本科毕业设计（论文）

外文翻译

Gallery of the Infinite

作者：Richard Evan Schwartz

国籍：America

出处：Gallery of the Infinite

If you think about walking along a number line, with the numbers set out in front of you one after the other, then INFINITY......seems to be a long way off, a point on a horizon you will never reach, a height to which you can never climb. Infinity seems to be a thing outside of our universe...a far edge you wonrsquo;t see no matter how hard you stare into space.

I wrote the book to explain how a typical mathematician thinks about infinity. The approach takes some getting used to, but yoursquo;ll see that a mathematical view of infinity leads to some breathtaking surprises.

The first order of business is to talk about SETS. A set is the name mathematicians have for collections of things. The things in the set are called the MEMBERS of the set. Traditionally, mathematicians write the members of a set in symbols, in between two brackets and separated by commas. The brackets and commas are not part of the set. They are like a frame that goes around the outside of the picture. I sometimes picture sets as things placed inside boxes, because then the box looks more clearly like a frame.

Informally, i like to picture the members of a set as all sorts of things, like playing cards...or cats...or aliens. Formally, the members of a mathematical set are not really cards or cats or aliens. They are sets themselves. This gives mathematics a certain beauty and purity, but

it does raise the question as to how the whole enterprise gets off the ground. Lets not get into these technical details just yet. For now, well think of sets as being all kinds of things.

Some sets are called FINTE. Here are some examples. The set of pancake spatulas with faces drawn on them. The set of windows in Manhattan. The set of tic-tac-toe games. The set of seagulls on the Rhode Island coast. Of course, i havenrsquo;t drawn all the members of these sets.

Intuitively, a set is finite if you can start counting its members and get to the end. But this isnt phrased quite right because sometimes you might not ACTUALLY be able to get all the way to the end. Consider the set of all chess games which last less than 200 moves...or the set of all molecules on Earth. It is hard give a formal definition of a finite set, but we certainly seem to recognize finite sets when we see them. Incidentally, one of these sets has WAY more members than the other. Which one? Sometimes you might want to compare sets without having to count them. Are there more people or chairs at a concert? Just have everyone pick a chair and sit down. See if you have extra chairs or extra people, or if there is a perfect match. Are there more children or gumballs? It would be crazy to count! Just give each kid a gumball and see if there are kids left over at the end, or gumballs.

When two sets match up perfectly, the matching between them is known as a BIJECTION. Here is a bijection between a set of cats and a set of cards. In a bijection, different members of one set are matched with different members of the other, and nothing is left over. If both cats got matched to the chicken, they might fight over it. Here is one of the many bijections between the set {A,B,C,D,E,F,G,H,l,J,K,L} and the set of hours on a clock...and here are few others. This bijection might remind you of binary numbers. Finding a bijection between two finite sets is a way of saying that they have the same size. This chart illustrates how the set

{0,1,2,3,4,5,6} is matched to the set{A,B,C,D,E,F,G}. You can read from the chart that 0 is matched to F and1 is matched to B, and so on. Here are a few more of the 5040 possible bijections between these two sets. As one last example, there is a bijection between the set of animal bodies and the set of there heads.

Let us talk more about sets. If the members of sets are sets themselves, how does the whole enterprise get off the ground? You can say that the whole theory of sets is founded on the existence of NOTHING. Is one thing to say that nothing exists and quite another to say that NOTHING exists. I sometimes imagine NOTHING as a blank red painting hanging in an art gallery that nobody visits. The mathematical concept for NOTHING is the empty set:{}. Is the set with no members. Once we have the empty set, we can form the set whose only member is the empty set. Next, we can form the set whose members are the empty set and the set whose member is the empty set. Next..., and so on. Now i want to say a word about how we can define numbers in terms of these sets. Think of 0 as another name for {}. That is, 0={}. Think of 1 as another name for {{}}. That is, 1={0}. Think of 2 as another name for {{},{{}}}. That is, 2={0,1}. The pattern continues:3={{},{{}},{{},{{}}}={0,1,2},etc. From this point of view, numbers are just organized emptiness!

At the risk of sounding a bit strange, let me confess something about my view of the world. Sometimes I think that everything is just organized emptiness. Consider a baby. If you look very closely at a baby, his recognizable features dissolve into bits of organic material. Organic material turns out to be highly organized chains of atoms, which we often picture as patterns of balls and rods. The 'balls' are mostly empty space, tiny protons and neutrons surrounded by a cloud of electrons. The 'rods' are shared electron clouds. The clouds are described by the same language that mathematicians invented to understand music. At this scale, physical reality blends into pure mathematics. At still smaller scales we have no experience of physical reality at all. We j

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本科毕业设计（论文）

外文翻译

Gallery of the Infinite

作者：Richard Evan Schwartz

国籍：America

出处：Gallery of the Infinite

If you think about walking along a number line, with the numbers set out in front of you one after the other, then INFINITY......seems to be a long way off, a point on a horizon you will never reach, a height to which you can never climb. Infinity seems to be a thing outside of our universe...a far edge you wonrsquo;t see no matter how hard you stare into space.

