The hottest Mathematics Substack posts right now

And their main takeaways
Category
Top Science Topics

A conversation with Claude

Noahpinion • 12529 implied HN points • 22 Mar 26
🔬 Science Mathematics
  1. AI will rapidly accelerate materials discovery and optimization, helping find candidates for things like room‑temperature superconductors, solid‑state batteries, novel catalysts, and topological or quantum materials while autonomous labs compress the loop from design to experiment.
  2. AI is most powerful where there’s a huge combinatorial search space, good simulation data, and fast experimental feedback (for example drugs, materials, climate parameterizations, and chip design), but it struggles where data are sparse, experiments are slow, or real progress requires new conceptual frameworks; and even when discoveries happen, manufacturability, testing, and regulatory inertia often dominate commercialization timelines.
  3. Beyond simple, teachable laws, AI can uncover complex but reproducible "Cloud Laws" that humans can’t easily compress or explain, potentially transforming biology, neuroscience, and social systems; these advances may function as powerful black‑box tools rather than neat, human‑readable theories.

All paths point downhill

arg min • 218 implied HN points • 31 Oct 24
🕹 Technology Mathematics
  1. In optimization, there are three main approaches: local search, global optimization, and a method that combines both. They all aim to find the best solution to minimize a function.
  2. Gradient descent is a popular method in optimization that works like local search, by following the path of steepest descent to improve the solution. It can also be viewed as a way to solve equations or approximate values.
  3. Newton's method, another optimization technique, is efficient because it converges quickly but requires more computation. Like gradient descent, it can be interpreted in various ways, emphasizing the interconnectedness of optimization strategies.

Natural Ordinal Addition

Infinitely More • 7 implied HN points • 15 Mar 26
🔬 Science Mathematics
  1. The natural sum and product (Hessenberg operations) make the ordinals into a commutative semiring, contrasting with standard ordinal arithmetic where addition and multiplication are not commutative.
  2. The natural ordinal operations match the operations on surreal numbers, so the ordinals under natural addition and multiplication form a subsemiring of the surreals.
  3. There are five independent, self-contained ways to define the natural sum and product—order-theoretic, computational, proof-theoretic, and others—and all five are equivalent, giving complementary perspectives and routes to generalization.

Basic Linear Algebra Subprogramming

arg min • 178 implied HN points • 29 Oct 24
🕹 Technology Mathematics
  1. Understanding how optimization solvers work can save time and improve efficiency. Knowing a bit about the tools helps you avoid mistakes and make smarter choices.
  2. Nonlinear equations are harder to solve than linear ones, and methods like Newton's help us get approximate solutions. Iteratively solving these systems is key to finding optimal results in optimization problems.
  3. The speed and efficiency of solving linear systems can greatly affect computational performance. Organizing your model in a smart way can lead to significant time savings during optimization.

A use-theory of testing

arg min • 436 implied HN points • 24 Oct 24
🔬 Science Mathematics
  1. Statistical tests are designed to help separate real signals from random noise. It's not just about understanding what they mean, but what they can do in practical situations.
  2. Many people misuse statistical tests, which can lead to misunderstandings about their purpose. Communities should establish clear guidelines on how to use these tests correctly.
  3. The main function of statistical tests is to regulate opinions and decisions in various fields like tech and medicine. They help ensure that important standards are met, rather than just preventing errors.
Get a weekly roundup of the best Substack posts, by hacker news affinity:

Approximation game

lcamtuf’s thing • 2244 implied HN points • 28 Feb 26
🔬 Science Mathematics
  1. Simple rational numbers (like p/q) have only a few very-close different fractions with small denominators; once denominators reach q or larger you can’t get new inexact approximations that beat the 1/b^2 error threshold.
  2. Irrational numbers, by contrast, admit infinitely many surprisingly accurate rational approximations; Dirichlet’s pigeonhole argument guarantees infinitely many fractions a/b with error on the order of 1/b^2 (for example 22/7 and 355/113 for π).
  3. Intuitively, rationals form a uniform grid so their gaps limit how close other fractions can get, while irrationals sit inside those gaps and repeated multiples plus the pigeonhole principle produce arbitrarily close rational hits, which is the essence of Diophantine approximation.

