The hottest Mathematics Substack posts right now

And their main takeaways
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Top Science Topics

Brilliant Simplicity

David Friedman’s Substack • 242 implied HN points • 22 Nov 24
🚌 Education Economics Science Philosophy Mathematics History
  1. Brilliant individuals can contribute to knowledge in two main ways: through challenging, complex work and by highlighting simple ideas that others may overlook. Simple ideas often seem obvious once recognized.
  2. Examples like the median voter theorem and Coase's theories show how simple concepts can explain complex phenomena, such as election outcomes or the functioning of firms, making them essential in economics.
  3. Even in biology, like Darwin's theory of evolution, simple ideas can lead to significant insights, changing how we understand life and its development over time.

Constructing space-filling curves

A Piece of the Pi: mathematics explained • 90 implied HN points • 30 Dec 24
🔬 Science Mathematics Physics Biology Chemistry Engineering
  1. Space-filling curves, like the Hilbert curve, can fill a whole area by connecting points in a specific way through iterations. They start small and grow by adding more points and connections at each step.
  2. Different seeds can lead to different types of curves. Each seed can be developed using two choices for how to connect the points, leading to many possible variations.
  3. The process used to create these curves can also be reversed. By looking at a curve and breaking it down, you can see how it was made step by step.

"Oliver Heaviside. An Appreciation : The Personal Equation : The Work of a Genius Elucidated"

Fields & Energy • 279 implied HN points • 10 Jun 24
🔬 Science Physics Engineering Mathematics History Biography
  1. Oliver Heaviside was a genius who contributed greatly to electrical science but was often misunderstood and neglected during his life. His work wasn't acknowledged until long after he had passed away.
  2. Heaviside developed important theories on cable signaling and electromagnetic waves, introducing many key terms that are still used today. His insights helped improve how signals could be transmitted over long distances, which was crucial for communication.
  3. Despite his brilliance, Heaviside lived a reclusive life and struggled financially. He preferred to work alone and only began to receive recognition later in life, which made him a complex figure in the world of science.

Nash's Invention of Non-Cooperative Game Theory (1949-50)

Cantor's Paradise • 221 implied HN points • 05 Nov 24
🚌 Education Game Theory Mathematics Economics Psychology History
  1. Nash developed his idea of non-cooperative game theory during his time at Princeton, focusing on how people can benefit from making decisions independently. His work changed the way games and competitive actions are analyzed.
  2. He introduced the concept of Nash equilibrium, where no player can improve their outcome by changing their strategy alone. This idea is crucial for understanding strategic interactions in economics and beyond.
  3. Despite initial indifference from established economists, Nash's theories gained recognition and eventually earned him a Nobel Prize. His insights made game theory relevant and valuable for various fields, including economics.

Where have the International Math Olympiad Gold Medallists Ended Up? Part One of Three

Neeloy’s Substack • 119 implied HN points • 24 Jul 24
🚌 Education Mathematics Competitions Career Paths Data Analysis
  1. Many International Math Olympiad gold medalists end up pursuing careers in different fields, not just in finance or academia. It's interesting to see how their paths vary after such early success.
  2. Data collection on these medalists shows a clear trend where China dominates in terms of gold medals, with a majority of their students achieving this top honor. This highlights the competitive environment in math education in that country.
  3. The dataset used to track these medalists has its limitations, particularly due to language and cultural barriers in finding information. However, the findings still provide valuable insights into the outcomes of these talented individuals.
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4.1 Heaviside, Poynting, & Energy Flow

Fields & Energy • 239 implied HN points • 12 Jun 24
🔬 Science Physics Energy Electromagnetism Theories Mathematics
  1. Poynting and Heaviside explained how energy moves through space, not just through wires. They believed that energy travels through the surrounding medium as it shifts from one spot to another.
  2. They challenged the traditional 'fluid' model of electricity, saying that while current flows through wires, the energy actually flows outside of them. This highlights the importance of electric and magnetic fields in energy transfer.
  3. The debate between the fluid model and the electromagnetic theory showed that although the latter was complex, it provided a more accurate understanding of how energy moves in electrical systems.

