Tandem+Paper

Our Last Writing Assignment

For this assignment, we're going to write a paper together, tandem style. A tandem story is a device where two people take turns writing parts of the story.

In case you've never seen it, there is a classic example which has been around for about ten years. I reproduce it here for your amusement.

Tandem Writing Assignment
The following is a true story received from an English professor. You know that book "Men are from Mars, Women from Venus"? Well, here's a prime example of that. This assignment was actually turned in by two of my English students: Rebecca (last name deleted) and Gary (last name deleted). First, the Assignment: English 44A

SMU

Creative Writing

Prof. Miller In-Class Assignment for Wednesday:

Today we will experiment with a new form called the tandem story. The process is simple. Each person will pair off with the person sitting to his or her immediate right. One of you will then write the first paragraph of a short story. The partner will read the first paragraph and then add another paragraph to the story. The first person will then add a third paragraph, and so on back and forth. Remember to re-read what has been written each time in order to keep the story coherent. The story is over when both agree a conclusion has been reached. And now, the Assignment as submitted by Rebecca & Gary: Rebecca starts: At first, Laurie couldn't decide which kind of tea she wanted. The camomile, which used to be her favorite for lazy evenings at home, now reminded her too much of Carl, who once said, in happier times, that he liked camomile. But she felt she must now, at all costs, keep her mind off Carl. His possessiveness was suffocating, and if she thought about him too much her asthma started acting up again. So camomile was out of the question. Gary: Meanwhile, Advance Sergeant Carl Harris, leader of the attack squadron now in orbit over Skylon 4, had more important things to think about than the neuroses of an air-headed asthmatic bimbo named Laurie with whom he had spent one sweaty night over a year ago. "A.S. Harris to Geostation 17," he said into his transgalactic communicator. "Polar orbit established. No sign of resistance so far...". But before he could sign off a bluish particle beam flashed out of nowhere and blasted a hole through his ship's cargo bay. The jolt from the direct hit sent him flying out of his seat and across the cockpit. Rebecca: He bumped his head and died almost immediately, but not before he felt one last pang of regret for psychically brutalizing the one woman who had ever had feelings for him. Soon afterwards, Earth stopped its pointless hostilities towards the peaceful farmers of Skylon 4. "Congress Passes Law Permanently Abolishing War and Space Travel." Laurie read in her newspaper one morning. The news simultaneously excited her and bored her. She stared out the window, dreaming of her youth -- when the days had passed unhurriedly and carefree, with no newspapers to read, no television to distract her from her sense of innocent wonder at all the beautiful things around her. "Why must one lose one's innocence to become a woman?" she pondered wistfully. Gary: Little did she know, but she had less than 10 seconds to live. Thousands of miles above the city, the Anu'udrian mothership launched the first of its lithium fusion missiles. The dim-witted wimpy peaceniks who pushed the Unilateral Aerospace Disarmament Treaty through Congress had left earth a defenseless target for the hostile alien empires who were determined to destroy the human race. Within two hours after the passage of the treaty the Anu'udrian ships were on course for Earth, carrying enough firepower to pulverize the entire planet. With no one to stop them, they swiftly initiated their diabolical plan. The lithium fusion missile entered the atmosphere unimpeded. The President, in his top-secret mobile submarine headquarters on the ocean floor off the coast of Guam, felt the inconceivably massive explosion which vaporized Laurie and 85 million other Americans. The President slammed his fist on the conference table. "We can't allow this! I'm going to veto that treaty! Let's blow 'em out of the sky!" Rebecca: This is absurd. I refuse to continue this mockery of literature. My writing partner is a violent, chauvinistic, semi-literate adolescent. Gary: Yeah? Well, you're a self-centered tedious neurotic whose attempts at writing are the literary equivalent of Valium. Rebecca: A-**.** Gary: B*.

Our Tandem Paper's Working Title: Fumbled Infinity
Section 1. Aquinas and infinity (Comments on the 2 pages from Summa on this wiki)

====Aquinas addresses the concept of infinity and its relationship to God in the section of the Summa entitled God exists without limit. He opens this section saying "God is not limited in any way." This brings to mind the sizes of infinity and how there is God, and the capabilities of God must be uncountably infinite. The possibilities of God are endless and there is no way for us to quantify this potential. As Aquinas goes on to explain, God is not limited by matter or the ways of this world. He is beyond these limitations as he set them, and therefore has the potential to change anything and everything if he desired. As God exists outside this world, our logic on the possibilities of God (as we see in the light of this worlds) can not begin to explain the uncountably infinite limit of God's potential. user:AlexandriaHopp ====

