Class Notes from 10/22/2013

I apologize if I paraphrased or got something else incorrect. I compiled the notes to the best of my knowledge :)

October 22, 2013

An easy guide to infinite chocolate (YouTube Video)

http://youtu.be/7Yo6mwSB1As

Manifestations:

1. Mirrors
2. Yin Yang
3. Cycle of Life
4. Can't Stop Time
5. Tech/Science
6. Germs/Bacteria
7. Time in the Present Tense
8. Ascending/Descending
9. Thoughts
10. Shapes
11. Galaxy “Adam's Quote”
12. Creativity
13. Light Rays
14. Evolution
15. Energy
16. Roman Columns
17. Droste
18. Sculpture in DC
19. Klein Bottle
20. Recycling Symbol
21. Music

How difficult was the categorization of the manifestations of infinity?

• Abstract
• Concrete
• Formal
• Measurement
• Life

It was interesting to note that many people felt that it was more easily categorized at home. However, there were differences in opinion on how they are categorized. For example, the naming of each category seemed subjective. John suggested, that he could have simplified further with less categories (e.g.: getting rid of the “formal” category), and some others agreed.

Two previous readings

Comparing the Infinite: Pairing up collections via a One-to-One Correspondence. (pp. 146-157)

To Infinity & Beyond: To Infinity and Beyond (pp. 61-67)

Try to summarize the concepts presented in the two articles mentioned above. What are the implications of a one-to-one relationship?

One big take-away was the way of solving the problem. In the hotel with an infinite number of buses for example, it provided a different way of looking at things. Instead of calculating the number of remaining rooms, or the number of floors in the “infinite hotel”, you could just figure out a system of organizing people from the buses linearly to file into the hotel.

Scenario: the hotel is infinitely large, and it's already full. With a one-to-one correspondence, we were always able to solve the problem. This is because of the one-to-one relationship between the number of buses, or people- and the number of hotel rooms.

In the second reading, Cantor proved essentially two different “sizes” of infinity. For example...

For the incremental value between 0 and 1 you could write:

1 → 0.713234092358092834

2 → 0.281043205984534098

3 → 0.500000000000000000

4 → 0.111535493485394859

This was monumental because it disproves the ability to calculate the value in between any two real numbers (right...?)

… So what do we know about infinity?

||N|| < ||R||

Continuum hypothesis – the idea that there is no infinite set with a cadinal number in between the smaller infinity, and the larger infitinty.

Professor Carter

Zeno – A Greek philosopher from mid-late 400 BC known for his paradoxes.

Dichotomy A – Kitty and the food.

At some point the kitty has to traverse past the halfway point of the room. Then another half, etc. until it becomes aparent that the kitty will never be able to reach the food.

Achilles and the Tortoise- Tortoise is given a head start, and even though Achilles is moving faster, the tortoise moves during the time that Achilles catches up to him. This continues through an incremental period until it shows that Achilles will never quite catch up to the tortoise.

The Stadium-

The Arrow-

Infinite Perspective

In our discussion on September 25, 2013, we spoke about the topic art of perspective. I view infinity to be completely on perspective because how one person can look at an object or idea one way, another person could be looking at it from a complete different angle. An interesting topic that Mr. Carter spoke about was congruence in paintings. Originally, artists just put people or objects together in any format, even if the ratios were completely off. Then, painters started to use different skills and practices to make paintings more flush and fit together with harmony. What also amazed me was the fact that drawing lines to a specific point, (the vanishing point) which meets with the horizon line, can create an optical illusion that seems to go on infinitely.

Another topic of discussion was the "Hamman Hotel" conundrum. Looking at the problem from different angles, we as class tried to find a way to get an infinite amount of people into the infinite roomed hotel. In the beginning, we started off with one infinite bus of people. The solution was not as difficult to solve, but as more buses were added to the equation, the solutions became harder to do. Finally, when we thought the riddle was solved we were hit with an infinite amount of buses. We were stumped until we visually saw a way to conquer the problem and simply concluding that when talking about infinite it is best to start with the instead of the beginning. It is an interesting way of viewing infinity when there is no theoretical end, right?

The (in)finite Universe

Mankind has always been intrigued about the mysteries of space. One of the most compelling questions we ask is whether space is infinite or finite. Does space have an end or does it continue to go on indefinitely? Einstein once said “only two things are infinite, the universe and human stupidity.” How right could have Einstein been? And is it truly possible to tell?

