The hottest Mathematics Substack posts right now

And their main takeaways
Category
Top Science Topics

Mathematics of Machine Learning official release announcement!

The Palindrome • 8 implied HN points • 29 Jan 25
🕹 Technology Mathematics
  1. The book 'Mathematics of Machine Learning' is set to be published soon and will be available in a physical version. You can pre-order it at a discounted price now.
  2. It focuses on important math concepts needed for machine learning, including linear algebra, calculus, and probability theory. Understanding these areas is crucial for building effective models in machine learning.
  3. The author shares a personal journey of creating the book, which was inspired by his experiences in the field. The book aims to bridge the gap between theory and practical applications.

The Gold Coin game

Infinitely More • 7 implied HN points • 12 Feb 25
🎭️ Culture Mathematics
  1. The Gold Coin game involves two players trying to get a valuable gold coin while moving other coins on a board. Players can either move a coin or take the leftmost coin in their turn.
  2. The game is strategic and requires understanding the best moves to win. Knowing winning moves can make the game easier to navigate.
  3. Practicing the game with a partner helps improve your skills and understanding of the rules, making it more enjoyable to play.

QF Abstract Mathematics 101 Bootcamp

Quantum Formalism • 2 HN points • 22 Mar 24
🚌 Education Mathematics
  1. The Quantum Formalism (QF) community is launching the 'QF Abstract Mathematics 101 Bootcamp' to bridge the mathematical gap between advanced and new members, offering foundational knowledge crucial for quantum computing.
  2. The boot camp will cover modules on set theory, point-set topology, abstract measures, abstract algebra, measure theory & functional analysis, and differential geometry, with live lectures and instructional videos by industry experts.
  3. Participants will receive certifications for completing each module and have the opportunity to learn abstract mathematics relevant to quantum computing and other areas like machine learning.

Twenty one

Infinitely More • 7 implied HN points • 23 Jan 25
📖 Philosophy Mathematics
  1. The game of Twenty-One involves two players counting to twenty-one by saying one to three numbers each turn. The goal is to be the one who says 'twenty-one' to win.
  2. Players can develop strategies to control the game and eventually win. It’s smart to think ahead about how many numbers to say.
  3. This game can help illustrate important ideas in game theory. It’s a fun way to explore how cooperation and strategy work together.

I’ve never truly understood the Softmax function

just learning data science • 3 HN points • 23 Jan 24
🕹 Technology Mathematics
  1. The Softmax function involves two simple steps: converting input values into positive ones using the exponential function and then normalizing them to fit in the range [0, 1] and add up to 1.
  2. Understanding the Softmax function becomes clearer when broken down into these two operations.
  3. By following the process of converting and normalizing values, the Softmax function can be easier to grasp.
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Sum of all integers from 1 upto n

Technology Made Simple • 19 implied HN points • 21 Mar 22
🚌 Education Mathematics
  1. Calculating sum of integers from 1 to n using the formula n*(n+1)/2 can save time and complexity in coding interviews.
  2. Practicing variations of the sum formula, like sum of even/odd numbers or multiples of specific numbers, can enhance problem-solving skills.
  3. Embedding the sum formula into memory will help in quick recall during interviews and lead to better performance.

Linear Algebra Is Not Enough!

Quantum Formalism • 59 implied HN points • 01 Jun 20
🔬 Science Mathematics
  1. Linear Algebra is foundational in quantum mechanics but may not be enough if dealing with infinite-dimensional Hilbert spaces.
  2. To fully understand quantum mechanics, one needs to delve into Functional Analysis along with other mathematical branches like Topology, Measure Theory, and Group Theory.
  3. The newsletter focuses on explaining the mathematical formalism of quantum mechanics through easily understandable posts on relevant topics.

The infinite monkey theorem

A Piece of the Pi: mathematics explained • 18 implied HN points • 11 Mar 24
🔬 Science Mathematics
  1. The infinite monkey theorem states that given enough time and randomness, a monkey could type out the complete works of Shakespeare on a keyboard.
  2. Generating longer phrases by random means, as shown in simulations, becomes exponentially more difficult as the phrase length increases.
  3. The famous infinite monkey paradox has been explored through history, including Cicero's speculation in 45 BC and modern computer simulations using actual monkeys with disappointing results.

