The hottest Mathematics Substack posts right now

And their main takeaways
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Top Science Topics

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
  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.

4.1 Heaviside, Poynting, & Energy Flow

Fields & Energy • 239 implied HN points • 12 Jun 24
🔬 Science 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.

3.4.1 Ohm's Law

Fields & Energy • 519 implied HN points • 03 Apr 24
🔬 Science 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.

3.4.8 Maxwell's Methods and Views

Fields & Energy • 219 implied HN points • 22 May 24
🔬 Science Mathematics
  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 Mathematics
  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.
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3.4 James Clerk Maxwell & His Equations

Fields & Energy • 359 implied HN points • 27 Mar 24
🔬 Science Mathematics
  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.

Ordinal arithmetic

Infinitely More • 30 implied HN points • 22 Jan 26
🔬 Science Mathematics
  1. The series develops the basics of ordinal arithmetic—standard addition, multiplication, and exponentiation—and then moves on to topics like indecomposable and irreducible ordinals, Cantor normal form, and binary ordinal representation.
  2. It introduces the natural (Hessenberg) ordinal operations, which are commutative and make the ordinals into a commutative semiring, and it will study the natural ring of ordinals ⟨Ord⟩ inside the surreal numbers, asking about expressions, algebraic properties, and unique factorization.
  3. This essay first lays a rigorous foundation by giving order-theoretic and recursive definitions of the standard ordinal operations, which the later, deeper investigations will rely on.

An Intuitive Guide to Black Holes

Photon-Lines Substack • 556 implied HN points • 06 Jul 25
🔬 Science Mathematics
  1. A black hole is an area in space where the gravity is so strong that nothing, not even light, can escape. Imagine needing to throw a ball so hard that it never comes back; that's what escaping a black hole is like.
  2. To escape Earth's gravity, you need to reach a speed of about 11 kilometers per second. That's much slower than the speed of light, but black holes need escape velocities even greater than that!
  3. Black holes form from the collapse of massive stars after they've used up their fuel. When the star runs out of energy and can no longer hold itself up, it collapses into a point called a singularity, creating a black hole.

"On Action At a Distance"

Fields & Energy • 219 implied HN points • 03 May 24
🔬 Science Mathematics
  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 Mathematics
  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 Mathematics
  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.

Gain of Function.

News Items • 471 implied HN points • 18 Jan 24
🔬 Science Mathematics
  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.

Malcolm in the middle (well, at the beginning)

Logging the World • 299 implied HN points • 07 Mar 24
🔬 Science Mathematics
  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.

Next phase, next stage, next craze, next wave

Logging the World • 538 implied HN points • 02 Dec 23
🔬 Science Mathematics
  1. Understanding exponential growth in infection rates can help predict future COVID trends.
  2. Individual growth rates of different strains impact the overall daily growth rate, following a weighted average principle.
  3. Market share of strains, not just reaching a specific percentage threshold, influences overall infection growth.

3.3 Potentials, Energy, & Fields

Fields & Energy • 239 implied HN points • 20 Mar 24
🔬 Science Mathematics
  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 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.

Maths Olympiad Update

By Reason Alone • 114 implied HN points • 08 Nov 25
🚌 Education Mathematics
  1. Stripe is now the main sponsor of the Irish Maths Olympiad, helping secure funding for the next three years. This is a huge boost for promoting maths in Ireland.
  2. More training camps and classes for students are being created, which means more young people will have opportunities to excel in maths. This includes new centres for junior maths enrichment across several locations.
  3. Ireland performed exceptionally well in the International Maths Olympiad this year, achieving its best results ever. This shows the positive impact of the recent support and funding.

2.6.3 The Misinterpretation of Newton

Fields & Energy • 299 implied HN points • 14 Feb 24
🔬 Science Mathematics
  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 Mathematics
  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.

Modular curve stitching

A Piece of the Pi: mathematics explained • 66 implied HN points • 10 Dec 25
🔬 Science Mathematics
  1. Connecting each point k to a·k (mod m) on an m-point circle produces lacy modular stitch patterns that, as m grows, converge to smooth cycloid curves; for positive a these limit curves are epicycloids with a−1 petals.
  2. The same curves arise from a continuous 'dancing planets' model where two bodies orbit an origin an integer number of times and a tether between them is sampled; reversing a planet's direction turns epicycloids into hypocycloids.
  3. Sampling can cause aliasing so different orbital parameters can produce identical sampled patterns when the sample size m equals |αδ−βγ|, which explains why seemingly different stitch parameters sometimes look the same.

Ultrafinitism as arithmetic potentialism

Infinitely More • 25 implied HN points • 12 Jan 26
📖 Philosophy Mathematics
  1. Ultrafinitism can be fruitfully seen as a form of potentialism, which clarifies its philosophical commitments and lets us give a formal treatment of ultrafinitist theories.
  2. Models of finite arithmetic naturally extend step by step to larger models (M+, M++, and so on), presenting arithmetic as a growing structure even without committing to a single completed limit model.
  3. The potentialist view highlights that mathematical truth can depend on how a theory develops, and it provides a natural framework for discussing and adjudicating different ultrafinitist positions about mathematical existence.

3.1 The Discovery of Electromagnetism

Fields & Energy • 239 implied HN points • 06 Mar 24
🔬 Science Mathematics
  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.

2.6.1 Newton’s Methods

Fields & Energy • 299 implied HN points • 31 Jan 24
🔬 Science Mathematics
  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.