The first step to determining whether or not the universe is infinite is by examining the universe itself. To solve this mystery we shall review what we already (think) we know about our universe. Astronomers and scientists estimate that the universe is 13.7 billion years old (Wright). This year is determined from the moment the big bang occurred. We also know that in theory the big bang was a large explosion that created and expanded the universe we know of today. The universe contains a massive amount of planets, including stars, which are grouped into unimaginable amount of galaxies. To put into retrospect how large our universe is, let’s look at the second nearest star to us. It is estimated that the second nearest star to us is 25 trillion miles away. Our galaxy, the Milky Way, measures 100,000 light years in diameter and is one galaxy within a group of 50 galaxies measuring ten million light years across. A light year is the distance traveled at speed of light for one year. This is roughly six trillion miles (Hincks). Those are certainly very large numbers. But does this mean the universe is truly infinite? There have been many theories on this matter that I wish to explore.

Ever since the beginning, early man has pondered and tried to answer the question of an infinite universe. Epicurus in the third century B.C. said “there are infinite worlds both like and unlike ours” (Chrichton). When we think about the extraordinary size the universe appears to be it is no wonder why we think of it as infinite.  However not everyone believes the universe is infinite. Jean-Pierre Luminet in his book “The Wrap Around Universe” argues that the universe is not infinite. Luminet states that the universe is shaped like a sphere much like a soccer ball. The sky acts as an array of mirrors that mimics multiple galaxies from the light of just one. In his theory, many images of galaxies are mere copies of one single galaxy (Geftner). Not only does this declare that the universe then is not infinite, but the universe is also limited and smaller than it appears.

            Another theory to explore examines the universe as infinite. However, to better understand this theory we must understand Einstein’s “Theory of Relativity.” In the 20^th century Einstein announced that time and space were not separate but connected. This is known as space-time. According to space-time, time is warped and curved rather than flat. The more mass a planet has, the more curve will be produced in space-time. The greater the curve or planets mass, the slower time moves. This theory was proven in 1962. Two atomic clocks, the most precise clocks in the world, were placed at the top and bottom of a water tower. The clock at the bottom of the water tower near the earth’s core moved slower than the clock at the top. Einstein called this difference “Time Dilution” (Fuller).

            Now that we have an idea of how space-time works we can examine the next theory. Edward Wright from the University of California explains the assumption since the universe is 13.7 billion years old means it can only have grown by 13.7 billion light years. Wright announced this assumption false. According to Wright, Einstein’s Theory of Relativity is the answer. A 13.7 billion year journey through the universe with an atomic clock would hardly tick at all according to Einstein’s theory of relativity. This means that before the universe was 13.7 billion years old it was already massive in size, and if you add inflation and exponential growth it adds even more size. Mathematics shows that even simple huge models are infinite in size. This means that if a model is infinite, for it to be truly infinite it would have to be infinite indefinitely before and after the big bang. For it to be infinite before the big bang it would have had to be born infinite. Wright states that this is logical and mathematics can simply prove it (Wright). For those of you who wish to explore the mathematics behind Wrights theory please visit: http://www.astro.ucla.edu/~wright/cosmo_02.htm.

In my opinion the question whether the universe is finite or infinite remains a mystery. We examined two different theories—both theories claim to be true; however, neither can ultimately be proven. This is the problem with this question. Man is a mere observer to the universe. Although we have been to space, and even to the moon, we have but faintly scratched the surface of the universe. Infinite or not, we are but a spec in a large painting.

In conclusion, until man can travel through the universe and explore space and its wonders more clearly, I don’t think we will ever have a definite answer on whether or not the universe is infinite. As we have discussed in class our perception determines our opinion. You either believe one way or the other. Maybe we will never truly know the answer. Nonetheless, our imaginations and curiosity will continue to set sail our ships in the voyage of discovery.

Works Cited.

Crichton-Miller, Emma. "Infinite Space." New Statesman 142.5171 (2013): 46. Academic

Search Complete. Web. 20 Sept. 2013.

Fuller, John.  "Does time change speed?"  27 May 2008.  HowStuffWorks.com

Geftner, Amanda. "The cosmic mollusc." New Scientist magazine. 16 02 2008: 46. Web. 20 Sep.

2013.

Hincks, Adam D. "Wonders of the universe: new scientific discoveries and old

truths."America 16 Apr. 2012: 10+. U.S. History In Context. Web. 19 Sept. 2013.