Giáo sư Ngô Bảo Châu: đào Pi đậm đà bản sắc dân tộc Việt

Thái | Hacker | Kỹ sư tin tặc • 39 implied HN points • 27 Feb 21
🚌 Education Mathematics
  1. Ngô Bảo Châu sees the interest in Pi mining in Vietnam as a positive sign for the country's mathematics.
  2. The process of mining Pi involves calculating its digits, an important and culturally significant mathematical problem.
  3. Despite criticisms, Ngô Bảo Châu highlights the cultural significance and national pride associated with solving the Pi calculation in Vietnam.

An algebra of orders

Infinitely More • 17 implied HN points • 04 Feb 24
🔬 Science Mathematics
  1. There is a rich algebra of orders involving operations like addition and multiplication.
  2. The disjoint sum operation creates a combined order without interactions between the two parts.
  3. The ordered sum operation combines two orders by placing one above the other, creating new orders with distinct properties.

Group Theory Crash Course

Quantum Formalism • 19 implied HN points • 20 Dec 21
🚌 Education Mathematics
  1. The crash course on Group Theory will cover essential concepts like cosets, generators, orbits, and stabilizers, catering both beginners and those interested in advanced topics like quantum computing.
  2. Prior to starting Module II on Lie Groups and Representations, it is recommended to review the Topology & Differential Geometry crash course to understand smooth manifolds.
  3. Lie Groups not only have a group structure but also an underlying smooth structure that is crucial in the theory, making it important to grasp these concepts before diving into Module II.

Community update | November 2021

Quantum Formalism • 19 implied HN points • 30 Nov 21
🚌 Education Mathematics
  1. Consider taking a crash course on Group Theory and Representation Theory before starting Module II for an easier understanding, especially for those interested in topics like quantum computing.
  2. There will be a session on an industry career fellowship scheme for those interested, and additional sessions on Differential Geometry to cover advanced concepts.
  3. The schedule includes planned activities such as the industry career fellowship session and extra sessions on Differential Geometry to cover advanced concepts.

A formal language for first-order predicate logic

Infinitely More • 15 implied HN points • 02 Mar 24
🚌 Education Mathematics
  1. A formal language for first-order predicate logic involves understanding the basic syntax, terms, variables, and structure interpretations.
  2. Signatures in structures specify the elements like relations, functions, and constants in a mathematical structure, detailing their features and meanings.
  3. Mathematics uses a wide array of first-order structures to study various concepts like orders, graphs, groups, and more, unifying different mathematical investigations.

Distinguishing mathematical structures by their theories

Infinitely More • 15 implied HN points • 24 Feb 24
🔬 Science Mathematics
  1. With first-order logic, subtle features can help distinguish mathematical structures from similar alternatives.
  2. Different mathematical structures can be differentiated by how symbols are interpreted in each structure, revealing unique properties.
  3. Finding statements in the language of orders that are true in one structure and false in others can help distinguish mathematical structures.

The continuum hypothesis

Infinitely More • 30 implied HN points • 04 Mar 23
🔬 Science Mathematics
  1. The continuum hypothesis suggests there is no infinity strictly between natural numbers and real numbers.
  2. The continuum problem has been a challenging and prominent open question in mathematics for over a century.
  3. David Hilbert included the continuum problem as the top question in his list of important open mathematical problems in 1900.

Representing integers as a sum

Infinitely More • 7 implied HN points • 27 Oct 24
🚌 Education Mathematics
  1. Every positive integer can be split into a sum in a specific number of ways. For any integer n, there are exactly 2^(n-1) ways to do this.
  2. To figure out how to split an integer, you can visualize it as a series of ones with spaces in between. Each space can either have a plus sign or not, giving rise to different sums.
  3. A common mistake in math is the 'fence-post error,' where people confuse the number of items with the number of spaces between them, leading to miscounts in things like days or numbers.

Influential Mathematicians: Henri Lebesgue

Quantum Formalism • 39 implied HN points • 03 Aug 20
🔬 Science Mathematics
  1. Henri Lebesgue is known as the father of modern integration theory for formulating the theory of measure and giving the definition of the Lebesgue integral, which expanded mathematical analysis.
  2. Lebesgue's work on integration theory was crucial to the development of the Hilbert space formalism in quantum mechanics, demonstrating its influence on mathematical foundations of quantum physics.
  3. Lebesgue made major contributions in various mathematical areas beyond integration theory, including topology, potential theory, calculus of variations, set theory, and dimension theory.

