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.
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.
The blog post explores the interconnectedness of various measurement systems and mathematical concepts, examining potential historical links that are likely coincidental. The author notes the near equivalence of a meter to a royal cubit times the golden ratio, and how this relates to the dimensions of the Great Pyramid of Giza. While acknowledging the established historical definition of the meter based on Earth's circumference, the post speculates on whether ancient Egyptians might have possessed a sophisticated understanding of these relationships, potentially incorporating the golden ratio and Earth's dimensions into their construction. However, the author ultimately concludes that the observed connections are likely due to mathematical happenstance rather than deliberate design.
HN commenters largely dismiss the linked article as numerology and pseudoscience. Several point out the arbitrary nature of choosing specific measurements and units (meters, cubits) to force connections. One commenter notes that the golden ratio shows up frequently in geometric constructions, making its presence in the pyramids unsurprising and not necessarily indicative of intentional design. Others criticize the article's lack of rigor and its reliance on coincidences rather than evidence-based arguments. The general consensus is that the article presents a flawed and unconvincing argument for a relationship between these different elements.
This interactive visualization explains Markov chains by demonstrating how a system transitions between different states over time based on predefined probabilities. It illustrates that future states depend solely on the current state, not the historical sequence of states (the Markov property). The visualization uses simple examples like a frog hopping between lily pads and the changing weather to show how transition probabilities determine the long-term behavior of the system, including the likelihood of being in each state after many steps (the stationary distribution). It allows users to manipulate the probabilities and observe the resulting changes in the system's evolution, providing an intuitive understanding of Markov chains and their properties.
HN users largely praised the visual clarity and helpfulness of the linked explanation of Markov Chains. Several pointed out its educational value, both for introducing the concept and for refreshing prior knowledge. Some commenters discussed practical applications, including text generation, Google's PageRank algorithm, and modeling physical systems. One user highlighted the importance of understanding the difference between "Markov" and "Hidden Markov" models. A few users offered minor critiques, suggesting the inclusion of absorbing states and more complex examples. Others shared additional resources, such as interactive demos and alternative explanations.
Terry Tao's blog post discusses the recent proof of the three-dimensional Kakeya conjecture by Hong Wang and Joshua Zahl. The conjecture states that any subset of three-dimensional space containing a unit line segment in every direction must have Hausdorff dimension three. While previous work, including Tao's own, established lower bounds approaching three, Wang and Zahl definitively settled the conjecture. Their proof utilizes a refined multiscale analysis of the Kakeya set and leverages polynomial partitioning techniques, building upon earlier advances in incidence geometry. The post highlights the key ideas of the proof, emphasizing the clever combination of existing tools and innovative new arguments, while also acknowledging the remaining open questions in higher dimensions.
HN commenters discuss the implications of the recent proof of the three-dimensional Kakeya conjecture, praising its elegance and accessibility even to non-experts. Several highlight the significance of "polynomial partitioning," the technique central to the proof, and its potential applications in other areas of mathematics. Some express excitement about the possibility of tackling higher dimensions, while others acknowledge the significant jump in complexity this would entail. The clear exposition of the proof by Tao is also commended, making the complex subject matter understandable to a broader audience. The connection to the original Kakeya needle problem and its surprising implications for analysis are also noted.
This post explores the complexities of representing 3D rotations, contrasting quaternions with other methods like rotation matrices and Euler angles. It highlights the issues of gimbal lock and interpolation difficulties inherent in Euler angles, and the computational cost of rotation matrices. Quaternions, while less intuitive, offer a more elegant and efficient solution. The post breaks down the math behind quaternions, explaining how they represent rotations as points on a 4D hypersphere, and demonstrates their advantages for smooth interpolation and avoiding gimbal lock. It emphasizes the practical benefits of quaternions in computer graphics and other applications requiring 3D manipulation.
