Infinitely More

Infinitely More explores the mathematics and philosophy associated with the concept of infinity, covering a diverse array of topics such as surreal numbers, lattices, the continuum hypothesis, transfinite numbers, mathematical structures, and paradoxes. It examines both the abstract theories and their practical implications, integrating logical analysis and historical perspectives.

Mathematics Philosophy Logic Number Theory Algebra Graph Theory Mathematical Induction Epistemology

Top posts of the year

And their main takeaways

What is the essential structure of the complex numbers?

38 implied HN points • 10 Nov 24
📖 Philosophy Mathematics Epistemology Ontology Logic
  1. There are different ways to think about complex numbers, like focusing on their algebraic or topological structures. Each viewpoint gives us unique insights into how complex numbers behave.
  2. Mathematicians don't all agree on what the essential structure of complex numbers is, leading to multiple interpretations. It shows us that understanding math can be quite flexible.
  3. The paper identifies four main perspectives on complex numbers, which can help clarify the discussions around their nature and engage with broader philosophical questions in mathematics.

Take my Infinity final exam!

35 implied HN points • 21 Dec 24
🚌 Education Philosophy Mathematics Learning Assessment Critical Thinking
  1. The Cantor-Hume principle connects with Euclid's principle, and there are different views on whether they agree or conflict. It's a topic worth exploring in depth.
  2. Understanding potential and actual infinity is important in calculus, especially when dealing with infinite series. This distinction affects how we solve mathematical problems.
  3. The continuum hypothesis and the axiom of choice raise interesting questions in philosophy and mathematics, showing how these concepts challenge our understanding of infinity and choice.

The zigzag theorem

33 implied HN points • 04 Jan 25
🚌 Education Mathematics Geometry Learning Teaching
  1. The zigzag theorem states that when you create a zigzag pattern in a rectangle, the triangles formed below this pattern take up exactly half the area of the rectangle.
  2. Even if the zigzag lines sometimes move backward without crossing, the triangles will still cover half the rectangle's area due to how the bases and heights of the triangles are calculated.
  3. This theorem is interesting because it holds true even if the zigzag involves an infinite number of lines.

Interpretability

28 implied HN points • 30 Nov 24
📖 Philosophy Logic Mathematics Interpretation Language Concepts
  1. In math, we can understand one idea by using another. It's like using different languages to explain the same thing.
  2. Sometimes, when we translate ideas back and forth, we lose some meaning, similar to playing a game of telephone.
  3. To make this work, we create special objects in a new system that can help us relate and understand the original idea better.

Abstraction in the function concept

25 implied HN points • 09 Jun 25
📖 Philosophy Mathematics Concepts Logic Theory Abstraction
  1. The function concept in mathematics has evolved a lot, allowing for more abstract definitions. This means mathematicians can explore complex ideas that go beyond simple rules and formulas.
  2. Examples like the Devil's staircase and space-filling curves challenge our understanding of functions. These unique functions have properties that seem strange and unexpected compared to our usual ideas of what a function should be.
  3. The Conway function shows how every real number can be linked to another number in a complex way. It helps to illustrate that functions don't always need a clear formula and can still be valid in mathematics.
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Chomp

20 implied HN points • 31 Jan 25
🎭️ Culture Games Logic Philosophy Analysis Creativity
  1. The game Chomp involves two players taking turns biting from a chocolate bar, and the goal is to avoid being the one to take the last bite. Players remove chocolate squares from the lower-left, taking away everything above and to the right of their chosen square.
  2. Winning strategies in Chomp can depend on whether you're going first or second, especially based on the size of the chocolate bar. Players need to think carefully about their moves to ensure they don't end up losing.
  3. Chomp is not just a fun game; it also teaches lessons about strategy and decision-making in game theory. Understanding how to analyze your options can give you an edge in winning.

How we might have viewed the continuum hypothesis as a fundamental axiom necessary for mathematics

20 implied HN points • 22 May 25
📖 Philosophy Mathematics Logic Metaphysics Epistemology History
  1. The continuum hypothesis (CH) is about understanding different sizes of infinity, particularly if there's a number between natural numbers and real numbers. Many assume its truth or falsehood is needed for math.
  2. If early mathematicians had been clearer about infinitesimals and different types of numbers, they might have accepted CH as a key part of math and calculus, making hyperreal numbers a standard concept.
  3. Whether CH is true or false is not just a technical question; it reflects deeper philosophical views about the nature of mathematics and how we interpret infinity and set theory.

Permutations and combinations

17 implied HN points • 17 Nov 24
🚌 Education Mathematics Theory Logic Combinatorics
  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.

Interpretability of theories

17 implied HN points • 11 Jan 25
📖 Philosophy Logic Theory Interpretation Models Semantics
  1. You can understand one theory by interpreting it through another theory. This means translating ideas from one set of concepts to another.
  2. Interpreting theories involves a consistent method to show how one theory fits within the framework of another. It connects the ideas and structures from both.
  3. The host theory provides a detailed explanation of how the interpreted theory operates, using only its own language and concepts. This helps clarify the relationships between different theories.

Mutual and bi-interpretation of models

17 implied HN points • 14 Dec 24
📖 Philosophy Logic Interpretation Mathematics Theories Models
  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.

Finding Fifteen

15 implied HN points • 20 Jan 25
🚌 Education Game Theory Mathematics Philosophy Logic
  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.

Infinite Connect Four

15 implied HN points • 17 Jun 25
🎭️ Culture Games Mathematics Philosophy Entertainment Education
  1. Connect Four is a game where players try to get four of their coins in a row, either horizontally, vertically, or diagonally. The game shows that the first player has a winning strategy if they play perfectly.
  2. The concept of an infinite version of Connect Four allows for interesting variations, like playing on an infinite board. This leads to questions about how long winning chains players can aim to create.
  3. With infinite possibilities, players might aspire to create very long winning combinations, even infinite chains. This expands the game beyond traditional limits and invites deeper strategic thinking.

Recursive Chess

12 implied HN points • 19 Oct 24
🎭️ Culture Games Concepts Philosophy Art Education
  1. Recursive chess is a new twist on the traditional game where pieces must play their own mini-games before capturing. This makes each capture more complex and interesting.
  2. The rules of recursive chess are still being debated, with no clear answer on how to play. Different interpretations could lead to many unique gameplay experiences.
  3. Exploring the rules of recursive chess invites deeper discussions about game theory and the nature of games themselves, making it a fascinating topic for anyone interested in strategy.

On going first

12 implied HN points • 30 May 25
📖 Philosophy Games Fairness Strategy Decision-making Mathematics
  1. In many games, going first can give a big advantage, which isn't always fair. It's important to find ways to balance the game for both players.
  2. Rules like the 'swap rule' in games like Hex can help make things fair, but they may not always work perfectly for every game.
  3. For games like Go, figuring out the right value to give the second player can be tricky, and the common values used might not actually be the best.

The interpretation translation

10 implied HN points • 07 Dec 24
🔬 Science Mathematics Logic Philosophy Education Computer Science
  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 game of Nim

10 implied HN points • 07 Feb 25
🚌 Education Mathematics Games Logic Strategy Learning
  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.

Twenty one

7 implied HN points • 23 Jan 25
📖 Philosophy Game Theory Logic Education Mathematics Culture
  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.

Representing integers as a sum

7 implied HN points • 27 Oct 24
🚌 Education Mathematics Learning Teaching Proofs Exercises
  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.

The Gold Coin game

7 implied HN points • 12 Feb 25
🎭️ Culture Games Strategy Logic Philosophy 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.