Unreal numbers

lcamtuf’s thing • 4489 implied HN points • 15 Feb 26
🔬 Science Mathematics
  1. Natural numbers can be built from a base element (zero) and a successor rule, and addition and multiplication follow from simple recursive definitions.
  2. Integers and rationals are formed by ordered pairs and equivalence classes so subtraction and division have in-system representations, and these extended sets remain countable.
  3. Computable numbers are those a Turing machine can approximate and are still countable, but the real numbers are uncountable (by diagonalization), so most reals cannot be computed.

“Lights Out” on a triangular grid

A Piece of the Pi: mathematics explained • 30 implied HN points • 22 Mar 26
🔬 Science Mathematics
  1. The triangular Lights Out game reduces to linear algebra over the field with two elements: pressing a button toggles bits mod 2, pressing a button twice cancels, order doesn’t matter, and any solution is a subset of buttons pressed once.
  2. Solvability and uniqueness depend on the kernel of the toggle map: if the kernel is only the empty set (ℓ=0) then every starting state has a unique solution, which occurs for certain side lengths such as 1, 3, 4, 7, 8, 9, 11, 15, 16, 17, 20, and 21.
  3. If the kernel is nontrivial (ℓ>0) there are nonzero button patterns that have no effect and some starting configurations cannot be solved; the kernel is a 2^ℓ-sized vector space over GF(2) and its patterns often form visually striking shapes like the Sierpiński triangle.

You gotta think outside the hypercube

lcamtuf’s thing • 5101 implied HN points • 26 Jan 26
🕹 Technology Mathematics
  1. You can build a tesseract wireframe by extending the same edge-construction rules from a square to a cube and then to 4D, which yields 16 vertices and 32 edges.
  2. Rotations in four dimensions are still planar operations that act on pairs of axes, so animations come from applying familiar 2D rotation formulas to axis pairs like XZ or Z🌀.
  3. There are many ways to project 4D to 2D with different tradeoffs—cavalier, cabinet, isometric, perspective, and fisheye—and a mixed approach (isometric for XYZ plus perspective or fisheye for the fourth axis) gives the clearest, most informative views.

The Shape of Stats to Come

arg min • 634 implied HN points • 10 Oct 24
🚌 Education Mathematics
  1. Statistics often involves optimizing methods to get the best results. Many statistical techniques can actually be viewed as optimization problems.
  2. Choosing a statistical method isn't just about the math—it's also based on beliefs about reality. This philosophical side is important but often overlooked.
  3. There's a danger in relying too much on tools and models we can solve. Sometimes, we force the data to fit our preferred methods instead of being open to the actual complexities.

Designed Interactions

arg min • 257 implied HN points • 15 Oct 24
🚌 Education Mathematics
  1. Experiment design is about choosing the right measurements to get useful data while reducing errors. It's important in various fields, including medical imaging and randomized trials.
  2. Statistics play a big role in how we analyze and improve measurement processes. They help us understand the noise in our data and guide us in making our experiments more reliable.
  3. Optimization is all about finding the best way to minimize errors in our designs. It's a practical approach rather than just seeking perfection, and we need to accept that some questions might remain unanswered.

Inverse Problems

arg min • 515 implied HN points • 03 Oct 24
🔬 Science Mathematics
  1. Inverse problems help us create images or models from measurements, like how a CT scan builds a picture of our insides using X-rays.
  2. A key part of working with inverse problems is using linear models, which means we can express our measurements and the related image or signal in straightforward mathematical terms.
  3. Choosing the right functions to handle noise and image characteristics is crucial because it guides how the algorithm makes sense of the data we collect.

What's the deal with Euler's identity?

lcamtuf’s thing • 5917 implied HN points • 08 Nov 25
🔬 Science Mathematics
  1. Euler's identity, which is e^(iπ) + 1 = 0, connects five important math constants: e, π, 0, 1, and i. It shows how complex numbers and trigonometry blend together in a fascinating way.
  2. The number i is known as the imaginary unit, and it allows us to represent two-dimensional rotations. When we multiply by i, it represents a 90° turn in the complex plane.
  3. Using Euler's formula, we can relate complex exponentials to trigonometric functions. This connection helps us understand circular motion in a mathematical way.

