The hottest Mathematics Substack posts right now

And their main takeaways
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Top Science Topics

2x2 games will change your life

Wyclif's Dust • 7 HN points • 24 Feb 24
🚌 Education Mathematics
  1. Mathematics can change the way you think by showing how words correspond to underlying structures.
  2. 2x2 games, like the Prisoner's Dilemma, are simple models that offer powerful insights into cooperation, trade, and decision-making.
  3. Understanding game theory, particularly 2x2 games, can help in analyzing real-world scenarios such as economics, politics, and social interactions.

Influential Mathematicians: Sophus Lie

Quantum Formalism • 19 implied HN points • 13 Aug 20
🔬 Science Mathematics
  1. Sophus Lie was a Norwegian mathematician who made significant contributions to mathematics, developing the theory of continuous transformation groups that later led to Lie groups and Lie algebras.
  2. Lie Groups and Lie Algebras, named after Sophus Lie, are essential in the Hilbert space formalism of quantum mechanics, specifically in understanding symmetry and operators in quantum systems.
  3. Although Sophus Lie did not directly contribute to quantum formalism, his mathematical work has had a profound influence on areas of mathematics that are crucial to understanding quantum mechanics.

Influential Mathematicians: Maurice René Fréchet

Quantum Formalism • 19 implied HN points • 23 Jul 20
🔬 Science Mathematics
  1. Maurice René Fréchet, a disciple of Jacques Hadamard, made significant contributions to mathematics through his work on metric spaces and abstract spaces, laying the groundwork for modern mathematical formalism, including quantum mechanics.
  2. Fréchet's research on functional analysis has influenced the development of the quantum formalism, allowing for the creation of abstract concepts crucial in understanding quantum mechanics.
  3. The Riesz–Fréchet representation theorem plays a key role in making mathematical sense of Dirac's bra-ket notation used in quantum mechanics, showcasing the impact of Fréchet's work in this field.

Influential Mathematicians: Jacques Hadamard

Quantum Formalism • 19 implied HN points • 17 Jun 20
🔬 Science Mathematics
  1. Jacques Hadamard's early struggles in mathematics did not deter him from becoming an influential mathematician of the 20th century.
  2. Hadamard's work on functions of a complex variable and entire functions laid significant groundwork for future mathematical developments.
  3. Although Hadamard did not directly contribute to quantum formalism, his work on functional analysis played a crucial role in its mathematical foundation.

Content structure survey

Quantum Formalism • 19 implied HN points • 04 Jun 20
🔬 Science Mathematics
  1. The content structure of the Quantum Formalism newsletter involves foundational content on mathematics concepts like set theory, abstract algebra, topology, and analysis.
  2. Specific Quantum Formalism content focuses on topics like the genesis of quantum formalism, functional analysis, measure theory, and the formalism of quantum mechanics.
  3. The newsletter team is open to subscriber feedback through a survey to adjust and tailor the content structure to meet the needs of their diverse audience.
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The Power of Mathematics: Can We Hack the Next-Gen Quantum Founders?

Deep-Tech Newsletter • 19 implied HN points • 22 Sep 20
🔬 Science Mathematics
  1. The free mathematical course by Zaiku Group is attracting professionals from diverse backgrounds, aiming to equip them with advanced mathematical knowledge for fields like quantum algorithms.
  2. The power of mathematical abstraction was showcased by Zaiku Group's co-founder, Bambordé Baldé, who reconstructed the notion of probability measure using basic set-theoretic concepts.
  3. The course covers foundational topics over 12 weeks, with plans for fireside chats involving industry and academia experts to provide guidance and answer questions from learners.

The orders of infinity

Infinitely More • 12 implied HN points • 11 Mar 23
🔬 Science Mathematics
  1. Real-valued functions can exhibit various behaviors as they approach infinity.
  2. Different functions can have the same behavior at infinity, based on their rates of growth.
  3. Defining an equivalence relation helps capture the idea of functions having the same behavior at infinity.

Applications of induction

Infinitely More • 5 implied HN points • 13 Mar 24
🚌 Education Mathematics
  1. Induction is about the impossibility of minimal counterexamples, and it comes in various forms like common induction and strong induction.
  2. Flexible use of induction is key - choose the valid form that best fits your proof.
  3. Differentiate between examples and proofs - examples can provide insight but don't prove universal statements.

Issue 13: Deep heat

The Works in Progress Newsletter • 6 implied HN points • 15 Nov 23
🔬 Science Mathematics
  1. Mathematics played a crucial role in shaping the modern world through geometry, algebra, and industrial machinery.
  2. Basic mathematics training in Europe from the 1200s to 1800s drove technological advancements in various fields.
  3. Competition between states, including war, was a key factor in the economic development of Europe, pushing states to improve governance.

Understanding The Monte Carlo Method

Photon-Lines Substack • 6 HN points • 20 Jul 23
🔬 Science Mathematics
  1. The Monte Carlo method uses random sampling to estimate complex mathematical results or simulate probabilistic events.
  2. It is applicable in various fields like finance, physics, engineering, risk analysis, environmental modeling, manufacturing, and artificial intelligence.
  3. The method involves generating a large number of random samples to approximate outcomes when exact solutions are difficult to obtain analytically.

Why I hate Pi Day

Logging the World • 1 HN point • 14 Mar 23
🔬 Science Mathematics
  1. Pi Day can be annoying for some mathematicians due to the overemphasis on the beauty of the Pi formula and memorizing digits of Pi.
  2. The beauty in mathematics is not just about formulas like Pi, but also in the precise form of logical arguments and the way pieces fit together like a complex mechanism.
  3. Fourier analysis, involving Fourier transform and harmonics, is a powerful tool used in various scientific fields beyond Pi Day celebrations.