Interestingly, Aquinas also stated in the same section of the //Summa// that not even God could create an unlimited object as it would lead to a contradiction. To create something is to give it form, and form imposes limits. He also states that the ability to understand does not have form and is not limited in the same way. Humans, do have a limited capacity to understand the world around them. Despite his best efforts Cantor was never able to prove the continuum hypothesis, and it was later proved that such a proof is impossible. As human understanding is inadequate to comprehend the natural world, it goes without saying that it would be unable to fully comprehend the supernatural. user:Abbeybowman05

Aquinas argues in the Summa Theologia that two kinds of infinity are actually impossible. The first infinity is infinity in size. The second is infinity in number. To better understands what he means we need to travel back to Aristotle and understand what he had to say about infinity. However, to understand Aristotle, we similarly, need to travel back to the pre-socratic philosophers as they debated about change and permanence. Heraclitus (553-475 BC) argued that everything was in a state of flux. He famously states that one "could not step into the same river twice." Parmenides argued, against Heraclitus, that nothing changed and our senses fooled us. Zeno, a disciple of Parmenides, offered a famous argument against the notion of change. This is one of Zeno's famous paradoxes. Zeno argued that to go from point A to point B (i.e. movement), we must first go half that distance, we will call this point C. However, before we can go half of that distance and reach point C, we must first go half of that distance, which we will call point D. However, before we can arrive at point D, we must first go half of that distance and so on infinity. Zeno concluded that we cannot transgress an infinite distance, even in a infinite amount of time and so all known instances of movement must, in actuality, be illusions. Zeno's argument is fallacious in many instances, one example would be in that fact that even accepting his argument would be an example of change, but that is for another day. Aristotle took this argument on and argued that no infinity can actually exist (except for space and time) (Aristotle for Everybody p. 173). Aristotle would have agreed with Zeno, we can divide a distance into ever smaller parts. However, that is only a potential infinity, not an actual infinity (Aristotle for everybody p. 173). This is the concept Aquinas is picking up on when he says that infinity in number can never be actualized. There will never be a moment in which one can say "Aha! I got it! I have gotten to infinity!. Just counted to it! I got it!" Infinity does exist, however, only potentially (showing movement is possible....duh). Aquinas simply means that we can always make numbers bigger or lines longer and in that sense infinity exists. The second infinity is also impossible, an object of infinite size. Anything that is made of matter - or has size cannot be infinite. For if it was, there would be nothing else. As Aquinas famously says, "If one of two contraries be infinite, the other would be altogether destroyed." Furthermore, material objects are a composite of matter and form and so are defined by their shape. However, shape implies limit. If a ball had no limit, it most certainly would not be a ball. What is Aquinas's point to all of this one might ask? Why does he talk about infinity in a section in which he discusses the attributes of God? His point is simple. God is otherly other. He is not a being among beings. He is not just the greatest being in a set of other beings. God, because he is infinite, cannot be an object in the world. He is an actual infinity and so cannot be like anything in the universe. This may seem simple to a Christian (well maybe....at least the "God not being in the world" bit) but it is something the likes of Richard Dawkins, Stephan Hawking, Daniel Dennett and the rest of the so-called and self-named "brights" do not understand. user:sethkreeger

Section 2. Other philosophers before Cantor (Famous phil who wrote about infinity. Galileo is ok)

Near the end of his life, Galileo began to play with the concept of infinity. He wrote about infinity in his book //Discourses and Mathematical Demonstrations Relating to Two New Sciences.// In one section of the book, Galileo introduced what has come to be known as Galileo’s Paradox. Galileo first stated that every natural number has a unique square number (nothing new here). Then he explained that the set of all natural numbers can be “mapped” onto the set of square numbers, thus creating a one-to-one correspondence between the natural numbers and the square numbers. For example, 1 maps to 1, 2 maps to 4, 3 maps to 9, etc. If there exists a natural number for every square number (and vice versa), would that not mean that the number of natural numbers is equal to the number of square numbers? When Galileo discovered this intriguing property, he called it a paradox because it goes against intuition. How could there be just as many square numbers as there are natural numbers if square numbers make up only a sliver of all natural numbers? Galileo thus believed that the relational concepts of a set being less than, greater than, or equal to another set could not be applied to infinite sets. Cantor would later disprove Galileo's claim about the relational operators. Still, the ideas in Galileo’s Paradox would later be extended to show that any set that forms a one-to-one correspondence with the natural numbers has the same cardinality as the set of natural numbers, a size that Cantor would later call “aleph null.” user:seankeast