An Inversion Cookbook

arg min • 297 implied HN points • 04 Oct 24
🚌 Education Mathematics
  1. Using modularity, we can tackle many inverse problems by turning them into convex optimization problems. This helps us use simple building blocks to solve complex issues.
  2. Linear models can be a good approximation for many situations, and if we rely on them, we can find clear solutions to our inverse problems. However, we should be aware that they don't always represent reality perfectly.
  3. Different regression techniques, like ordinary least squares and LASSO, allow us to handle noise and sparse data effectively. Tuning the right parameters can help us balance accuracy and manageability in our models.

Great scientists follow intuition and beauty, not rationality

The Intrinsic Perspective • 33817 implied HN points • 30 Dec 24
🔬 Science Mathematics
  1. Great scientists often rely on their gut feelings and a sense of beauty rather than just cold hard logic. This mix of intuition leads to important discoveries.
  2. Famous scientists aren't just rational thinkers; they have quirky beliefs and passions that drive their creativity. This uniqueness helps them come up with groundbreaking ideas.
  3. There's a complex balance between formal science and the imaginative, intuitive side. Embracing both can push the boundaries of what we understand about the universe.

Counting to Epsilon Naught

Infinitely More • 35 implied HN points • 04 Mar 26
🔬 Science Mathematics
  1. Counting ordinals continues past the finite numbers to ω, then ω+1, ω+2, and onward through blocks like ω·2, ω·3, … so that each new limit ordinal begins a new ω-long era.
  2. By iterating these constructions and forming longer and longer exponential towers—ω, ω^ω, ω^(ω^ω), …—we reach ever higher ordinals, and the supremum of all finite such towers is the ordinal ε0.
  3. ε0 is the first ordinal fixed point of exponentiation by ω (so ω^ε0 = ε0), and there is a computable notation system for all ordinals below ε0 with important applications like Goodstein’s theorem and the Hydra game.

Trò chơi của Diophantus

Thái | Hacker | Kỹ sư tin tặc • 2037 implied HN points • 27 Jun 24
🔬 Science Mathematics
  1. The game of Diophantus, an ancient Greek mathematician, has had a lasting impact on cryptography and internet security, with the basis of elliptic curve cryptography originating from his mathematical puzzles.
  2. Diophantus's famous book 'Arithmetica' went missing for centuries but resurfaced to contribute to the advancements in mathematics, leading to significant discoveries like Fermat's Last Theorem.
  3. The study of elliptic curves, inspired by concepts like Kepler's study of ellipses, has become a central focus in mathematics, intersecting various branches like number theory, algebra, and geometry, and even impacting modern technology such as Bitcoin security.

Genomic prediction of IQ is modern snake oil

The Infinitesimal • 1298 implied HN points • 06 Jul 24
🔬 Science Mathematics
  1. Genetic tests claiming to predict IQ are not reliable. They often rely on complex methods that mostly just lead to guesswork.
  2. The accuracy of these genetic predictions is very low, explaining only a tiny fraction of variations in IQ scores. In fact, other factors like age and social environment play a much bigger role.
  3. Many of these predictions confuse people about how genetics really work. It's important to understand that these scores should be treated more like entertainment than serious assessments.

Gödel's beavers, or the limits of knowledge

lcamtuf’s thing • 7958 implied HN points • 30 Jun 25
🔬 Science Mathematics
  1. Gödel's incompleteness theorem shows that in any consistent mathematical system, there are truths that cannot be proven within that system. This means no system can fully capture all mathematical truths.
  2. The busy beaver problem illustrates how there are limits to what we can compute; some functions can't be determined, just like how we can't always know if an algorithm will stop running.
  3. Even though we can create programs that seem powerful, like those that could prove big math ideas, there are inherent limitations to knowledge and computation due to the nature of math itself.

