This blog post explores methods for proving false statements within formal systems like logic and mathematics. It focuses on proof by contradiction, where you assume the statement is true and then demonstrate that this assumption leads to a logical inconsistency, thereby proving the original statement false. The post uses the example of proving the irrationality of √2, illustrating how assuming its rationality (expressibility as a fraction) ultimately contradicts the fundamental theorem of arithmetic. It highlights the importance of clearly defining the terms and axioms of the system within which the proof operates.
Mathematicians are exploring the boundaries of provability using large language models (LLMs) and other automated theorem provers. While these tools can generate novel and valid proofs, they often rely on techniques too complex for human comprehension. This raises questions about the nature of mathematical truth and understanding. If a proof is too long or intricate for any human to verify, does it truly constitute "knowledge"? Researchers are investigating ways to make these computer-generated proofs more accessible and understandable, potentially revealing new insights into mathematical structures along the way. The ultimate goal is to find a balance between the power of automated proving and the human need for comprehensible explanations.
HN commenters discuss the implications of Gödel's incompleteness theorems and the article's exploration of concrete examples in Ramsey theory and Diophantine equations. Some debate the philosophical significance of undecidable statements, questioning whether they represent "true" mathematical facts or merely artifacts of formal systems. Others highlight the practical limitations of proof assistants and the ongoing search for more powerful automated theorem provers. The connection between computability and the physical universe is also raised, with some suggesting that undecidable statements could have physical implications, while others argue for a separation between abstract mathematics and the concrete world. Several commenters express appreciation for the article's clarity in explaining complex mathematical concepts to a lay audience.
This paper explores the potential of Large Language Models (LLMs) as tools for mathematicians. It examines how LLMs can assist with tasks like generating conjectures, finding proofs, simplifying expressions, and translating between mathematical formalisms. While acknowledging current limitations such as occasional inaccuracies and a lack of deep mathematical understanding, the authors demonstrate LLMs' usefulness in exploring mathematical ideas, automating tedious tasks, and providing educational support. They argue that future development focusing on formal reasoning and symbolic computation could significantly enhance LLMs' capabilities, ultimately leading to a more symbiotic relationship between mathematicians and AI. The paper also discusses the ethical implications of using LLMs in mathematics, including concerns about plagiarism and the potential displacement of human mathematicians.
Hacker News users discussed the potential for LLMs to assist mathematicians, but also expressed skepticism. Some commenters highlighted LLMs' current weaknesses in formal logic and rigorous proof construction, suggesting they're more useful for brainstorming or generating initial ideas than for producing finalized proofs. Others pointed out the importance of human intuition and creativity in mathematics, which LLMs currently lack. The discussion also touched upon the potential for LLMs to democratize access to mathematical knowledge and the possibility of future advancements enabling more sophisticated mathematical reasoning by AI. There was some debate about the specific examples provided in the paper, with some users questioning their significance. Overall, the sentiment was cautiously optimistic, acknowledging the potential but emphasizing the limitations of current LLMs in the field of mathematics.
This blog post explores using eqn, a pre-troff equation formatter, to generate MathML for static websites. The author details their process of converting eqn syntax to MathML using geqn and groff, emphasizing its simplicity and suitability for static site generation. They highlight eqn's ease of use compared to LaTeX and demonstrate the process with a practical example, showcasing the conversion steps and the resulting rendered MathML embedded in HTML. This approach allows for maintainable, human-readable equation source files while leveraging standard tools to produce accessible and widely compatible mathematical expressions on the web.
HN users largely praised the simplicity and elegance of using eqn for math rendering, particularly in contrast to MathJax or LaTeX. Several appreciated the author's demonstration of a straightforward, lightweight approach for static sites, avoiding JavaScript dependencies. Some discussed the limitations of eqn, such as lack of support for more complex equations and symbols, while others offered alternative tools like KaTeX and MathML. One user pointed out potential accessibility issues and suggested investigating MathML for better screen reader compatibility. The overall sentiment favored eqn for its ease of use in simple scenarios, but acknowledged the need for more robust solutions for complex mathematical content.
