. While not as efficient as more advanced methods like the general number field sieve, it provides a relatively simple approach to factorization and serves as a stepping stone towards understanding more complex techniques.
Hacker News users discuss Lehmer's algorithm, mostly focusing on its impracticality despite its mathematical elegance. Several commenters point out the exponential complexity, making it slower than trial division for realistically sized numbers. The discussion touches upon the algorithm's reliance on finding small quadratic residues, a process that becomes computationally expensive quickly. Some express interest in its historical significance and connection to other factoring methods, while others question the article's claim of it being "simple" given its actual complexity. A few users note the lack of practical applications, emphasizing its theoretical nature. The overall sentiment leans towards appreciation of the mathematical beauty of the algorithm but acknowledges its limited real-world use.
"Matrix Calculus (For Machine Learning and Beyond)" offers a comprehensive guide to matrix calculus, specifically tailored for its applications in machine learning. It covers foundational concepts like derivatives, gradients, Jacobians, Hessians, and their properties, emphasizing practical computation and usage over rigorous proofs. The resource presents various techniques for matrix differentiation, including the numerator-layout and denominator-layout conventions, and connects these theoretical underpinnings to real-world machine learning scenarios like backpropagation and optimization algorithms. It also delves into more advanced topics such as vectorization, chain rule applications, and handling higher-order derivatives, providing numerous examples and clear explanations throughout to facilitate understanding and application.
Hacker News users discussed the accessibility and practicality of the linked matrix calculus resource. Several commenters appreciated its clear explanations and examples, particularly for those without a strong math background. Some found the focus on differentials beneficial for understanding backpropagation and optimization algorithms. However, others argued that automatic differentiation makes manual matrix calculus less crucial in modern machine learning, questioning the resource's overall relevance. A few users also pointed out the existence of other similar resources, suggesting alternative learning paths. The overall sentiment leaned towards cautious praise, acknowledging the resource's quality while debating its necessity in the current machine learning landscape.
Creating accessible open textbooks, especially in math-heavy fields, is challenging due to the complexity of mathematical notation. While LaTeX is commonly used, its accessibility features are limited, particularly for screen reader users. Converting LaTeX to accessible formats like HTML requires significant manual effort and often compromises semantic meaning. The author explores MathML as a potential solution, highlighting its accessibility advantages and integration possibilities with HTML. However, MathML also presents challenges including limited browser support and authoring difficulties. Ultimately, creating truly accessible math content necessitates a shift towards semantic encoding and tools that prioritize accessibility from the outset, rather than relying on post-hoc conversions.
Hacker News users discussed the challenges and potential solutions for creating accessible open textbooks, particularly in math-heavy fields. Commenters highlighted the complexity of converting LaTeX, a common tool for math typesetting, into accessible formats. Some suggested focusing on HTML-first authoring, using tools like MathJax and Pandoc, or exploring MathML. The need for semantic tagging and robust tooling for image descriptions also emerged as key themes. Several users pointed to specific projects and resources like PreTeXt, which aims to facilitate accessible textbook creation. Concerns about funding and institutional support for these initiatives were also raised, as was the question of whether creating truly accessible math content requires a fundamental shift away from current publishing workflows.
"The Matrix Calculus You Need for Deep Learning" provides a practical guide to the core matrix calculus concepts essential for understanding and working with neural networks. It focuses on developing an intuitive understanding of derivatives of scalar-by-vector, vector-by-scalar, vector-by-vector, and scalar-by-matrix functions, emphasizing the denominator layout convention. The post covers key topics like the Jacobian, gradient, Hessian, and chain rule, illustrating them with clear examples and visualizations related to common deep learning scenarios. It avoids delving into complex proofs and instead prioritizes practical application, equipping readers with the tools to derive gradients for various neural network components and optimize their models effectively.
