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Elements of Programming: A Treatise on the Foundations of Computing
6 months ago
- Elements of Programming presents programming as a mathematical discipline built on structure, logic, and proof.
- The book is structured into six books, each focusing on different aspects of programming and computation.
- Book I: Foundations of computation (identity, transformation, composition).
- Book II: Introduces algebraic structures (categories, functors, monads).
- Book III: Unites operational and denotational semantics for correctness.
- Book IV: Formalizes capability-based security and verification.
- Book V: Connects type theory with formal assurance.
- Book VI: Extends ideas into philosophy and ethics of software.
- Leanpub offers a 60-day 100% happiness guarantee with no-risk refunds.
- Authors earn 80% royalties on sales, with examples of significant earnings.
- Leanpub books come with free updates, DRM-free, and in multiple formats (PDF, EPUB).
Emily Riehl is rewriting the foundations of higher category theory (2021)
6 months ago
- Emily Riehl compares her work in higher category theory to playing the viola, seeing both as foundational 'glue' in their fields.
- Category theory, introduced by Eilenberg and Mac Lane in 1945, replaces equality with the concept of equivalence, revolutionizing mathematics.
- Higher category theory extends this by considering equivalences between equivalences, captured in structures called infinity categories.
- Riehl and Dominic Verity are finalizing a book to make higher category theory more accessible, aiming to rewrite its technical foundations.
- Riehl won the 2021 AWM-Joan and Joseph Birman Research Prize for her contributions to topology and geometry.
- She discusses her early career inspiration from 'Jurassic Park' and her passion for abstract algebra and category theory.
- Riehl highlights the importance of community in mathematics, likening her role to a 'categorical consultant' supporting others.
- She co-founded Spectra, an association for LGBTQ mathematicians, to provide visibility and support within the community.
- Riehl advocates for social justice in mathematics, addressing issues like inclusivity and the impact of language on learning.
- Her work with Mike Shulman aims to simplify higher category theory for future undergraduate education.
Could a 50-year-old maths problem have kept the Louvre museum safe?
6 months ago
- Thieves stole priceless Napoleonic-era crown jewels from the Louvre in broad daylight in an eight-minute heist.
- The Louvre's security was found lacking, with only one misdirected camera covering the balcony used by the thieves and one in three rooms in the Denon wing lacking security cameras.
- The 'museum problem' or 'art gallery problem' is a 50-year-old mathematical question about the minimum number of guards or cameras needed to monitor a museum.
- The solution involves dividing the number of corners in a room by three to determine the number of cameras needed, with adjustments for non-divisible numbers.
- Steve Fisk's elegant proof in 1978 involved dividing the gallery into triangles and using a three-coloring method to determine optimal camera placement.
- The problem has applications beyond museums, including robotics, urban planning, disaster management, and computer vision.
- The Louvre's perimeter cameras do not cover all external walls, a known weakness acknowledged by the museum's director.
- Museums also face threats from internal theft, vandalism, and other forms of destruction, requiring comprehensive security strategies.
What Is a Manifold?
6 months ago
- Manifolds are shapes that appear flat locally but may have complex global structures, introduced by Bernhard Riemann in the 19th century.
- Riemann's work generalized geometry to higher dimensions, leading to modern topology and influencing fields like physics and data analysis.
- Manifolds allow mathematicians to study spaces intrinsically, using charts and atlases to translate complex shapes into Euclidean patches for analysis.
- Examples include circles (1D manifolds) and Earth's surface (2D manifold), while non-examples include figure eights and double cones.
- Manifolds are foundational in physics (e.g., Einstein's space-time) and applied to problems like pendulum motion, fluid dynamics, and data dimensionality reduction.
Mathematical Exploration and Discovery at Scale
6 months ago
- Collaboration with Google Deepmind using AlphaEvolve tool for mathematical exploration.
- AlphaEvolve evolves computer code via LLM to generate inputs for optimization problems.
- Tested on 67 math problems in analysis, combinatorics, and geometry.
- Advantages include scalability, robustness, and interpretability of solutions.
- AlphaEvolve found exact solutions in some cases, like the Talenti function.
- Struggled with analytic number theory problems but excelled in algebraic structures.
- Discovered new constructions for finite field Kakeya and Nikodym sets.
- Potential for systematic recording of negative results in mathematical research.
Math Books
6 months ago
- List of mathematical textbooks covering various topics such as probability, linear algebra, calculus, and geometry.
- Includes works by notable authors like E.S. Wentzel, Walter Rubin, Richard Courant, and A.N. Kolmogorov.
