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100 Years to Solve an Integral (2020)

Posted: 2025-04-20 03:16:42

The blog post details the surprisingly difficult journey to find the exact value of the integral of the secant cubed function, a problem that stumped mathematicians for a century. While the integral's relevance to the Mercator projection made an approximate solution necessary for navigation since the 1600s, the precise antiderivative remained elusive. The post highlights the unexpected complexity arising from seemingly simple trigonometric functions and celebrates how a collaborative effort using symbolic computation software finally cracked the problem in 2020, yielding a closed-form solution involving complex logarithms. This solution, though exact, is less practical for computation than previously known series approximations. The post ultimately underscores the persistent challenge of symbolic integration and the power of modern computational tools.

This blog post, titled "100 Years to Solve an Integral (2020)," delves into the fascinating history and eventual solution of a seemingly simple integral involving the secant function, specifically ∫sec(x) dx. The author begins by highlighting the integral's connection to the Mercator projection, a crucial map projection technique developed in the 16th century by Gerardus Mercator. This projection revolutionized navigation by allowing sailors to plot straight-line courses, known as rhumb lines, which corresponded to constant compass bearings. The mathematical underpinnings of this projection, however, involved understanding the distortion introduced when projecting a sphere onto a flat surface. This distortion, as the post explains, is directly related to the integral of the secant function.

The author then meticulously outlines the historical journey of solving this integral. Beginning with Edward Wright's numerical tables for the Mercator projection, which implicitly contained the solution, the post details how Henry Bond noticed a connection between these tables and the logarithmic tangent function in the late 17th century. This observation, though initially met with skepticism, was rigorously proven by James Gregory in 1668, marking a significant step towards a closed-form solution.

The post proceeds to elaborate on the actual derivation of the integral, showcasing two distinct methods. The first method, considered the "standard" approach found in modern textbooks, involves a clever multiplication by a form of one, specifically (sec(x) + tan(x))/(sec(x) + tan(x)). This manipulation, followed by a u-substitution, leads to the familiar logarithmic form of the integral: ln|sec(x) + tan(x)| + C. The author emphasizes the ingenuity required to identify this seemingly "out-of-the-blue" multiplication factor.

The second method presented is a more geometrically motivated approach attributed to Isaac Barrow. This method leverages the relationship between the secant function and the lengths of specific segments within a unit circle and hyperbola. Through a careful geometric construction and manipulation of ratios, Barrow elegantly demonstrates the connection between the integral and the hyperbolic areas, ultimately leading to the same logarithmic solution. This geometric approach, the author suggests, offers a more intuitive and visually appealing understanding of the integral's derivation.

Finally, the post concludes by reflecting on the historical context of this mathematical discovery. The author marvels at the fact that a seemingly simple integral, prevalent in introductory calculus courses today, took a century to solve, highlighting the evolution of mathematical thought and the collaborative nature of mathematical progress. The author also notes the profound connection between seemingly abstract mathematical concepts, like the integral of the secant function, and practical real-world applications, such as mapmaking and navigation.

Summary of Comments ( 42 )
https://news.ycombinator.com/item?id=43741273

HN users discuss the surprising difficulty of integrating the secant function, historically important for mapmaking. Several commenters highlight the multiple approaches to the solution, appreciating the blog post's clear explanation of the "circular reasoning" some methods employ. Some debate the true "difficulty" of the integral, pointing out that with knowledge of complex logarithms, it's straightforward. Others note the historical context, where these tools were not yet available, emphasizing the ingenuity of earlier mathematicians. A few commenters dive into specific techniques, like differentiating under the integral sign and the usefulness of Weierstrass substitutions. The overall sentiment is one of appreciation for the blog post shedding light on a seemingly simple yet historically significant integral.

The Hacker News post "100 Years to Solve an Integral (2020)" discussing the article about the secant integral has several comments exploring various aspects of the topic.

Some commenters delve into the historical context of the problem, highlighting the contributions of Isaac Newton and James Gregory in developing methods to calculate the integral, and emphasizing the long-standing interest in this mathematical puzzle. They discuss the importance of the integral in navigation and map projection, linking it to the Mercator projection used in nautical charts.