Hacker News users generally praised the article for its clarity and accessibility in explaining matrix calculus for deep learning. Several commenters appreciated the visual explanations and step-by-step approach, finding it more intuitive than other resources. Some pointed out the importance of denominator layout notation and its relevance to backpropagation. A few users suggested additional resources or alternative notations, while others discussed the practical applications of matrix calculus in machine learning and the challenges of teaching these concepts effectively. One commenter highlighted the article's helpfulness in understanding the chain rule in a multi-dimensional context. The overall sentiment was positive, with many considering the article a valuable resource for those learning deep learning.
Terry Tao explores the problem of efficiently decomposing a large factorial n! into a product of factors of roughly equal size √n. He outlines several approaches, including a naive iterative method that repeatedly divides n! by the largest integer below √n, and a more sophisticated approach leveraging prime factorization. The prime factorization method cleverly groups primes into products close to the target size, offering significant computational advantages. While both methods achieve the desired decomposition, the prime factorization technique highlights the interplay between the smooth structure of factorials (captured by their prime decomposition) and the goal of obtaining uniformly large factors. Tao emphasizes the efficiency gains from working with the prime factorization, and suggests potential generalizations and connections to other mathematical concepts like smooth numbers and the Dickman function.
Hacker News users discussed the surprising difficulty of factoring large factorials, even when not seeking prime factorization. One commenter highlighted the connection to cryptography, pointing out that if factoring factorials were easy, breaking RSA would be as well. Another questioned the practical applications of this type of factorization, while others appreciated the mathematical puzzle aspect. The discussion also touched upon the computational complexity of factoring and the effectiveness of different factoring algorithms in this specific context. Some commenters shared resources and further reading on related topics in number theory. The general sentiment was one of appreciation for the mathematical curiosity presented by Terry Tao's blog post.
Francis Bach's "Learning Theory from First Principles" provides a comprehensive and self-contained introduction to statistical learning theory. The book builds a foundational understanding of the core concepts, starting with basic probability and statistics, and progressively developing the theory behind supervised learning, including linear models, kernel methods, and neural networks. It emphasizes a functional analysis perspective, using tools like reproducing kernel Hilbert spaces and concentration inequalities to rigorously analyze generalization performance and derive bounds on the prediction error. The book also covers topics like stochastic gradient descent, sparsity, and online learning, offering both theoretical insights and practical considerations for algorithm design and implementation.
HN commenters generally praise the book "Learning Theory from First Principles" for its clarity, rigor, and accessibility. Several appreciate its focus on fundamental concepts and building a solid theoretical foundation, contrasting it favorably with more applied machine learning resources. Some highlight the book's coverage of specific topics like Rademacher complexity and PAC-Bayes. A few mention using the book for self-study or teaching, finding it well-structured and engaging. One commenter points out the authors' inclusion of online exercises and solutions, further enhancing its educational value. Another notes the book's free availability as a significant benefit. Overall, the sentiment is strongly positive, recommending the book for anyone seeking a deeper understanding of learning theory.
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.
Edsger Dijkstra argues that array indexing should start at zero, not one. He lays out a compelling case based on the elegance and efficiency of expressing slices or subsequences within an array. Using half-open intervals, where the lower bound is inclusive and the upper bound exclusive, simplifies calculations and leads to fewer "off-by-one" errors. Dijkstra demonstrates that representing a subsequence from element 'i' through 'j' becomes significantly more straightforward when using zero-based indexing, as the length of the subsequence is simply j-i. This contrasts with one-based indexing, which necessitates more complex and less intuitive calculations for subsequence lengths and endpoint adjustments. He concludes that zero-based indexing offers a more natural and consistent way to represent array segments, aligning better with mathematical conventions and ultimately leading to cleaner, less error-prone code.
Hacker News users discuss Dijkstra's famous argument for zero-based indexing. Several commenters agree with Dijkstra's logic, emphasizing the elegance and efficiency of using half-open intervals. Some highlight the benefits in loop constructs and simplifying calculations for array slices. A few point out that one-based indexing can be more intuitive in certain contexts, aligning with how humans naturally count. One commenter notes the historical precedent, mentioning that Fortran used one-based indexing, influencing later languages. The discussion also touches on the trade-offs between conventions and the importance of consistency within a given language or project.
