Reason & Reflection

Virtual Reality (VR) allows people to step into digital worlds that feel surprisingly real. By wearing special headsets or using controllers, users can move around and interact with objects that only exist as computer code. Some find these virtual experiences exciting and full of potential. Others worry about the impact they might have on our ideas about right and wrong, as well as on our understanding of what is truly “real.” In this post, we will explore how VR challenges our sense of reality, examine moral issues that come up in virtual worlds, and consider how spending time in simulations could change our identities in everyday life. Apple Vision Pro, a glimpse into the next generation of immersive virtual and augmented experiences. "Introducing Apple Vision Pro: Apple’s first spatial computer", Apple, https://www.apple.com/newsroom/2023/06/introducing-apple-vision-pro/ What Counts as “Real” in Virtual Worlds? One of the biggest questions about VR is whether experience...

When we look at the universe, it seems almost perfectly set up for the existence of life. Many of the laws of physics work in just the right way to allow stars to form, planets to exist, and complex life to develop. This idea that our universe is “fine-tuned” for life has led to many discussions about what it really means. Some believe it might be just a lucky accident, while others think there could be a deeper reason. These debates bring us to the Anthropic Principle, which is a way of explaining why we see the universe as so well suited for living things. The Puzzle of Fine-Tuning Scientists have found that if certain physical laws or constants—such as the strength of gravity or the charge on the electron—were slightly different, stars might not form or atoms might not stay together. If that happened, life as we know it would not be possible. The universe’s seeming “perfect fit” for life is sometimes called the “fine-tuning” problem, because it is as though these constants were set ...

In my AP Calculus BC class, we recently explored Taylor series—a topic that combines mathematical precision with philosophical depth. As we worked through how derivatives at a single point can approximate a function's behavior nearby, I started thinking about how this process reflects larger ideas about knowledge and prediction. While Taylor series are tools for solving problems in calculus, they are also windows into how local information can illuminate broader patterns and behaviors. What Is a Taylor Series? The Taylor series uses information about a function's derivatives at a single point to approximate that function near the point. The formula for a Taylor series expansion around a point a  is: f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) + f ′ ′ ( a ) 2 ! ( x − a ) 2 + f ( 3 ) ( a ) 3 ! ( x − a ) 3 + ⋯ This formula is built step by step, incorporating more terms to reflect higher-order behaviors of the function. The zeroth term, f ( a ) f(a) , gives the function's value at...

Cryptography is often seen as a way to eliminate trust. It secures our communications, safeguards sensitive data, and even enables decentralized systems like cryptocurrencies (shoutout Dogecoin). At its heart, cryptography is a promise: you don’t need to trust people or institutions; you just need to trust the math. But this promise, while compelling, isn’t as simple as it seems. Trust doesn’t disappear in cryptography—it shifts. It moves from people and systems to algorithms, keys, and the humans who design them. This shift raises questions about the nature of trust in a world increasingly mediated by encryption. Trustless Systems: The Ideal Vision One of cryptography’s primary goals is to create systems that don’t rely on trust in third parties. End-to-end encryption (E2EE) is a clear example of this ambition. With E2EE, only the sender and recipient can read the content of a message. Not even the service provider has access, which means users don’t need to trust their data is safe—t...

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...