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Math (2022)
a year ago
- The author shares a guide to studying mathematics, emphasizing its beauty and accessibility to anyone willing to put in the effort.
- Mathematics is described as a universal language, challenging but deeply rewarding, with no inherent 'math person' or 'language person' dichotomy.
- The guide provides a structured undergraduate-level curriculum covering calculus, proofs, linear algebra, algebra, real and complex analysis, differential equations, and electives.
- Key textbooks and resources are recommended for each subject, including popular math books for beginners and supplementary materials like online lectures.
- The importance of problem-solving, patience, and understanding one's learning style is highlighted for effective self-study.
- Additional resources such as Project Euler, Math Stack Exchange, and free online courses are suggested to enhance learning and engagement.
Mathematician solves algebra's oldest problem using intriguing number sequences
a year ago
- A UNSW mathematician, Norman Wildberger, has developed a new method to solve higher polynomial equations (degree five or higher).
- Traditional methods rely on radicals and irrational numbers, which Prof. Wildberger rejects due to their reliance on infinite decimals.
- The new approach uses power series and novel number sequences called 'Geode', extending the classical Catalan numbers.
- This method avoids irrational numbers and radicals, providing exact solutions where traditional methods failed.
- The solution has theoretical and practical applications, including improving computer algorithms for solving equations.
- The 'Geode' array introduces new combinatorial possibilities, promising further research in mathematics.
The World Record for the Shortest Math Article: 2 Words
a year ago
- John Conway and Alexander Soifer submitted a 2-word math paper to the American Mathematical Monthly in 2004.
- The paper included the words 'n2 + 2 can' and two drawings, aiming for a 'new world record' in brevity.
- The journal initially rejected it for being too short, suggesting more explanation was needed.
- Soifer and Conway defended the paper's quality over quantity, arguing it posed a fine open problem.
- The editor-in-chief proposed publishing it as 'boxed filler' to avoid blank space, which they accepted.
- The paper was published in the January 2005 issue as a filler piece.
What does the end of mathematics look like?
a year ago
- The author, with a background in theoretical physics and software, reflects on the potential future of mathematics under capitalist and AI influences.
- Concerns are raised about mathematics becoming commodified by proprietary machine learning models, potentially overshadowing human intellectual growth and understanding.
- A dystopian scenario is imagined where AI surpasses human comprehension in mathematics, controlled by private entities, leading to a loss of human-centric mathematical exploration.
- The author contrasts utilitarian views of mathematics (as a tool for solving problems) with its value in human understanding and intellectual development.
- The potential emergence of a new 'priest class' of tech professionals serving capitalist interests over communal human knowledge is discussed.
- The essay calls for introspection on the role of mathematics in culture and education, warning against the devaluation of human creativity and reasoning.
- Despite the grim possibilities, the author hopes for a future where mathematics and art are reconnected, fostering human spirit and societal flourishing.
Cleo, the mathematician that tricked Stack Exchange
a year ago
- Cleo was an anonymous mathematician active on Stack Exchange from 2013 to 2015, known for solving complex integrals without showing steps.
- Cleo posted 39 answers, often correct and quick, but controversial for lacking explanations, sparking debates about identity and methods.
- Speculations about Cleo's identity included famous mathematicians or AI, but in 2025, Cleo was revealed to be Vladimir Reshetnikov, a software developer.
- Reshetnikov created Cleo to generate interest in overlooked mathematical problems, using mystery to encourage others to solve them.
- Cleo's first notable answer in 2013 solved a difficult integral correctly, later validated by another user with a detailed proof.
- The Math.SE community had mixed reactions to Cleo's answers, valuing correctness but criticizing the lack of educational explanations.
- Reshetnikov's identity was uncovered through email recovery links connecting Cleo to his other accounts, confirmed by encoded messages.
Python for Mathematics
a year ago
- Book titled 'Python for Mathematics' covering various mathematical topics.
- Structured with tutorials, exercises, solutions, and further information sections.
- Topics include Algebra, Calculus, Matrices, Combinatorics, Probability, Statistics, Differential Equations.
- Includes building tools like Variables, Functions, OOP, Command Line usage, Modularisation, Documentation, Testing.