HN users generally praised the article for its clear explanation of quaternions and their application to 3D rotations. Several commenters appreciated the visual approach and interactive demos, finding them helpful for understanding the concepts. Some discussed alternative representations like rotation matrices and axis-angle, comparing their strengths and weaknesses to quaternions. A few users pointed out the connection to complex numbers and offered additional resources for further exploration. One commenter mentioned the practical uses of quaternions in game development and other fields. Overall, the discussion highlighted the importance of quaternions as a tool for representing and manipulating rotations in 3D space.
The blog post details a formal verification of the standard long division algorithm using the Dafny programming language and its built-in Hoare logic capabilities. It walks through the challenges of representing and reasoning about the algorithm within this formal system, including defining loop invariants and handling edge cases like division by zero. The core difficulty lies in proving that the quotient and remainder produced by the algorithm are indeed correct according to the mathematical definition of division. The author meticulously constructs the necessary pre- and post-conditions, and elaborates on the specific insights and techniques required to guide the verifier to a successful proof. Ultimately, the post demonstrates the power of formal methods to rigorously verify even relatively simple, yet subtly complex, algorithms.
Hacker News users discussed the application of Hoare logic to verify long division, with several expressing appreciation for the clear explanation and visualization of the algorithm. Some commenters debated the practical benefits of formal verification for such a well-established algorithm, questioning the likelihood of uncovering unknown bugs. Others highlighted the educational value of the exercise, emphasizing the importance of understanding foundational algorithms. A few users delved into the specifics of the chosen proof method and its implications. One commenter suggested exploring alternative verification approaches, while another pointed out the potential for applying similar techniques to other arithmetic operations.
The Simons Institute for the Theory of Computing at UC Berkeley has launched "Stone Soup AI," a year-long research program focused on collaborative, open, and decentralized development of foundation models. Inspired by the folktale, the project aims to build a large language model collectively, using contributions of data, compute, and expertise from diverse participants. This open-source approach intends to democratize access to powerful AI technology and foster greater transparency and community ownership, contrasting with the current trend of closed, proprietary models developed by large corporations. The program will involve workshops, collaborative coding sprints, and public releases of data and models, promoting open science and community-driven advancement in AI.
HN commenters discuss the "Stone Soup AI" concept, which involves prompting LLMs with incomplete information and relying on their ability to hallucinate missing details to produce a workable output. Some express skepticism about relying on hallucinations, preferring more deliberate methods like retrieval augmentation. Others see potential, especially for creative tasks where unexpected outputs are desirable. The discussion also touches on the inherent tendency of LLMs to confabulate and the need for careful evaluation of results. Several commenters draw parallels to existing techniques like prompt engineering and chain-of-thought prompting, suggesting "Stone Soup AI" might be a rebranding of familiar concepts. A compelling point raised is the potential for bias amplification if hallucinations consistently fill gaps with stereotypical or inaccurate information.
Terence Tao's blog post explores how "landscape functions," a mathematical tool from optimization and computer science, could improve energy efficiency in buildings. He explains how these functions can model the complex interplay of factors affecting energy consumption, such as appliance usage, weather conditions, and occupancy patterns. By finding the "minimum" of the landscape function, one can identify the most energy-efficient operating strategy for a given building. Tao suggests that while practical implementation presents challenges like data acquisition and model complexity, landscape functions offer a promising theoretical framework for bridging the "green gap" – the disparity between predicted and actual energy savings in buildings – and ultimately reducing electricity costs for consumers.
HN commenters are skeptical of the practicality of applying the landscape function to energy optimization. Several doubt the computational feasibility, pointing out the complexity and scale of the power grid. Others question the focus on mathematical optimization, suggesting that more fundamental issues like transmission capacity and storage are the real bottlenecks. Some express concerns about the idealized assumptions in the model, and the lack of consideration for real-world constraints. One commenter notes the difficulty of applying abstract mathematical tools to complex real-world systems, and another suggests exploring simpler, more robust approaches. There's a general sentiment that while the math is interesting, its impact on lowering electricity costs is likely minimal.
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https://news.ycombinator.com/item?id=43434503
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.