Reason & Reflection

Infinity represents something that has no end, something boundless and limitless, yet it’s a concept that we struggle to fully grasp within our usual frameworks of thought. In philosophy, infinity raises deep questions about the nature of reality, mathematics, and even human understanding. Infinity in Mathematics: Not All Infinities Are Equal In mathematics, infinity takes on many forms, like the endless set of natural numbers and the uncountable infinity of real numbers (all the decimal numbers between integers). Mathematician Georg Cantor proved that some infinities are “larger” than others by introducing the idea of cardinality. For instance, the set of real numbers is a “larger” infinity than the set of natural numbers, because there are simply more real numbers between any two whole numbers than there are whole numbers themselves. This discovery challenged the way we think about numbers and set the stage for modern mathematical theories. Paradoxes of Infinity Infinity also introdu

Claude Shannon, often called the "father of information theory," developed a groundbreaking way to understand communication. His theory, created in the 1940s, showed how information could be transmitted efficiently, whether through telegraphs, radios, or computers. Shannon introduced the idea of entropy , which measures uncertainty in a message. For example, a completely random message has high entropy, while a predictable one has low entropy. Shannon’s work also addressed how noise, or interference, can affect communication and how redundancy can help correct errors. The formula for Shannon's Entropy illustrates how the probability of each symbol contributes to the overall uncertainty or "information" in a system. This foundational equation in information theory has broad implications in both technology and philosophy, raising questions about the nature of knowledge and reality. (Najera, Jesus. “Intro To Information Theory.” Setzeus, 18 March 2020,  https://www

Imagine you have a heap of sand. If you remove a single grain of sand, you’d still call it a heap, right? But what if you keep removing grains, one by one? At some point, it seems like you’d be left with just a few grains—and surely, that’s no longer a heap. But where exactly does the heap stop being a heap? This puzzling question is at the heart of the Sorites Paradox, also known as the paradox of the heap. This paradox highlights the challenges of dealing with vague concepts, which can be tricky not just in everyday life but also in science. What Is the Sorites Paradox? The Sorites Paradox comes from the Greek word "soros," which means heap. The paradox arises when we try to apply precise logic to vague concepts. In its simplest form, it goes like this: A heap of sand is still a heap if you remove one grain. If you keep removing grains, eventually you’ll be left with just one grain. But according to the first point, even one grain less than a heap should still be a heap, wh

This summer, I had the incredible opportunity to attend the Wolfram High School Summer Research Program. Interested in ruliology, I focused my project on mobile automata, a type of simple program similar to cellular automata. Mobile Automata with Non-Local Rules In cellular automata, all cells update in parallel according to a set of rules, whereas mobile automata feature a single active cell that updates at each iteration. The rules for mobile automata dictate the new state of the active cell and its movement. These rules consider the states of the active cell and its immediate neighbors, determining the new color of the active cell and whether it moves to the left or right. Traditionally, mobile automata involve the active cell interacting with its immediate left and right neighbors. However, in my project, I explored the effects of non-local interactions, where the dependent cells are farther away from the active cell. For instance, I examined scenarios where the dependent cells wer

Following up on my post about the philosophy of ruliology and cellular automata, I decided to look more into the concept of ruliology. And what better way to explore ruliology than by reading the writings of the term’s inventor Stephen Wolfram? So, over the past month, I have read "A New Kind of Science" by Steven Wolfram . After diving into the book, however, the last chapter left me pondering the Principle of Computational Equivalence. And as I head into the Wolfram High School Summer Research Program this summer, I thought I might write some philosophical questions down about this concept. This principle proposes a profound idea that may reshape our thoughts about the universe and our place within it. But what does this principle really mean, and how does it connect to broader philosophical questions? What is the Principle of Computational Equivalence? Wolfram suggests that systems capable of doing computations, whether they are human brains or simple algorithms, have equa