Napkin Math Tool is a web-based calculator designed for quick, back-of-the-envelope estimations and explorations. It emphasizes natural language input, allowing users to type expressions like "2 apples + 3 oranges" or "10% of 1 million." It handles unit conversions, uncertainties (e.g., "10±1"), and supports variables for building more complex calculations. The tool aims to be a versatile scratchpad for thinking through quantitative problems, offering a more flexible and expressive alternative to traditional calculators.
Hacker News users generally praised the Napkin Math Tool for its simplicity and ease of use, finding it a handy alternative to a full spreadsheet program for quick calculations. Several commenters appreciated the clean interface and the focus on keyboard navigation. Some suggested improvements, such as the ability to copy calculated results, a dark mode, and support for variables and functions. One user pointed out the potential benefit for teaching basic math principles, while another highlighted its usefulness for estimating cloud computing costs. There was also a discussion comparing it to other similar tools like Tydlig and Soulver.
An undergraduate student, Noah Stephens-Davidowitz, has disproven a longstanding conjecture in computer science related to hash tables. He demonstrated that "linear probing," a simple hash table collision resolution method, can achieve optimal performance even with high load factors, contradicting a 40-year-old assumption. His work not only closes a theoretical gap in our understanding of hash tables but also introduces a new, potentially faster type of hash table based on "robin hood hashing" that could improve performance in databases and other applications.
Hacker News commenters discuss the surprising nature of the discovery, given the problem's long history and apparent simplicity. Some express skepticism about the "disproved" claim, suggesting the Kadane algorithm is a more efficient solution for the original problem than the article implies, and therefore the new hash table isn't a direct refutation. Others question the practicality of the new hash table, citing potential performance bottlenecks and the limited scenarios where it offers a significant advantage. Several commenters highlight the student's ingenuity and the importance of revisiting seemingly solved problems. A few point out the cyclical nature of computer science, with older, sometimes forgotten techniques occasionally finding renewed relevance. There's also discussion about the nature of "proof" in computer science and the role of empirical testing versus formal verification in validating such claims.
The blog post explores the exceptional Jordan algebra, a 27-dimensional non-associative algebra denoted 𝔥₃(𝕆), built from 3x3 Hermitian matrices with octonion entries. It highlights the unique and intricate structure of this algebra, focusing on the Freudenthal product, a key operation related to the determinant. The post then connects 𝔥₃(𝕆) to exceptional Lie groups, particularly F₄, the automorphism group of the algebra, demonstrating how transformations preserving the algebra's structure generate this group. Finally, it touches upon the connection to E₆, a larger exceptional Lie group related to the algebra's derivations and the structure of its projective space. The post aims to provide an accessible, though necessarily incomplete, introduction to this complex mathematical object and its significance in Lie theory.
The Hacker News comments discuss the accessibility of the blog post about the exceptional Jordan algebra, with several users praising its clarity and the author's ability to explain complex mathematics in an understandable way, even for those without advanced mathematical backgrounds. Some commenters delve into the specific mathematical concepts, including octonions, sedenions, and their connection to quantum mechanics and string theory. One commenter highlights the historical context of the algebra's discovery and its surprising connection to projective geometry. Others express general appreciation for the beauty and elegance of the mathematics involved and the author's skill in exposition. A few commenters mention the author's other work and express interest in exploring further.
A Brown University undergraduate, Noah Solomon, disproved a long-standing conjecture in data science known as the "conjecture of Kahan." This conjecture, which had puzzled researchers for 40 years, stated that certain algorithms used for floating-point computations could only produce a limited number of outputs. Solomon developed a novel geometric approach to the problem, discovering a counterexample that demonstrates these algorithms can actually produce infinitely many outputs under specific conditions. His work has significant implications for numerical analysis and computer science, as it clarifies the behavior of these fundamental algorithms and opens new avenues for research into improving their accuracy and reliability.
