Timothy Gowers shows that GPT‑5.5 Pro can solve open additive‑number‑theory problems in minutes, prompting a deep analysis of how AI will reshape mathematical research, PhD training, publishing norms, and the need for new collaborative workflows such as DeepMind's AI Co‑Mathematician.
A 23‑year‑old amateur named Liam Price teamed up with ChatGPT to solve the 60‑year‑old Erdős primitive‑sets conjecture in just 80 minutes, using a novel “vibe mathing” approach that bypassed traditional number‑theoretic methods and surprised leading mathematicians including Terence Tao.
The article surveys recent observations from leading mathematicians who report that AI breakthroughs—ranging from solving most IMO problems in 2025 to accelerating research with systems like AlphaEvolve—signal a decisive turning point in how mathematics is explored, proved, and taught.
This article breaks down DeepSeek‑V4's six core capability categories—knowledge, reasoning, programming, math, long‑context, and agent—showing how each benchmark works, presenting concrete scores that place V4 first or second against leading models, and explaining the hidden efficiency gains that make V4 up to 13.7× cheaper to run.
GPT-5.5, co‑designed with Nvidia hardware, breaks the traditional scaling‑law trade‑off by delivering higher intelligence while keeping token latency similar, achieves over 20% faster token generation, outperforms competitors across coding, knowledge‑work, and math benchmarks, and even proves new Ramsey‑number results verified by Lean.
OpenAI’s GPT-5.5, co‑designed with Nvidia’s GB200/GB300 hardware, matches GPT‑5.4’s latency while delivering higher efficiency, beating Claude Opus 4.7 across coding, knowledge‑work and math benchmarks, and even autonomously optimizes its own inference infrastructure for a 20% speed gain.
In an interview, Fields Medalist Michael Freedman explains how his new paper shows that mathematical reasoning is fundamentally about compression, presents statistical analysis of the Lean mathlib library revealing astronomically large unfolded statements versus compact token representations, and argues that understanding this compression is key to future human‑AI collaboration in mathematics.
The 2026 Breakthrough Prize announced its New Horizons in Mathematics winners—Wang Hong, Yunqing Tang, Otis Chodosh and Vesselin Dimitrov—highlighting Wang's solution of the Kakeya conjecture, Tang and Dimitrov's breakthroughs on the unbounded denominators conjecture and a new irrationality result, and Chodosh's advances in differential geometry, with a total prize pool of $18.75 million.
The article examines the history and criteria of Chinese mathematical terminology, highlighting well‑translated terms like "微积分" and "向量", exposing misleading translations such as "数学归纳法", and offering guidelines for creating clear, concise, and unambiguous mathematical names.
In just an hour, Claude Opus 4.6 cracked a 30‑year‑old combinatorial problem posed by Donald Knuth, showcasing a leap from pattern‑recognition to symbolic logical reasoning and suggesting that AI may become a core driver of fundamental scientific discovery rather than merely a supporting tool.
The Riemann Hypothesis, a 1859 conjecture linking the zeros of the ζ(s) function to the exact distribution of prime numbers, remains the most celebrated unsolved problem in mathematics, influencing number theory, cryptography, physics, and countless downstream theorems.
This article explains how to implement three complex visual effects—Strange Attractor, Fibonacci Sphere, and Galaxy animations—in Flutter using only Dart code, covering the underlying differential equations, Euler integration, 3D‑to‑2D projection, rotation, perspective, performance optimizations, and solutions to common GPU tile‑artifact issues.
This article curates twelve classic textbooks that connect core mathematical concepts—linear algebra, calculus, probability, statistics, and optimization—with machine‑learning theory and practice, offering concise descriptions, author information, and direct links for readers seeking a solid mathematical foundation for AI.
This article explains the definitions, mathematical foundations, classic examples, solving methods, and practical differences between recurrence relations and recursion, illustrating how each approach works, their implementation details, and how they can be transformed into one another.