The Ramanujan Machine

A Piece of the Pi: mathematics explained • 90 implied HN points • 23 Dec 24
🔬 Science Mathematics Technology Artificial Intelligence Research
  1. Srinivasa Ramanujan was a brilliant mathematician known for his unique insights and identities, many of which he discovered in unconventional ways.
  2. The Ramanujan Machine is an AI project that helps generate new mathematical conjectures, making it easier to discover complex equations related to fundamental constants.
  3. The odd double factorial is a useful concept in pairing problems and can be calculated by multiplying all odd numbers up to a certain point, making it easier to understand how to pair off groups.

3.4.1 Ohm's Law

Fields & Energy • 519 implied HN points • 03 Apr 24
🔬 Science Physics Education History Engineering Mathematics
  1. Ohm's Law shows that voltage is equal to current times resistance, which is key to understanding how electrical circuits work.
  2. Georg Simon Ohm faced a lot of criticism during his time for his ideas, but later scientists recognized his important contributions to physics.
  3. Henry Cavendish had discovered concepts similar to Ohm's Law before Ohm, but much of Cavendish's work went unnoticed because he rarely published his findings.

Finding Fifteen

Infinitely More • 15 implied HN points • 20 Jan 25
🚌 Education Game Theory Mathematics Philosophy Logic
  1. Finding Fifteen is a game where two players try to pick numbers that add up to 15. It's a fun way to learn about strategy and competition.
  2. Players take turns choosing numbers between 1 and 9, and they can't repeat numbers. The first player to use three numbers that sum to 15 wins.
  3. Some moves can be forced, meaning players may have to make certain choices to avoid losing immediately. This adds a layer of strategy to the game.

The zigzag theorem

Infinitely More • 33 implied HN points • 04 Jan 25
🚌 Education Mathematics Geometry Learning Teaching
  1. The zigzag theorem states that when you create a zigzag pattern in a rectangle, the triangles formed below this pattern take up exactly half the area of the rectangle.
  2. Even if the zigzag lines sometimes move backward without crossing, the triangles will still cover half the rectangle's area due to how the bases and heights of the triangles are calculated.
  3. This theorem is interesting because it holds true even if the zigzag involves an infinite number of lines.

Number Theoretic Transform with RVV

Fprox’s Substack • 83 implied HN points • 07 Dec 24
🕹 Technology Algorithms Cryptography Computing Software Development Mathematics
  1. The Number Theoretic Transform (NTT) helps speed up polynomial multiplication, which is important in cryptography. It uses a smart method to do complicated calculations faster than traditional methods.
  2. Using RISC-V Vector (RVV) technology can further improve the speed of NTT operations. This means that by using special hardware instructions, operations can be completed much quicker.
  3. Benchmarks show that a well-optimized NTT using RVV can be substantially faster than basic polynomial multiplication, making it crucial for applications in secure communications.

3.4.8 Maxwell's Methods and Views

Fields & Energy • 219 implied HN points • 22 May 24
🔬 Science Physics Mathematics History Theories Research
  1. Maxwell used physical analogies and models to understand complex electrical and magnetic behaviors. This helped him discover important concepts like the displacement current.
  2. He believed that energy is linked to electromagnetic fields, not just to electric charges. This was a key part of his theory of electromagnetism.
  3. Despite his great contributions, some of Maxwell's ideas were not recognized during his time. His work on gases faced rejection, showing how science can overlook important discoveries.

3.4.2 Maxwell’s Original Equations

Fields & Energy • 339 implied HN points • 10 Apr 24
🔬 Science Physics Mathematics Electromagnetism Equations History
  1. Maxwell's equations describe how electric and magnetic fields interact. They show the principles of electromagnetism in a clear way.
  2. Heaviside simplified Maxwell's original equations, reducing them from twenty to four. This makes them easier to understand and use today.
  3. The concepts of electric displacement and charge continuity are central to these equations. They help us understand how electricity flows and behaves in various situations.

3.4 James Clerk Maxwell & His Equations

Fields & Energy • 359 implied HN points • 27 Mar 24
🔬 Science Physics Mathematics Electromagnetism History Mechanics
  1. James Clerk Maxwell was a key figure in understanding electricity and magnetism. He linked these topics together, showing how they relate to light.
  2. Maxwell created a set of equations that describe how electric and magnetic fields behave. These are known today as Maxwell's equations.
  3. Maxwell built on the ideas of earlier scientists, like Gauss and Faraday, and later, Heaviside simplified his work into the four equations used today.