The Tensor Cookbook (2024) is a free online resource offering a practical, code-focused guide to tensor operations. It covers fundamental concepts like tensor creation, manipulation (reshaping, slicing, broadcasting), and common operations (addition, multiplication, contraction) using NumPy, TensorFlow, and PyTorch. The cookbook emphasizes clear explanations and executable code examples to help readers quickly grasp and apply tensor techniques in various contexts. It aims to serve as a quick reference for both beginners seeking a foundational understanding and experienced practitioners looking for concise reminders on specific operations across popular libraries.
Hacker News users generally praised the Tensor Cookbook for its clear explanations and practical examples, finding it a valuable resource for those learning tensor operations. Several commenters appreciated the focus on intuitive understanding rather than rigorous mathematical proofs, making it accessible to a wider audience. Some pointed out the cookbook's relevance to machine learning and its potential as a quick reference for common tensor manipulations. A few users suggested additional topics or improvements, such as including content on tensor decompositions or expanding the coverage of specific libraries like PyTorch and TensorFlow. One commenter highlighted the site's use of MathJax for rendering equations, appreciating the resulting clear and readable formulas. There's also discussion around the subtle differences in tensor terminology across various fields and the cookbook's attempt to address these nuances.
This post explores the connection between quaternions and spherical trigonometry. It demonstrates how quaternion multiplication elegantly encodes rotations in 3D space, and how this can be used to derive fundamental spherical trigonometric identities like the spherical law of cosines and the spherical law of sines. Specifically, by representing vertices of a spherical triangle as unit quaternions and using quaternion multiplication to describe the rotations between them, the post reveals a direct algebraic correspondence with the trigonometric relationships between the triangle's sides and angles. This approach offers a cleaner and more intuitive understanding of spherical trigonometry compared to traditional methods.
The Hacker News comments on Tao's post about quaternions and spherical trigonometry largely express appreciation for the clear explanation of a complex topic. Several commenters note the usefulness of quaternions in applications like computer graphics and robotics, particularly for their ability to represent rotations without gimbal lock. One commenter points out the historical context of Hamilton's discovery of quaternions, while another draws a parallel to using complex numbers for planar geometry. A few users discuss alternative approaches to representing rotations, such as rotation matrices and Clifford algebras, comparing their advantages and disadvantages to quaternions. Some express a desire to see Tao explore the connection between quaternions and spinors in a future post.
"Slicing the Fourth" explores the counterintuitive nature of higher-dimensional rotations. Focusing on the 4D case, the post visually demonstrates how rotating a 4D cube (a hypercube or tesseract) can produce unexpected 3D cross-sections, seemingly violating our intuition about how rotations work. By animating the rotation and showing slices at various angles, the author reveals that these seemingly paradoxical shapes, like nested cubes and octahedra, arise naturally from the higher-dimensional rotation and are consistent with the underlying geometry, even though they appear strange from our limited 3D perspective. The post ultimately aims to provide a more intuitive understanding of 4D rotations and their effects on lower-dimensional slices.
HN users largely praised the article for its clear explanations and visualizations of 4D geometry, particularly the interactive slicing tool. Several commenters discussed the challenges of visualizing higher dimensions and shared their own experiences and preferred methods for grasping such concepts. Some users pointed out the connection to quaternion rotations, while others suggested improvements to the interactive tool, such as adding controls for rotation. A few commenters also mentioned other resources and tools for exploring 4D geometry, including specific books and software. Some debate arose around terminology and the best way to analogize 4D to lower dimensions.
