A Further Look at Series

When Professor Hamman first brought up the topic of series, and pointed out how not all series need add up to one, he reminded me of something I had heard a few years ago. When I was in Calculus II, and Prof. Wheatley was explaining series, he said that he could take a series and have it add up to whatever he wanted it to. However, he could only do this with certain series. Prof. Wheatley's argument was thus: Because we are adding up all of the terms of the series, we should be able to add them up in any order we want to and have the correct value.

So let's look at the series {1/2, 1/3, 1/4, 1/5, ...} again. As Prof. Hamman showed, this series should add up to infinity. So let's change the order around some; let's move the 1/4 in front of the one third. We now have {1/2, 1/4, 1/3, 1/5, 1/6, 1/7, ...}. As you can see, the sum should still be the same. However, let's move the 1/8 in front of the 1/3 also. And let's do the same thing with all of the powers of two we can find. The series now looks like {1/2, 1/4, 1/8, 1/16, 1/32, 1/64, ...}. Intuition would say that this should add up to the same number, but Prof. Hamman showed that this series adds up to one

So what went wrong here? If we count the series in that order, we will only ever be counting powers of two. In doing so, we will be skipping a large part of the series, which will never get counted. This is like Hilbert's Hotel, where we had a bus with an infinite number of people onboard.  If we gave the people on the bus rooms in normal order, everyone would get their own room. However, if we were to try giving all of the odd-numbered people rooms first, not everyone would get a room, as we would have no end to the number of odd-numbered people on the bus.

So what does this mean? Just like before, if we take a set with a finite number of terms, no matter what order we count in, we will count them all with no exception.  However, if we take an infinite set, we can count in some order so that we can skip terms and never count them, even if we do not stop counting.  This also relates to the idea of one-to-one correspondence.  Just as we can take an infinite set and form a one-to-one correspondence with a subset of itself, we can do the same with the terms of a series.  The important thing to note here, though, is that doing so with a series changes the series, changing what it will add up to.  So, in the end, we cannot have a series add up to whatever we want it to without changing the series itself. Sorry if I got your hopes up.

Infinity...More Personalities Than Cybil

"The known is finite, the unknown infinite; intellectually we stand on an island in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land."--Thomas Huxley

After clarifying some fuzzy spots on our old friend Zeno, and discussed our mid term paper; we delved into the topic of series that shed new light on Zeno's paradoxes. It made me see them in a different way, a better way.

Tear a piece of paper in half. Then tear one half into quarters, then that quarter into eighths, etc. When will you have the entire piece of paper elsewhere? When will the piece of paper you are tearing disappear from your hands? The answer of course; lies within the study of the infinite.

Professor Hamman introduced us to the concept of series: basically it is a sum of a bunch of numbers. If we take the series {1/2+1/4+1/8+1/16+...} and never stop adding terms this series will come to equal one. If you never stop, you will never reach the number one. Another series goes like this: {1/2+1/3+1/4+1/5+...} If you continue to add these terms you do not reach one. If you never stop adding; the sum is -----infinity. You may ask,"How can one series add up to a finite number and another series add up to infinity?" I could say "because Prof. Hamman says its so" or "it just does." But I implore you to think of reasons why this is. Intuition leads us there but it is hard to put into words. Here lies the hidden beauty within the math.

We also discussed the concept of a repetend equaling a finite number, such as .9 rep. =1

If we assign "x" to .9rep. then 10x=9.9rep. If we set these two equations up as a subtraction problem such as 10x-x, and 9.99999999-.9999999; we get 9x=9. Therefore x=1. But we assigned x=.9rep. It is both! So it follows that x=1 AND .9rep.. Another way of saying this is an infinite decimal can equal a finite number. But not ALL infinite decimals equal a finite number. An example of this would be if we set x=.01001000100001... no equation would be possible to use and get a finite number.

I hope this helps everyone with their paper!