- Textbooks range from introductory to advanced levels, suitable for students and professionals.
- Topics include probability theory, mathematical analysis, differential equations, and combinatorial mathematics.
- Some books are problem-solving oriented, providing exercises and solutions.
- Also includes physics textbooks and applied mathematics books for engineering students.
Think in Math. Write in Code
6 months ago
- Programmers often integrate programming languages into their identities, debating their merits and associating them with personal values.
- Programming languages are primarily tools for instructing machines, not for expressing thoughts, which are better suited to free and flexible mediums like mathematics.
- Mathematics, often misunderstood as rigid and symbolic, is actually about creating logical models to solve real-world problems through definitions and deductions.
- Effective programming involves solving logical problems first (steps 1 and 2) before implementation (step 3), with math providing clarity and better code outcomes.
- Programming languages are burdened by implementation concerns, distracting from the core problem-solving process.
- Abstraction in programming (black boxes) hides details but is rigid and leaks, unlike mathematical abstraction which is flexible and adjusts to the right level of problem-solving.
- Math allows reasoning about problems from multiple perspectives (e.g., formulaic, geometric, algebraic), enhancing understanding and solution design.
- Programming languages' rigid data representation forces early commitment to specific structures, whereas math allows problem-solving before choosing representations.
- Example: Graph representations vary widely based on use cases, making reusable libraries impractical without forcing inappropriate structures.
- Modern language features like async/await succeed by addressing practical implementation issues without introducing unnecessary theoretical complexity.
- Thinking in math encourages a 'C style' of programming—tailored solutions with carefully chosen trade-offs, avoiding unnecessary abstraction layers.
- A practical example of mathematical thinking in programming: designing a cryptocurrency pricing API, where rigorous definitions and theorems clarify assumptions and improve outcomes.
The Math of Ocean Waves
6 months ago
- Alberto Maspero enjoys a beautiful view of the Adriatic Sea from his office in Trieste, Italy.
- Trieste is known for its 'bora' wind, which can reverse the direction of waves in the bay.
- Euler's equations describe fluid flow but are notoriously difficult to solve, even for simple wave patterns.
- Mathematicians have long struggled to prove instabilities in wave patterns, known as Benjamin-Feir instabilities.
- Maspero, Ventura, Berti, and Corsi have provided a proof showing when these instabilities occur.
- Stokes waves, once thought stable, can be disrupted by certain disturbances, leading to instability.
- Deconinck and Oliveras discovered alternating intervals of stability and instability in wave patterns, called 'isole'.
- Maspero's team used computational and theoretical techniques to prove the existence of these instabilities.
- The findings open new avenues for understanding wave behavior and solving other problems in fluid dynamics.
- The research enhances the mathematical understanding of water waves, combining computational tools with pen-and-paper proofs.
What's the Deal with Euler's Identity?
6 months ago
- Euler's identity combines five special mathematical constants: e, π, 0, 1, and the imaginary unit i.
- The identity is a special case of Euler's formula, derived by setting the angle α = π radians (180°).
- Complex numbers can be used to model 2D geometry, with i representing a 90° rotation.
- Rotation in the complex plane can be expressed using trigonometric functions or powers of i.
- Euler's formula connects exponential functions with trigonometric functions, showing e^(iα) = cos(α) + i·sin(α).
- The constant e is chosen as the base because it provides a 1:1 correspondence between the parameter and the distance traveled in the unit circle.
- The article provides an intuitive explanation rather than a rigorous proof, appealing to geometric interpretations of complex numbers.
M.C. Escher Prints Digitized and Put Online by the Boston Public Library
6 months ago
- M.C. Escher's work has appeared in various media, including album covers and speculative fiction.
- His prints became popular with 1960s counterculture, though he discouraged mystical interpretations.
- Escher aimed to visually represent intellectual concepts, not metaphysical ideas.
- His architectural background influenced his focus on buildings and impossible structures.
- Escher's art gained academic and mathematical interest in the 1950s and 60s.
- He collaborated with scientists, despite lacking formal training in exact sciences.
- Escher's fame expanded beyond academia through countercultural interest.
- Boston Public Library has digitized a collection of Escher's prints for public viewing.
To Have Machines Make Math Proofs, Turn Them into a Puzzle
6 months ago
- Marijn Heule uses SAT (satisfiability) to solve complex mathematical problems in geometry and combinatorics.
- SAT is a form of symbolic AI that relies on binary logic (true/false) to build airtight proofs.