Hacker News commenters generally expressed excitement and praise for the undergraduate student's achievement. Several questioned the "40-year-old conjecture" framing, pointing out that the problem, while known, wasn't a major focus of active research. Some highlighted the importance of the mentor's role and the collaborative nature of research. Others delved into the technical details, discussing the specific implications of the findings for dimensionality reduction techniques like PCA and the difference between theoretical and practical significance in this context. A few commenters also noted the unusual amount of media attention for this type of result, speculating about the reasons behind it. A recurring theme was the refreshing nature of seeing an undergraduate making such a contribution.
This post presents a simplified, self-contained proof of a key lemma used in proving the Fundamental Theorem of Galois Theory. This lemma establishes a bijection between intermediate fields of a Galois extension and subgroups of its Galois group. The core idea involves demonstrating that for a finite Galois extension K/F and an intermediate field E, the fixed field of the automorphism group fixing E (denoted as Inv(Gal(K/E)) is E itself. The proof leverages the linear independence of field automorphisms and constructs a polynomial whose roots distinguish elements within and outside of E, thereby connecting the field structure to the group structure. This direct approach avoids more complex machinery sometimes used in other proofs, making the fundamental theorem's core connection more accessible.
Hacker News users discuss the linked blog post explaining a lemma used in the proof of the Fundamental Theorem of Galois Theory. Several commenters appreciate the clear explanation of a complex topic, with one pointing out how helpful the visualization and step-by-step breakdown of the proof is. Another commenter highlights the author's effective use of simple examples to illustrate the core concepts. Some discussion revolves around different approaches to teaching and understanding Galois theory, including alternative proofs and the role of intuition versus rigor. One user mentions the value of seeing multiple perspectives on the same concept to solidify understanding. The overall sentiment is positive, praising the author's pedagogical approach to demystifying a challenging area of mathematics.
This blog post explores the geometric relationship between the observer, the sun, and the horizon during sunset. It explains how the perceived "flattening" of the sun near the horizon is an optical illusion, and that the sun maintains its circular shape throughout its descent. The post utilizes basic geometry and trigonometry to demonstrate that the sun's lower edge touches the horizon before its upper edge, creating the illusion of a faster setting speed for the bottom half. This effect is independent of atmospheric refraction and is solely due to the relative positions of the observer, sun, and the tangential horizon line.
HN users discuss the geometric explanation of why sunsets appear elliptical. Several commenters express appreciation for the clear and intuitive explanation provided by the article, with some sharing personal anecdotes about observing this phenomenon. A few question the assumption of a perfectly spherical sun, noting that atmospheric refraction and the sun's actual shape could influence the observed ellipticity. Others delve into the mathematical details, discussing projections, conic sections, and the role of perspective. The practicality of using this knowledge for estimating the sun's distance or diameter is also debated, with some suggesting alternative methods like timing sunset duration.
Mathematicians have finally proven the Kakeya conjecture, a century-old problem concerning the smallest area required to rotate a unit line segment 180 degrees in a plane. The collaborative work, spearheaded by Nets Katz and Joshua Zahl, builds upon previous partial solutions and introduces a novel geometric argument. While their proof technically addresses the finite field version of the conjecture, it's considered a significant breakthrough with strong implications for the original Euclidean plane problem. The techniques developed for this proof are anticipated to have far-reaching consequences across various mathematical fields, including harmonic analysis and additive combinatorics.
HN commenters generally express excitement and appreciation for the breakthrough proof of the Kakeya conjecture, with several noting its accessibility even to non-mathematicians. Some discuss the implications of the proof and its reliance on additive combinatorics, a relatively new field. A few commenters delve into the history of the problem and the contributions of various mathematicians. The top comment highlights the fascinating connection between the conjecture and seemingly disparate areas like harmonic analysis and extractors for randomness. Others discuss the "once-in-a-century" claim, questioning its accuracy while acknowledging the significance of the achievement. A recurring theme is the beauty and elegance of the proof, reflecting a shared sense of awe at the power of mathematical reasoning.
