MAT 1575 Calculus II

Infinity is often thought of as something vast and unending, existing beyond our reach. We are used to finding infinity in something abstract, but it also surrounds us in the natural process, especially the water cycle. This photograph, taken during the winter, caught a moment of that infinite cycle. The frozen lake, the snow-covered landscape, reminds us of the continuing state of transition that water is always undergoing: from vapor to liquid, from liquid to ice, and back again. This is a cycle without end, a natural depiction of infinity.

The water cycle is the infinity in math, because it works in a circle constantly. Just as an endless series of numbers without terminus, so too the perpetual motion of nature’s water: change of form, flow, and return onto itself. This has been going on for millennia and shall continue-independently of seasons or climate change. Each constituent part of the cycle-precipitation, freezing, melting, evaporation-is a link in an endless chain which reinforces nature’s own version of infinity.

The cycle somehow reminds us that infinity exists in tangible ways, even which we are able to see. It connects the photograph to that endless concept-that infinity isn’t just numbers or equations, but a part of life woven into the natural world and repeating endlessly before one’s eyes. But in this picture I would like to capture, not only the beauty of winter, but even something more transcendent, something like infinite, eternal nature.

1. What is infinity? Explain in your own words.

Infinity is the state of something that has no bounds or end. It always outgrows any number that one can ever imagine and goes further. In math, it’s often used to describe things that have no end or processes that can be continued forever. But it isn’t a big number; it’s an entirely different kind of concept with no boundary or endpoint.

2. Who first introduced you to the idea of infinity? How old were you? How did this new idea make you feel?

I think the first time I ever thought of infinity was in elementary school, where we learned about the number system and how numbers could just keep getting larger. I was only seven or eight at that time, but it was something that really made me curious, though a little bit overwhelmed. Understanding that there is no “biggest number” made me realize that some concepts are really beyond what we are fully able to grasp.

3. Describe either a conversation you had about infinity or an argument you had about infinity. Give us the details!

I once had a debate with my friend as to whether or not there’s a “biggest” infinity. My friend said that infinity is only one thing, whereas I had heard about the different “sizes” or types of infinity, such as the infinity of real numbers being “bigger” than the infinity of whole numbers. We went back and forth trying to explain it, and I remember realizing that infinity was more complex than I originally thought.

4. Find a reference to infinity in popular culture (a song, a movie, a product, a book, etc.). What does infinity mean in this context? Does it mean the same thing that it means in mathematics? Explain.

One of the most famous pop references to infinity is when, in the movie Toy Story, it says, “To infinity and beyond!” There, infinity symbolizes an idea of limitless possibilities and courage. It doesn’t mean infinity in a severe mathematical sense; in math, there is no actual “beyond” of infinity. Instead, it’s rather an inspirational catchphrase to encourage people to go beyond their limits.

5. Is infinity an important idea in your religion? Do you think that infinity in a religious setting refers to the same idea as infinity in mathematics? Why or why not?

In most religions, infinity has to do more with the eternity of the universe, soul, or god. It is, in some ways, more of a philosophical or spiritual kind of infinity that generally speaks to time without end or unlimited existence. While the mathematical concept of infinity is exact and applied in formulae, religious infinity is more abstract and deals with ideas beyond human comprehension.

1. An infinity is a concept that represents the quantity of something that has no limit. It can be either the lifespan of a living thing, durability of a non-living thing or from the moment of existing to never seize to exist at all. An infinity is a concept that we apply to many spheres of our lives, including religion, science and even math. Throughout the history we tried to grasp the idea of infinity, but everything on Earth seems to be finite, as we know it.
2. I cannot pinpoint the exact location or the time that I was introduced to the idea of infinity; however, I would assume that around middle school I was visualizing the concepts outside the school programs. I used to sketch and draw fighting scenes on my free time during school hours, so entertainment elements such as comics and games characters seem very fun to me. Some of those comics characters have bigger lifespan than others due to the power they possess, therefore I probably started thinking about infinity when characters were immortal or there were some universal types of scale which makes you ponder upon the idea how small we are in an astronomical scale compared to outer planets, stars and even galaxies. Overall, I was probably around 12 years old when infinity came to my mind and this idea made me feel intrigued, because how can a person’s life be finite and yet, most of the religions have a concept of a God that is immortal, who created us and yet, there were never a mention of the ultimate end except for a human life. In my opinion, it really starts to deal with an abstract mind in a philosophical and even psychological way, since there is no definite answer to whether infinity really exists.
3. I cannot recall any conversations I had about an infinity; however, I have an argument that I have been thinking about recently, is that how can our universe expand without any limits into an infinity if anything we ever know on the Earth and even within our galaxy, such as planets and stars have a certain lifespan they can co-exist relatively to their creation. For example, my main theory on that, is that even if the universe is expanding, it needs a certain outside space to do it, it cannot just bend the fabric of reality and form new stars, planets and even galaxies as it wishes out of nothing. As one of the principles that governs our physics is the “Law of conservation of energy”, which states that matter cannot be created or destroyed, which implies that even the universe itself, in theory, cannot just create new matter out of nowhere and disperse the energy as it wishes, there has to be a limit at some point, and I believe that the concept of physics that we have discovered throughout the history should be applicable anywhere else outside of our scope of astronomical scale. And based on science, the idea of infinity is arbitrary/abstract, instead of something that has a value. Therefore, my conclusion on this argument, is that either there is a being like God that governs over the universe and the way it behaves, or the universe does have a limit even if it expands beyond our human understanding as of right now.
4. One of the references to the infinity found in pop-culture that I can relate to is the endless, internal battle between good and evil within the Star Wars franchise. Even since I was a kid, I really enjoyed playing Lego Star Wars, where I build my own minifigures, battles and scenarios as an entertainment. What really intrigued me about Star Wars, is that there is an always ongoing rivalry between the good and evil forces that somehow need to stay in balance, or the entire civilizations will collapse if the either side would go out of hands. It really makes you think philosophically about how even in real life, we have positive/negative emotions, law-abiding and criminal activities, as well as good and evil intentions from our overall surroundings. This made me wonder, how we as humans cannot be fully good or fully bad either, we are a combination of both. This brings my topic to an idea of Yin and Yang, a Chinese philosophy on how the light and dark sides of the circle are in balance with each other, which shows two opposite flowing substances which collide into one shape. However, this philosophical notion about good and evil concept isn’t the same as in mathematics. Because as a referenced before, the Star Wars franchise is portraying light and darks sides of the force as individuals who have the ability to wield the force and even lightsaber as a way to fight their opponents, but it is up to a person to decide which sides he is on, based on his experience in life. But in mathematics, we use infinity to represent points on the graph within the curve that has no limits, either towards the negative or positive directions. (Star Wars pop-culture link: https://toxigon.com/the-philosophy-of-the-force ); (Yin and Yang link: What Is the Meaning of Yin and Yang? ).

• A drawing that illustrates the concept of infinity: