The hottest Mathematics Substack posts right now

And their main takeaways
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Top Science Topics

John F. Nash Jr.'s Letter to J. Robert Oppenheimer (1957)

Cantor's Paradise β€’ 205 implied HN points β€’ 17 Jan 25
🎭️ Culture Mathematics
  1. John F. Nash Jr. was very bold in reaching out to famous scientists like Einstein and von Neumann. He wasn't afraid to discuss his ideas with them, even at a young age.
  2. Nash had limited formal education in physics but still engaged deeply with complex ideas. He wasn't shy about diving into new topics and sharing his thoughts.
  3. His interactions with these great minds show that having confidence and curiosity can lead to meaningful discussions, even with experts in the field.

The edge of reason

Logging the World β€’ 219 implied HN points β€’ 28 Dec 22
πŸ”¬ Science Mathematics
  1. When adding numbers, there are basic properties like getting another number, having a special zero that doesn't change sums, and having partners that return to zero when added.
  2. Mathematicians use abstraction to find essential properties, like in groups, to study various systems efficiently and effectively.
  3. Seeking historical analogies in current events can be misleading; it's important to understand the limitations of models and not be overconfident in applying mathematical rules to real-world situations.

Book review: Math-ish and Math Mind

The Bell Ringer β€’ 3 HN points β€’ 30 Aug 24
🚌 Education Mathematics
  1. Kids may not be learning math deeply if they only focus on concepts without practicing basic skills. Just like music, understanding math requires more than just thinking about itβ€”it needs practice and foundational knowledge.
  2. Two new books suggest that creativity and a positive mindset toward math can help kids learn effectively. However, it’s important to also teach the necessary skills and techniques for actually doing math.
  3. There's a concern that these books might promote a 'think system' approach to math education, which could overlook the important learning processes. Skills in math, as in music, come from practice and mastering the basics.

The drunkard and the lady

inexactscience β€’ 79 implied HN points β€’ 23 Jan 24
πŸ“– Philosophy Mathematics
  1. Having a vision can help you make significant progress in life. Instead of just wandering aimlessly, a clear goal can lead you to where you want to go.
  2. Without a direction, your progress will be limited and unpredictable. You might only cover a small distance instead of reaching your true potential.
  3. In life, it's important to develop a sense of direction. The more focused and goal-oriented you are, the further you'll move towards your aspirations.
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Brilliant Simplicity

David Friedman’s Substack β€’ 242 implied HN points β€’ 22 Nov 24
🚌 Education Mathematics
  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.

Berkson's Paradox Is Everywhere

Outlandish Claims β€’ 19 implied HN points β€’ 12 Jun 24
πŸ”¬ Science Mathematics
  1. Berkson's Paradox applies to various situations where multiple factors influence outcomes, leading to counterintuitive results.
  2. Applying Berkson's Paradox to different scenarios can reveal hidden correlations and insights, such as in medical studies, card games, or economic policies.
  3. The essence of Berkson's Paradox lies in understanding that when focusing on a specific subcategory, the causes of membership in that category can be more negatively correlated than in the broader category.

Strong Students Aren't Always Ready for More

Pershmail β€’ 137 implied HN points β€’ 07 Aug 23
🚌 Education Mathematics
  1. Strong students may not always be ready for more challenging material.
  2. Mathematics education is not a one-size-fits-all journey, some students have specific areas of interest and may not be ready for broader mathematical growth.
  3. Kids may have peaks of interest in specific mathematical topics, and that's completely normal, parents should be aware of this and educator should offer real challenges to help them grow.

The Equation That Outsmarts AI: y=mx+b

Shrek's Substack β€’ 4 HN points β€’ 19 Aug 24
πŸ•Ή Technology Mathematics
  1. The way you ask questions and set the model's temperature can really affect how well AI solves math problems. Clear prompts and specific instructions can help improve its accuracy.
  2. AI like GPT-4o struggles with big numbers and can make mistakes about half the time when calculating linear equations. It works better with smaller numbers.
  3. It's important to be careful when using AI for math, especially in education. Using other tools to double-check results can help avoid mistakes.

Math for Software Engineering[Math Mondays]

Technology Made Simple β€’ 139 implied HN points β€’ 21 Mar 23
πŸ•Ή Technology Mathematics
  1. Linear Algebra is crucial for software engineers, especially for operations involving vector and matrix operations. Understanding the basics is key for most developers.
  2. Probability and Statistics play a significant role in analyzing data, and even non-AI professionals can benefit from grasping concepts like causal inference. Focus on foundational principles before diving deeper.
  3. Calculus, though important, may not be essential for all software engineers. Studying up to Calc-2 is generally adequate, as it appears in various other topics.

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

Cantor's Paradise β€’ 221 implied HN points β€’ 05 Nov 24
🚌 Education Mathematics
  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.

Yet Another Ball and Urn Problem

Metarational β€’ 59 implied HN points β€’ 13 Feb 24
πŸ”¬ Science Mathematics
  1. The problem involves repeatedly selecting balls from an urn, inspecting their color, putting them back, and adding another of the same color. The goal is to find the probability that the majority of balls in the urn will be white after a large number of repetitions.
  2. To solve the problem, it was analyzed that there must be at least half white draws to achieve a white majority. Calculations led to a final result of 11/16 as the probability limit.
  3. The solution involved understanding the probabilities of different color sequences and using Riemann sums to simplify and find the answer, showcasing an intricate application of mathematics to a probability riddle.

Why is the Golden Ratio Hiding in the Fibonacci Sequence?

The Palindrome β€’ 4 implied HN points β€’ 30 Jan 26
🚌 Education Mathematics
  1. The Fibonacci sequence has a simple closed-form (Binet's) formula that uses the golden ratio phi (Ο†) and its conjugate psi (ψ) to compute Fn directly. It yields exact integers even though Ο† and ψ are irrational.
  2. Generating functions turn the recurrence into the rational function F(x)=x/(1-x-x^2), and partial fraction decomposition expresses F(x) as a sum of two geometric series, which leads to Binet's formula.
  3. The recurrence can also be encoded by a 2Γ—2 Fibonacci matrix whose eigenvalues are Ο† and ψ; diagonalizing that matrix and raising it to the nth power gives the same closed-form result, and computing matrix powers is often numerically more stable than directly evaluating Binet's formula.

Review of "Mathematica" by David Bessis

Silicon Reckoner β€’ 117 implied HN points β€’ 09 Mar 23
πŸ“š Literature Mathematics
  1. The book 'Mathematica' by David Bessis emphasizes the idea that everyone is already an accomplished mathematician, promoting awakening and emancipation from misconceptions about math.
  2. Mathematics is viewed as a sensual and carnal experience by mathematicians like Bill Thurston, focusing on understanding over logical reasoning.
  3. The concept of 'elephantitude' in the book highlights the importance of human understanding in mathematics, contrasting with the focus on reasoning in technology like deep learning.

WELCOME!

Silicon Reckoner β€’ 117 implied HN points β€’ 03 Jul 23
πŸ”¬ Science Mathematics
  1. There has been a surge in newsletter subscriptions after being mentioned in a New York Times article.
  2. There are concerns about the relationship between AI, mathematics, and industries like tech and defense.
  3. Articles in the newsletter cover topics such as the conflict between mathematics and computing, the goals of mathematics, and AI's impact on mathematical reasoning.

Introducing QF's Advanced Mathematics GPT

Quantum Formalism β€’ 59 implied HN points β€’ 02 Feb 24
πŸ•Ή Technology Mathematics
  1. QF has introduced an Advanced Mathematics GPT as a learning assistant for graduate-level mathematics.
  2. The new GPT covers advanced branches of pure mathematics like Abstract Algebra, Algebraic Geometry, and Differential Geometry, allowing users to ask questions on various topics.
  3. Access to the Advanced Mathematics GPT is currently restricted to users with ChatGPT Plus or higher subscription plans.

Some results on income distributions and total utility

Splitting Infinity β€’ 59 implied HN points β€’ 28 Jan 24
πŸ”¬ Science Mathematics
  1. The type of income distribution models used like Pareto or lognormal can impact total utility calculations in economics
  2. There is an interesting relationship observed where the degree of inequality doesn't directly correlate with total utility in certain scenarios
  3. Introducing more risk-averse utility functions can bring the focus back on the importance of inequality in calculations

Introducing QF GPT

Quantum Formalism β€’ 59 implied HN points β€’ 26 Jan 24
🚌 Education Mathematics
  1. QF GPT is a learning assistant tool for those studying quantum mechanics and quantum information science, especially those struggling with mathematical definitions.
  2. QF GPT utilizes ChatGPT's capabilities to provide accurate responses most of the time and encourages users to correct errors to solidify their understanding.
  3. Access to QF GPT is currently limited to users with ChatGPT Plus or higher subscription plans, but the hope is to make it accessible to the entire QF community in the future for advanced mathematical learning in QM and QIS.

Chess and the number e

A Piece of the Pi: mathematics explained β€’ 163 implied HN points β€’ 16 Dec 24
πŸ”¬ Science Mathematics
  1. The number e, around 2.718, plays a big role in math, especially in combinatorial problems like derangements. This is when items are arranged so that none are in their original position.
  2. In chess, setting up nonattacking rooks can be related to derangements. The chance that none of them land on the main diagonal equals about 36.8%, which links back to the number e.
  3. Recent studies have also looked at how many safe squares remain on a chessboard when placing random pieces. As more pieces are added, the proportion of safe squares follows certain patterns connected to e.