How to turn a number into a rooted forest

A Piece of the Pi: mathematics explained • 90 implied HN points • 01 Mar 26
🔬 Science Mathematics
  1. Matula arborification is a recursive recipe that turns any positive integer into a rooted forest: 1 is the empty forest, 2 is a single node, primes become trees by attaching a new root to the forest of their index, and composites are represented by juxtaposing the trees of their prime factors.
  2. This correspondence is useful in number theory and combinatorics — it can help prove relationships between primes and encodes integer sequences (for example the primeth sequence appears as vertical chains of trees).
  3. The idea also has practical applications in chemistry for canonically labeling alkane structures (with valence limits ruling out some forests), and there are online tools that generate and visualize Matula trees for given integers.

Analyze FIR filters using just high-school algebra

filterwizard • 39 implied HN points • 23 Sep 24
🕹 Technology Mathematics
  1. FIR filters have a finite impulse response, meaning they only remember a limited amount of past input. This makes them predictable and stable, especially for applications needing fast settling times.
  2. You can think of FIR filter coefficients as a polynomial, which allows you to use algebra to analyze and create filters. This approach helps in understanding how changing coefficients affects the filter's behavior.
  3. By factoring the polynomial of an FIR filter, you can create smaller filters that combine to produce the same overall effect. This technique allows for a deeper exploration of filter design, giving you more control over the filter's characteristics.

4.4.1 How Do Transmission Lines Work?

Fields & Energy • 319 implied HN points • 14 Aug 24
🔬 Science Mathematics
  1. Transmission lines work by sending electrical signals through wires, where one wire gets a negative charge and the other gets a positive charge. This creates electric fields that help move energy along the line.
  2. To avoid signal loss and distortion, it's important to balance the electric and magnetic energies in transmission lines. If they are not balanced, the signal can get messed up over long distances.
  3. Oliver Heaviside developed key equations that describe how signals travel through transmission lines. His work highlighted the importance of using both electric and magnetic energies to achieve clear signal propagation.

The Story of the Telegrapher's Equations

Fields & Energy • 259 implied HN points • 16 Aug 24
🔬 Science Mathematics
  1. Oliver Heaviside was a young scientist who created the Telegrapher's Equations in 1876. His work helped connect theories of electromagnetism to practical applications in telecommunication.
  2. Before Heaviside, the diffusion model was the main idea for how signals traveled. Heaviside improved this by showing that signals could travel as waves instead of just spreading out slowly.
  3. The development of these equations was influenced by earlier mathematicians like Fourier and scientists like Lord Kelvin. Heaviside's contribution built on their ideas and advanced the understanding of signal transmission over long distances.

Meta: 5,000

lcamtuf’s thing • 8366 implied HN points • 27 Feb 25
🕹 Technology Mathematics
  1. Reaching 5,000 subscribers is a big deal for a project that went against the usual trends. It's great to see growth, even if it seems small compared to others.
  2. Writing a newsletter is unique because you don't get much direct feedback from readers. It's interesting to see who signs up or leaves but hard to know what they really think.
  3. Three articles worth revisiting cover complex topics: discrete Fourier transforms, fractals, and core concepts in electronic circuits. They offer in-depth discussions that are easy to understand, even for beginners.

Now synthesize your FIR filters using high-school algebra

filterwizard • 19 implied HN points • 27 Sep 24
🕹 Technology Mathematics
  1. You can create FIR filters by breaking them down into smaller parts using simple math. This makes it easier to understand how each piece works together.
  2. The sharp notches or deep points in a filter's response happen because of certain factors in the polynomial. Each notch can be traced back to specific frequencies based on these factors.
  3. To improve a filter's performance, you can add more mathematical pieces to make the response smoother in certain areas. This way, you can customize how the filter behaves at different frequencies.

The Birth of Transmission Line Theory

Fields & Energy • 279 implied HN points • 09 Aug 24
🔬 Science Mathematics
  1. The first Transatlantic Telegraph Cable in 1858 was crucial for developing transmission line theory. It helped researchers understand how to send messages over long distances.
  2. Lord Kelvin created an early model for long cables, focusing on how to evenly spread resistance and capacitance. This helped explain why the first cable failed.
  3. Oliver Heaviside later added the concept of inductance to the equations, which improved the understanding of transmission lines even further.