Renowned mathematician Terence Tao experiments with ChatGPT‑5 Pro on a bounded‑curvature sphere problem, revealing how AI excels at detailed calculations, offers concise proofs, yet struggles with strategic guidance and subtle geometric assumptions, highlighting both the promise and limits of AI in advanced mathematical research.
This article explores ten fundamental mathematical inequalities—AM‑GM, Jensen, Bernoulli, Chebyshev, Markov, Cauchy‑Schwarz, Triangle, Mean‑Inequality Chain, Information, and Hoeffding—explaining their formal statements and illustrating how each offers practical insights for personal growth, risk management, and decision‑making.
The article traces the rise and fall of the Khwarezmian empire, the life of mathematician Al‑Khwārizmī, his pioneering work in algebra, the origin of the term “algorithm”, and how his legacy shaped medieval Islamic scholarship and modern computing.
This article introduces mathematics as a thinking tool and explains its three fundamental structures—topology, algebra, and order—detailing their key properties, illustrative examples such as city road networks, social graphs, groups, rings, and natural number ordering, and showing how they model relationships in both abstract theory and real‑world contexts.
This article traces the historical transformation of mathematics—from ancient empirical methods through medieval axiomatization and the rise of calculus to modern structural and categorical approaches—highlighting the shift from numerical computation to a science of structures, with emphasis on geometry, group theory, and isomorphism.
Exploring how the evolution of mathematical thought—from ancient Egyptian land surveys to Newtonian mechanics—mirrors the core steps of problem‑driven modeling, this article shows how historical breakthroughs illustrate the problem‑method‑result chain, offering modern modelers a rich source of examples and insights.
This article dissects the claim "handsome people must have girlfriends" using formal logic, Bayesian probability, and fuzzy set theory, showing why the statement lacks necessity, how probability raises but does not guarantee the likelihood, and why real relationships are far more nuanced.
The article explores the mathematical definition of prime numbers, their cultural characterization as possessing a masculine, indestructible quality in Chinese tradition, explains their rigid structure, discusses the unpredictable distribution of primes, and connects these ideas to the profound Riemann Hypothesis that seeks a hidden order in prime patterns.
Discover how Polya’s classic four‑step problem‑solving framework—understand, plan, execute, review—combined with nine versatile strategies can transform tackling mathematics, engineering, and everyday logical challenges into a systematic, transferable thinking process, illustrated with concrete examples and visual guides.
This article examines the long‑standing philosophical question of whether mathematics qualifies as a science, discussing viewpoints from Gauss, Einstein, Popper, and modern scholars, and highlighting the evolving relationship between mathematics and the natural sciences.
This article outlines practical techniques for crafting mathematical proofs that are logically sound, easy to follow, and aesthetically pleasing, emphasizing clear planning, linear flow, careful symbol use, structured long proofs, and thoughtful conclusions to enhance both readability and rigor.
From a wartime interview where Feynman urged a novelist to learn calculus as the language of God, through Newton's plague‑ridden youth and his hidden use of infinitesimals, to the Leibniz‑Newton controversy and modern applications, this article traces how calculus became the indispensable tool of science and engineering.
Mathematics, often seen merely as a logical tool or language, is argued to be a profound cultural force that shapes human civilization, from its historical development and cross‑cultural transmission to its aesthetic beauty, creative processes, and the need for reflective, culturally‑aware education.
DeepMind's AlphaEvolve, a Gemini-powered coding agent, combines large language models with an evolutionary framework to automatically discover, optimize, and evolve advanced algorithms, achieving notable gains in data‑center scheduling, TPU design, matrix multiplication, and even solving longstanding mathematical problems.
Tao Zhe Xuan’s new YouTube channel demonstrates how an AI‑driven workflow using GitHub Copilot and the Lean proof assistant produced a 33‑minute “blind” proof of a Magma equation, highlighting the potential of semi‑automated theorem proving while preserving human readability.
During the May Day holiday, Fields Medalist Tao Zhuoxuan used ChatGPT to develop an open‑source Python tool that automatically verifies asymptotic estimates involving arbitrary positive parameters, shared the GitHub repository, and then promoted his AI community, paid courses, and related services.