Maxima, a powerful computer algebra system (CAS), is now accessible directly in web browsers thanks to a project leveraging Embedded Common Lisp (ECL) compiled to WebAssembly (WasM). This allows users to perform complex symbolic computations, including algebra, calculus, and numerical analysis, without any local installation. The browser-based interface provides a REPL (read-eval-print loop) for interactive calculations and utilizes MathJax for displaying formatted mathematical expressions. This project makes Maxima's capabilities more readily available, eliminating the need for dedicated software or server-side setups.
Commenters on Hacker News express excitement about Maxima running in the browser via WASM and ECL. Several highlight the potential for educational uses and interactive symbolic computation in web environments. Some discuss the performance overhead of WASM and suggest improvements, like pre-compilation for faster startup. The ability to share computational documents easily and integrate with other web technologies is praised. A few users mention other similar projects, including one using ClojureScript and another involving a Python CAS in the browser. The general sentiment is positive, with commenters intrigued by the possibilities this opens up for accessibility and collaborative mathematical work. One commenter expresses interest in building symbolic computation directly into a browser rather than running it as a VM.
The "Taylorator" is a Python tool that efficiently generates Taylor series approximations of arbitrary Python functions. It leverages automatic differentiation to compute derivatives and symbolic manipulation with SymPy to construct the series representation. This allows for a faster and more versatile alternative to manually deriving Taylor expansions, especially for complex functions, and provides a symbolic representation that can be further manipulated or evaluated. The post demonstrates its capabilities with examples like approximating sine and a more intricate function involving exponentials and logarithms. It also highlights the trade-offs between accuracy and computational cost as the number of terms in the series increases.
Hacker News users discussed the Taylorator's practicality and limitations. Some questioned its usefulness beyond simple sine wave generation, highlighting the complexity of real-world signals and the difficulty of obtaining precise Taylor series coefficients. Others were concerned about the computational cost of evaluating high-order polynomials in real-time. However, several commenters appreciated the project's educational value, viewing it as a clever demonstration of Taylor series and a potential starting point for more sophisticated signal processing techniques. A few users suggested alternative approaches like wavetable synthesis, pointing out its computational efficiency and prevalence in music synthesis. Overall, the reception was mixed, with some intrigued by the concept while others remained skeptical of its practical applications.
This post explores the problem of uniformly sampling points within a disk and reveals why a naive approach using polar coordinates leads to a concentration of points near the center. The author demonstrates that while generating a random angle and a random radius seems correct, it produces a non-uniform distribution due to the varying area of concentric rings within the disk. The solution presented involves generating a random angle and a radius proportional to the square root of a random number between 0 and 1. This adjustment accounts for the increasing area at larger radii, resulting in a truly uniform distribution of sampled points across the disk. The post includes clear visualizations and mathematical justifications to illustrate the problem and the effectiveness of the corrected sampling method.
HN users discuss various aspects of uniformly sampling points within a disk. Several commenters point out the flaws in the naive sqrt(random()) approach, correctly identifying its tendency to cluster points towards the center. They offer alternative solutions, including the accepted approach of sampling an angle and radius separately, as well as using rejection sampling. One commenter explores generating points within a square and rejecting those outside the circle, questioning its efficiency compared to other methods. Another details the importance of this problem in ray tracing and game development. The discussion also delves into the mathematical underpinnings, with commenters explaining the need for the square root on the radius to achieve uniformity and the relationship to the area element in polar coordinates. The practicality and performance of different methods are a recurring theme, including comparisons to pre-calculated lookup tables.
This post explores Oliver Heaviside's crucial role in developing the theory of transmission lines. It details how Heaviside simplified Maxwell's equations, leading to the "telegrapher's equations" which describe voltage and current behavior along a transmission line. He introduced the concepts of inductance, capacitance, conductance, and resistance per unit length, enabling practical calculations for long-distance telegraph cables. Heaviside also championed the use of loading coils to compensate for signal distortion, significantly improving long-distance communication, despite initial resistance from prominent physicists like William Preece. The post highlights Heaviside's often-overlooked contributions and emphasizes his practical, results-oriented approach, contrasting it with the more theoretical perspectives of his contemporaries.