Your Brain is Wrong About Probability

The Palindrome • 2 implied HN points • 01 Jul 25
🔬 Science Mathematics
  1. Our brains often misunderstand probability, leading us to make poor decisions. We think past events can change future outcomes, but each event is independent.
  2. In games like poker, winning one hand might be luck, but winning consistently is about skill and understanding the odds.
  3. Chasing losses, like believing you're 'due' for a win after losing, can lead to financial problems. It's important to recognize that bad luck doesn't influence future chances.

Chaos train and monkey madness—fun with quantifiers

Infinitely More • 12 implied HN points • 19 Feb 24
🚌 Education Mathematics
  1. First-order predicate logic provides a formal language and semantics capable of expressing fine distinctions and shades of meaning.
  2. Understanding quantifiers, such as ∃ and ∀, is crucial in first-order logic as they allow one to make statements like 'there is an x such that φ' or 'every x has property φ.'
  3. Engaging in logic puzzles and practice can help in developing a deeper comprehension of first-order logic concepts and their applications.

Corridor numbers, Fibonacci numbers, and Pascal’s triangle

A Piece of the Pi: mathematics explained • 12 implied HN points • 25 Feb 24
🔬 Science Mathematics
  1. Corridor numbers count ways to take diagonal steps down a corridor with fixed width. The numbers in each box form Fibonacci numbers when summed vertically.
  2. Fibonacci sequence is generated by summing the previous two terms. In the context of corridor numbers, Fibonacci numbers represent different routes to specific boxes.
  3. Pascal's triangle has rows starting and ending with 1, where each entry is the sum of two nearest entries from the row above. Circular Pascal arrays relate to corridor numbers and can produce Fibonacci numbers when subtracting specific entries.

Transfinite recursive constructions

Infinitely More • 23 implied HN points • 19 Mar 23
🔬 Science Mathematics
  1. Recursions can express fundamental relations in various contexts like mathematical sequences and modelings like the Fibonacci sequence.
  2. Many familiar arithmetic functions can be defined by recursion, showing a deeper fundamental aspect of their operations.
  3. Defining functions by recursion can be legitimate for natural numbers, but may fail for real numbers due to multiple or no solutions.

The method of exhaustion

Infinitely More • 23 implied HN points • 14 Feb 23
🚌 Education Mathematics
  1. Archimedes used the method of exhaustion to find the area of a parabolic segment, a concept that predicted calculus ideas by 2000 years.
  2. Archimedes focused on calculating the area between a parabola and a linear chord, exploring the geometry of parabolas in his work.
  3. To read more about the method of exhaustion and key mathematical concepts, subscribe to Infinitely More for a 7-day free trial.

Should we teach children to memorize the multiplication tables?

The Science of Learning • 4 HN points • 26 Jun 23
🚌 Education Mathematics
  1. Children benefit from memorizing multiplication tables because it helps them solve math problems more easily. When students know their math facts, they can focus on more complex thinking instead of getting stuck on basic calculations.
  2. Research shows that students who memorize math facts do better in math overall. This memorization builds a strong foundation for advanced math skills later on.
  3. It's important to strike a balance between memorization and understanding in math education. Teaching kids to remember math facts can actually support their overall learning and make problem-solving easier.

Foreword: Lost in Math

Critical Mass • 20 implied HN points • 20 Apr 23
🔬 Science Mathematics
  1. Science is driven by careful observations and making predictions.
  2. In physics, the tension between truth and beauty can lead to debates and different perspectives.
  3. Evaluating theories in physics should prioritize empirical data over mathematical elegance.

Module II Crash Course Prerequisite: Basic Real Analysis

Quantum Formalism • 19 implied HN points • 10 May 21
🚌 Education Mathematics
  1. Understanding basic real analysis, like open intervals and closed intervals, is important for the upcoming crash course on Point-Set Topology and standard topology.
  2. Being challenged to work with abstract concepts can help in feeling comfortable before starting Module II.
  3. Learning sophisticated mathematics can better prepare individuals for making theoretical contributions in physics or quantum information.