A few comments focus on the technical details of the solution, discussing different approaches to solving the integral, including the use of complex analysis and power series expansions. They also touch upon the computational aspects of evaluating the integral numerically, and the challenges involved in obtaining accurate results.

Several commenters express appreciation for the clear and accessible explanation provided in the linked article, praising the author's ability to break down a complex mathematical concept into understandable terms. They find the connection between seemingly abstract mathematics and practical applications like mapmaking particularly fascinating.

There's a discussion about the significance of the integral's solution, with some commenters emphasizing its historical importance and its contribution to our understanding of calculus. Others point out the practical implications of the integral in modern applications, including navigation and surveying.

A few commenters share their own experiences with the integral, recounting their attempts to solve it and the challenges they encountered. They also discuss the educational value of exploring such problems, highlighting the importance of mathematical curiosity and problem-solving skills.

Some commenters draw parallels between this integral and other mathematical puzzles, discussing the broader theme of unsolved problems in mathematics and the ongoing quest for solutions. They also touch upon the role of collaboration and communication in advancing mathematical knowledge.

Finally, some comments offer additional resources and references related to the integral and its applications, providing further avenues for exploration and learning. They also suggest alternative approaches to understanding the integral and its implications.

The Lost Art of Logarithms

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Posted: 2025-03-13 19:05:17

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.

The article, "The Lost Art of Logarithms," laments the decline in practical understanding and utilization of logarithms, particularly within the realm of software engineering. The author posits that while logarithms remain a crucial underpinning of computer science, their direct application in day-to-day programming tasks has diminished, leading to a gap in many developers' intuitive grasp of their power and utility. This "lost art," as the author terms it, is not about rote memorization of logarithmic identities or complex manipulations, but rather a nuanced appreciation for the logarithmic scale and its inherent relationship to phenomena like algorithm complexity and dynamic range.

The author argues that this decline stems from several factors. Firstly, the increasing abstraction layers within modern software development tools and libraries often obscure the underlying mathematical principles. Programmers can manipulate data structures and algorithms without needing to explicitly engage with the logarithmic relationships that govern their efficiency. Secondly, the emphasis in many computer science curricula has shifted towards higher-level concepts, potentially at the expense of a deep dive into fundamental mathematical tools like logarithms. Finally, the sheer power of modern computing hardware can mask inefficiencies that a keen understanding of logarithms would quickly expose. In simpler terms, brute force computation can sometimes compensate for a lack of algorithmic elegance rooted in logarithmic principles.

The article proceeds to illustrate the continued relevance of logarithms through a series of concrete examples. It elucidates how logarithms play a critical role in analyzing algorithm performance, particularly in Big O notation, where logarithmic time complexity represents a highly desirable characteristic. The author further demonstrates how logarithms are intrinsically linked to concepts like binary search, demonstrating their practical utility in optimizing search operations. The piece also touches upon the importance of logarithms in understanding and managing data with vastly varying magnitudes, highlighting their application in areas such as sound processing and computer graphics, where decibels and color representation often rely on logarithmic scales.

Ultimately, the author advocates for a renewed focus on the intuitive understanding of logarithms within the software engineering community. The article concludes with a call to action, urging developers to rediscover the elegance and power of this "lost art" and to integrate a deeper appreciation for logarithmic thinking into their daily practice. This, the author suggests, will lead to more efficient, elegant, and ultimately more effective software solutions.

Summary of Comments ( 166 )
https://news.ycombinator.com/item?id=43356314

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.

The Hacker News post titled "The Lost Art of Logarithms" generated several comments discussing the article's premise and related concepts. Here's a summary of the discussion:

Agreement with the core premise: Several commenters concurred with the author's point about the declining emphasis on logarithmic thinking in modern education and programming. They noted that while logarithms are still taught, their deeper significance and intuitive understanding often get lost. Some attributed this to the increasing abstraction layers in software development, where developers rarely need to grapple with low-level details involving logarithmic scales. One commenter lamented that even computer science graduates often lack a strong grasp of logarithms, despite their importance in various algorithms and data structures.
Logarithms in specific domains: Commenters highlighted areas where logarithmic thinking remains crucial. Examples included understanding complexity analysis (Big O notation), working with databases and indexing, comprehending information theory and entropy, and appreciating the logarithmic nature of human perception (e.g., sound intensity, brightness). One commenter specifically mentioned the importance of logarithms in music theory.
Calculators and the loss of intuition: The discussion touched upon how the readily available calculators might have contributed to the decline of mental estimation skills and a deeper intuitive understanding of logarithms. One commenter argued that while calculators are helpful tools, over-reliance on them can hinder the development of a "feel" for numbers and their relationships.
Historical perspective and slide rules: Some comments reflected on the historical significance of logarithms, particularly their use in slide rules before the advent of electronic calculators. This sparked a brief tangent discussing the elegance and tactile experience of using slide rules, and how they fostered a different kind of numerical literacy.
Teaching approaches and practical examples: Commenters suggested ways to improve the teaching of logarithms, emphasizing the need for more engaging and practical examples that resonate with students. They proposed scenarios involving exponential growth, compound interest, and the Richter scale to illustrate the power and relevance of logarithms.
Alternative explanations of logarithms: A couple of commenters offered their own concise explanations of logarithms, aiming to provide a clearer understanding of the concept. One commenter framed logarithms as a way to express large numbers compactly, while another described them as the inverse of exponentiation.
Beyond base 10 and base e: While the article and most comments focused on base 10 and base e logarithms, one commenter highlighted the importance of base 2 logarithms in computer science, especially in the context of binary representation and bit manipulation.

In summary, the Hacker News discussion largely agreed with the article's observation about the declining appreciation for logarithms. The comments explored the reasons behind this trend, highlighted the continued relevance of logarithms in various fields, and offered suggestions for improving the way logarithms are taught and understood. The thread offered a nuanced perspective on the topic, enriched by the diverse backgrounds and experiences of the commenters.

Mathematics in the 20th century, by Michael Atiyah [pdf] (2002)

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Posted: 2025-02-09 08:54:37

Michael Atiyah's "Mathematics in the 20th Century" provides a sweeping overview of the field's progress during that period, highlighting key trends and breakthroughs. He emphasizes the increasing abstraction and unification within mathematics, exemplified by the rise of algebraic topology and category theory. Atiyah discusses the burgeoning interplay between mathematics and physics, particularly quantum mechanics and general relativity, which spurred new mathematical developments. He also touches upon the growing influence of computers and the expansion of specialized areas, while noting the enduring importance of core subjects like analysis and geometry. The essay concludes with reflections on the evolving nature of mathematical research, acknowledging the challenges of specialization while expressing optimism for future discoveries driven by interdisciplinary connections and new perspectives.

Sir Michael Atiyah's 2002 lecture, "Mathematics in the 20th Century," provides a sweeping overview of the remarkable advancements and evolving nature of mathematics throughout the 1900s. Atiyah structures his analysis around three key themes: the rise of abstraction, the increasing interconnectedness within mathematics, and the expanding relationship between mathematics and the sciences, particularly physics.

He begins by elaborating on the growing trend of abstraction, highlighting how mathematical concepts shifted from concrete objects and problems to more general and abstract structures. This shift, he argues, was crucial for unifying diverse mathematical fields and fostering new discoveries. He illustrates this trend through examples like the development of abstract algebra, where the study of specific numerical systems evolved into the study of groups, rings, and fields, providing a common framework for analyzing seemingly disparate mathematical objects. He also discusses the increasing focus on axiomatic systems and the rise of set theory as a foundational language for mathematics.

Atiyah then explores the growing interconnectedness within the discipline. He portrays the 20th century as a period of significant unification, where previously distinct branches of mathematics began to converge and cross-fertilize. He emphasizes the role of concepts like topology and algebraic geometry in bridging the gap between algebra and geometry, leading to profound new insights in both fields. He specifically mentions the development of homological algebra as a prime example of this interdisciplinary trend, allowing algebraic tools to be applied to topological problems. Atiyah also highlights the importance of category theory in providing a unifying language for diverse mathematical structures and relationships.