The blog post "The Lost Art of Logarithms" argues that logarithms are underappreciated and underutilized in modern mathematics education and programming. While often taught purely as the inverse of exponentiation, logarithms possess unique properties that make them powerful tools for simplifying complex calculations, particularly those involving multiplication, division, powers, and roots. The author emphasizes their practical applications in diverse fields like finance, music theory, and computer science, citing examples such as calculating compound interest and understanding musical intervals. The post advocates for a shift in how logarithms are taught, focusing on their intuitive understanding and practical uses rather than rote memorization of formulas and identities. Ultimately, the author believes that rediscovering the "lost art" of logarithms can unlock a deeper understanding of mathematical relationships and enhance problem-solving skills.
Hacker News users generally praised the article for its clear explanation of logarithms and their usefulness, particularly in understanding scaling and exponential growth. Several commenters shared personal anecdotes about how a proper grasp of logarithms helped them in their careers, especially in software engineering and data science. Some pointed out the connection between logarithms and music theory, while others discussed the historical context and the importance of slide rules. A few users wished they had encountered such a clear explanation earlier in their education, highlighting the potential of the article as a valuable learning resource. One commenter offered a practical tip for remembering the relationship between logs and exponents. There was also a short thread discussing the practical applications of logarithms in machine learning and information theory.
Daniel Chase Hooper created a Sudoku variant called "Cracked Sudoku" where all 81 cells have unique shapes, eliminating the need for row and column lines. The puzzle maintains the standard Sudoku rules, requiring digits 1-9 to appear only once in each traditional row, column, and 3x3 block. Hooper generated these puzzles algorithmically, starting with a solved grid and then fracturing it into unique, interlocking pieces like a jigsaw puzzle. This introduces an added layer of visual complexity, making the puzzle more challenging by obfuscating the traditional grid structure and relying solely on the shapes for positional clues.
HN commenters generally found the uniquely shaped Sudoku variant interesting and visually appealing. Several praised its elegance and the cleverness of its design. Some discussed the difficulty of the puzzle, wondering if the unique shapes made it easier or harder to solve, and speculating about solving techniques. A few commenters expressed skepticism about its solvability or uniqueness, while others linked to similar previous attempts at uniquely shaped Sudoku grids. One commenter pointed out the potential for this design to be adapted for colorblind individuals by using patterns instead of colors. There was also brief discussion about the possibility of generating such puzzles algorithmically.
The blog post "The Cultural Divide Between Mathematics and AI" explores the differing approaches to knowledge and validation between mathematicians and AI researchers. Mathematicians prioritize rigorous proofs and deductive reasoning, building upon established theorems and valuing elegance and simplicity. AI, conversely, focuses on empirical results and inductive reasoning, driven by performance on benchmarks and real-world applications, often prioritizing scale and complexity over theoretical guarantees. This divergence manifests in communication styles, publication venues, and even the perceived importance of explainability, creating a cultural gap that hinders potential collaboration and mutual understanding. Bridging this divide requires recognizing the strengths of both approaches, fostering interdisciplinary communication, and developing shared goals.
HN commenters largely agree with the author's premise of a cultural divide between mathematics and AI. Several highlighted the differing goals, with mathematics prioritizing provable theorems and elegant abstractions, while AI focuses on empirical performance and practical applications. Some pointed out that AI often uses mathematical tools without necessarily needing a deep theoretical understanding, leading to a "cargo cult" analogy. Others discussed the differing incentive structures, with academia rewarding theoretical contributions and industry favoring impactful results. A few comments pushed back, arguing that theoretical advancements in areas like optimization and statistics are driven by AI research. The lack of formal proofs in AI was a recurring theme, with some suggesting that this limits the field's long-term potential. Finally, the role of hype and marketing in AI, contrasting with the relative obscurity of pure mathematics, was also noted.