Why does gradient descent work?

The Palindrome • 5 implied HN points • 06 Apr 23
🔬 Science Mathematics
  1. In machine learning, gradient descent is used to find local extrema by following the direction of steepest ascent or descent.
  2. Understanding derivatives helps us interpret the rate of change, such as speed in physics.
  3. Differential equations provide a mathematical framework to understand gradient descent and optimization, showing how systems flow towards equilibrium.

The Computer Scientist's Guide to Graph Theory, ep. 02

The Palindrome • 3 implied HN points • 13 Dec 23
🔬 Science Mathematics
  1. Matching problems can be modeled using bipartite graphs where no edges go between vertices of the same type.
  2. In graph theory, a full matching of one partition of a bipartite graph implies that every vertex in that partition has at least as many neighbors in the other partition.
  3. Hall's theorem provides a necessary and sufficient condition for determining the existence of a full matching in a bipartite graph.

Inventing Mathematics as a Kid

Kids Who Love Math • 3 HN points • 17 Nov 23
🚌 Education Mathematics
  1. Teaching math by involving kids to solve practice problems first can lead to deeper understanding
  2. Encouraging kids to struggle through problem-solving helps them build important skills and confidence
  3. Presenting problems that require critical thinking before introducing techniques can enhance learning and creativity

Is probability frequentist or Bayesian?

The Palindrome • 3 implied HN points • 14 Aug 23
🔬 Science Mathematics
  1. Probability is a number that quantitatively measures the likelihood of events, always between 0 and 1.
  2. Probability is a well-defined mathematical concept, separate from how probabilities are assigned.
  3. The frequentist and Bayesian schools of thought differ in how they assign probabilities, but each has its own advantages in different situations.

The mathematics of optimization for deep learning

The Palindrome • 2 implied HN points • 12 Feb 24
🕹 Technology Mathematics
  1. The post discusses the mathematics of optimization for deep learning - essentially minimizing a function with many variables.
  2. The author reflects on their progression since 2019, highlighting growth and improvement in their writing.
  3. Readers can sign up for a 7-day free trial to access the full post archives on the topic of math and machine learning.

Potential versus actual infinity

Infinitely More • 3 HN points • 14 Apr 23
📖 Philosophy Mathematics
  1. Mathematicians and philosophers may disagree on the nature of existence of infinite collections or infinite objects.
  2. According to potentialism, natural numbers are potentially infinite, allowing for more to be added continuously.
  3. Consider exploring potentialism and actualism for different perspectives on the concept of infinity.

Epsilons, no. 3: The LU decomposition

The Palindrome • 3 implied HN points • 27 Mar 23
🔬 Science Mathematics
  1. Matrix factorizations are a key part of linear algebra, used for inverting matrices and simplifying determinants.
  2. The LU decomposition method involves breaking a matrix into upper and lower triangular forms.
  3. Linear algebra helps in solving systems of linear equations by transforming them into echelon form using operations like multiplying by scalars and adding equations.

Epsilons, no. 1: The geometric series

The Palindrome • 3 implied HN points • 08 Mar 23
🔬 Science Mathematics
  1. The geometric series is a key concept in mathematics with many practical applications.
  2. Deriving the closed-form expression of the geometric series involves understanding its partial sums and limiting behavior.
  3. The geometric series is convergent for |q| < 1 and has a simple closed-form expression.

Busy Beaver, the current BB(5) conjecture and bbchallenge.org

Fikisipi • 1 HN point • 24 Jun 24
🕹 Technology Mathematics
  1. The Busy Beaver function is a mathematical concept related to Turing Machines that aims to find the machine performing the most operations without entering an endless loop. It's a fun way to think about extremely large numbers.
  2. Professor Scott Aaronson made a conjecture that the value of BB(5) is 47,176,870, which is a big number in the context of the Busy Beaver problem. This means trying to determine how many steps the best machine with 5 states can make.
  3. A group called bbchallenge.org is working together to solve this conjecture and make progress on understanding BB(5). They've made some recent updates and are excited about their upcoming findings.

Machine learning is not just statistics

The Palindrome • 2 implied HN points • 09 Aug 23
🔬 Science Mathematics
  1. Machine learning heavily utilizes statistics, but it is not just applied statistics.
  2. Probability enables reasoning about uncertainty, while statistics quantifies and explains it.
  3. Probability theory provides tools to deal with missing information and formulate models with likelihood measures.

Announcing posetree.py: Wrangling Timestamps and Transforms for Robots

General Robots • 2 HN points • 10 Jul 23
🕹 Technology Mathematics
  1. Posetree.py is a library for dealing with poses and transforms in robotics, making code more readable and reducing common bugs.
  2. Understanding the distinction between transforms, poses, and frames is crucial for clarity in robotics code.
  3. The 'timestamps' capability of posetree.py allows for expressing powerful ideas with simple code by automatically handling frame motion.

Steelmanning Anti-Utilitarianism

Superb Owl • 1 HN point • 04 Mar 24
📖 Philosophy Mathematics
  1. Quantifying morality through Utilitarianism can be limiting, as it may lead to extremist views and overlook other sources of moral guidance.
  2. Trying to quantify morality using mathematical frameworks can obscure the complexity of human well-being and lead to oversimplified moral judgements.
  3. Mathematizing ethics can allow for biases to be disguised as objective truths, potentially leading individuals to act against their own moral compass.