The Monty Hall problem: The golden goat variation

Simplicity is SOTA β€’ 131 implied HN points β€’ 03 Feb 25
πŸ”¬ Science Mathematics
  1. The Monty Hall problem has a new twist, focusing on a valuable goat instead of a car. In this version, knowing which goat is valuable affects your choice.
  2. Using Bayes' theorem can help calculate the probabilities in this variation. After a goat is revealed, you can reassess your chances to make a better decision.
  3. The essential lesson is to update your beliefs with new information. Recognizing how new clues impact your choices is key to making smarter decisions.

Christmas chocolates, plus UK COVID data, week starting 5th December 2022

Logging the World β€’ 159 implied HN points β€’ 08 Dec 22
πŸ₯ Health & Wellness Mathematics
  1. The author discusses using mathematical and statistical concepts to understand real-world situations, including analyzing Christmas chocolates.
  2. The UK COVID data from late 2022 shows a notable increase in hospital admissions, but the rate of growth is not as alarming as in previous waves.
  3. Various data plots highlight slow increases in COVID cases and hospital admissions, indicating a less rapid growth compared to past variant-driven waves.

The binary sphere packing problem

A Piece of the Pi: mathematics explained β€’ 54 implied HN points β€’ 12 Jul 25
πŸ”¬ Science Mathematics
  1. The best way to pack spheres to use space efficiently is known thanks to a theory called the Kepler conjecture. It shows that no arrangement can be denser than stacking spheres in a certain structured way.
  2. When packing two types of spheres together, it’s possible to fill more space than just using one size. An ideal ratio of the sizes can help maximize how much space is used.
  3. Researchers are still working on the binary packing problem to determine how densely two sizes of spheres can fill space. They have found hints that a specific size ratio might help achieve the best packing.

Links for 2023-06-01

Axis of Ordinary β€’ 98 implied HN points β€’ 01 Jun 23
πŸ•Ή Technology Mathematics
  1. Model training can be improved by rewarding each correct step of reasoning in mathematical problem solving.
  2. New fMRI-to-image approach called MindEye retrieves and reconstructs images from brain activity.
  3. Probabilistic AI can assess its own performance effectively.

Turning a triangle into a square

A Piece of the Pi: mathematics explained β€’ 115 implied HN points β€’ 11 Jan 25
πŸ”¬ Science Mathematics
  1. Henry Dudeney showed in 1902 that you can cut an equilateral triangle into four pieces and rearrange them into a square with the same area. This is a fun example of how shapes can transform while keeping their total area the same.
  2. The Wallace–Bolyai–Gerwien theorem explains how you can rearrange two shapes with the same area into each other through cutting, but Dudeney's method is unique because the pieces stay connected during the transformation.
  3. Recent research proved that you can't turn a triangle into a square using fewer than four pieces without flipping any. This shows how specific and tricky these geometric dissections can be.

The Network Effect- The phenomenon enabling trillion dollar social media industry [Math Mondays]

Technology Made Simple β€’ 79 implied HN points β€’ 20 Jun 23
πŸ•Ή Technology Mathematics
  1. The Network Effect refers to a concept where the value of a product/service increases as more people use it, making the network more valuable for each participant.
  2. The power of the Network Effect can be understood mathematically; as more individuals join a network, the connections exponentially increase, making the system more useful for outsiders.
  3. Businesses/systems built around the Network Effect are powerful due to factors like increased value with more users, a growing network, and the ability to reshape industries and drive innovation.

Antiprisms

Pershmail β€’ 78 implied HN points β€’ 24 Apr 23
πŸ”¬ Science Mathematics
  1. Antiprisms are shapes made by connecting triangles to the edges of bases, twisting one base to fit them together
  2. An antiprism has specific properties like vertices, edges, and faces, which can be calculated using Euler's polyhedron formula
  3. Square antiprisms, or 'squaps', have intriguing features like cross-sections of octagons and can be understood with geometry toys

Links for 2023-05-09

Axis of Ordinary β€’ 78 implied HN points β€’ 09 May 23
πŸ•Ή Technology Mathematics
  1. A call for a Manhattan Project for AI safety and alignment
  2. New AI method called AdaSubS that adapts the planning horizon based on subgoals
  3. Various research papers on AI, language models, and mathematics discussed

The Toughest Math Benchmark Ever Built

TheSequence β€’ 133 implied HN points β€’ 17 Nov 24
πŸ•Ή Technology Mathematics
  1. Frontier Math is a really tough math test designed for AI. It has new, unique problems that are hard for AI to solve, testing deeper reasoning skills.
  2. Many AI models do well on easier math problems but struggle with Frontier Math. They often can't combine ideas creatively like a human can.
  3. This benchmark shows the big gap between current AI abilities and true mathematical understanding, highlighting the need for better AI reasoning.