Furthermore, Atiyah examines the evolving relationship between mathematics and the sciences, with a particular focus on physics. He notes that while the 19th century saw a strong interplay between mathematics and classical physics, the 20th century witnessed an even deeper integration with the advent of relativity and quantum mechanics. He elucidates how these groundbreaking physical theories spurred the development of new mathematical tools and concepts, such as Riemannian geometry and functional analysis, which in turn provided a more rigorous framework for theoretical physics. He also discusses how the rise of quantum field theory and string theory further cemented the relationship between mathematics and physics, leading to a continuous exchange of ideas and methods.

Throughout his lecture, Atiyah stresses the dynamic nature of mathematics, emphasizing its constant evolution and adaptation to new challenges and influences. He portrays the 20th century as a period of remarkable progress and transformation, culminating in a highly sophisticated and interconnected mathematical landscape. He concludes by expressing optimism for the future of mathematics, anticipating continued advancements and deeper integration with other scientific disciplines in the 21st century. He also emphasizes the importance of maintaining a balance between specialized research and the pursuit of unifying principles, acknowledging the tension between these two driving forces in the development of mathematics.

Summary of Comments ( 5 )
https://news.ycombinator.com/item?id=42989419

Hacker News commenters discuss Atiyah's lecture, praising its clarity, accessibility, and broad yet insightful overview of 20th-century mathematics. Several highlight the interesting connections Atiyah draws between seemingly disparate fields, particularly geometry and physics. Some commenters reminisce about Atiyah's lectures, describing him as a brilliant and engaging speaker. Others share anecdotes or additional resources related to the topics discussed, including links to other writings by Atiyah and recommendations for further reading. A few express disappointment that the lecture doesn't delve deeper into certain areas, but the overall sentiment is one of appreciation for Atiyah's insightful and inspiring presentation.

The Hacker News post titled "Mathematics in the 20th century, by Michael Atiyah [pdf] (2002)" contains several comments discussing Atiyah's lecture. Many commenters praise Atiyah's clarity and ability to synthesize complex ideas into an accessible overview. Several highlight his insights into the interplay between different mathematical fields and the cyclical nature of mathematical progress.

One commenter appreciates Atiyah's emphasis on the "trinity" of geometry, algebra, and analysis, and how these areas both diverge and converge throughout the 20th century. They find this framework helpful for understanding the overall trajectory of mathematical development.

Another comment focuses on Atiyah's discussion of the "grand unification" in mathematics, mirroring the search for unification in physics. This commenter also touches upon the contrast between the "classical" and "romantic" styles of mathematics, with Atiyah representing a more classical approach.

A different user picks up on the "romantic" versus "classical" theme, mentioning how Atiyah's lecture highlights the shift towards abstraction in 20th-century mathematics. This user also points out Atiyah's observation that physics became more abstract while mathematics became, in a sense, more concrete due to its increasing use of computation.

The thread also includes some discussion of specific mathematical concepts mentioned in the lecture, like K-theory and index theory. One commenter expresses particular interest in Atiyah's explanation of how index theory connects analysis, geometry, and topology.

A few commenters share personal anecdotes about attending Atiyah's lectures and express admiration for his communication skills. They describe his ability to convey complex ideas with enthusiasm and make them understandable to a broader audience.

Overall, the comments reflect a deep appreciation for Atiyah's lecture and its insightful perspective on the development of mathematics in the 20th century. Commenters value his ability to connect different branches of mathematics and present a coherent narrative of its progress, highlighting both the unifying trends and the contrasting styles within the field. They also acknowledge his remarkable talent for communicating complex mathematical ideas clearly and engagingly.

Weierstrass's Monster

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Posted: 2025-01-24 03:02:10

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.

In the realm of mathematical analysis, a field dedicated to the study of change and continuity, there exists a peculiar function known as the Weierstrass function, often dramatically dubbed "Weierstrass's Monster." This function, formulated by the eminent German mathematician Karl Weierstrass in 1872, presented a groundbreaking challenge to the conventional understanding of calculus, specifically the concept of differentiability. It possesses a truly remarkable property: while being undeniably continuous, meaning its graph can be drawn without lifting the pen, it is nowhere differentiable. This means that at no point on the curve can a tangent line be drawn, as the function experiences infinitely rapid oscillations at every scale.