Shape, Symmetries, and Structure: The Changing Role of Mathematics in Machine Learning Research

The Gradient • 87 implied HN points • 16 Nov 24
🕹 Technology Machine Learning Mathematics Research AI
  1. Mathematics is playing a bigger role in machine learning by connecting with fields like topology and geometry. This helps researchers create better tools and methods.
  2. It's not just about scaling up current methods; there's a need for new approaches based on mathematical theories. This can lead to more innovative solutions in machine learning.
  3. Mathematicians should view advancements in machine learning as chances to explore and deepen their theoretical work, not as threats to their field. Embracing these changes can lead to new discoveries.

"On Action At a Distance"

Fields & Energy • 219 implied HN points • 03 May 24
🔬 Science Physics Research Mathematics Electricity Magnetism
  1. There are debates about how forces act over distances. Some people think there's a hidden connection, while others believe that objects can directly affect each other without any medium.
  2. Here’s a fun example: when you ring a bell using a wire, the movement happens gradually, showing that actions often involve a series of connections, not just instant forces.
  3. Scientists like Faraday introduced the idea of 'lines of force' to visualize these actions. Instead of just thinking about pushes and pulls, we can now understand force as stretching and pressing through a medium.

3.4.3 Gauss’s Laws

Fields & Energy • 259 implied HN points • 17 Apr 24
🔬 Science Physics Mathematics Electromagnetism History
  1. Johann Carl Friedrich Gauss was a brilliant mathematician known for his early talent, like solving a tricky addition problem in second grade. He made significant contributions to math and physics, including the development of formulas to calculate important dates, like Easter.
  2. Gauss's Law describes how electric fields and charges relate to each other. For instance, electric field lines begin at positive charges and end at negative ones, while magnetic field lines always form loops.
  3. Gauss and Wilhelm Weber worked together to measure the Earth's magnetic field. They created detailed maps of magnetic intensity that are still referenced today, showing the long-lasting impact of Gauss's work in science.

3.4.4 Ampère’s Law

Fields & Energy • 239 implied HN points • 24 Apr 24
🔬 Science Physics Electromagnetism Mathematics Research Theory
  1. Ampère’s Law explains how electric currents create magnetic fields. You can use the right-hand rule to find the direction of the magnetic field around a current.
  2. We visualize magnetic fields using 'dot-x' notation. A 'dot' shows current coming toward you, while an 'x' shows it going away, helping to understand how fields form around currents.
  3. Maxwell introduced the idea of displacement current, which means a changing electric field can create a magnetic field. This is important for understanding how electromagnetic waves travel.

Take my Infinity final exam!

Infinitely More • 35 implied HN points • 21 Dec 24
🚌 Education Philosophy Mathematics Learning Assessment Critical Thinking
  1. The Cantor-Hume principle connects with Euclid's principle, and there are different views on whether they agree or conflict. It's a topic worth exploring in depth.
  2. Understanding potential and actual infinity is important in calculus, especially when dealing with infinite series. This distinction affects how we solve mathematical problems.
  3. The continuum hypothesis and the axiom of choice raise interesting questions in philosophy and mathematics, showing how these concepts challenge our understanding of infinity and choice.

Chutes, ladders, and Markov chains

A Piece of the Pi: mathematics explained • 72 implied HN points • 04 Dec 24
🚌 Education Mathematics Statistics Game Theory Probability Teaching
  1. The game of Chutes and Ladders is a fun example of a Markov chain. It shows how the next move depends only on where you are now, not on how you got there.
  2. There are different types of game boards, some allow for winning while others can trap players forever. Ultimately winnable boards guarantee that a player can reach the end if they keep playing.
  3. On average, players need about 39 spins to win the game, and surprisingly, most random boards created will still offer a winning chance.

Twenty one

Infinitely More • 7 implied HN points • 23 Jan 25
📖 Philosophy Game Theory Logic Education Mathematics Culture
  1. The game of Twenty-One involves two players counting to twenty-one by saying one to three numbers each turn. The goal is to be the one who says 'twenty-one' to win.
  2. Players can develop strategies to control the game and eventually win. It’s smart to think ahead about how many numbers to say.
  3. This game can help illustrate important ideas in game theory. It’s a fun way to explore how cooperation and strategy work together.

Gain of Function.

News Items • 471 implied HN points • 18 Jan 24
🔬 Science AI Mathematics Reasoning Data generation Training
  1. AlphaGeometry AI system solves complex geometry problems as well as a human Olympiad gold-medalist.
  2. AlphaGeometry combines neural language model with a rule-bound deduction engine for reasoning.
  3. Development of AlphaGeometry highlights AI's logic reasoning progress and ability to discover and verify new knowledge.

