Final Class Summary

On our December 17 meeting, only three students were able to attend the class. Therefore, only three presentations were made.

    Orsi, the first student to present, chose an Escher-like repeated tessellation. She explained that she first drew a shape on a paper, and then cut out that shape to use as a tracing tool to repeat the pattern. After the repeated pattern was complete, Orsi gave some of the shapes faces. Then she colored her entire artwork and gave particular colors to those shapes with the face. Some of us saw Cookie monster in those shapes with the face, while others saw Marge from the Simpsons, without the necklace and the tall blue hair.

Professors Hamman and Carter asked questions about Orsi’s artwork. One of the questions they asked her was what she would differently if she had to redo her work, and Orsi replied that she would have made the shapes a bit smaller.

    I was the second person to present after Orsi. I chose to use one point perspective. My drawing depicted a road to infinity beneath a repeated chain of floating boxes. The boxes seemed to be stacked in front of one another and disappeared into the horizon line as they got further away. Professor Carter asked why I left out the tiles from my previous sketch, and I explained that I had tried to do one drawing with the tiles, but that the proportions of the tiles did not come right and the final drawing had looked more like a cobweb. Professor Carter then demonstrated how to connect lines when trying to make tiles.

    Finally, Sara, the third person to present, demonstrated her art work. She also chose a one point perspective. She drew a box with lines that were getting smaller in the middle. She then colored her art work four different colors, and manipulated the colors to be darker as they got closer to the middle. This way, the box appeared to get deeper and deeper infinitely. Sara also shared how she drew her lines and all the steps she took in completing her drawing, including photocopying her drawing onto a different paper and coloring the final copy.

    After we finished our presentation, we ended the class by enjoying the snack Professor Hamman brought. I left the class feeling like Norton Juster when he said about trying to reach infinity:

 “You know that it's there, you just don't know where-but just because you can never reach it doesn't mean that it's not worth looking for.”

Extension November 19 Meeting.

                                                        Zeno, a Man of Mystery.

Just like Vincent Van Gogh and Emily Dickenson, Zeno of Elea and his paradoxes were appreciated after his death. In fact, many mathematicians today are picking up where he left off. Zeno proposed the idea that you can divide the finite distance between two points infinitely. He was the first person in history to shed light on the problem with infinity (Dowden).

His most famous example was that of a race between a tortoise and the legendary warrior Achilles. Zeno claimed that you can continue halving the distance between the tortoise and Achilles, causing Achilles never to reach the tortoise, because he must cross half of the distance, and then half of that distance, and so on, ad infinitum.

However, this paradox does not pass the test of our daily experience. Following Zeno’s logic, there would be no such thing as motion, because to get somewhere, a car would have to cross infinite halves of halves of halves. However, in our world, cars do make it to their destinations, and the faster the car can move, the sooner it will reach its goal.  This car example fits our common sense, even though we may not be able to mathematically explain how the car traveled through infinity – that is, the infinite number of points between the departure and the destination.

If you think that those ideas are wild, then check out these crazy stories written about him. Not much is known about his life. It is assumed that he was born around 490 BCE, in a city called Elea in southern Italy.  He was the student of Parmenides, and Plato stated that Zeno’s paradoxes all came from his teacher’s ideas. Parmenides emphasized the distinction between appearance and reality, and he believed that our perception of reality is deceptive and does not report what is real (Dowden).

Plato also claimed that Zeno was caught aiding the rebellion against the tyrant of Elea. Zeno was eventually arrested, and when questioned about his supplying weapons to the opposition, he tricked the tyrant into getting closer to him, whereupon he bit him and wouldn’t let go until he was stabbed (Dowden). After reading all these crazy rumors, do you wonder about Plato’s credibility in reporting Zeno’s life?

 

Dowden, Bradley. "Internet Encyclopedia of Philosophy." Zeno’s Paradoxes []. N.p., 7 June 2009. Web. 25 Nov. 2013. <http://www.iep.utm.edu/zeno-par/>.

 

Class Summary November 19, 2013

Class Summary- November 19, 2013

“If you stare too long into an abyss, the abyss will stare back into you”

At the beginning of the class, we first had a discussion about a number line and that no matter what number you choose on it, it will always be in the middle. This statement was made based on the concept of transformation, the shifting of coordinates, in mathematics. 