Hacker News users discuss Heaviside's contributions to transmission line theory and his difficult personality. Several commenters highlight his impressive ability to intuitively grasp complex concepts and perform calculations, despite lacking formal mathematical rigor. One notes Heaviside's development of operational calculus, which was later formalized by mathematicians. Others discuss his conflicts with the scientific establishment, attributed to his unconventional methods and abrasive personality. His insistence on using vectors and his operational calculus, initially viewed with skepticism, ultimately proved crucial for understanding electromagnetic phenomena. Some lament the lack of recognition Heaviside received during his lifetime. The discussion also touches upon his eccentric lifestyle and social isolation.
Desmos is a free online graphing calculator that supports a wide range of mathematical expressions, from simple lines and parabolas to complex inequalities and implicit equations. Its intuitive interface allows for easy manipulation and exploration of graphs, including animations and interactive parameters. Beyond basic calculations, Desmos offers features like lists, regressions, tables, and statistics, making it a powerful tool for visualizing and analyzing data. It's accessible on various devices and is designed for both educational and professional use.
Hacker News users praised Desmos's graphing calculator for its ease of use, clean interface, and powerful features, comparing it favorably to other options like GeoGebra. Several commenters highlighted its educational value, particularly for visualizing complex concepts and making math more engaging for students. Some discussed its use in specific fields like engineering and data visualization, while others shared personal anecdotes about using it for various projects. The fast rendering speed and accessibility were also mentioned as significant advantages. A few comments delved into technical aspects like the underlying technology and potential improvements, including the desire for offline functionality and more advanced features like 3D graphing.
This article dissects the structure of a formal mathematical proof, illustrating it with a simple example about even and odd numbers. It emphasizes the distinction between informal proofs aimed at human understanding and formal proofs designed for automated verification. Formal proofs meticulously lay out every logical step, referencing specific axioms and inference rules within a chosen formal system. This detailed approach, while tedious for humans, enables computer-assisted verification and eliminates ambiguity, ensuring absolute rigor. The article highlights the importance of choosing appropriate axioms and the role of proof assistants in constructing and checking these complex formal structures, ultimately increasing confidence in mathematical results.
HN commenters discuss the accessibility of formal proof systems, particularly referencing Lean. Some express excitement about the potential of formal proofs to revolutionize mathematics, while others are more skeptical, citing the steep learning curve and questioning the practical benefits for most mathematicians. Several commenters debate the role of intuition versus rigor in mathematical practice, with some arguing that formalization can enhance understanding and others suggesting it might stifle creativity. The feasibility of formalizing existing mathematical knowledge is also discussed, with varying opinions on the timescale and resources required for such a project. Some users highlight the potential of AI in assisting with formalization efforts, while others remain cautious about its current capabilities. The overall tone is one of cautious optimism, acknowledging the challenges but also recognizing the potential transformative power of formal proof systems.
A new algorithm for the "pancake sorting problem" — sorting a disordered stack by repeatedly flipping sections of it — has achieved near-optimal efficiency. While the minimal number of flips required to sort any stack remains unknown, the new algorithm, developed by researchers at MIT and other institutions, guarantees completion within 1.375 times the theoretical minimum. This represents a significant improvement over previous algorithms, edging closer to a perfect solution for a problem that has puzzled computer scientists for decades. The researchers employed a recursive strategy that breaks down large stacks into smaller, more manageable substacks, optimizing the flipping process and setting a new benchmark for pancake sorting efficiency.
Hacker News users discussed the practicality and significance of the new book-sorting algorithm. Some questioned the real-world applicability given the specialized constraints, like pre-sorted sections and a single robot arm. Others debated the definition of "perfection" in sorting, pointing out that minimizing the arm's travel distance might not be the only relevant metric. The algorithm's novelty and mathematical elegance were acknowledged, but skepticism remained about its potential impact beyond theoretical computer science. Several commenters highlighted the existing highly optimized solutions for real-world sorting problems and suggested that this new algorithm is more of an interesting theoretical exercise than a practical breakthrough. There was also discussion about the difference between this algorithm and existing techniques like Timsort, with some arguing the new algorithm addresses a distinctly different problem.