A polynomial with Rubik’s cube symmetry

A Piece of the Pi: mathematics explained • 78 implied HN points • 25 Nov 24
🔬 Science Mathematics Symmetry Group Theory
  1. Rubik’s Cube has a huge number of ways it can be scrambled, around 43 quintillion, which shows its interesting symmetry in math. It can be thought of as not just a puzzle, but a complex mathematical object.
  2. There are specific rules about how the pieces of the Rubik’s Cube can be rearranged, which creates a lot of interesting patterns and symmetries. This helps mathematicians understand how groups of movements relate to each other.
  3. Recent research has shown that it's possible to find polynomials that have the same symmetries as the Rubik’s Cube. This connects the world of puzzles to deeper mathematical concepts, making it a fun area to explore.

Malcolm in the middle (well, at the beginning)

Logging the World • 299 implied HN points • 07 Mar 24
🔬 Science Communication Mathematics Education
  1. Using interesting anecdotes or 'Malcolms' at the beginning can engage a wider audience and make complex topics more appealing.
  2. Balancing academic style writing with engaging storytelling can make science communication more effective and impactful.
  3. Integrating rhetorical tricks and interesting facts can drive curiosity and encourage broader audiences to explore complex subjects.

3.3 Potentials, Energy, & Fields

Fields & Energy • 239 implied HN points • 20 Mar 24
🔬 Science Physics Mathematics Theory Electricity Energy
  1. There's a debate in science about how we understand forces, like whether they act at a distance or through fields in space. Two main theories exist: one says forces happen instantly, while the other suggests they spread out gradually.
  2. George Green, a self-taught baker turned mathematician, made important contributions to the math behind electromagnetism. His work, which included ideas about electric potential and field theory, changed how we study these forces.
  3. Fields and potentials are two simple ways to describe how electricity and magnetism work. They help us understand how energy moves and behaves in different situations, like around charges or between capacitor plates.

What the world needs isn't "Artificial Intelligence" but Optimized Computational Methodologies

Top Carbon Chauvinist • 79 implied HN points • 21 Jun 24
🕹 Technology Artificial Intelligence Machine Learning Engineering Mathematics
  1. We should focus on making smarter tools instead of trying to make machines think like humans. Real progress comes from solving practical problems, not imitating nature.
  2. Copying how living things work is often a bad approach. Nature is full of flaws, and we don't need to mimic those to create better designs.
  3. It's important to clearly define the problems we want machines to solve. Without a clear goal, projects will struggle and waste resources on unnecessary tasks.

2.6.3 The Misinterpretation of Newton

Fields & Energy • 299 implied HN points • 14 Feb 24
🔬 Science Physics Mathematics History Philosophy Theories
  1. Newton did not explain why gravity exists. He focused on describing what gravity does instead of offering guesses about its cause.
  2. Many scientists after Newton misinterpreted his ideas, leading to a belief that gravity was an essential quality of matter, even though Newton disagreed with such views.
  3. Over time, Newton's concepts became viewed as abstract ideas rather than being connected to real evidence from the physical world.

A Bridging of Dimensions

ᴋʟᴀᵾs • 628 implied HN points • 15 Jun 23
🔬 Science Physics Mathematics Biology Neuroscience Consciousness
  1. Former government officials have revealed details about UFO crash retrieval programs involving non-human intelligence and advanced materials.
  2. The use of topological materials in UFO technology could explain exotic properties, like strange isotopes and materials able to deform into higher dimensions.
  3. Connections between the human brain's multi-dimensional functions and UFO phenomena could suggest a link between consciousness and unexplained aerial phenomena.

Mathematics of Machine Learning official release announcement!

The Palindrome • 8 implied HN points • 29 Jan 25
🕹 Technology Machine Learning Mathematics Education Publishing Programming
  1. The book 'Mathematics of Machine Learning' is set to be published soon and will be available in a physical version. You can pre-order it at a discounted price now.
  2. It focuses on important math concepts needed for machine learning, including linear algebra, calculus, and probability theory. Understanding these areas is crucial for building effective models in machine learning.
  3. The author shares a personal journey of creating the book, which was inspired by his experiences in the field. The book aims to bridge the gap between theory and practical applications.

