The hottest Mathematics Substack posts right now

And their main takeaways
Category
Top Science Topics

Which derivative are you?

Sunday Letters β€’ 59 implied HN points β€’ 10 Jul 22
πŸ•Ή Technology Mathematics
  1. Derivatives help us understand how things change over time. The first derivative tells us how fast something is changing, while the second shows how that rate of change is changing.
  2. In conversations about technical issues, we often focus on the current situation. Instead, we should look at how we can improve that situation over time.
  3. Thinking about changes as derivatives can help us tackle complex problems. By considering not just the current state but how to change it over time, we can better manage projects.

Derivatives in One, Two, and Billion Variables

The Palindrome β€’ 2 implied HN points β€’ 25 Nov 25
🚌 Education Mathematics
  1. Derivatives help us understand how a function changes. They're key to training models, especially in machine learning.
  2. To minimize errors in models, we use gradient descent, which relies on finding the gradient using derivatives.
  3. Computational graphs represent our mathematical models visually, making it easier to track how inputs lead to outputs.

What's on your mind?

Kids Who Love Math β€’ 83 implied HN points β€’ 16 Aug 23
🚌 Education Mathematics
  1. The author is curious about your thoughts and questions related to teaching math to kids
  2. The author shares information about their kids' math education and current activities
  3. The author is open to sharing book highlights and is interested in knowing if you'd like to read about them

Undecidability Isn't So Bad

Bzogramming β€’ 22 implied HN points β€’ 07 Dec 24
πŸ•Ή Technology Mathematics
  1. Some problems in computing are called undecidable, which means we can't find a definite solution for them. However, that doesn’t mean we can’t approach them creatively and get some useful results.
  2. When working with programs, understanding their behavior can often reveal hidden bugs. If a program doesn't behave the way we expect, it might be a sign that something is wrong in the code.
  3. There are smarter ways to analyze code than just throwing our hands up and saying it’s impossible. Advanced tools are already in place in many programming environments, but they often work behind the scenes without us being aware of them.

Hexagonal knot mosaics

A Piece of the Pi: mathematics explained β€’ 24 implied HN points β€’ 03 Nov 24
πŸ”¬ Science Mathematics
  1. Hexagonal knot mosaics are a way to represent knots on a hexagonal board. You can use different types of tiles to create them.
  2. There are three categories of hexagonal mosaics: standard, semi-enhanced, and enhanced. Each type has different rules about how crossings can occur.
  3. Research has shown the maximum number of crossings you can achieve in these mosaics. Enhanced mosaics can have the most crossings, while standard ones allow the least.
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The Role of Advanced Mathematics in AGI: Could Abstract Theories Pave the Way?

Deep-Tech Newsletter β€’ 1 HN point β€’ 12 Jul 24
πŸ”¬ Science Mathematics
  1. The AI industry is investing heavily in large language models and AGI, but faces financial challenges and uncertainty in meeting high expectations.
  2. To achieve AGI, more advanced mathematical techniques beyond current ML algorithms like gradient descent may be needed, with Category Theory showing promise.
  3. Barriers exist in understanding Category Theory for AGI due to its abstract nature, but efforts are being made to empower AI researchers and engineers with necessary mathematical knowledge.

The 10 Most Popular Posts of the Palindrome in 2025

The Palindrome β€’ 1 implied HN point β€’ 23 Dec 25
🚌 Education Mathematics
  1. The most-read posts emphasize math and foundational CS for machine learning, covering topics like a mathematics roadmap, algorithmic analysis, graph theory, and practical skills such as coding on paper and representing graphs.
  2. A holiday promotion offers a 30% lifetime discount on the annual paid subscription, which unlocks paid-only content and helps fund more math and machine learning material for the community.
  3. Subscriber-count milestones will unlock community perks (mini-courses, a dedicated Manim animator, and a full-time writer), and the publication invites feedback while planning to expand and reinvest in 2026.

The 10 Most Important Lessons 20 Years of Mathematics Taught Me

The Palindrome β€’ 7 implied HN points β€’ 25 Jun 25
🚌 Education Mathematics
  1. Mastery takes time and effort. There are no shortcuts, so you need to be willing to put in the work to really understand something.
  2. Breaking rules can lead to new discoveries, but you need to know the rules well first. It's like finding a tiny crack in a tough problem to solve it.
  3. Learning is often best done by solving problems yourself. Getting hands-on experience helps deepen your understanding and skills.

Finding Fifteen

Infinitely More β€’ 15 implied HN points β€’ 20 Jan 25
🚌 Education Mathematics
  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.

Mutual and bi-interpretation of models

Infinitely More β€’ 17 implied HN points β€’ 14 Dec 24
πŸ“– Philosophy Mathematics
  1. Mutual interpretation means that two models can understand each other. Each model can be explained using the features of the other.
  2. When you interpret one model within another, it creates a loop of understanding. You can go back and forth between the two models, revealing deeper connections.
  3. Bi-interpretability is when both models not only understand each other but are actually related in a stronger way. This offers even more insights into their structure.

Real Morality?

Orbis Tertius β€’ 71 implied HN points β€’ 29 May 23
πŸ“– Philosophy Mathematics
  1. Morality can be viewed as a decision-support system, not an absolute truth.
  2. Mathematics is a created formal system based on chosen axioms.
  3. In both math and morality, the axioms chosen shape the system and guide evaluations.

Tips for new community members

Quantum Formalism β€’ 39 implied HN points β€’ 09 Mar 22
🚌 Education Mathematics
  1. Start with the 'Foundation Module' YouTube playlist for basics on finite-dimensional Hilbert spaces and quantum mechanics postulates
  2. Consider auditing crash courses on topics like Topology & Differential Geometry for Lie Groups and Group Theory for advanced knowledge
  3. Exploring topics like smooth manifolds and Group Theory can be valuable not just in quantum computation but also in applied fields like ML and Cryptography

Permutations and combinations

Infinitely More β€’ 17 implied HN points β€’ 17 Nov 24
🚌 Education Mathematics
  1. A permutation is just a way to rearrange a list of objects. For example, with three letters like 'a', 'b', and 'c', you can arrange them in six different ways.
  2. The factorial of a number shows how many ways you can arrange that many objects. For example, 5! equals 120 because it's 5 times 4 times 3 times 2 times 1.
  3. When choosing items from a group without caring about the order, we use combinations. The formula for this is called 'n choose k', which helps calculate how many ways you can select items.

The Magic of Monads[Math Mondays]

Technology Made Simple β€’ 19 implied HN points β€’ 01 Nov 22
πŸ•Ή Technology Mathematics
  1. Monads in programming are a software engineering pattern for code reusability and writing pipelines at scale.
  2. Monads are great for chaining functions together, abstracting implementation details for clear code.
  3. Using monads is helpful when dealing with a group of objects that behave similarly, allowing for reusable code and easier implementation of common behaviors.

The surreal numbers

Infinitely More β€’ 41 implied HN points β€’ 06 Jan 24
πŸ”¬ Science Mathematics
  1. The surreal numbers unify various number systems into one comprehensive system.
  2. Surreal numbers are generated through a recursive process of completion and ordering.
  3. The surreal number generation rule involves separating existing numbers into lower and upper sets to create new numbers.

Why Deep Learning is everywhere [Math Mondays]

Technology Made Simple β€’ 19 implied HN points β€’ 25 Oct 22
πŸ•Ή Technology Mathematics
  1. Deep Learning is a subset of Machine Learning that uses Neural Networks with many layers, introducing non-linearity in functions which is crucial for its success.
  2. Deep Networks work well because they can approximate any continuous function by combining non-linear functions, allowing them to tackle complex problems.
  3. The widespread use of Deep Learning is driven by its trendiness and efficiency, appealing to many due to its ability to provide results without extensive data analysis or training.

Community update | January 2022

Quantum Formalism β€’ 39 implied HN points β€’ 25 Jan 22
🚌 Education Mathematics
  1. The Group theory crash course will start soon, beginning with basic concepts like binary operations and moving to more complex ones over time.
  2. The course will cover monoids, submonoids, monoid homomorphisms, and eventually groups in a pedagogically friendly manner.
  3. To adjust the course level accordingly, participants are requested to complete a short survey about their technical background.

The Game of Diophantus

ThΓ‘i | Hacker | Kα»Ή sΖ° tin tαΊ·c β€’ 1 HN point β€’ 26 Jun 24
πŸ”¬ Science Mathematics
  1. Diophantus, an ancient Greek mathematician, is known for solving equations with two unknowns and his work formed the basis for modern cryptographic algorithms like ECC.
  2. The lost works of Diophantus, preserved during historical upheavals, resurfaced over centuries, impacting later mathematicians like Fermat, who posed the famous Last Theorem.
  3. The journey from Tycho, who pioneered data science with precise astronomical observations, to Kepler discovering that planetary orbits are ellipses, connects to the development of elliptic curves in mathematics.

Probability Distributions 101

Olshansky's Newsletter β€’ 68 implied HN points β€’ 20 Feb 23
πŸ”¬ Science Mathematics
  1. Probability is about the likelihood of events happening.
  2. Probability distributions are functions describing random events over a sample space.
  3. Key concepts include random variables, probability mass function, probability density function, cumulative distribution function, Gaussian distribution, and Bernoulli distribution.

Under the microscope: Skeleton Warriors (Saturn, PlayStation)

Rings of Saturn β€’ 14 implied HN points β€’ 08 Dec 24
πŸ•Ή Technology Mathematics
  1. The article talks about finding cheat codes for the game Skeleton Warriors on the Saturn and PlayStation consoles. It highlights how some of these codes were previously unknown and required research and coding to uncover.
  2. The author used a combination of brute force methods and mathematical approaches to determine the missing cheat codes. They collaborated with a cryptographer to solve complex hashing problems involved in the game.
  3. Mick West, a cofounder of Neversoft, contributed to the findings by sharing source code and insights about how the cheats work in the game. This shows a connection between game development and player discovery.

The game of Nim

Infinitely More β€’ 10 implied HN points β€’ 07 Feb 25
🚌 Education Mathematics
  1. The game of Nim is based on a smart mathematical strategy that lets informed players almost always win against those who don't know the trick.
  2. In Nim, players take turns removing coins from piles, and the goal is to take the last coin to win.
  3. Anyone, even kids, can learn the winning strategy and easily beat more experienced players who don't know it.

Lattices

Infinitely More β€’ 33 implied HN points β€’ 17 Jan 24
πŸ”¬ Science Mathematics
  1. A lattice is an order relation where every pair of elements has a least upper bound and a greatest lower bound.
  2. In lattices, the join of two elements is the larger of them and the meet is the smaller of them.
  3. Every linear order, set of positive integers, Boolean algebra, and field of sets can be considered lattices.

The 15 Hats Problem

Metarational β€’ 59 implied HN points β€’ 23 Apr 21
πŸ”¬ Science Mathematics
  1. The 15 Hats Problem is a complex brain teaser that involves guessing the color of hats based on a strategy involving bitwise XOR operations.
  2. Strategies for simpler versions of the problem, like 3 hats, helped in understanding and developing a winning strategy for the 15 hats scenario.
  3. The key to solving the 15 Hats Problem involves clever observations about modular arithmetic, powers of 2, and the use of bitwise XOR operations.

Measure Theory and Functional Analysis

Quantum Formalism β€’ 19 implied HN points β€’ 05 Aug 22
🚌 Education Mathematics
  1. The upcoming lecture series on Measure Theory and Functional Analysis requires some prerequisite reading, especially understanding sets, functions, logic, proofs, least upper bound, greatest lower bound, density of rationals in reals, and cardinality of sets.
  2. If you don't have access to the recommended book, you can still find explanations of the key topics online.
  3. A live Q&A session will be held on August 20th at 11:00am Eastern time for further engagement and clarification.

The eventual domination order

Infinitely More β€’ 30 implied HN points β€’ 11 Jan 24
πŸ”¬ Science Mathematics
  1. The eventual domination order involves comparing functions in a specific way
  2. In the eventual domination order, every countable sequence of functions is strictly bounded
  3. This order prohibits one from climbing a simple ladder to reach the top

Lecture 09 Replay + LOGML 2022

Quantum Formalism β€’ 19 implied HN points β€’ 28 Jun 22
🚌 Education Mathematics
  1. The lecture 09 replay covering the application of cyclic groups and DLP in cryptography is available on YouTube.
  2. Future plans include a mini-course series on cryptography focusing on DLP cryptographic systems for security assurance.
  3. Joining the Discord Community is encouraged for organizing group study sessions and accessing live study session links.

Picturing symmetry

A Piece of the Pi: mathematics explained β€’ 24 implied HN points β€’ 04 Mar 24
πŸ”¬ Science Mathematics
  1. The order in which symmetries are applied can significantly affect the final result, as shown through reflections and rotations of a square.
  2. Using Cayley graphs can help visualize and calculate products of symmetries.
  3. In symmetry operations, combining reflections and rotations follows specific rules, similar to adding odd and even numbers. Grouping rotations and reflections can simplify understanding complex symmetries.

The interpretation translation

Infinitely More β€’ 10 implied HN points β€’ 07 Dec 24
πŸ”¬ Science Mathematics
  1. You can interpret one mathematical structure using another, which helps express features of the first in terms of the second. This means you find a way to connect different types of math using a common language.
  2. There are many examples of this interpretation, like placing integers inside natural numbers or examining complex numbers through real numbers. These examples show how different math concepts relate to each other.
  3. Understanding how to interpret structures can help us explore logic more deeply, opening up new ways of thinking in math, philosophy, and computer science.

The Science of Science

The Palindrome β€’ 3 implied HN points β€’ 30 Jul 25
πŸ”¬ Science Mathematics
  1. Thinking in terms of probabilities helps us make better judgments when we are not certain. Unlike absolute truths, we can measure how likely something is to be true instead.
  2. Bayes' theorem allows us to update our beliefs based on new evidence. This means we can make smarter decisions by adjusting our understanding as we gather more information.
  3. To figure out causes from effects, we can use conditional probabilities. This helps us connect symptoms, like a headache and sore throat, to possible underlying issues, like the flu, in a more accurate way.

Edge colourings and Archimedean lattices

A Piece of the Pi: mathematics explained β€’ 24 implied HN points β€’ 18 Feb 24
πŸ”¬ Science Mathematics
  1. Edge colorings of graphs are not just recreational, but have practical applications in quantum technology.
  2. Graphs can be colored either by edges or by vertices, with different requirements for each coloring approach.
  3. Vizing's Theorem states that a graph can be edge colored with either the maximum degree or the maximum degree plus one colors.

Do Large Language Models have a "Reasoning Gap"?

The Irregular Voice β€’ 2 HN points β€’ 01 Apr 24
πŸ•Ή Technology Mathematics
  1. Large Language Models (LLMs) may not always exhibit true reasoning abilities, with a potential reliance on memorization instead of learning general techniques.
  2. Synthetic data generation systems like MATH() can be used to explore the reasoning capabilities of LLMs, but may introduce biases if not carefully analyzed and corrected for errors.
  3. Fine-tuning LLMs on specific problem areas can reveal insights into their reasoning abilities, but challenges with longer solutions and complex problem sets may impact performance.