Wright, Edward. "The infinite cosmos?" Astronomy Nov. 2012: 57. Science In Context. Web. 19

Sept. 2013.

How comfortable are you with the idea of infinity?

The concept of the infinite is a potentially daunting idea. We can often approach it in a very analytical fashion and speculate on it's theoretical implications without allowing the idea to really "hit home" emotionally.  If, on the other hand, we consider our own existence in a context of infinite time and space - if we really think about it, we run the risk of dissolving the comfortable sense of reality we call home.  After all, who among us can really integrate the idea of infinity into practical, daily life?      

One of your assigned readings (The Art of Infinity by Robert Huddleston) began with the contrasting emotional reactions of two 17th century characters when they considered the notion that the universe may be infinite ins size.  The first person, (Blaise Pascal),  finds the idea frightening. He says:

When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill . . . engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened, and am astonished at being here rather than there. 

Nevertheless, the second person, a fictional character created by author Bernard le Bovier de Fontanelle in 1686, has a very different reaction:

I feel entirely at ease.  When the sky was only a blue vault, with the stars nailed to it, the universe seemed small and narrow to me; I felt oppressed by it. Now that they've given infinitely greater breadth and depth to this vault . . . it seems to me that I breath more freely, that I'm in a larger atmosphere, and certainly the universe has a greater magnificence. 

How do YOU feel?  If you consider the infinite as a reality that is not "just" and abstract idea, are you comfortable with it?  Are you "astonished at being here rather than there"?  Do you find the notion of infinite space and time threatening at any level?   Or, do you identify more readily with the second example?  Does the proposition of a finite universe, regardless of how vast, make you feel confined or somehow disappointed?  Just the idea that musical possibilities were somehow finite was enough to send John Stuart Mill into depression - even if, in all practicality, the possibilities of music are virtually endless.  Or does it seem that, in the end, Fontanelle's confident character just doesn't get it?

Is infinity comfortable for you?


 

Manifestations of Infinity

   During the second meeting of Infinity and Beyond, each of us discussed what we believe is our two best manifestations of infinity. Some had unique perspectives and some had similar perspectives on their belief of infinity. First, our professors defined manifestation as, “an event, action or object that clearly shows or embodies something, especially a theory or abstract idea.” Second, each of us had to explain our justifications of why we chose a specific manifestation to be infinite. We listened to each others ideas and discussed our perspectives to figure out if our manifestations could really be infinite or not. Towards the end of class, our perspectives have broadened to another level, maybe knowing something being infinite had to or doesn't have be a known unknown. It seems infinity had to be an indefinite manifestation that can be difficult to be put into words because some of the manifestations are so strong that there were no definite objections against it being infinite, (e.g., Time in the Present Tense, Reflection of Two Mirrors, Cycle of Life, Energy, Mobius Strip, Droste: Picture within a picture, Penrose Staircase). It will be interesting to see what type of categories we can create for these examples of infinity.

What Is Infinity?

At the beginning of class, our first group discussion was to introduce ourselves and state why we were interested in taking this course. For many, the interest laid in infinity and what connections it had in both mathematics and art. Next, the most basic question was asked; “what is infinity?” Everyone had an opinion. We could all agree the infinity was something large scale, something massive, that doesn't have an end. When asked for examples of infinity the universe was brought up. This seemed logical at first, yet we were stumped. Our critical thinking skills began to take place as everyone pondered at this question. Seeing our confusion, Professor Hamman asked us what defined something finite. Surely, if we can determine what makes something finite, then we must be able to determine what is infinite! As a class, we began throwing out examples of things finite: Oil, an inch, life, roads. What makes these things finite? We determined that if something has an end, it must be finite. Therefore, something that does not have an end must be infinite. We began to look back at the universe, as we debated whether this is really an example of something infinite or not. Part of the class argued yes, while others disagreed. After much debate, it was decided that if you believed in the big bang theory, then all the matter in the universe is the same as it has always been, just more widespread. This meant that the universe wasn't infinite after all. However, this is only true if you believe in the big bang theory, thus; your perception plays a part in what’s infinite or not. Could something be infinite? What about the number of books that could be written? We examined this notion. We found that if you wrote a book with one letter, or two letters, or three or four, the number could go on indefinitely creating an infinite library. We began to look at things differently. We thought of infinity as something large, but what about something small? Can the space between two items be divided infinity? One half of an inch is a half inch. Half of a half inch is a quarter inch and so forth. This theory shows that space can divide infinitely. Have we discovered what infinity is? Or are we just as lost as we were? 