This article shares a lecture for middle‑school math teachers on integrating AI into everyday teaching, presenting three AI roles, a four‑step teaching workflow, and a four‑element prompting model to boost efficiency, collaboration, and professional insight.
This article reviews five landmark mathematical proofs—from Euclid’s infinite primes to the computer‑assisted Four‑Color Theorem—highlighting their elegant logic, historical impact, and the enduring beauty that makes proof a central, inspiring pursuit in mathematics.
This article explores how mathematical thinking—through concrete puzzles, probability paradoxes, and the Fermi estimation method—helps us overcome intuitive traps, make rigorous calculations, and apply logical reasoning to everyday problems and complex estimations.
The article explains how feedback transforms mere repetition into meaningful iteration by comparing recursive sequences and linear functions, illustrating the concept with learning cycles and Kandinsky’s artistic process, and showing when feedback matters most for effective growth.
This guide outlines the essential foundations for becoming a data science professional, debunking common myths, highlighting three core skill areas—mathematics, machine learning, and programming—and detailing key concepts such as linear algebra, probability, calculus, and essential Python libraries.
This article explores how differential equations serve as the mathematical language that describes phenomena ranging from planetary orbits and cosmic expansion to atmospheric dynamics, cellular reactions, ecological interactions, electrical circuits, and financial markets, highlighting their universal role in modeling change.
Exploring the essential role of mathematics, this article traces its evolution from ancient survival tools to a universal language of rational thought, examines how its applications shape modern life and technology, highlights its limitations, and envisions its future as a bridge across disciplines.
In a December 2024 online event titled “o1 Reasoning and the Future of Mathematics,” UCLA professor Terence Tao, OpenAI senior vice president Mark Chen, and policy lead James Donovan discuss how advanced AI reasoning models could transform mathematical research, problem solving, collaboration, and education.
This article introduces WolframAlpha, the computational knowledge engine, outlines its core features—from advanced mathematics and scientific calculations to data analysis and natural-language queries—explains how to use it, compares it with large language models like ChatGPT, and discusses its strengths, limitations, and practical applications across research, education, and daily life.
π’s endless, non‑repeating decimal expansion behaves like a universal information reservoir: any finite digit sequence—birthdates, historic years, or arbitrary data—appears somewhere, and if the unproven normal‑number hypothesis holds, π could serve as an infinite random number generator for cryptography, art, and scientific modeling.
This article explores core mathematical thinking—from induction proofs and set theory to bijective functions, combinatorial counting, and Cantor’s proof of uncountable reals—illustrating how rigorous logic and abstract concepts provide clear solutions to complex problems across various contexts.
This article introduces Python's built‑in math module, explains its most commonly used functions—including basic arithmetic, trigonometry, logarithms, constants, and utility helpers—and provides clear code examples that demonstrate how to perform each operation.
This article explores how mathematics originated from humanity's need to simplify complex real‑world problems—starting with basic counting for food and trade, evolving through geometry and algebra, and becoming the language that underpins modern science and everyday decision‑making.
This article explains PHP's ceil() function, shows its syntax, and provides three code examples demonstrating how to round floating‑point numbers and integers upward, illustrating the returned smallest integer greater than or equal to the given value.
Based on a 1988 survey of 68 readers who scored 24 famous mathematical theorems, this article presents a ranked list with average scores and brief explanations, revealing which results are considered the most beautiful in mathematics.
The article uses the mathematical concept of Taylor series as a metaphor for human cognition, describing how initial perception, detailed analysis, and pattern recognition progressively refine our understanding of the world, while also reflecting on the historical origins of the series and its philosophical implications.
John von Neumann’s pioneering work—from the stored‑program computer architecture and the famous fly‑between‑trains paradox to his foundational contributions in game theory, quantum mechanics, and self‑replicating automata—illustrates how his mathematical brilliance drove breakthroughs that still underpin today’s computing, economics, and artificial life research.