- Heule believes combining SAT with large language models (LLMs) can solve problems beyond human capability.
- SAT solvers work by searching for combinations of true/false values that satisfy given constraints, akin to solving puzzles.
- LLMs can generate plausible mathematical lemmas, while SAT solvers verify their correctness or provide counterexamples.
- Automated proofs can be extremely long and incomprehensible to humans, but Heule argues trust in correctness is more important than understanding.
- Heule envisions a collaborative future where mathematicians, LLMs, and automated reasoning work together to solve open problems.
- Human intuition and creativity remain essential in mathematics, with automation serving as a tool rather than a replacement.
The Duties of John von Neumann's Assistant in the 1930s (2020)
6 months ago
- Edgar R. Lorch, a 26-year-old mathematician, secured a postdoctoral fellowship at Harvard in 1933 and later applied to work under John von Neumann at Princeton.
- Lorch's Ph.D. thesis focused on Liouville's results about the impossibility of solving certain integrals and differential equations with elementary functions.
- Under Marshall H. Stone's supervision, Lorch aimed to study Hilbert and Banach spaces, leading him to seek guidance from von Neumann and Frederick Riesz.
- Lorch was offered a position as von Neumann's assistant, which included duties like note-taking, translating papers, editing the Annals of Mathematics, and teaching typesetting.
- Von Neumann's lectures were notoriously fast-paced, and his assistant had to keep up with his rapid delivery and extensive notes.
- Lorch declined the demanding role and instead chose to study with Frederick Riesz in Szeged, Hungary, where he had a fruitful and harmonious mentorship.
- The essay is based on Lorch's writings about his experiences, compiled by Reuben Hersh and published in 'The American Mathematical Monthly' in 1993.
Tetrahedral Analog of the Pythagorean Theorem
6 months ago
- A tetrahedron has four triangular faces, with three meeting at a right angle.
- De Gua's theorem states: A0² = A1² + A2² + A3² for the areas of the faces.
- Python code demonstrates De Gua's theorem numerically, confirming the theorem holds.
- The theorem generalizes to higher dimensions, such as a 4-simplex in a hypercube.
- For a 4-simplex, V0² = V1² + V2² + V3² + V4², extending De Gua's theorem.
- Python code for the 4-simplex shows the theorem holds, with minor floating-point errors.
- Floating-point arithmetic limitations cause precision loss when subtracting nearly equal numbers.
New Proofs Probe Soap-Film Singularities
6 months ago
- Joseph Plateau's experiments with soap films in the mid-19th century led to the discovery of area-minimizing surfaces.
- Mathematicians Jesse Douglas and Tibor Radó proved Plateau's conjecture in the 1930s, showing that for any closed curve, a minimizing surface exists.
- Minimizing surfaces are smooth up to dimension seven, but in higher dimensions, singularities (folds, pinches, or intersections) can occur.
- In 1985, mathematicians showed singularities in eight dimensions can be 'wiggled away,' but progress stalled in higher dimensions.
- Recent work by Chodosh, Mantoulidis, Schulze, and Wang proved generic regularity in dimensions nine, ten, and eleven, allowing smoother minimizing surfaces.
- The findings help resolve long-standing problems in geometry, topology, and general relativity, such as the positive mass theorem.
- Future research aims to extend these results to even higher dimensions or uncover new mysteries about singularities.
The curious case of broken theorems
6 months ago
- Andrew Wiles announced a proof of Fermat’s Last Theorem in 1993, but a gap was later found, requiring a fix.
- Mathematics often contains errors in proofs and theorems, yet the field continues to progress despite these issues.
- Amnon Neeman disproved a 1961 'theorem' in homological algebra, highlighting how errors can persist unnoticed for decades.
- Formal logic suggests that a single contradiction should invalidate a mathematical theory, but in practice, mathematics remains robust.
- Mathematicians like Raphaël Rouquier and Terry Tao have published errata, showing that errors are common but manageable.
- Efforts to formalize mathematics, such as using proof assistants like Lean, aim to reduce errors but reveal existing gaps in proofs.
- Kevin Buzzard's team encountered a broken lemma in crystalline cohomology, but the issue was fixable due to the theory's extensive use.
- Mathematicians often rely on intuitive meaning attached to symbols, blending methodological Platonism and formalism in their work.
- Conceptualism views mathematics as a human activity combining formal logic with intuitive concept-building.
- Mathematics breaks down like pottery (kintsugi) rather than shattering, with errors being fixable and contained.
AsciiMath
6 months ago
- AsciiMath is an easy-to-write markup language for mathematics.
- Two options to get started: using MathJax (recommended) or loading the AsciiMath JavaScript file (limited browser support).
- MathJax is a full-fledged JavaScript display engine for mathematics, compatible with all browsers.
- AsciiMath JavaScript file converts AsciiMath notation to Presentation MathML, but only works in Firefox and Safari.