By the end of class I think we all understood why the concept of infinity could drive a person crazy. Professor Carter introduced us to John Stuart Mill, a musician who believed that music was finite and went into depression. So if the idea of infinity will drive you mad, and finite make you sad, I ask us as a class; what have we gotten ourselves into? Although our class extended past our release time, we still only seemed to scratch the surface. 

Finite Music

Jade Ballard

Finite Music

When we began to discuss the concept of infinity during HP 102, one of the first things that came to my mind was, “Music is infinite.” After further exploration, I began to understand the point of view of that music, when looked at as a series of possible combinations, is actually finite. Although it was very hard for me grasp this concept, there is a mathematical way to measure the possibilities of music based on using a formula. Although music is considered finite from a technical standpoint, I think if you’re too concerned about the possibility that we might run out of music or that there will be a time when the musical combinations will be exhausted, you’re missing out on the beauty of music.

In the Autobiography of John Stuart Mill, he writes about his great love for music but after discovering the idea of music being finite, he becomes depressed. For instance, he writes, “The only one of the imaginative arts in which I had from childhood taken great pleasure, was music; the best effect of which (and in this it surpasses perhaps every other art) consists in exciting enthusiasm; in winding up to a high pitch those feelings of an elevated kind which are already in the character, but to which this excitement gives a glow and a fervour, which, though transitory at its utmost height, is precious for sustaining them at other times” (7). These words express how music makes Mill feel joyful, enthusiastic and excited from hearing music. Furthermore, Mill writes, “And it is very characteristic both of my then state, and of the general tone of my mind at this period of my life, that I was seriously tormented by the thought of the exhaustibility of musical combinations” (8). This idea of “the exhaustibility of musical combinations” refers to Mill’s his discovery that music is finite.

However, Mill perceives this idea of finiteness as a terrible, depressing concept. He elaborates by writing, “The octave consists of only five tones and two semi-tones, which can be put together in only a limited number of ways, of which but a small proportion are beautiful: most of these, it seemed to me, must have been already discovered, and there could not be room for a long succession of Mozarts and Webers, to strike out, as these had done, entirely new and surpassingly rich veins of musical beauty” (8). These words illustrate how Mill feels that because music is finite, there are only a limited number of ways in which music could be expressed. His tone of voice implies that the fact that music is finite is a hopeless thing, because there would no longer be any musical greats such as Mozart or Weber to arise. I would argue that Mill has a limited point of view when it comes to music and he has a distorted perception of what musical beauty is. I believe Mill is solely basing his ideas on musical beauty from a Western standpoint, and keeping a narrow minded perspective when it comes to music. Many years have passed since this piece of work was written and there are still thousands of musicians creating music today. Although the number of musical combinations is considered finite, it doesn’t stop thousands of musicians to continue to create works of art.

I feel that music is a subjective experience, and that being said, everyone has their own unique interpretation of what music means to them. I think this is also true in how you perceive the idea of music being finite. Just because music is finite and has a set number of possibilities does not mean it is not valuable or important as a whole. Even though there is a way to measure the large number of possibilities of ways to create musical combinations, does that mean everyone should give up making music or enjoying music? I have to disagree with Mills and I don’t believe this is true at all. To me, the number of possibilities for musical combinations is so vast it is very difficult for my mind to even comprehend this idea. It seems like it is such a large number to even fathom. Even if you were to calculate the number of possibilities for music, the number would surpass the possibilities to be created during my lifetime. And besides, music means so much more to me than a mathematical formula. I feel that we imbue meaning onto the music we love and can relate to. Listening to music, actively engaging in music and experiencing music is hard to define, but I feel it would be most akin to a spiritual experience. To me, the meaning of music is something that is difficult to pin down, but I know it something that stirs the emotions and is beyond a limited definition of a mathematical formula. If we are too concerned with the idea of “exhausting musical combinations” in the future, we fail to recognize and acknowledge the musical beauty in our present lifetime. Creating meaning is part of what it means to be human and I feel that the joy which resonates from music can be universally experienced by everyone.

Bibliography
Stuart Mill, John. Autobiography. N.p.: n.p., 1873. Print.