This article surveys the various kinds of numbers—natural, integers, rational, irrational, real, complex, hypercomplex, vectors, matrices, and tensors—explaining their definitions, mathematical significance, and practical applications in fields such as computer science, engineering, physics, graphics, and machine learning.
The article invites readers to revive the mathematician, physicist, economist and other scholarly personas within themselves, showing how interdisciplinary concepts such as Voronoi diagrams, graph theory, knot theory, non‑Euclidean geometry, fuzzy logic and fractal dimensions can enrich everyday problem‑solving and inspire a love of mathematics.
This article explains how diffusion equation methods can be applied to digitally repair spotted paintings, covering the mathematical representation of images, the underlying heat‑transfer analogy, step‑by‑step inpainting procedures, and improvements such as total variation flow to preserve edges.
This article explains Jeff Tupper’s 2001 formula—a simple inequality that, by selecting an appropriate integer range, can reproduce its own bitmap and generate virtually any two‑dimensional picture, with examples and a handy online tool.
Fourier series, introduced by Joseph Fourier to solve the heat equation, uses sine and cosine functions to represent complex phenomena, enabling the decomposition of problems into simpler parts and finding applications across wave motion, physics, finance, and modern digital signal processing.
Discover the mysterious silver ratio—a lesser‑known counterpart to the golden ratio—by learning how dividing a line into three parts yields a value of about 2.414, and see how both fit into the broader family of metallic means that reveal endless mathematical beauty.
This article examines the definition of a square, explains why in Euclidean plane a square must have four equal sides and right angles, and then explores how curvature on spherical and hyperbolic surfaces allows shapes with equal edges and right angles to have three, two, or more sides, illustrating concepts of curvature and geodesics.
The article explains Liu Qian’s last magic performance by modeling the card shuffling as a periodic ABCD pattern, interpreting it through group theory, and ultimately solving the underlying Josephus problem with a binary‑based formula that predicts the surviving card.
This article explores the fundamental physics and mathematics behind game development, covering object motion models, scrolling techniques, collision detection, ray casting, screen transition effects, and provides Python code examples to illustrate how these concepts are implemented in practice.
Exploring the interplay of chaos and order, this article examines their mathematical foundations—from chaotic dynamics and the butterfly effect to geometric regularities—and extends the discussion to everyday life, showing how recognizing patterns amid uncertainty can bring balance, meaning, and stability.
Long before the West embraced negatives, Chinese mathematicians of the Han era employed symbolic notation and sophisticated methods—documented in the Nine Chapters on the Mathematical Art—to solve debts, land surveys, and resource allocations, linking the concept to yin‑yang philosophy and later spreading via the Silk Road to the world.
This essay explores why mathematics, while crucial in specialized fields, is not indispensable for daily activities, highlighting its quantitative nature, alternative expressions in art, personal relevance, and how recognizing its limited everyday role can reveal its true value.
The article explores Shing‑Tung Yau’s classification of mathematicians into three groups—those who create theory, those who uncover patterns from phenomena, and those who solve major problems—highlighting their distinct contributions to mathematical innovation, problem solving, and the pursuit of beauty.
This article examines whether the world is governed by chance or necessity, discussing quantum uncertainty, classical determinism, the dual role of knowledge, and how mathematics, probability, AI, and big‑data analytics together shape our ability to understand and forecast complex systems.
This article explores ten well‑known motivational sayings by expressing each with a mathematical formula—ranging from probability and integrals to logistic and exponential functions—offering a fresh, analytical perspective on how effort, success, knowledge, and happiness can be quantified.
This article explores five of the most iconic mathematical symbols—e, π, i, 0, and =—detailing their definitions, historical origins, and profound impact across calculus, physics, engineering, computer science, and beyond, illustrating how each symbol bridges abstract theory and real‑world applications.
This article explores how intensive mathematical thinking consumes significant brain energy—about 20% of total daily expenditure—and uses a simple metabolic model to show that prolonged mental work can contribute to a negative energy balance and potential weight loss.