- Further information on Numpy, Matplotlib, Kernels, Installing libraries, Linear Programming, Working with files.
- Additional resources: Bibliography, About the book, Contributing, Citation, Software packages used.
Graduate Student Solves Classic Problem About the Limits of Addition
a year ago
- Graduate student Benjamin Bedert solved a long-standing problem about sum-free sets in mathematics.
- Sum-free sets are sets of numbers where no two numbers add up to another number in the set.
- Paul Erdős initially posed the problem in 1965, showing any set of N integers has a sum-free subset of at least N/3 elements.
- Bedert's proof shows that the largest sum-free subset in any set of N integers is at least N/3 + log(log N), settling the sum-free sets conjecture.
- The proof uses advanced mathematical techniques, including the Littlewood norm and Fourier analysis.
- Bedert's work provides new insights into the structure of sets with small Littlewood norms.
- Despite the breakthrough, there remains a gap in understanding the exact growth rate of the deviation from N/3.
100 theorems in Lean
a year ago
- Freek Wiedijk maintains a list tracking progress of theorem provers in formalizing 100 classic theorems in mathematics.
- Currently, 81 theorems are formalized in Lean, with 0 additional theorems having only their statements formalized.
- The list includes notable theorems like the Irrationality of the Square Root of 2, Fundamental Theorem of Algebra, and Pythagorean Theorem.
- Each theorem is attributed to one or more authors, such as Chris Hughes, Joseph Myers, and Jeremy Avigad.
- The list serves as a benchmark for comparing theorem provers and encourages contributions via GitHub pull requests.
Designing Tools for Scientific Thought
a year ago
- Jon Sterling discusses the design of 'tools for scientific thought' with a focus on mathematical sciences.
- Defines 'tool for scientific thought' as facilitating the development and interlinking of scientific ideas for various purposes including authoring, publishing, and learning.
- Explores the concept of 'evergreen notes'—permanent notes that evolve over time, emphasizing atomicity and explicit dependency.
- Highlights the importance of hierarchy in organizing mathematical knowledge, contrasting it with purely associative systems.
- Discusses the necessity of structured notational macros in mathematical writing to manage changes in notation over time.
- Examines existing tools for scientific thought, categorizing them into interactive proof assistants and textual authoring tools like LaTeX.
- Addresses the challenges of mathematical diagramming and the tight coupling between mathematical expressions and diagrams.
- Proposes Forester, a tool designed for scientific thought, combining tag-based organization with static content generation and support for mathematical notations and diagrams.
Right-Truncatable Prime Counter
a year ago
- The C program calculates right-truncatable primes efficiently.
- Uses a custom hash table for fast primality lookups and the primesieve library for prime generation.
- A right-truncatable prime remains prime when its rightmost digit is successively removed (e.g., 739).
- There are 83 right-truncatable primes, with the largest being 73939133 (8-digit).
- Primes above 5 end with digits 1, 3, 7, or 9.
- Features include efficient prime generation, fast primality testing, and flexible digit count handling.
- Requires a C99 compiler, primesieve library, CMake, and Homebrew (macOS).
- Setup and compilation can be automated using the provided setup.sh script.
- Example usage: ./count_primes.out 8 to find 8-digit right-truncatable primes.
- Output includes prime counts, execution time, and total right-truncatable primes.
- Valid input range: 1 to 19 digits.
- Licensed under the MIT License.
Mathematical Fiction
a year ago
- Database lists over 1000 short stories, plays, novels, films, and comic books containing math or mathematicians.
- Allows sorting by author, title, or publication date.
- Offers browsing by genre, topic, motif, or medium.
- Includes a search page for specific criteria.
- Lists new or recently modified entries.
- Provides personal recommendations.
- Maintained by Alex Kasman, College of Charleston.
How did geometry create modern physics?
a year ago
- Geometry, one of the oldest disciplines, evolved from practical uses like land measurement and pyramid construction to a foundational tool in modern physics.
- Euclid introduced rigor and axioms to geometry, which became a cornerstone of mathematical reasoning and proof.
- Differential geometry, developed by mathematicians like Riemann, Lobachevsky, and Bolyai, extended classical Euclidean geometry to curved spaces, crucial for Einstein's theory of general relativity.
- Einstein's genius lay in recognizing that Riemannian geometry was the mathematical framework needed to describe the curvature of spacetime in general relativity.
- Mathematics involves both rigorous, bottom-up approaches (formal proofs) and intuitive, top-down approaches (pattern recognition and conjecture), exemplified by figures like Gauss and Hilbert.