How to count

Infinitely More • 20 implied HN points • 22 Feb 23
🚌 Education Mathematics
  1. Ordinal numbers extend natural numbers beyond infinity.
  2. The first infinite ordinal is ω, pronounced as omega.
  3. To understand ordinals, focus on their own order structure rather than embedding them into our perception of time.

The countable random graph

Infinitely More • 10 implied HN points • 10 Feb 24
🔬 Science Mathematics
  1. A countable random graph is a graph where you flip a coin to decide the edges between vertices in an infinite set, and the result is the same graph almost every time.
  2. Graph theory is a complex subject with beautiful theorems, and different notions of graphs exist, such as directed graphs and simple graphs.
  3. In mathematics, there are variations in graph definitions, such as allowing reflexivity or multiple edges, but in simpler contexts, graphs are typically referred to as simple graphs.

Siêu trí tuệ: tìm số nguyên tố hay là tăng minh bạch cho ngành đánh đề

Thái | Hacker | Kỹ sư tin tặc • 39 implied HN points • 27 Dec 19
🔬 Science Mathematics
  1. When faced with challenges involving prime numbers, clever algorithms can help quickly eliminate composite numbers and pinpoint the secret numbers.
  2. The difficulty of a problem depends on the randomness of number selection within a matrix and the position of prime numbers.
  3. Designing a fair random number generation system is crucial for ensuring transparency, not only in intellectual competitions but also in traditional gambling industries.

Mathematical induction

Infinitely More • 10 implied HN points • 30 Jan 24
🚌 Education Mathematics
  1. Mathematical induction is a fundamental principle in mathematics, used to prove many fundamental facts in arithmetic and number theory.
  2. The common induction principle states that if a set of natural numbers contains 0 and whenever n is in the set, n+1 is also in the set, then every natural number is in the set.
  3. Strong induction allows the induction step to use multiple smaller numbers to prove a statement, and can be proven from the least-number principle.

Evidence Belongs on the Canvas Not the Scale

Metarational • 19 implied HN points • 20 Apr 21
🔬 Science Mathematics
  1. Evaluating evidence like weighing it on a balance scale can be an elegant metaphor but may not be mathematically correct, as evidence doesn't always work that way.
  2. The scenario with two judges deliberating on a statement showcases how evidence overlap matters, revealing flaws in the scale metaphor and emphasizing the need for a more nuanced model.
  3. Imagining evidence on a canvas with shaded regions for different hypotheses can better capture the complexity of multiple evidence lines overlapping, offering a more accurate representation than a simple scale.

One MILLION Stories!

Something interesting • 4 implied HN points • 28 Nov 24
🔬 Science Mathematics
  1. Building a skyscraper with a million stories would be huge. It would have to cover a massive area and be taller than any building we have today.
  2. Elevators in such a tall building would take forever to reach the top. They might need to be super advanced and comfortable, like mini-hotels.
  3. The universe is really big, and we are very small. Even the tallest buildings are tiny compared to the whole Earth and space around us.

Session 16 Reminder

Quantum Formalism • 19 implied HN points • 21 Jan 21
🔬 Science Mathematics
  1. The next Quantum Axioms & Operators session is coming up soon - mark your calendars!
  2. Newcomers are welcome, but it's advised to have some background in physics/math to fully follow along.
  3. A special surprise gift was received from the community, showing appreciation and support for the course.

Shut up and Calculate!

Quantum Formalism • 19 implied HN points • 12 Dec 20
🔬 Science Mathematics
  1. Focus on learning how to use quantum mechanics as a toolkit without worrying too much about the foundational meaning at first.
  2. Some physicists advocate for the 'Shut Up and Calculate' philosophy, emphasizing the importance of actively engaging with conceptual issues in physics.
  3. The 'Shut Up and Calculate' approach is viewed as a necessary and respectful way to tackle profound questions in fundamental physics.

Lecture #1

Quantum Formalism • 19 implied HN points • 19 Sep 20
🚌 Education Mathematics
  1. The foundation module of the lecture series will focus on finite-dimensional complex Hilbert spaces, aligning with standard quantum computing textbooks.
  2. The mathematics session in Lecture #1 included topics like natural numbers, integers, maps between sets, and discussion on cardinality of sets.
  3. Attendees are encouraged to participate in the live sessions for interaction and asking questions rather than solely watching replays.