- AsciiMath syntax mimics rendered symbols, with many symbols having TeX alternatives.
- Includes a wide range of mathematical symbols, operations, and notations.
- Supports fractions, roots, integrals, sums, products, and more.
- Provides symbols for relations, logic, brackets, arrows, accents, and Greek letters.
- Special cases include matrices, column vectors, augmented matrices, and complex subscripts.
- Standard functions like sin, cos, tan, log, etc., are supported.
- Grammar definition provided in Backus-Naur form for parsing AsciiMath expressions.
When 1+1+1 Equals 1
6 months ago
- Mathematical operations that are involutions return to the original state when applied twice, such as taking the negative or reciprocal of a number.
- Near-involutions are operations where applying them three times is equivalent to applying them once, and four times is equivalent to twice, etc.
- Examples of near-involutions include orthocomplementation in linear algebra, intuitionistic negation in logic, and a network operation in graph theory.
- Orthocomplementation in geometry involves finding perpendicular lines, where applying the operation three times returns the same result as applying it once.
- Intuitionistic negation in constructivist logic behaves such that triple negation is equivalent to single negation, differing from classical logic.
- In social network analysis, an operation K(S) defines people who know everyone in set S, and applying K three times returns the original K(S).
- The article explores how these operations, though distinct, share the property of 'thrice equals once' and connects them to broader mathematical concepts.
Mathematics in Movies
6 months ago
- A collection of movie clips featuring Mathematics, started in 2006 and now includes about 500 clips.
- The site transitioned from Flash to HTML5 video in 2010, and since 2016, movies are added as separate pages for easier maintenance.
- The page banner is a 360-degree panorama GIF rendered in 2006 using the open-source Povray raytracer.
- The collection includes movies from various genres and years, such as 'Futurama', 'The Big Short', 'Hidden Figures', and 'A Beautiful Mind'.
- Each movie clip is accompanied by a brief description of the mathematical content featured in it.
- The site allows teachers to grab and include clips in presentations, with direct video links provided.
- The collection is maintained by Oliver Knill from the Department of Mathematics at Harvard University.
- The page has been featured in a BBC podcast and acknowledges contributions from over 160 individuals.
How to Identify a Prime Number Without a Computer
6 months ago
- Édouard Lucas proved the 39-digit number 170,141,183,460,469,231,731,687,303,715,884,105,727 (2^127 - 1) is prime without computers.
- Prime numbers are divisible only by 1 and themselves and form the building blocks of mathematics.
- Mersenne primes have the form 2^p - 1, where p is a prime number, but not all such numbers are primes.
- Lucas developed the Lucas-Lehmer primality test to verify Mersenne primes, reducing the computational effort significantly.
- The test involves generating a sequence where the (p-2)nd term must be divisible by the Mersenne number for it to be prime.
- Finite number fields, studied by Galois, underpin the Lucas-Lehmer test, leveraging their symmetric properties when p is prime.
- The largest known prime as of October 2025 is 2^13,627,984 - 1, a Mersenne prime with over 41 million digits.
My Favorite Math Problem
6 months ago
- The problem involves covering a mutilated chessboard (with two opposing diagonal corners removed) using 2x1 blocks.
- The question is a yes or no question about the possibility of such a covering, not about how to do it.
- The problem is simple to state but difficult to solve, illustrating the nature of combinatorial problems.
- It serves as an introduction to higher mathematics concepts like existence proofs and abstract definitions.
- Modern mathematics is highly abstract, with proofs and definitions requiring human intelligence, but computers are starting to contribute.
- Proofs can be seen as types, which are common in programming languages, enabling computers to understand mathematical theory.
- Projects like Microsoft's aim to formalize mathematical knowledge into a computer-readable form.
- AI and LLMs are beginning to generate type-theoretic formulations of mathematical statements, potentially changing mathematical research.

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#mathematics

Elements of Programming: A Treatise on the Foundations of Computing

Emily Riehl is rewriting the foundations of higher category theory (2021)

Could a 50-year-old maths problem have kept the Louvre museum safe?

What Is a Manifold?

Mathematical Exploration and Discovery at Scale

Math Books

Think in Math. Write in Code

The Math of Ocean Waves

What's the Deal with Euler's Identity?

M.C. Escher Prints Digitized and Put Online by the Boston Public Library

To Have Machines Make Math Proofs, Turn Them into a Puzzle

The Duties of John von Neumann's Assistant in the 1930s (2020)

Tetrahedral Analog of the Pythagorean Theorem

New Proofs Probe Soap-Film Singularities

The curious case of broken theorems

AsciiMath

When 1+1+1 Equals 1

Mathematics in Movies

How to Identify a Prime Number Without a Computer

My Favorite Math Problem