3.1 The Discovery of Electromagnetism

Fields & Energy • 239 implied HN points • 06 Mar 24
🔬 Science Physics Mathematics Electromagnetism History Experimentation
  1. Hans Christian Örsted proved that electricity and magnetism are connected by running a current near a compass, making them part of the same field called electromagnetism.
  2. André-Marie Ampère built on Örsted's work, showing that moving electric currents can attract or repel each other just like magnets do.
  3. Many scientists assumed forces acted at a distance, but Michael Faraday later suggested that closer particles must interact to create these forces.

Inside FunSearch: Google DeepMind’s LLM that Discovered New Math and Computer Science Algorithms

TheSequence • 1106 implied HN points • 18 Jan 24
🕹 Technology AI Algorithms Computer Science Mathematics Machine Learning
  1. Discovering new science is a significant challenge for AI models.
  2. Google DeepMind's FunSearch model can generate new mathematics and computer science algorithms.
  3. FunSearch uses a Language Model to create computer programs and iteratively search for solutions in the function space.

Philosophy is the mathematics of language

Wood From Eden • 816 implied HN points • 23 Dec 23
📖 Philosophy Language Mathematics Concepts Logic Epistemology
  1. Philosophy is the art of clarifying concepts and finding links between them.
  2. Philosophy is similar to mathematics in that it explores relationships between concepts, just as mathematics explores relationships between numbers.
  3. Concepts in philosophy change over time, making it a field that evolves constantly unlike mathematics which is built on stable concepts.

2.6.1 Newton’s Methods

Fields & Energy • 299 implied HN points • 31 Jan 24
🔬 Science Physics Philosophy Mathematics History Methodology
  1. Newton believed that geometry should be connected to real-world observations, rather than just logical deductions from axioms. He saw math as a tool to understand the physical world.
  2. He emphasized that we should always seek the simplest explanation for natural phenomena, following the principle of parsimony. If a simpler explanation fits the facts, it should be preferred.
  3. Newton argued that conclusions drawn from experiments should be regarded as generally true, even if new evidence could change our understanding later on. This highlights the importance of adapting our views as we gather more information.

The Dragon Every Math Teacher Has to Slay Eventually

Mathworlds • 569 implied HN points • 22 Jun 23
🚌 Education Teaching Mathematics
  1. Students often feel worse about math class compared to other subjects because of the pressure to only have one correct answer for each question.
  2. Math should be taught as a creative discipline that embraces human subjectivity, not just a set of memorized steps.
  3. Teachers can help students deconstruct the idea of one right way to do math by introducing activities that show multiple paths lead to the same solution.

1.0 On Generation & Corruption

Fields & Energy • 459 implied HN points • 25 Oct 23
🔬 Science Physics Mathematics Technology Philosophy History
  1. In physics, our understanding has greatly improved over time, but some concepts can still feel confusing or counterintuitive. We often have to rely on complex math that works well, even if it doesn't make total sense at first.
  2. Michael Faraday challenged the common ideas of his time by introducing the concept of 'fields' instead of just focusing on point particles. This helped explain how forces work in a way that made more sense to him.
  3. Today, we still face similar questions about our understanding of reality in physics. As we develop new mathematical tools, we should ask if we need to rethink our basic ideas about how things work, just like Faraday did.

Reading List for Chapter 1: On Generation and Corruption

Fields & Energy • 359 implied HN points • 07 Dec 23
🚌 Education Literature Science Mathematics Physics Philosophy
  1. Reading is important for understanding complex topics like calculus and physics. Books like 'Calculus Made Easy' can help beginners grasp the basics more easily.
  2. Narratives and storytelling are essential in both fiction and non-fiction writing. They shape how we understand and connect with concepts.
  3. Scientific revolutions often depend on the context of ideas rather than just rational evidence. This means new theories can take time to be accepted.

Teaching math with John Mighton

The Bell Ringer • 99 implied HN points • 10 May 24
🚌 Education Teaching Learning Mathematics Pedagogy Curriculum
  1. Kids can get confused easily when we push them too hard with complex ideas. It's important to teach in a way that builds understanding step by step.
  2. Real learning happens when we focus on what students can grasp, not just on covering a lot of content. It's better to let them understand the basics well.
  3. Using evidence from research helps improve how we teach math. This can help solve the ongoing debates about the best ways to learn math.