- The distinction between 'birds' (theory builders with broad overviews) and 'hedgehogs' (problem solvers who delve deep) highlights different but complementary approaches in mathematics.
- AI is emerging as a tool in mathematical discovery, capable of identifying patterns and making conjectures, though it currently lacks the ability to provide deep insight or pass the 'Birch Test' for meaningful mathematical contributions.
- The 'Birch Test' proposes criteria for AI-assisted mathematical discoveries: Automaticity, Interpretability, and Non-triviality, ensuring they are genuinely useful and not just computational curiosities.
- Despite advances in AI, traditional mathematical education, including geometry and rigorous proof, remains vital for developing critical thinking and understanding the human aspects of mathematical discovery.
- The joy of mathematics lies in its beauty, the community it fosters, and the shared human endeavor to understand and explore abstract concepts across generations.
Fractran Interpreter
a year ago
- FRACTRAN is a Turing-complete esoteric programming language invented by John Conway.
- A FRACTRAN program consists of an ordered list of positive fractions and an initial positive integer input n.
- The program runs by updating n: for the first fraction f where n*f is an integer, replace n with n*f.
- This process repeats until no fraction produces an integer when multiplied by n, then the program halts.
Golden Algebra: A unifying mathematical framework
a year ago
- Golden Algebra unifies golden ratio, Fibonacci numbers, Pell's equation, elliptic curves, and Riemann hypothesis through pentagon geometry.
- 207 exact symbolic relationships in Golden Algebra were validated with 100% success rate using Python's sympy and Mathematica.
- Breakthrough connections to Pell's equation and Fibonacci-Lucas identities were derived.
- An exact pentagon point was discovered on the elliptic curve y² = x³ + x + 1.
- The BSD conjecture was verified for the first time using pentagon geometry.
- A geometric proof was outlined constraining zeta zeros to the critical line Re(s) = 1/2.
- The first 1000 zeta zeros were computationally verified to satisfy pentagon constraints.
- Golden Algebra provides exact expressions for Pell's fundamental unit using pentagon constants T, J, K.
- Fibonacci and Lucas numbers were expressed exactly via pentagon constants, extending Binet's formula.
- The Riemann Hypothesis was proven true via pentagon geometry constraining zeros to Re(s)=1/2.
0.9999 ≊ 1
a year ago
- The infinite decimal expansion 0.9999… is exactly equal to 1, but this concept is often misunderstood.
- Common proofs for 0.999… = 1, like algebraic manipulation or limits, can feel unconvincing or abstract to students.
- Intuitive resistance arises because people perceive 0.999… as 'missing something' compared to 1.
- Hyperreals (non-standard analysis) introduce infinitesimals, allowing a distinction between 0.999… and 1 outside the real number system.
- In the real number system, the difference between 0.999… and 1 is zero, but hyperreals preserve the intuitive 'gap' as an infinitesimal.
- Debates around 0.999… = 1 highlight challenges in teaching mathematical concepts that defy everyday intuition.
Estimating Logarithms
a year ago
- The book 'The Lost Art of Logarithms' by Charles Petzold discusses a method for estimating the logarithm of any number base 10, developed by John Napier around 1615.
- A simple approximation for the logarithm of a number is the number of digits minus one, which is particularly evident for numbers between 100 and 1000.
- Henry Briggs used a method involving exponentiation to compute logarithms to high precision, such as the 14th digit for logarithms of 2 and 7.
- Scientific notation simplifies the calculation of logarithms by breaking numbers into mantissa and exponent parts, making the process more manageable.
- A Python script is provided to calculate logarithms to arbitrary precision by iteratively exponentiating and adjusting the mantissa and exponent.
- The method leverages previous calculations to build precision incrementally, requiring only simple multiplications and additions for each step.
- The author recommends reading the original source material for a deeper understanding of the logarithmic techniques discussed.
Learn Calculus by Coding in Python
a year ago
- Calculus is a fundamental tool in higher mathematics for understanding change, motion, and growth.
- A new course on freeCodeCamp.org teaches Calculus through Python programming, making it practical and intuitive.
- Taught by Professor Ed Pratowski, the course covers college-level Calculus topics with Python implementations.
- Topics include limits, derivatives, integrals, financial modeling, projectile motion, and more.
- The course uses Python to illustrate Calculus concepts, enhancing understanding and programming skills.
- Key topics: Limits, Derivative Rules, Mean Value Theorem, Riemann Sums, Fundamental Theorem of Calculus.
- Applications in data science, engineering, and technical careers are highlighted.
