Stories with Tag history of mathematics

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

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

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  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.

  • mathematics
  • 20th Century Mathematics
  • history of mathematics
  • Michael Atiyah
  • Mathematical Fields
  • Mathematical Development
  • Mathematics PDF
  • Academic Paper
  • Math History

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

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  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

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  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.

  • mathematics
  • calculus
  • analysis
  • Weierstrass function
  • continuous function
  • nowhere differentiable
  • fractal
  • history of mathematics
  • Karl Weierstrass
  • Infinite Series
  • mathematical analysis

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

  Verbose tl;dr

  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.