Ultrafinitism

Infinitely More • 48 implied HN points • 12 Dec 25
📖 Philosophy Mathematics
  1. Ultrafinitism is the view that only relatively small or computationally accessible numbers truly exist, and extremely large numbers conventionally discussed by mathematicians are denied.
  2. This stance is different from general anti-realism because it accepts small numbers as unproblematic while treating very large numbers as ontologically different or nonexistent.
  3. A central challenge is the 'draw the line' objection: it’s hard to specify where feasible numbers stop and huge ones begin, and this makes concrete questions about enormous expressions difficult or undecidable.

The Dragon Every Math Teacher Has to Slay Eventually

Mathworlds • 569 implied HN points • 22 Jun 23
🚌 Education 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.

The Intensive Margin

David Friedman’s Substack • 260 implied HN points • 29 Jul 25
🚌 Education Mathematics
  1. Many economics courses focus too much on math, making it less about real economic concepts. This can turn students away who expect more practical learning.
  2. Doing new research on topics that have been studied for a long time is tough because it's hard to say something fresh. It's often easier to use new math tools on old problems.
  3. To make meaningful contributions in economics, it's better to apply existing ideas to new areas rather than just trying to add more math to classic studies.

1.0 On Generation & Corruption

Fields & Energy • 459 implied HN points • 25 Oct 23
🔬 Science Mathematics
  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 Mathematics
  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 Mathematics
  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.

2.6.2 Newton's Legacy

Fields & Energy • 219 implied HN points • 07 Feb 24
🔬 Science Mathematics
  1. Newton's laws of motion were groundbreaking but took time to be fully understood and accepted. People did not immediately grasp his ideas about forces and motion.
  2. Many later scientists built on Newton's work, refining and developing his theories. Newton laid the groundwork, but others were key in shaping what we now know as classical physics.
  3. Newton's scientific approach set a high standard for future research. His methods are still considered a model for how scientific investigations should be conducted.

Thank you grocery store man

Disaffected Newsletter • 1019 implied HN points • 28 Feb 23
🚌 Education Mathematics
  1. People enjoy simple, friendly interactions, like sharing a tote bag at the grocery store, which can brighten their day.
  2. Many young people struggle with basic skills, such as math, due to a lack of foundational education.
  3. The current teaching environment focuses more on social issues than on essential subjects like math and history, leaving students unprepared for real-life challenges.

Emerging Stars

Compounding Quality • 216 implied HN points • 08 Feb 24
🔬 Science Mathematics
  1. Isaac Newton is famous for his Laws of Motion in physics and mathematics.
  2. Newton's Laws of Motion are fundamental to classical mechanics and still widely used today.
  3. This post about Newton's Laws of Motion is for paid subscribers.

Colouring identical cuboids

A Piece of the Pi: mathematics explained • 30 implied HN points • 30 Dec 25
🔬 Science Mathematics
  1. The number of colours needed depends on the cuboids' sizes and orientations; even with all pieces aligned, some stacks need five colours (2×2×1), others need four (3×1×1) or three (2×1×1).
  2. Letting identical cuboids meet at right angles in the same plane can raise the colour count — for example, 2×1×1 blocks in mixed planar orientations can require five colours.
  3. Allowing arbitrary orientations makes the problem harder: some constructions with 4×1×1 cuboids force at least six colours, there is a proven upper bound of 12 in that case, and it’s unknown whether six is the true maximum overall.

Maths until 18

Logging the World • 418 implied HN points • 15 Aug 23
🚌 Education Mathematics
  1. The proposal for compulsory math education until age 18 in the UK received mixed reactions, highlighting the importance of making math appealing and accessible to a wide audience.
  2. Implementing math education until 18 requires consideration of factors like shortage of math teachers and effective delivery methods such as leveraging online resources.
  3. Math education should cover areas such as practical number skills, understanding uncertainty and randomness, and exploring connections between math and other subjects like art and music.

BA.2.86 growth rates

Logging the World • 418 implied HN points • 23 Aug 23
🔬 Science Mathematics
  1. New COVID variant BA.2.86 has mutations that suggest fast growth, but estimating its growth rate is tricky.
  2. Statisticians use models and likelihood functions to estimate parameters like growth rates, but uncertainty exists in the estimates.
  3. The work of statistician C.R. Rao, like the Fisher information, shows fundamental limits to parameter estimation and the role of geometry in statistics.

Genius in turbulent times

Logging the World • 418 implied HN points • 05 Jul 23
🔬 Science Mathematics
  1. Genius can be found in lesser-known figures like Kolmogorov, who made significant contributions to mathematics and other fields.
  2. Kolmogorov's work on probability theory and the Kolmogorov-Arnold theorem had a lasting impact on mathematics and even underpins modern AI algorithms.
  3. Kolmogorov's life was not only marked by academic achievements but also by navigating personal challenges, such as opposing Lysenkoism and living as an openly gay man in Stalinist Russia.

2.4 The Copernican Revolution & the “Horror Vacui”

Fields & Energy • 239 implied HN points • 10 Jan 24
💾 History Mathematics
  1. Nicolaus Copernicus suggested that the Earth orbits the sun, which was a big change from the earlier belief that everything revolves around the Earth. This idea helped set the stage for modern astronomy.
  2. Competing theories like heliocentrism and geocentrism can both be useful in explaining observations. Sometimes even incorrect models are used because they make calculations easier.
  3. Galileo and other scientists built on Copernicus' ideas, leading to a deeper understanding of motion, gravity, and the nature of vacuums. This helped shift thinking from old beliefs to observations and experiments.