This article explores how mathematics provides precise, universal language, science offers empirical investigation, and philosophy supplies foundational questioning, illustrating their interconnections and distinctions through examples like Euclidean geometry, Newton’s law of gravitation, and Plato’s theory of forms, and highlighting their shared pursuit of discovery.
The article distills nine key learning principles from Shing‑Tong Yau’s experiences, covering academic attitude, foundational skill building, interdisciplinary study, independent thinking, and scholarly communication, offering practical guidance for anyone aiming to excel in mathematics and related fields.
From prime numbers harmonizing musical tones to Fibonacci patterns in sunflowers, from map projections to insurance risk models, this article explores eleven real-world examples showing how mathematics underpins music, nature, engineering, security, economics, and everyday decision-making.
Explore how various mathematical equations—from parametric and polar forms to implicit and animated functions—can be used to generate beautiful heart-shaped graphics, illustrating the romantic symbolism of love through visualized curves and dynamic patterns.
This article explains the mathematical definition of convolution, walks through discrete and continuous examples, demonstrates its use in signal smoothing with moving averages, and surveys its wide-ranging applications in signal processing, communications, computer vision, seismology, medical imaging, and statistics.
Pick’s Theorem offers a surprisingly simple way to compute the area of any simple lattice polygon by counting its boundary and interior integer points, and this article explains the theorem’s statement, proof sketch, and a step‑by‑step rectangle example illustrating its practical use.
From Euler’s identity to Einstein’s mass‑energy equation, this article explores ten celebrated mathematical formulas—detailing their concise expressions, underlying concepts, and wide‑ranging applications across geometry, number theory, physics, and probability—illustrating why they’re hailed as the most beautiful in mathematics.
From its appearance as a rescue signal in a thriller film to its role as a triangular number, a sum of primes, and a culturally charged symbol, this article explores the many mathematical properties and cultural meanings of the number 666.
The article explains PHP's built‑in pi() function, shows how to retrieve the approximate value of π, demonstrates its use in simple output and in calculating a circle’s circumference and area with code examples, and notes the function’s limited precision.
This article reviews Microsoft’s extensive 155‑page work on early experiments with GPT‑4, exploring how the model approaches artificial general intelligence, its testing methodology, multimodal capabilities, programming and mathematical performance, interaction with tools and humans, limitations, societal impact, and future research directions.
In his Spark Award address, Huawei’s founder Ren Zhengfei outlined the company’s R&D investment, the role of mathematics and AI, strategies for university collaboration, talent cultivation, open platforms, and the balance between scientific curiosity and practical technology development.
A recent 154‑page Microsoft paper titled "Sparks of Artificial General Intelligence: Early Experiments with GPT‑4" argues that GPT‑4, despite being an early prototype, already exhibits many capabilities—multimodal reasoning, programming, mathematics, and human‑like interaction—suggesting it may be an early form of AGI, though experts highlight significant limitations and ongoing debates.
The article argues that while you can start programming without deep math, mastering mathematical concepts—from physics simulations to linear algebra in search algorithms and abstract algebra in functional programming—significantly boosts problem‑solving ability, code safety, and career growth for developers.
Derivatives, the core concept of calculus, originated from Newton’s instantaneous speed, Leibniz’s tangent, and Fermat’s extremum problems, and are defined as the limit of a function’s change ratio, linking continuity, geometry, and physics, while finding extensive applications across science, economics, and biology.
This article explores the classic catenary curve—its definition, historical development, and mathematical derivation—while examining whether it can model a kite’s string under wind, highlighting the assumptions of the model and suggesting more realistic approaches such as advanced derivations or simulation.
The article argues that while mathematics is essential for modeling, solving real-world problems requires a blend of disciplines—biology, computation, writing, and more—highlighting that interdisciplinary collaboration, rather than any single field, is the true driver of effective solutions.
This article introduces the definition of sequences, explains the concept of limits with formal criteria, demonstrates how to prove limit properties through examples, and outlines the basic arithmetic rules for limits of sequences.