Adam Kucharski: The Uncertain Science of Certainty

Ground Truths • 3718 implied HN points • 29 Jun 25
🔬 Science Mathematics
  1. Science is about understanding uncertainty and the limits of what we know. It's important to recognize that truth can change as new evidence comes in.
  2. Different types of proof, such as randomized trials or natural experiments, all have their pros and cons. It's crucial to evaluate what type is best for the situation at hand.
  3. Repetition can affect our belief in something, even if it's not true. It's essential to stay open to different viewpoints and challenge our own beliefs.

This week: Bacteria anticipate the seasons! Placebos activate the same neurons as anesthesia! And don’t touch radio towers with hot dogs.

Niko McCarty • 79 implied HN points • 07 Sep 24
🔬 Science Mathematics
  1. Bacteria can sense changes in seasons and adapt to prepare for colder weather. This helps them survive better when temperatures drop.
  2. Placebos work by activating the same brain neurons as pain relief drugs like anesthesia. This shows how our mind can influence our body’s responses.
  3. A fun fact: touching a hot dog to a radio tower can turn it into a speaker. Just a quirky reminder to be careful with food and electronics!

GDP is Inevitable

Cremieux Recueil • 477 implied HN points • 17 Dec 25
🔬 Science Mathematics
  1. When you add up many positively correlated variables with positive weights, different composite scores tend to become very similar because shared covariance grows faster than unique variance, so the sums converge toward perfect correlation as components increase.
  2. GDP will naturally correlate highly with lots of other measures since it aggregates overlapping components (and is sometimes included in other indexes), and aggregation reduces within-group noise which mechanically inflates between-group correlations.
  3. Adding items to make a composite more reliable often makes it harder to distinguish from other composites, so improving reliability can reduce discriminant validity (for example, measures like grit can converge with conscientiousness).

A 15-minute intro to involute gears

lcamtuf’s thing • 6938 implied HN points • 17 Nov 24
🕹 Technology Mathematics
  1. Involute gears are used in many everyday items like toys and cars. Their special shape helps them work smoothly and efficiently.
  2. These gears have specific properties that reduce friction and vibrations, allowing them to transfer motion without problems. This ensures they work together seamlessly.
  3. Understanding the design of involute gears helps in creating various gear types, even unique shapes, making it easier to innovate in mechanical engineering.

A Nonconstructive Existence Proof of Aligned Superintelligence

Transhuman Axiology • 99 implied HN points • 12 Sep 24
🕹 Technology Mathematics
  1. Aligned superintelligence is possible, despite some people thinking it isn't. This idea shows proof that it can exist without needing complicated construction.
  2. Desirable outcomes for AI mean producing results that people think are good. We define these outcomes based on what humans can realistically accomplish.
  3. While the concept of aligned superintelligence exists, it faces challenges. It's hard to create, and even if we do, we can't be sure it will work as intended.

The Power of Limit Thinking

Asimov Press • 515 implied HN points • 06 Nov 25
🔬 Science Mathematics
  1. Limit Thinking helps us figure out the best possible performance of a system. It focuses on the essential features and gives a clear measure of efficiency.
  2. This way of thinking has driven major improvements in technology, like in engines and information theory, by establishing concrete limits to what can be achieved.
  3. In biology, applying Limit Thinking can lead to new discoveries by helping scientists understand the fundamental processes, even in complex systems.

AlphaGeometry2: Impressive accomplishment, but still a long path ahead

Marcus on AI • 3161 implied HN points • 17 Feb 25
🕹 Technology Mathematics
  1. AlphaGeometry2 is a specialized AI designed specifically for solving tough geometry problems, unlike general chatbots that tackle various types of questions. This means it's really good at what it was built for, but not much else.
  2. The system's impressive 84% success rate comes with a catch: it only achieves this after converting problems into a special math format first. Without this initial help, the success rate drops significantly.
  3. While AlphaGeometry2 shows promising advancements in AI problem-solving, it still struggles with many basic geometry concepts, highlighting that there's a long way to go before it can match high school students' understanding in geometry.