This paper provides a comprehensive overview of percolation theory, focusing on its mathematical aspects. It explores bond and site percolation on lattices, examining key concepts like critical probability, the existence of infinite clusters, and critical exponents characterizing the behavior near the phase transition. The text delves into various methods used to study percolation, including duality, renormalization group techniques, and series expansions. It also discusses different percolation models beyond regular lattices, like continuum percolation and directed percolation, highlighting their unique features and applications. Finally, the paper connects percolation theory to other areas like random graphs, interacting particle systems, and the study of disordered media, showcasing its broad relevance in statistical physics and mathematics.
HN commenters discuss the applications of percolation theory, mentioning its relevance to forest fires, disease spread, and network resilience. Some highlight the beauty and elegance of the theory itself, while others note its accessibility despite being a relatively advanced topic. A few users share personal experiences using percolation theory in their work, including modeling concrete porosity and analyzing social networks. The concept of universality in percolation, where different systems exhibit similar behavior near the critical threshold, is also pointed out. One commenter links to an interactive percolation simulation, allowing others to experiment with the concepts discussed. Finally, the historical context and development of percolation theory are briefly touched upon.
"Snapshots of Modern Mathematics from Oberwolfach" presents a collection of short, accessible articles showcasing current mathematical research. Each "snapshot" offers a glimpse into a specific area of active study, explaining key concepts and motivations in a way understandable to a broader audience with some mathematical background. The project aims to bridge the gap between cutting-edge research and the public's understanding of mathematics, illustrating its beauty, diversity, and relevance to the modern world through vivid examples and engaging narratives. The collection covers a broad spectrum of mathematical topics, demonstrating the interconnectedness of the field and the wide range of problems mathematicians tackle.
Hacker News users generally expressed appreciation for the Snapshots of Modern Mathematics resource, finding it well-written and accessible even to non-mathematicians. Some highlighted specific snapshots they found particularly interesting, like those on machine learning, knot theory, or the Riemann hypothesis. A few commenters pointed out the site's age (originally from 2014) and suggested it could benefit from updates, while others noted its enduring value despite this. The discussion also touched on the challenge of explaining complex mathematical concepts simply and praised the project's success in this regard. Several users expressed a desire to see similar resources for other scientific fields.
A new mathematical framework called "next-level chaos" moves beyond traditional chaos theory by incorporating the inherent uncertainty in our knowledge of a system's initial conditions. Traditional chaos focuses on how small initial uncertainties amplify over time, making long-term predictions impossible. Next-level chaos acknowledges that perfectly measuring initial conditions is fundamentally impossible and quantifies how this intrinsic uncertainty, even at minuscule levels, also contributes to unpredictable outcomes. This new approach provides a more realistic and rigorous way to assess the true limits of predictability in complex systems like weather patterns or financial markets, acknowledging the unavoidable limitations imposed by quantum mechanics and measurement precision.
Hacker News users discuss the implications of the Quanta article on "next-level" chaos. Several commenters express fascination with the concept of "intrinsic unpredictability" even within deterministic systems. Some highlight the difficulty of distinguishing true chaos from complex but ultimately predictable behavior, particularly in systems with limited observational data. The computational challenges of accurately modeling chaotic systems are also noted, along with the philosophical implications for free will and determinism. A few users mention practical applications, like weather forecasting, where improved understanding of chaos could lead to better predictive models, despite the inherent limits. One compelling comment points out the connection between this research and the limits of computability, suggesting the fundamental unknowability of certain systems' future states might be tied to Turing's halting problem.
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 project introduces lin-alg, a Rust library providing fundamental linear algebra structures and operations with a focus on performance. It offers core types like vectors and quaternions (with 2D, 3D, and 4D variants), alongside common operations such as addition, subtraction, scalar multiplication, dot and cross products, normalization, and quaternion-specific functionalities like rotations and spherical linear interpolation (slerp). The library aims to be simple, efficient, and dependency-free, suitable for graphics, game development, and other domains requiring linear algebra computations.