Then, we transitioned into the significance of a paradox and defined it. Professor Hamman gave an example of a simple paradox: “This Statement is False”. This is a paradox because this sentence is technically written correctly, yet the content within the sentence makes it false. Therefore, a contradiction occurs making it neither true nor false.

Afterwards, we then discussed Zeno’s Paradoxes in depth. Many of us were confused about the Arrow and Stadium paradox. Professor Carter provided the class with an analogy of a camera for the Arrow paradox to help the class understand it. He said that when you capture a moment with a camera-the faster the shutter speed, the less time the object has to move. Eventually, the object will not move. However, there is always time between the moment of when it is taken and when it is being exposed.

 

Next, we looked at possible solutions for Zeno’s paradoxes, but then realized that when one solution is made, it would contradict with another paradox, which then made the paradoxes unsolvable. For the dichotomy and Achilles paradoxes, we said that there could be a solution by creating an ending point to the dividing. However, when we made a solution for the arrow and stadium paradoxes, taking away the “Zeno” or time/space in between the object, it would contradict with the dichotomy and Achilles paradox.  

Lastly, we viewed the various manifestations and compared them to Zeno’s paradoxes. One of the manifestations that we mentioned was the mirror image and how it is similar to the dichotomy because the mirrors appear to be getting smaller (half the size) in the distance. 

Class Summary For Tuesday Nov. 5th 2013

We started the class discussing about our previous homework which was to take a stand on how we prefer to view infinity, either from a rationalistic point of view or from an empirical point of view. Prof Hamman asked if anyone had difficulties choosing a stand. 50% of the class preferred to view infinity from a rationalistic point of view and the other 50% preferred the empirical.

Then, Prof. Carter continued on the topic of the different paradoxes. He started with the paradox of the arrow. He explained that the arrow can only travel as far as its length. And when it gets down to one instance, the arrow will be occupying a hundred percent of its space. And so this means the arrow is not moving at all for every given instance. Prof. Hamman tried to explain the concept behind Zeno’s paradox that there is no subdivision of time. This revealed a big difference between the first two paradoxes that suggest that, we cannot keep subdividing and the next two paradoxes that says if we cannot continue to subdivide, then we’ll get into trouble.

Another paradox Prof. Carter talked about was the stadium. The shortest distance the dogs can move is one length. They cannot move half a length. If the pandas are not moving while the dogs are moving next to each other passed the pandas, it will be discovered that we cannot tell when the last dog passed the second panda. From Prof. Hamman’s point of view, it was hard for him to understand this concept because we cannot think of a smallest unit of time. The next question that came up was whether we can divide time? If there is not a smallest unit to divide time, then we are going to have problem resolving this concept. At this point, it became clear that the paradox of the stadium got everyone more confused. The major challenge was unveiled, the rational is saying one thing and the empirical is saying another. Just as in the case of the dichotomy, it makes no sense theoretically that before an object can travel a given distance, it must travel half its distance. We all concluded that our individual perspectives still count in how we interpret these various concepts of infinity.

Prof. Hamman commented that Zeno was really clever to have imagined all these paradoxes. The class became curious to know how Zeno was able to come up with these concepts. Prof. Carter pointed out that Zeno might have been influenced by his friend who believed there is no space and there is no time.

In conclusion, Prof. Hamman commented on how Aristotle misinterpreted Zeno’s paradoxes and the great impact it had on the knowledge of infinity for a long period of time. At the end of the class, we were assigned our next assignment which is to create something that is a best representation of infinity. What a good way to end the class!