Games in projective space

A Piece of the Pi: mathematics explained β€’ 48 implied HN points β€’ 16 Jun 25
πŸ”¬ Science Mathematics
  1. Projective geometry removes the concepts of distance and parallel lines, which changes how we think about shapes and space. It's a unique way to understand geometry differently.
  2. In projective space, there are still points, lines, and planes, but the rules are different from traditional geometry. This can lead to interesting and complex interactions.
  3. Games can be explored in the context of projective space, allowing for creative new strategies and outcomes based on its unique properties.

The curvature of space

Infinitely More β€’ 33 implied HN points β€’ 27 Jul 25
πŸ“– Philosophy Mathematics
  1. Spherical geometry has positive curvature, which means circles on a sphere are smaller than expected compared to flat surfaces.
  2. In hyperbolic space, there are way more locations nearby than in regular space, making it easier to get lost or have many places to explore.
  3. Although spherical and hyperbolic geometries are quite different, they can seem similar to a person at a small scale, just like how our everyday experience seems like flat geometry.

The maximum number of SETs for 12 cards

A Piece of the Pi: mathematics explained β€’ 90 implied HN points β€’ 10 Feb 25
πŸ”¬ Science Mathematics
  1. The game SET uses 81 cards that have four qualities: quantity, shading, color, and shape. Players look for sets of three cards where each quality is either all the same or all different.
  2. SET can be understood through linear algebra, where each card is represented as a four-dimensional vector. If the vectors for three cards add up to zero, they form a valid set.
  3. Recent research showed that with 12 cards, a maximum of 14 sets can be formed, and they provided proofs for similar results with fewer cards. This reveals interesting mathematical properties of the game.

Infinitesimals revisited

Infinitely More β€’ 38 implied HN points β€’ 04 Jul 25
πŸ“– Philosophy Mathematics
  1. Infinitesimals were once thought to be nonsense in calculus but actually led to important mathematical breakthroughs. They help us understand changes in functions in a very effective way.
  2. Nonstandard analysis, introduced in the 1960s, provides a solid way to use infinitesimals rigorously through hyperreal numbers. This helps to connect the old and modern approaches in calculus.
  3. Different perspectives on nonstandard analysis can lead to various mathematical ideas and research directions, showing that there's not just one correct way to approach mathematical concepts.

Fast Biology

Asimov Press β€’ 270 implied HN points β€’ 20 Feb 24
πŸ”¬ Science Mathematics
  1. The concept of viewing time differently through the lens of the Minute Man and the Millennium Man prompts questions about our understanding of speed and time in the world.
  2. Biological processes at the cellular level can occur at astonishing speeds, with enzymes performing millions of chemical reactions per second and protein 'motors' spinning thousands of times a minute.
  3. Scientists use innovative experiments to directly observe rapid biological processes, such as watching ATP synthase spin or tracking ribosomes moving along messenger RNA strands, to gain a deeper understanding of the intricate workings of life.

Introducing QF Cities

Quantum Formalism β€’ 39 implied HN points β€’ 20 Jan 24
🚌 Education Mathematics
  1. QF Cities is an initiative by Quantum Formalism to create local hubs for learning math and quantum information science.
  2. They are looking for volunteers to run QF programs in their cities and require proof of at least ten participants to request sponsorship.
  3. The timelines for the QF Cities program include an application deadline of March 22 and a kickstart date of June 21.

Summer haze comes with ink

Vesuvius Challenge β€’ 34 implied HN points β€’ 04 Jul 25
πŸ”¬ Science Mathematics
  1. Researchers are using advanced scanning techniques to read ancient carbonized scrolls. They hope to find ways to read more ink that isn't visible to the naked eye.
  2. They are experimenting with different scanning methods and technologies to better capture the details of the scrolls.
  3. The research team is committed to sharing their findings more often to keep the community updated on their progress.

Bell's Inequality - Guide and Walkthrough

Donkeyspace β€’ 6 implied HN points β€’ 08 Dec 25
πŸ”¬ Science Mathematics
  1. Bell's theorem shows that the universe is fundamentally non-local, meaning particles can be connected no matter how far apart they are. This idea challenges our traditional understanding of space and distance.
  2. The CHSH game illustrates how entangled particles can outperform classical strategies by showing that Alice and Bob can get better results by measuring angles differently. This surprising outcome demonstrates the strange nature of quantum mechanics.
  3. Understanding Bell's inequality reshapes how we see physical laws; it's more like a set of logical rules rather than forces acting on objects. This perspective changes how we think about the universe and its fundamental nature.

Constructing space-filling curves

A Piece of the Pi: mathematics explained β€’ 90 implied HN points β€’ 30 Dec 24
πŸ”¬ Science Mathematics
  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.