Hacker News users generally praised the Rust vector and quaternion library for its clear documentation, beginner-friendly approach, and focus on 2D and 3D graphics. Some questioned the practical application of quaternions in 2D, while others appreciated the inclusion for completeness and potential future use. The discussion touched on SIMD support (or lack thereof), with some users highlighting its importance for performance in graphical applications. There were also suggestions for additional features like dual quaternions and geometric algebra support, reflecting a desire for expanded functionality. Some compared the library favorably to existing solutions like glam and nalgebra, praising its simplicity and ease of understanding, particularly for learning purposes.
Anime fans inadvertently contributed to solving a long-standing math problem related to the "Kadison-Singer problem" while discussing the coloring of anime character hair. They were exploring ways to systematically categorize and label hair color palettes, which mathematically mirrored the complex problem of partitioning high-dimensional space. This led to mathematicians realizing the fans' approach, involving "Hadamard matrices," could be adapted to provide a more elegant and accessible proof for the Kadison-Singer problem, which has implications for various fields including quantum mechanics and signal processing.
Hacker News commenters generally expressed appreciation for the approachable explanation of Kazhdan's property (T) and the connection to expander graphs. Several pointed out that the anime fans didn't actually solve the problem, but rather discovered an interesting visual representation that spurred further mathematical investigation. Some debated the level of involvement of the anime community, arguing that the connection was primarily made by mathematicians familiar with anime, rather than the broader fanbase. Others discussed the surprising connections between seemingly disparate fields, highlighting the serendipitous nature of mathematical discovery. A few commenters also linked to additional resources, including the original paper and related mathematical concepts.
The "inspection paradox" describes the counterintuitive tendency for sampled observations of an interval-based process (like bus wait times or class sizes) to be systematically larger than the true average. This occurs because longer intervals are proportionally more likely to be sampled. The blog post demonstrates this effect across diverse examples, including bus schedules, web server requests, and class sizes, highlighting how seemingly simple averages can be misleading. It explains that the perceived average is actually the average experienced by an observer arriving at a random time, which is skewed toward longer intervals, and is distinct from the true average interval length. The post emphasizes the importance of understanding this paradox to correctly interpret data and avoid drawing flawed conclusions.
Hacker News users discuss various real-world examples and implications of the inspection paradox. Several commenters offer intuitive explanations, such as the bus frequency example, highlighting how our perception of waiting time is skewed by the longer intervals between buses. Others discuss the paradox's manifestation in project management (underestimating task completion times) and software engineering (debugging and performance analysis). The phenomenon's relevance to sampling bias and statistical analysis is also pointed out, with some suggesting strategies to mitigate its impact. Finally, the discussion extends to other related concepts like length-biased sampling and renewal theory, offering deeper insights into the mathematical underpinnings of the paradox.
This post introduces rotors as a practical alternative to quaternions and matrices for 3D rotations. It explains that rotors, like quaternions, represent rotations as a single action around an arbitrary axis, but offer a simpler, more intuitive geometric interpretation based on the concept of "geometric algebra." The author argues that rotors are easier to understand and implement, visually demonstrating their geometric meaning and providing clear code examples in Python. The post covers basic rotor operations like creating rotations from an axis and angle, composing rotations, and applying rotations to vectors, highlighting rotors' computational efficiency and stability.
Hacker News users discussed the practicality and intuitiveness of using rotors for 3D rotations. Some found the rotor approach more elegant and easier to grasp than quaternions, especially appreciating the clear geometric interpretation and connection to bivectors. Others questioned the claimed advantages, arguing that quaternions remain the superior choice for performance and established library support. The potential benefits of rotors in areas like interpolation and avoiding gimbal lock were acknowledged, but some commenters felt the article didn't fully demonstrate these advantages convincingly. A few requested more comparative benchmarks or examples showcasing rotors' practical superiority in specific scenarios. The lack of widespread adoption and existing tooling for rotors was also raised as a barrier to entry.