In Aristotle’s Physica III, as discussed by Jaakko Hintikka in " the article "Aristotelian Infinity", Aristotle discusses Infinity. At the time there was an ongoing debate as to the existence of infinity. Arguing for its existence, Aristotle states that infinity does not exist in the same way that individual items do, In an attempt to counter the argument that infinity did not exist because it cannot be held or studied in the same way that a physical object. Aristotle argued that while infinity did exist it gradually and continuously unfolded, unlike physical objects. He likens infinity to a day; the entire day does not exist at any given time, but unfolds a moment at a time. Aquinas made a similar statement in //Summa//, commenting that “potential infinities are actualized…bit by bit.”(Aquinas, pg 21)user:Abbeybowman05

Leonard Euler (1707-1783) considered [countable] infinity as just another number, because it is made from adding the natural numbers. This is different from most mathematicians and philosophers that we have been studying in the class. If Euler considered infinity as just another number, then different sizes of infinity would be difficult to handle. People would not be able to use it as a number, because we don’t know which number it is. Euler argued later went on to argue that -1 is larger than infinity because of a series summation that he performed. He thought that infinity separates the positive and negative numbers much like 0. Because of these theories, infinity is the reciprocal of 0 according to Euler. None of his claims on infinity appear to be justified with proofs, so it is hard to determine the validity of his claims, but it seems like he was stretching a bit far with his assertions on infinity. user:bethherdegen

=Aristotelian Infinity =

Section 3. Cantor's Detractors (Poincare and Kroneker)

One of Cantor’s greatest detractors was French mathematician Jules Henri Poincaré. Poincaré was known to vocalize his thoughts, claiming that the views of Cantor and his followers were riddled with contradiction. While Cantor tried to prove that there are different sizes of infinity, none of this mattered to Poincaré because Poincaré did not believe in infinity to begin with. Specifically, Poincaré did not believe in something called “actual infinity” (Think of an infinite number of jellybeans. How could they all fit in the universe if the universe is of finite size?). He thought of actual infinity as separate from human reality. Since Poincaré did not believe in infinity, sizes of infinity obviously could not be applied, making Cantor’s work meaningless. Poincaré despised Cantor’s ideas so strongly that he referred to them as a disease in mathematics. user:seankeast

When Cantor first tried to publish his finding in //Crelle’s Journal,// Cantor's former teacher, Leopold Kronecker blocked the article's publication. Kronecker believed that the integers were the work of God and that any other numbers were the work of man. This view caused Kronecker to reject all of Cantor's reasoning. Kronecker felt so strongly toward Cantor's findings that Kronecker stopped Cantor's appointment as a professor at the University of Berlin. The backlash from established mathematicians greatly hurt Cantor's mental health and prevented him from achieving success in his lifetime. Fortunately, Cantor's friend and colleague, Richard Dedekind was one of the few willing to listen to Cantor's theories. Despite this, Cantor received skepticism from his peers throughout his life.user:craigjensen13

Section 4. Consequences (Did we waste time on dumb ideas? Did we have bad math? Paradoxes?)