I wrote the book to explain how a typical mathematician thinks about infinity. The approach takes some getting used to, but yoursquo;ll see that a mathematical view of infinity leads to some breathtaking surprises.

The first order of business is to talk about SETS. A set is the name mathematicians have for collections of things. The things in the set are called the MEMBERS of the set. Traditionally, mathematicians write the members of a set in symbols, in between two brackets and separated by commas. The brackets and commas are not part of the set. They are like a frame that goes around the outside of the picture. I sometimes picture sets as things placed inside boxes, because then the box looks more clearly like a frame.

Informally, i like to picture the members of a set as all sorts of things, like playing cards...or cats...or aliens. Formally, the members of a mathematical set are not really cards or cats or aliens. They are sets themselves. This gives mathematics a certain beauty and purity, but

it does raise the question as to how the whole enterprise gets off the ground. Lets not get into these technical details just yet. For now, well think of sets as being all kinds of things.

Some sets are called FINTE. Here are some examples. The set of pancake spatulas with faces drawn on them. The set of windows in Manhattan. The set of tic-tac-toe games. The set of seagulls on the Rhode Island coast. Of course, i havenrsquo;t drawn all the members of these sets.

Intuitively, a set is finite if you can start counting its members and get to the end. But this isnt phrased quite right because sometimes you might not ACTUALLY be able to get all the way to the end. Consider the set of all chess games which last less than 200 moves...or the set of all molecules on Earth. It is hard give a formal definition of a finite set, but we certainly seem to recognize finite sets when we see them. Incidentally, one of these sets has WAY more members than the other. Which one? Sometimes you might want to compare sets without having to count them. Are there more people or chairs at a concert? Just have everyone pick a chair and sit down. See if you have extra chairs or extra people, or if there is a perfect match. Are there more children or gumballs? It would be crazy to count! Just give each kid a gumball and see if there are kids left over at the end, or gumballs.

When two sets match up perfectly, the matching between them is known as a BIJECTION. Here is a bijection between a set of cats and a set of cards. In a bijection, different members of one set are matched with different members of the other, and nothing is left over. If both cats got matched to the chicken, they might fight over it. Here is one of the many bijections between the set {A,B,C,D,E,F,G,H,l,J,K,L} and the set of hours on a clock...and here are few others. This bijection might remind you of binary numbers. Finding a bijection between two finite sets is a way of saying that they have the same size. This chart illustrates how the set

{0,1,2,3,4,5,6} is matched to the set{A,B,C,D,E,F,G}. You can read from the chart that 0 is matched to F and1 is matched to B, and so on. Here are a few more of the 5040 possible bijections between these two sets. As one last example, there is a bijection between the set of animal bodies and the set of there heads.

Let us talk more about sets. If the members of sets are sets themselves, how does the whole enterprise get off the ground? You can say that the whole theory of sets is founded on the existence of NOTHING. Is one thing to say that nothing exists and quite another to say that NOTHING exists. I sometimes imagine NOTHING as a blank red painting hanging in an art gallery that nobody visits. The mathematical concept for NOTHING is the empty set:{}. Is the set with no members. Once we have the empty set, we can form the set whose only member is the empty set. Next, we can form the set whose members are the empty set and the set whose member is the empty set. Next..., and so on. Now i want to say a word about how we can define numbers in terms of these sets. Think of 0 as another name for {}. That is, 0={}. Think of 1 as another name for {{}}. That is, 1={0}. Think of 2 as another name for {{},{{}}}. That is, 2={0,1}. The pattern continues:3={{},{{}},{{},{{}}}={0,1,2},etc. From this point of view, numbers are just organized emptiness!

At the risk of sounding a bit strange, let me confess something about my view of the world. Sometimes I think that everything is just organized emptiness. Consider a baby. If you look very closely at a baby, his recognizable features dissolve into bits of organic material. Organic material turns out to be highly organized chains of atoms, which we often picture as patterns of balls and rods. The 'balls' are mostly empty space, tiny protons and neutrons surrounded by a cloud of electrons. The 'rods' are shared electron clouds. The clouds are described by the same language that mathematicians invented to understand music. At this scale, physical reality blends into pure mathematics. At still smaller scales we have no experience of physical reality at all. We j

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