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?