This class has been most beneficial. It has also inspired these thoughts and reflections: Science is dependent upon there being reasons for the existence of things. A scientist sees at a volcano that has just irrupted and looks for reasons as to why it irrupted. Indeed reasons permeate everything we do as human beings. It seems then that it follows that if we are to be consistent as knowledge-seekers, me must posit a reason for the existence of the Universe, the Big Bang or a Singularity. This is what we mean by God (God must be outside of time and space to be the cause of time and space, for nothing in the Universe can be the cause of the Universe for that would simply beg the question). God must transcends time and space and be the answer to the question "why is there something rather than nothing?" God is be a necessary being, or a being that simply was his own reason for existence, and so would simple be existence itself (calling God a being is misleading). One might here object and argue that matter or energy could just be this necessary reality. However, as soon as one accepts this, one has a problem. Matter by its very nature is what scholastic philosophers would characterize as potential, it is in this shape or that shape and so cannot explain itself. Thus even if matter were eternal and had no beginning, it is still simply potential and in need of an actualizing agent. Furthermore, the best science we have suggests that the universe came into being 13 billion years ago. Now if something came into being at a particular moment, it most certainly cannot be a necessary reality. Therefore, simply to say that the Universe "just is" is to raise a huge problem. These two words would be devastating to all categories of human knowledge. Why? Simple because we have allowed one instance of something being the case without need of an explanation, one case where something just is, with not reason for its existence. However, in mathematics, philosophy, chemistry, biology and so on we seek reasons for the discoveries we find and the phenomena we seek to explain. If, as human knowledge does, we seek reasons for the existence of things we observe, but at the same time allow one instance of something not having a reason for its existence, we have just allowed a contradiction and it is a basic law of logic that anything can follow a contradiction. Therefore, if we allow this contradiction of seeking explanations for things, but allowing one instance of something not needing an explanation for its existence, then the whole enterprise of human knowledge is inconsistent. Therefore, to be consistent we need to posit the existence of God (remember God is his own reason for existence and so asking "who caused God?" is just silly. God's essence and existence would be the same. Also, remember matter cannot be this necessary reality). To further show my point I will reference an argument made by Edward Feser in his book //Scholastic Metaphysics//. Suppose we see an object we will call A. Now the reason for A is B. However, the reason for B is C. If C has no reason for its existence then neither does A, for A derives its existence from B, who in turn derives its existence from C, but if C has no reason for its existence, then really there is no reason for A or B either. For example, if a book is only on a table because of gravity, an gravity only exists because of the earth's relationship to the sun, and the sun only exists because of the Universe, but the Universe has no reason for its existence, then there is really no reason the book should be on the table at all. This is just an argument that this course has helped form.user:sethkreeger

It could be thought that God and our understanding of him could be compared to a paradox. When we spent time looking at some mathematical paradoxes, ideas bases on human "logic" (more likely common sense), seem to contradict the mathematical logic which created the paradox. As humans, we create ideas on how things should be and if they don't seem to make sense with our logic than they must be a paradox. This same thought can be applied to God's logic and the view we have of him. The bible and our experiences have given us a picture of who God is, however sometimes in life God presents us with situations that seem to contradict this image. Our human view of how God should interact with us logically, sometimes is contradicted by reality. In a way this could be looked at like a paradox. We can not fully comprehend God but just like paradoxes this lack of understanding of the logic behind the story does not make it false, but rather makes us aware that we, as humans do not have the capability to truly understand God and the world. user:AlexandriaHopp

One of the consequences of Cantor's work was the continued focus on the Continuum Hypothesis. Cantor himself had quite a bit of trouble with this. Sometimes he would think that he had proved the Hypothesis true, only to prove (falsely) the Hypothesis was not true, but indeed false. A sane person would think that Cantor ought to have spent an extra few days looking over his completed proofs rather than proclaiming his triumph every time he thought he had an answer. Of course, Cantor wasn't completely sane at the time, so we can probably give him a pass. However, Cantor's obsession with the Continuum Hypothesis continued on in other mathematicians. It wasn't until Cohen and Gödel came along that an answer to the Hypothesis became available. These two mathematicians proved that in the existing models, the Continuum Hypothesis cannot be proved true or false and is therefore undecidable. Of course, this hasn't stopped mathematicians from trying to come up with a better model that can prove the Continuum Hypothesis true or false once and for all. A proof of this kind could potentially open up more types of math previously denied to us. However, mathematicians may never find a model that decides the Continuum Hypothesis. Hence, Cantor's struggle with the Continuum Hypothesis is carried on today through modern mathematicians.user:craigjensen13

Even though Cantor’s Continuum Hypothesis is unable to be proven, it is known that there are higher degrees of infinity. From this we get lots of pop culture references, the most common being “To infinity and beyond!” Which is actually correct, there is something bigger than infinity, namely the power set of the cardinality of the original infinity. In the popular book and box office hit movie John Green’s The Fault in Our Stars, there is a scene on infinity, “There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a //bigger// infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities.… I cannot tell you how grateful I am for our little infinity. You gave me forever within the numbered days, and I’m grateful.” While this is a sweet sentiment, Hazel is incorrect. There is the same amount of numbers between 0 and 1, 0 and 2, and 0 and a million. She is right in the sense that there are bigger sizes of infinity, but not in the explanation. There is much debate as to whether or not John Green meant to be incorrect in his book. Many people are uneducated in infinity and because of this they believe these cultural references that they are given. There is an infinite number of infinites and the limit does not exist for the infinites. Sorry Mariah Carey, there is not an end to infinity.user:bethherdegen

Each student must contribute a paragraph to two sections. Please sign your paragraph with three tildas ~ ~ ~ because that will turn into your username name. Your grade depends on spelling, grammar and accuracy. Humor is most definitely allowed.

user:mcdaniel30