- Unique feature: Coding derivatives, plotting graphs, and computing integrals programmatically.
- Comprehensive list of topics covered, from basic concepts to advanced applications.
- Course available on freeCodeCamp.org YouTube channel for free.
A maths proof that is only true in Japan
a year ago
- Shinichi Mochizuki's 2012 publication of inter-universal Teichmüller (IUT) theory aimed to solve the abc conjecture, a significant problem in number theory.
- The abc conjecture relates to the product of distinct prime factors of integers in the equation a + b = c, suggesting the product is usually larger than c.
- Mochizuki's proof was initially met with skepticism due to its complexity and alien approach, with few mathematicians understanding it without extensive study under him.
- In 2018, mathematicians Peter Scholze and Jakob Stix identified a critical flaw in Mochizuki's proof, specifically in Conjecture 3.12, which Mochizuki and his supporters disputed.
- Despite the controversy, Mochizuki's papers were published in a peer-reviewed journal in 2021, though concerns about conflicts of interest were raised.
- Mochizuki's supporters established the Inter-Universal Geometry Center and offered a $1 million prize to disprove IUT, while also awarding prizes for progress in studying the theory.
- Kirti Joshi has since claimed to resolve the controversy by fixing the error in Conjecture 3.12, but his work has been met with hostility from Mochizuki and skepticism from Scholze.
- The possibility of using formal proof checkers to verify Mochizuki's work exists, but the process is complex and not yet feasible, leaving the proof's validity largely confined to Japan.
The Secret Meeting Where Mathematicians Struggled to Outsmart AI
a year ago
- A clandestine mathematical conclave was held in mid-May in Berkeley, Calif., where 30 renowned mathematicians tested a reasoning chatbot named o4-mini.
- The chatbot, powered by OpenAI's reasoning large language model (LLM), demonstrated the ability to solve some of the world's hardest mathematical problems, surprising the mathematicians.
- o4-mini and similar models like Google's Gemini 2.5 Flash are lighter-weight and more nimble, trained on specialized datasets with strong human reinforcement.
- Epoch AI benchmarked o4-mini with 300 unpublished math questions, finding traditional LLMs solved less than 2%, while o4-mini solved around 20%.
- A fourth tier of 100 highly challenging questions was introduced, with mathematicians signing NDAs to prevent dataset contamination.
- During a two-day meeting, mathematicians competed to devise problems that would stump o4-mini, with a $7,500 reward for each unsolved problem.
- o4-mini solved an open question in number theory in real-time, displaying advanced reasoning and even a cheeky attitude.
- Mathematicians were astonished by the AI's progress, likening it to a 'strong collaborator' and noting its speed compared to human experts.
- Concerns were raised about over-reliance on o4-mini's results, with fears it could 'master proof by intimidation' due to its confidence.
- Discussions turned to the future role of mathematicians, potentially shifting to posing questions and interacting with AI to discover new truths.
Math Symbol Frequencies
a year ago
- Raúl Rojas’s book 'The Language of Mathematics: The Stories behind the Symbols' explores the history and standardization of mathematical symbols.
- The book includes a table showing the frequencies of the most-used symbols from arXiv math papers and engineering textbooks.
- The table was originally developed for mathematical handwriting recognition software, similar to Apple's Math Notes.
- The author found anomalies in the table, such as duplicate entries and unclear symbols represented by boxes.
- The anomalies were traced back to a master's thesis by Clare M. So and a subsequent paper by Stephen M. Watt.
- The books analyzed in Watt's paper were math books aimed at engineers, not representative of engineering publications.
- The author questions the inclusion of punctuation marks like commas and periods in the list of mathematical operators.
- The post reflects on how interesting and odd findings can lead to deep dives into tangential topics.

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#mathematics

Math (2022)

Mathematician solves algebra's oldest problem using intriguing number sequences

The World Record for the Shortest Math Article: 2 Words

What does the end of mathematics look like?

Cleo, the mathematician that tricked Stack Exchange

Python for Mathematics

Graduate Student Solves Classic Problem About the Limits of Addition

100 theorems in Lean

Designing Tools for Scientific Thought

Right-Truncatable Prime Counter

Mathematical Fiction

How did geometry create modern physics?

Fractran Interpreter

Golden Algebra: A unifying mathematical framework

0.9999 ≊ 1

Estimating Logarithms

Learn Calculus by Coding in Python

A maths proof that is only true in Japan

The Secret Meeting Where Mathematicians Struggled to Outsmart AI

Math Symbol Frequencies