Karl Weierstrass’s function revolutionized mathematics by demonstrating a curve that is continuous everywhere but differentiable nowhere. This “monster” function, built from an infinite sum of cosine waves with increasingly higher frequencies and smaller amplitudes, visually appears jagged and chaotic at every scale. Its existence challenged the prevailing assumption that continuous functions were mostly smooth, with only isolated points of non-differentiability. Weierstrass's discovery exposed a deep rift between intuition and mathematical rigor, ushering in a new era of analysis focused on precise definitions and rigorous proofs, impacting fields from calculus to fractal geometry.
HN users generally express fascination with the Weierstrass function and its implications for calculus. Several comments dive into the history and significance of the function, appreciating its role in challenging intuitive notions of continuity and differentiability. Some discuss its relation to fractals and Brownian motion, while others highlight the beauty of mathematical discoveries that defy expectations. A few commenters provide additional resources, including links to visualizations and related mathematical concepts like space-filling curves. Some debate the accessibility of the original Quanta article, suggesting ways it could be more easily understood by a broader audience. A recurring theme is the wonder inspired by such counterintuitive mathematical objects.
This blog post explores creating spirograph-like patterns by simulating gravitational orbits of multiple bodies. Instead of gears, the author uses Newton's law of universal gravitation and numerical integration to calculate the paths of planets orbiting one or more stars. The resulting intricate designs are visualized, and the post delves into the math and code behind the simulation, covering topics such as velocity Verlet integration and adaptive time steps to handle close encounters between bodies. Ultimately, the author demonstrates how varying the initial conditions of the system, like the number of stars, their masses, and the planets' starting velocities, leads to a diverse range of mesmerizing orbital patterns.
HN users generally praised the Orbit Spirograph visualization and the clear explanations provided by Red Blob Games. Several commenters explored the mathematical underpinnings, discussing epitrochoids and hypotrochoids, and how the visualization relates to planetary motion. Some users shared related resources like a JavaScript implementation and a Geogebra applet for exploring similar patterns. The potential educational value of the interactive tool was also highlighted, with one commenter suggesting its use in explaining retrograde motion. A few commenters reminisced about physical spirograph toys, and one pointed out the connection to Lissajous curves.
The post explores how the seemingly simple problem of calculating the equivalent capacitance of an infinite ladder network of capacitors can be elegantly solved using the concept of geometric series. By recognizing the self-similar nature of the circuit as sections are added, the problem is reduced to a quadratic equation where the equivalent capacitance of the infinite network is expressed in terms of the individual capacitances. This demonstrates a practical application of mathematical concepts to circuit analysis, highlighting the interconnectedness between seemingly disparate fields.
HN commenters generally praised the article for its clear explanation of how capacitors work, particularly its use of the geometric series analogy to explain charging and discharging. Some appreciated the interactive diagrams, while others suggested minor improvements like adding a discussion of dielectric materials and their impact on capacitance. One commenter pointed out a potential simplification in the derivation by using the formula for the sum of a geometric series directly. Another highlighted the importance of understanding the underlying physics rather than just memorizing formulas, praising the article for facilitating this understanding. A few users also shared related resources and alternative explanations of capacitor behavior.
SudokuVariants.com lets you play and create a wide variety of Sudoku puzzles beyond the classic 9x9 grid. The website offers different grid sizes, shapes, and rule sets, including variations like Killer Sudoku, Irregular Sudoku, and even custom rule combinations. Users can experiment with existing variants or design their own unique Sudoku challenges using a visual editor, and then share their creations with others via a generated link. The site aims to provide a comprehensive platform for both playing and exploring the vast possibilities within the Sudoku puzzle format.