Roger Penrose argues that Gödel's incompleteness theorems demonstrate that human mathematical understanding transcends computation and therefore, strong AI, which posits that consciousness is computable, is fundamentally flawed. He asserts that humans can grasp the truth of Gödelian sentences (statements unprovable within a formal system yet demonstrably true outside of it), while a computer bound by algorithms within that system cannot. This, Penrose claims, illustrates a non-computable element in human consciousness, suggesting we understand truth through means beyond mere calculation.
Hacker News users discuss Penrose's argument against strong AI, with many expressing skepticism. Several commenters point out that Gödel's incompleteness theorems don't necessarily apply to the way AI systems operate, arguing that AI doesn't need to be consistent or complete in the same way as formal mathematical systems. Others suggest Penrose misinterprets or overextends Gödel's work. Some users find Penrose's ideas intriguing but remain unconvinced, while others find his arguments simply wrong. The concept of "understanding" is a key point of contention, with some arguing that current AI models only simulate understanding, while others believe that sophisticated simulation is indistinguishable from true understanding. A few commenters express appreciation for Penrose's thought-provoking perspective, even if they disagree with his conclusions.
The Hacker News post presents a betting game puzzle where you predict the sum of your neighbors' bets, with the closest guess winning. The challenge is to calculate this sum efficiently when dealing with a large number of players, each choosing a bet from 0 to 9. The author shares a clever algorithm that achieves this in linear time, utilizing a frequency array to avoid redundant calculations. This approach significantly improves performance compared to a naive quadratic solution, making the game scalable for a substantial number of participants.
Hacker News users discussed the efficiency and practicality of the presented algorithm for the betting game puzzle. Some questioned the "linear time" claim, pointing out the algorithm's reliance on a precomputed lookup table, the creation of which would not be linear. Others debated the best way to construct such a table efficiently. A few commenters suggested alternative approaches, including using Gray codes or focusing on bit manipulation tricks. There was also discussion about the problem's framing, with some arguing it's more of a dynamic programming exercise than a puzzle. Finally, some users explored variations of the puzzle, such as changing the allowed bet sizes or considering non-integer bets.
Summary of Comments ( 9 )
https://news.ycombinator.com/item?id=43538986
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.
The Hacker News post titled "AI/Math Puzzle" linking to an article about an unsolved math problem related to AI generated text has a moderate number of comments, sparking a discussion around the puzzle's difficulty, potential approaches, and the nature of the challenge itself.
Several commenters discuss the ambiguity of the problem, particularly focusing on the interpretation of "random" and its implications for solving the puzzle. One commenter suggests the problem is ill-defined because the concept of "random text generated by a large language model" lacks a precise mathematical definition. They argue that without specifying the underlying distribution of the LLM's output, the problem becomes intractable. This point is echoed by other users who highlight that the inherent complexity and evolving nature of LLMs make it challenging to establish a fixed probabilistic framework for analysis.
Another thread of discussion revolves around the computational feasibility of brute-force approaches. Some commenters suggest that the vast search space makes it impractical to solve the puzzle by simply enumerating all possible strings and checking if they satisfy the given conditions. One user proposes a more targeted approach by focusing on shorter strings, arguing that the probability of finding a solution increases with decreasing string length.
A few commenters also touch upon the philosophical implications of the puzzle, pondering the nature of randomness and its relationship to AI-generated text. One user raises the question of whether LLM output can be considered truly random, given its deterministic nature. Another commenter speculates about the potential connection between this problem and other areas of mathematics, such as Kolmogorov complexity.
Finally, some comments express skepticism about the puzzle's originality and significance. One commenter questions whether the problem is genuinely novel or simply a repackaged version of existing mathematical concepts. Another expresses doubt about the practical value of solving the puzzle, suggesting that it may be more of a recreational challenge than a significant scientific endeavor. Despite some negativity, several users express interest in the problem and share ideas for potential solutions, demonstrating the engaging nature of the puzzle.