Actual Versus Potential Infinity- extension Oct. 22,2013

Actual versus Potential Infinity

Extension: Oct. 22, 2013

The Ancient Greeks generally referred to infinity as “formless, characterless, indefinite, indeterminate, chaotic, and unintelligible” (Dowden). At the time, the term was perceived negatively because it was vague and it had no clear criteria to differentiate finite from infinite. Among various philosophers, Aristotle was one who made an attempt to this confusion by creating terms for infinity, “actual infinite” and “potential infinite” (Dowden). Aristotle developed this clarification through Zeno’s paradoxes about infinite divisibility. His observations led him to defining actual infinity as “an endlessness fully realized at some point in time” (Daring). Furthermore, he defined potential infinity as something that could be “manifested in nature [and could be used in a technical sense]” (Dowden). For example, a potential swimmer can learn to become an actual swimmer, but a potential infinity cannot become an actual infinity. According to Dowden, Aristotle argued, “all the problems involving reasoning with infinity are really problems of improperly applying the incoherent concept of actual infinity instead of the coherent concept of potential infinity” (Dowden). During that era, viewing infinity through these terms was a way to comprehend Zeno’s paradoxical statements such as the Dichotomy paradox. Within the Dichotomy paradox, suggesting that a fixed destination could be infinitely divided, conveys how actual and potential infinity creates a contradicting statement (Waterfield). However, Zeno made the mistake, according to Aristotle, of supposing that “this infinite process needs completing when it really doesn’t; the finitely long path from start to finish exists undivided… and it is the mathematician who is demanding the completion of such a process…Without that concept of a completed infinite process there is no paradox” (Dowden).

Even though Aristotle promoted the belief that “the idea of the actual infinite−of that whose infinitude presents itself all at once−was close to a contradiction in terms…” (Moore, 40) various philosophers and theorist contradicted Aristotle’s idea. Archimedes, Duns Scotus, William of Ockham, Gregory of Rimini, and Leibniz each had there own ideas and questions concerning Aristotle theory (Dowden). Therefore, it almost seems impossible to define infinity because there are various perspectives that it can be viewed from.

Through looking at Aristotle connotations of infinity, it shows how there can be another dimension to defining infinity among various other theories that we have already studied. Moreover, do you think Aristotle’s actual and potential infinity would make a difference in how we categorized the manifestations that we produced as a class?

Works Cited:

Darling, David. "Infinity." Encyclopedia of Science. The Worlds of David Darling, n.d. Web. 01 Nov. 

            2013.

Dowden, Bradley. "The Infinite." Internet Encyclopedia of Philosophy. ISSN 2161-0002, 6 Sept.           

           2013. Web. 28 Oct. 2013.

Moore, A. W. The Infinite. Second edition, New York: Routledge, 2001. Print. 

Waterfield, Robin, trans. Aristotle Physics. Comp. David Bostock. London: Oxford UP, 1996. Print.

           ISBN # 0-1995-4028-4.

Extension of Class 7th October, 2013

In the article “The Intuition of Infinity,” Fischbein, E., D. Tirosh and P. Hess explain the problem of categorizing infinity based on the contradictory process of our thinking. They explain that this conflict is deep and that we cannot completely avoid it even with the most sophisticated mathematical tool. They made reference to Cantor’s argument that more than one kind of infinity exists. This they pointed out is definitely against our intuition of the nature of infinity.

The results of their studies show the difficulty in the process of understanding the concept of infinity based on our individual perception of infinity. Categorizing Cantor’s idea of the “Hierarchy of infinity” as abstract or as concrete is therefore based on our perception and thought process. They explain that physically performing the various examples of manifestations presented can make us categorize them as concrete- what we can see, and our mental ability of the process can lead us to categorize as abstract.

Citations:

Fischbein, E., Tirosh, D.  & Hess, P. (1979). The intuition of infinity. Educational Studies In Mathematics. JSTOR.

Manfreda Kolar, V., & Hodnik Cadez, T. (2012). Analysis of factors influencing the understanding of the concept of infinity. Educational Studies In Mathematics. Academic Search Complete.

Summary of our Oct. 7th meeting

My blog summary is about our meeting on Oct. 7^th. 

 For that Tuesday our homework was to think of 3-4 categories, which we could use in class to categorize the different manifestations of infinity.

During the first part of class we wrote our categories on the board and by the time everyone finished we ended up with 22 categories. Surprisingly, some of them were different from one another but others were quite similar.

 During the second part of class we tried to group the categories to limit their numbers and as it turned out the organization of them was quite difficult. We argued about which ones are similar, which ones should we keep and which ones are subcategories of others.  One of the major argument we had involved nature and life. Some of us thought nature is not a necessary category, because manifestations which belong to nature can be part of others as well. However, others in the class thought that nature and life are important categories because they represent certain manifestations better than any other categories.

At the end of class we ended up with only five categories; abstract, concrete, formal, measure and life. As our next homework we had to categorize the manifestations of infinity using the five different categories. As it turns out the five categories might be still too much, because most of us could fit some manifestations of infinity into more than one category.

Should we limit our categories maybe only to 3? And if yes, which categories should we keep and which ones we should not?

 

                                                      

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.