Hacker News users generally expressed interest in the SudokuVariants website. Several praised its clean design and the variety of puzzles offered. Some found the "construct your own variant" feature particularly appealing, and one user suggested adding a difficulty rating system for user-created puzzles. A few commenters mentioned specific variant recommendations, including "Killer Sudoku" and a variant with prime number constraints. There was also a brief discussion about the underlying logic and algorithms involved in generating and solving these puzzles. One user pointed out that some extreme variants might be NP-complete, implying significant computational challenges for larger grids or complex rules.
The blog post explores the origin of seemingly arbitrary divisibility problems often encountered in undergraduate mathematics courses. It argues that these problems aren't typically plucked from thin air, but rather stem from broader mathematical concepts, particularly abstract algebra. The post uses the example of proving divisibility by 7 using a specific algorithm to illustrate how such problems can be derived from exploring properties of polynomial rings and quotient rings. Essentially, the apparently random divisibility rule is a consequence of working within a modular arithmetic system, which connects to deeper algebraic structures. The post aims to demystify these types of problems and show how they offer a glimpse into richer mathematical ideas.
The Hacker News comments discuss the origin and nature of "divisibility trick" problems often encountered in introductory number theory or math competitions. Several commenters point out that these problems often stem from exploring properties within modular arithmetic, even if not explicitly framed that way. Some suggest the problems are valuable for developing intuition about number systems and problem-solving skills. However, others argue that they can feel contrived or "magical," lacking connection to broader mathematical concepts. The idea of "casting out nines" is mentioned as a specific example, with some commenters highlighting its historical significance for checking calculations, while others dismiss it as a niche trick. A few commenters express a general appreciation for the linked blog post, praising its clarity and exploration of the topic.
Certain prime numbers possess aesthetically pleasing or curious properties that make them stand out and become targets for "prime hunters." These include palindromic primes (reading the same forwards and backwards), repunit primes (consisting only of the digit 1), and Mersenne primes (one less than a power of two). The rarity and mathematical beauty of these special primes drive both amateur and professional mathematicians to seek them out using sophisticated algorithms and distributed computing projects, pushing the boundaries of computational power and our understanding of prime number distribution.
HN commenters largely discussed the memorability and aesthetics of the listed prime numbers, debating whether the criteria truly made them special or just reflected pattern-seeking tendencies. Some questioned the article's focus on base 10 representation, arguing that memorability is subjective and base-dependent. Others appreciated the exploration of mathematical beauty and shared their own favorite "interesting" numbers. Several commenters noted the connection to Smarandache sequences and other recreational math concepts, with links provided for further exploration. The practicality of searching for such primes was also questioned, with some suggesting it was merely a curiosity with no real-world application.
This post explores the Hilbert curve, a continuous fractal space-filling curve. The author visualizes its construction through iterative rotations and connections of smaller, U-shaped segments, demonstrating how this process generates increasingly complex patterns that effectively fill a square grid. The post further examines how points in 2D space can be mapped to a 1D position along the curve and vice-versa, highlighting the curve's applications in image processing and data organization by providing Python code examples for these conversions. The intricate visuals and detailed explanations offer a compelling portrait of the Hilbert curve's properties and practical utility.
Hacker News users generally praised the visualization and explanation of Hilbert curves in the linked blog post. Several appreciated the interactive nature and clear breakdown of the curve's construction. Some comments delved into practical applications, mentioning its use in mapping and image processing due to its space-filling properties and locality preservation. A few users pointed out its relevance to Morton codes (Z-order curves) and their applications in databases. One commenter linked to a Python implementation for generating Hilbert curves. The overall sentiment was positive, with users finding the post educational and well-presented.
This blog post presents a different way to derive Shannon entropy, focusing on its property as a unique measure of information content. Instead of starting with desired properties like additivity and then finding a formula that satisfies them, the author begins with a core idea: measuring the average number of binary questions needed to pinpoint a specific outcome from a probability distribution. By formalizing this concept using a binary tree representation of the questioning process and leveraging Kraft's inequality, they demonstrate that -∑pᵢlog₂(pᵢ) emerges naturally as the optimal average question length, thus establishing it as the entropy. This construction emphasizes the intuitive link between entropy and the efficient encoding of information.
Hacker News users discuss the alternative construction of Shannon entropy presented in the linked article. Some express appreciation for the clear explanation and visualizations, finding the geometric approach insightful and offering a fresh perspective on a familiar concept. Others debate the pedagogical value of the approach, questioning whether it truly simplifies understanding for those unfamiliar with entropy, or merely offers a different lens for those already versed in the subject. A few commenters note the connection to cross-entropy and Kullback-Leibler divergence, suggesting the geometric interpretation could be extended to these related concepts. There's also a brief discussion on the practical implications and potential applications of this alternative construction, although no concrete examples are provided. Overall, the comments reflect a mix of appreciation for the novel approach and a pragmatic assessment of its usefulness in teaching and application.
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https://news.ycombinator.com/item?id=42939312
Hacker News users discuss the potential misuse of zero-knowledge proofs (ZKPs), expressing concern that they could be used to convincingly lie or create fraudulent attestations. Some commenters highlight the importance of distinguishing between a ZKP verifying a computation versus verifying a real-world fact. They argue that while ZKPs can prove the correct execution of a program on given inputs, they cannot inherently prove the veracity of those inputs. Others discuss the "garbage in, garbage out" principle in this context, suggesting the need for robust, real-world verification methods alongside ZKPs to prevent their misuse. The trustworthiness of the prover remains crucial, and ZKPs alone cannot bridge the gap between computation and reality. A few comments also touch upon the complexity of understanding and implementing ZKPs correctly, potentially leading to vulnerabilities.
The Hacker News post titled "How to prove false statements? (Part 1)" linking to a blog post about cryptography and zero-knowledge proofs generated several comments discussing the technical details and implications of the presented concepts.
One commenter highlights the importance of distinguishing between provably false statements within a formal system and statements that are false in the "real world". They emphasize that a formal system can only work with what's defined within its axioms and rules, and thus "false" refers to inconsistency within that system, not necessarily a reflection of external reality. This commenter also points out the challenge of bridging the gap between a formal system and the real world, especially when dealing with real-world data and measurements that might be inherently imprecise or subject to error.
Another commenter delves into the specifics of zero-knowledge proofs, particularly the concept of a "simulation trapdoor". They explain how this trapdoor allows a simulator to create convincing "proofs" even for false statements, which is crucial for demonstrating the soundness of the zero-knowledge system. This comment also mentions the use of non-interactive zero-knowledge proofs and how they enhance the efficiency and practicality of the system.
Several commenters discuss the practical applications and limitations of zero-knowledge proofs. One user raises the issue of computational complexity and the potential for proof generation or verification to be computationally expensive. Another commenter mentions the importance of trusting the setup phase of the zero-knowledge system, as a compromised setup could undermine the entire system's security.
The topic of using zero-knowledge proofs for authentication and authorization also receives attention. One commenter points out the benefits of using these proofs to selectively disclose information without revealing unnecessary details, enhancing privacy and security. However, another commenter counters that this approach relies on having agreed-upon facts in the first place, which might be challenging to establish in certain scenarios.
Finally, there's a brief discussion on the relation between these cryptographic concepts and philosophical ideas about truth and provability, with one commenter drawing parallels to Gödel's incompleteness theorems.
Overall, the comments on the Hacker News post delve into the technical intricacies of zero-knowledge proofs, their practical implications, and even their philosophical connections. They provide valuable insights and perspectives beyond the original blog post, highlighting both the potential and the limitations of this fascinating area of cryptography.