This article compiles a comprehensive list of previously published WeChat posts from the “Model Perspective” series covering core calculus topics such as single‑variable functions, derivatives, integrals, and multivariable calculus, providing readers with quick access to each detailed tutorial.
This article explains that the gradient of a multivariable function is a vector indicating the direction of greatest increase, shows how directional derivatives are computed, and demonstrates that the maximum rate of change occurs when moving along the gradient direction.
By applying logarithmic transformation and division, the method constructs a function that exploits monotonicity to compare magnitudes, thereby reducing the problem to evaluating intervals, as illustrated in a concise Zhihu answer on mathematical problem solving.
This article explains the chain rule of conditional probability, also known as the multiplication rule, by illustrating how successive events' probabilities combine, providing a clear, intuitive example that demonstrates the step‑by‑step derivation of the formula for calculating joint probabilities.
This article compiles a comprehensive list of links to past posts from the "Model Perspective" public account covering probability and statistics, calculus, linear algebra, and mathematical modeling, while noting upcoming collections on Python programming, academic writing, and competition analysis.
This article compiles a series of WeChat posts covering calculus fundamentals—including limits, derivatives, integration techniques, gradient descent, and multivariable concepts—providing direct links to each detailed tutorial for learners to explore.
This article explains the concepts of neighborhoods, interior, exterior, and boundary points of sets, introduces accumulation points, defines open and closed regions, and describes n‑dimensional Euclidean spaces and multivariable functions, illustrating each with diagrams.
This article explains the integration by parts technique, derived from the product rule, and walks through five detailed examples that show how to rewrite integrals as products of functions, apply the formula, and solve the resulting equations.
This article outlines the fundamental properties of integrals—including the sum rule, constant factor rule, interval splitting, and limit‑based computation—and introduces differentials, their basic properties, and how they relate to integral calculus.
This article explains how the Fundamental Theorem of Calculus bridges derivatives and integrals, provides a straightforward method for evaluating integrals via antiderivatives, illustrates the concept with telescopic sums, and walks through several concrete examples to demonstrate its practical use.
This article traces the pivotal role of mathematics from the 18th‑century golden age of classical mechanics through Maxwell's equations, Einstein's relativity, biological modeling, game‑theoretic economics, the birth of modern computers, and the emergence of chaos and fractal theory, highlighting key figures and breakthroughs that linked abstract theory to real‑world applications.
This article introduces the concept of sets, their definitions, cardinality, and key properties, then explores various common sets and provides a comprehensive overview of functions—including definitions, classifications such as constant, linear, polynomial, rational, exponential, logarithmic, and trigonometric functions—along with illustrative diagrams.
This article introduces the concept of numerical sequences, explains the definition of sequence limits, demonstrates how to prove limits through detailed examples, and outlines the basic arithmetic operations on limits, providing a concise summary of these fundamental mathematical principles.
This article explains the classic Nim game where two players alternately remove 1‑3 stones from a pile, presents the LeetCode‑style problem of determining if the first player can force a win, analyzes the pattern of losing positions, provides a concise Go solution, and offers a formal proof based on game theory.
The PHP pi() function returns the approximate value of the mathematical constant π as a float, with precision set by the php.ini precision directive (default 14), and can also be accessed via the predefined M_PI constant, illustrated with a simple code example.
An AI‑driven project called the Ramanujan Machine generates novel mathematical formulas—including continued‑fraction representations of constants like π, e, and the Catalan constant—sparking interest in automated conjecture making, while highlighting both the promise and limits of machine‑assisted mathematics.
The PHP log() function computes the natural logarithm of a number or, when a base is supplied, the logarithm with that base, and the article explains its signature, parameters, return value, and provides example code with expected output.
The article explains PHP’s log1p() function, which accurately computes log(1 + number) even for values near zero, details its syntax, parameters, return value, and provides example code with sample outputs demonstrating its precision across different inputs.
The PHP log10() function returns the base‑10 logarithm of a floating‑point argument, detailing its signature, parameter description, return value, and providing example code with expected output for various inputs.
