#mathematics

This post explains why the component definition (sum of products of corresponding coordinates) and the geometric definition (product of magnitudes times the cosine of the angle) of the dot product in Euclidean space are equivalent.

The rational approximation 2721/1001 for e is notable because it achieves seven to eight significant figures of accuracy, which is far more precise than the trivial truncation 2718/1000 given its denominator size.

The article discusses catastrophe theory and draws parallels between geniuses and maniacs, exploring how small changes in circumstances can lead to sudden, dramatic shifts in behavior or outcomes. It reflects on the fine line between brilliance and madness, suggesting that both states may emerge from similar underlying conditions.

This video explains the concept of vector embeddings in machine learning, detailing how mathematical vectors are used to represent data like words, images, or items in a continuous space. It covers how these embeddings capture semantic relationships and similarities, enabling algorithms to process and compare complex information efficiently.

Stephen Wolfram explores the life and work of Gottfried Leibniz, examining his contributions to mathematics, philosophy, and computing. Wolfram reflects on Leibniz's vision of a universal language and symbolic reasoning, connecting his ideas to modern computational thinking and Wolfram's own work on the Wolfram Language.

Mathematicians are working to establish formal guidelines and rules for the use of artificial intelligence in their field, aiming to set standards for how AI tools should be integrated into mathematical research and practice.

A 1966 BBC programme explores Alan Turing's theoretical "universal machine" concept, demonstrating how a single machine could be programmed to perform any computational task. The film shows a working model of Turing's machine in action, highlighting his foundational contributions to computer science and artificial intelligence.

The article reveals the mathematical and geometric principles behind Gaudí's Sagrada Familia, showing how hyperboloids, paraboloids, and other complex shapes created its unique structure while also providing structural efficiency, allowing thinner columns and lighter load-bearing elements.

A Gaussian function, named after Carl Friedrich Gauss, is a specific type of mathematical function shaped like a symmetric bell curve. It is widely used in statistics, physics, signal processing, and image processing, with the normal distribution being its most famous application. The function is defined by parameters for its center, peak height, and width.

The Ramanujan Challenge for AI is a project that invites artificial intelligence systems to discover new mathematical identities and formulas, inspired by the intuitive genius of mathematician Srinivasa Ramanujan. The challenge aims to test and advance AI's ability to find patterns and generate conjectures in mathematics.

Kevin Hartnett explores how artificial intelligence is transforming mathematical research, with AI systems like DeepMind's AlphaTensor and OpenAI's o1 model discovering new conjectures and proofs. While AI can generate insights and tackle combinatorial problems, it also raises questions about the nature of mathematical understanding and whether human intuition remains essential.

The author walks through a formal proof of the fundamental theorem of arithmetic (unique prime factorization) using the Agda proof assistant and its standard library.

The Math Behind NUMB3RS Archive is an educational resource from Wolfram Research that explains the real mathematics featured in the CBS crime drama NUMB3RS. Each page corresponds to an episode and provides accessible explanations of mathematical concepts like probability, graph theory, and statistics used in the show's storylines.

This paper examines Alfred Tarski's foundational contributions to computer science, particularly his work on model theory, decision procedures, and semantics, which influenced areas such as database theory, programming language semantics, and automated theorem proving.

This page provides lecture notes and resources on optimal transport theory and its applications in machine learning, covering topics like the Wasserstein distance, computational methods (Sinkhorn algorithm), and use cases in domain adaptation, generative models, and gradient flows.

In this 1994 essay, William Thurston reflects on the nature of mathematical proof, how mathematicians communicate and understand ideas, and the gap between formal proofs and the deep, intuitive grasp that drives mathematical progress. He argues that mathematics is ultimately a human, social activity where understanding—not just logical rigor—is the true goal.

David Bessis argues that mathematics has been corrupted by a "theorem economy" prioritizing quantity over insight, stifling creativity and genuine understanding.

This paper develops a mathematical theory for modeling twisted balloon animals as polyhedral structures. It introduces the concept of balloon polyhedra, where balloons represent edges that meet at vertices formed by twisting, and provides algorithms for automatically determining whether a given polyhedron can be constructed from a single balloon.

This guide offers practical advice for studying mathematics effectively, emphasizing regular practice, reading textbooks actively, and working through problems without skipping steps. It highlights the importance of understanding concepts deeply rather than memorizing procedures, and recommends consistent, focused study habits to build mathematical skills.

The author explores the subjective experience of doing mathematics, describing it as "reading" mathematical structures through felt constraints rather than just manipulating symbols.

Researchers demonstrate that altering the mathematical foundations of AI computations can significantly reduce hardware demands. By replacing traditional floating-point arithmetic with alternative number formats, AI models can maintain accuracy while requiring less computational power and memory, potentially lowering costs and energy consumption.

The article connects Delannoy numbers—which count the ways a king can move across a chessboard—to DNA sequence alignment problems, showing how combinatorial mathematics applies to bioinformatics.

The article argues that AI will not make mathematicians obsolete, as mathematical research requires deep conceptual understanding, creativity, and intuition that current AI systems lack. While AI can assist with calculations and pattern recognition, it cannot replace the human ability to formulate novel problems and discover fundamentally new mathematical structures.

The On-Line Encyclopedia of Integer Sequences (OEIS) is a comprehensive, free online database of integer sequences, serving as a vital reference for researchers and enthusiasts in mathematics and related fields. It allows users to search for sequences by terms, keywords, or sequence ID, and provides formulas, references, and cross-references for each entry.

The blog post addresses the common confusion between variables and parameters in mathematics. It highlights how teachers often define a parameter as a "constant that can vary," leading students to question the distinction. The author notes this is a frequently recurring source of confusion in mathematical instruction.

Grant Sanderson discusses how AI is transforming mathematics, enabling new forms of visual and intuitive understanding while also posing challenges to traditional mathematical rigor. He explores the potential for AI to assist in mathematical discovery and education, and considers how the field might evolve as machine learning tools become more integrated into mathematical practice.

Grant Sanderson discusses how mathematics will be the first field to exhibit superintelligent AI, exploring what this transformation might look like and its implications for the future of mathematical discovery and human collaboration with AI.

The article explains the mathematical properties of silver rectangles, which are similar to golden rectangles but based on a different ratio. Whereas a golden rectangle can be subdivided into a square and a smaller golden rectangle, a silver rectangle can be divided into two similar silver rectangles of different sizes.

The article analyzes whether the financial returns of space exploration can justify the massive investments required, finding that current economic models often rely more on faith than solid math.

TheoremGraph is a searchable graph of over 18 million mathematical statements from the Formal Abstracts dataset, showing dependencies between theorems across arXiv, Wikipedia, ProofWiki, and Stack Exchange. It enables users to explore citation links and the logical structure of mathematical knowledge.