Before Weierstrass's discovery, mathematicians generally believed that continuous functions were differentiable almost everywhere, meaning they possessed derivatives at all but a few exceptional points, such as sharp corners or cusps. Weierstrass's function shattered this assumption, demonstrating the existence of functions that are continuous everywhere yet differentiable nowhere. This revelation sent shockwaves through the mathematical community, prompting a re-evaluation of the foundations of calculus and a deeper exploration of the subtle relationship between continuity and differentiability.

The Weierstrass function is typically represented as an infinite sum of cosine functions with carefully chosen frequencies and amplitudes. These individual cosine waves, while smooth and differentiable on their own, combine in a way that produces the function's characteristic jaggedness and non-differentiability. This intricate construction demonstrates the power of infinite series to generate highly complex and counterintuitive mathematical objects.

The discovery of the Weierstrass function was a pivotal moment in the history of mathematics. It exposed limitations in the intuitive understanding of continuity and differentiability, highlighting the importance of rigorous mathematical proof. It also spurred the development of more precise definitions and a deeper understanding of the subtleties of real analysis. Furthermore, the Weierstrass function served as a catalyst for the development of new branches of mathematics, such as fractal geometry, which deals with complex shapes exhibiting self-similarity at different scales. The "monster" ultimately helped mathematicians tame the wild and intricate landscape of functions, leading to a richer and more nuanced understanding of the mathematical universe. While initially perceived as a pathological example, it ultimately enriched and expanded the field, demonstrating the beauty and sometimes surprising nature of mathematical exploration.

Summary of Comments ( 81 )
https://news.ycombinator.com/item?id=42810103

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.

The Hacker News post titled "Weierstrass's Monster," linking to a Quanta Magazine article about the Weierstrass function, has a moderate number of comments discussing various aspects of the function and its historical context.

Several commenters focus on the function's counterintuitive properties. One commenter highlights the disconnect between intuitive notions of smoothness and the mathematical definition of continuity, noting that the Weierstrass function is continuous everywhere but differentiable nowhere. This leads to a discussion about how mathematical definitions can sometimes defy our intuition based on real-world experience. Another user echoes this sentiment by explaining how the function subverts the expectation that a continuous curve should have a tangent at every point, pointing out the "infinite jaggedness" at all scales.

Another thread discusses the historical significance of the function. A commenter notes that the Weierstrass function challenged the prevailing assumptions of 19th-century mathematicians about the relationship between continuity and differentiability, opening up new avenues of research in analysis. This sparked a brief exchange about other mathematical "monsters" and their role in advancing mathematical understanding. Someone further elaborates on the historical impact, mentioning that the function's discovery contributed to a more rigorous and formal approach to mathematical analysis, moving away from purely intuitive arguments.

A few comments delve into the practical implications or applications of the function. While some acknowledge that direct real-world applications might be limited, one commenter suggests its potential relevance to modeling phenomena exhibiting fractal-like behavior, where roughness persists at arbitrarily small scales.

Some comments offer additional resources or related concepts. One user points to a 3Blue1Brown video that visually explains the concept of the Weierstrass function. Another commenter connects the function to Brownian motion, mentioning the similar nowhere-differentiable nature of Brownian paths.

Finally, some comments provide technical clarifications or corrections. One commenter points out a subtle distinction between nowhere differentiability and infinite jaggedness, emphasizing that infinite jaggedness is not a formally defined mathematical concept. Another user clarifies the historical context, pointing out that while Weierstrass's example is the most famous, he was not the first to discover such a function.

Stories with Tag history of mathematics

100 Years to Solve an Integral (2020)

Summary of Comments ( 42 ) https://news.ycombinator.com/item?id=43741273

The Lost Art of Logarithms

Summary of Comments ( 166 ) https://news.ycombinator.com/item?id=43356314

Mathematics in the 20th century, by Michael Atiyah [pdf] (2002)

Summary of Comments ( 5 ) https://news.ycombinator.com/item?id=42989419

Weierstrass's Monster

Summary of Comments ( 81 ) https://news.ycombinator.com/item?id=42810103

Summary of Comments ( 42 )
https://news.ycombinator.com/item?id=43741273

Summary of Comments ( 166 )
https://news.ycombinator.com/item?id=43356314

Summary of Comments ( 5 )
https://news.ycombinator.com/item?id=42989419

Summary of Comments ( 81 )
https://news.ycombinator.com/item?id=42810103