On the greats and mathematical style

Infinitely More • 15 implied HN points • 22 Feb 26
🔬 Science Mathematics
  1. Greatness in mathematics is hard to rank because insights can come from many people and eras, and being the first often involves luck since ideas are sometimes "in the air."
  2. Simple, clear, easy-to-understand arguments are especially valued because they are easier to check and to learn from, and playful thought experiments or metaphors help visualize problems and reveal strategies.
  3. There are different successful working styles—long solitary grinds on one problem versus collaborative, social approaches that switch problems—and many practitioners pursue mathematics for the love of the subject rather than for prizes, with online collaboration regularly sparking new work.

Mathematical Maturity in Elementary School

Kids Who Love Math • 419 implied HN points • 14 Nov 25
🚌 Education Mathematics
  1. Mathematical maturity starts with curiosity, connection, and persistence. Kids need to explore math by asking questions and playing with ideas to develop a deeper understanding.
  2. There are four stages of mathematical maturity: Exposure, Pattern Recognition, Internalization, and Creative Mastery. Each stage helps kids build their confidence and skills in math.
  3. It's important to normalize getting stuck and encourage kids to embrace challenges. Learning is about asking better questions and exploring different ways to solve problems, not just getting the right answers.

Tormented by an Urn

rachaelmeager • 535 implied HN points • 04 Jun 24
🚌 Education Mathematics
  1. The Polya urn model, though simple at first glance, reveals the complexity of statistics and emphasizes the importance of understanding problems deeply before attempting to solve them.
  2. Teaching and learning in math are not just about facts; they require creativity and passion to engage students, much like how poets perceive deeper meanings in their art.
  3. There is a strong connection between the arts and sciences, where both disciplines can benefit from understanding each other, and students should learn foundational concepts in both to grasp the complexities of the world.

Surfaces and sequence graphs

A Piece of the Pi: mathematics explained • 66 implied HN points • 31 Jan 26
🔬 Science Mathematics
  1. You can build a graph by placing n vertices in a cycle and linking them according to the rank order of the first n terms of a real sequence, and as n grows these sequence graphs reveal striking geometric patterns.
  2. Graphs coming from the Kronecker sequence (multiples of the golden ratio mod 1) can be drawn on a torus without crossings, typically after removing the edge from n−1 to 0.
  3. Graphs from the van der Corput sequence embed into the Chamanara surface — a highly singular, infinite‑handle (“Loch Ness monster”) surface made by identifying shrinking boundary segments of a square — and finite approximations avoid the worst singularities so they can be visualized.

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

Fields & Energy • 279 implied HN points • 10 Jun 24
🔬 Science Mathematics
  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.

Cantor Normal Form

Infinitely More • 23 implied HN points • 12 Feb 26
🔬 Science Mathematics
  1. Cantor normal form gives every ordinal a unique, canonical representation, acting like a numeral system built on base ω instead of base ten.
  2. The notation is as powerful and convenient for working with ordinals as the decimal system is for ordinary numbers, so it makes representing and comparing ordinals systematic and clear.
  3. Using Cantor normal form simplifies ordinal arithmetic because many terms cancel or "disappear," and it provides a foundation for further topics like the surreal numbers.

Indecomposable Ordinals

Infinitely More • 38 implied HN points • 01 Feb 26
🔬 Science Mathematics
  1. An ordinal λ>0 is additively indecomposable when the sum of any two smaller ordinals is still below λ; equivalently, it cannot be expressed as a sum of two smaller ordinals.
  2. Concrete examples: ω is additively indecomposable, ω·2 is not, and the next indecomposable after ω is ω^2, so such ordinals must exceed every finite multiple of ω.
  3. There are analogous notions for multiplication and exponentiation — multiplicative or exponential indecomposability raises similar questions about whether indecomposable equals irreducible and motivates a full characterization via tools like Cantor normal form.