Don’t Go Near the Water Until You Learn How to Swim

Before we get into the “oooh” and the “aaahh,” or maybe even “huh? I don’t get it,” we took a little of time to talk about what is it mean by “countable?” A couple of our classmates argued that if infinity is boundless and has no limit, there is no way it’s countable. However, at the end we came to an agreement that countable is a special kind of infinity and that in order to be countable, it has to have one-to-one correspondence, and it can be put in order. For instance, natural numbers (N) are considered to be countable because we can say that 1 comes after 0, 2 comes after 1, 3 comes after 2, and so on, whereas P (N) or even the real numbers (r) are uncountable because they do not have one-to-one correspondence, nor can they be put in order. We don’t exactly know what comes after 1.278563; is it 1.278564 or is it 1.2785631 or is it 1.27856305?

After understanding that the idea of infinity has the first, second, third, and so on is the essence of it being countable, we went on to talk about paradox. What is paradox? It’s not just a contradiction, but it’s a statement that could be false and true at the same time, which does not suit very well with Aristotle’s “law of excluded middle.” There are different types of paradoxes, one of which is the self-referential type that can be shown in a famous Russell’s paradox, stated “R be the set of all sets that are not members of themselves.”  There is no right or wrong answer here, thus in order to “resolve” this problem, Russell decided that a huge set is not a set but a class.

Regarding the paradoxes, we went ahead and looked at 4 of the Zeno’s paradoxes that Aristotle tried to compile things to give an answer to each of them.  Each paradox does not work individually; hence they work as a group. The four paradoxes are:

1.      Dichotomy: in order to reach a certain location, you first have to get to half of that distant, and to get to that half of that distant, you have to get to half of the half of that distant, and so on. This is the infinite half-way points, which has no end, yet we have to pass all these infinite steps to get to that certain location.

2.      Achilles: By letting his tortoise starts out first, it doesn’t matter how fast Achilles runs, he will never catch his tortoise. This is because before he can overtake the tortoise, he first has to reach p1, but when he reaches p1, the tortoise is still ahead, reaching p2. Again, when Achilles reaches p2, the tortoise is still ahead, reaching p3, and so on.

When looking at these two situations, one might think that this is easy; if time and space has the smallest unit and is not infinitely divisible, of course, we will be able to move one unit of space at one unit of time to reach that certain location and catch that tortoise. However, let move on to the third and the fourth paradoxes:

3.      Arrows: clearly, we can definitely see the movement if the arrow moves from point A to point B. Assume that time is indivisible like we mentioned above, we will use the word “instant” as the smallest unit of time. If the word instant cannot be divisible, there wouldn’t be a start or an end, and thus the arrow never moves at an instant. The entire of time that the arrow moves from point A to point B will just consist of a bunch of “instant” moments. This means that the arrow has never moved, yet we see movement as the arrow goes from point A to point B.

 

4.Stadium:  The picture below assumes A is stationary, B is moving on unit to the right of A, and C is moving one unit to the left of A at the same time. If it takes one smallest unit of time to move the purple box of B to match with the red box of A, the time that it will take for the black box of C to match with the blue box of A is also one smallest unit of time. However if time can be atomic like one assumes in the first two paradoxes, then how long will it take to have a black box of C match with the purple box of B?

On Zeno and His Paradoxes- 9 total

Zeno of Elea was born around 490 B.C. and unfortunately none of his original writings have survived. Most of his writings we get from Aristotle's "Physics". It is speculated that he had over 40 paradoxes on pluralism alone, but only two survived that are definitely his. There are also the four paradoxes of motion that we discussed in class, and finally two more: the paradox of place, and the paradox of the millet seed. As Professor repeatedly stated last class, to be wary of my sources. I read from the Standford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/paradox-zeno/) for my information.
Zeno was rumored by Plato to be a lover of Parmenides (A philosopher and Zeno's teacher); and wrote these paradoxes to defend his teachings. Parmenides was against pluralism;" a belief in the existence of many things rather than only one" and Zeno set out to show that absurd conclusions can come out of this belief. (Talk about checking your sources)

PARADOXES OF PLURALITY---

1. The Argument from Denseness:

"If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited. (Simplicius(a) On Aristotle's Physics, 140.29)"

this can be summarized as follows: If you have a finite number of objects, then there is an infinite amount of "stuff" between these objects. Therefore the "limited collection is also unlimited." I keep thinking of the Reals when I hear this argument.

2. The Argument from Finite Size:

"… if it should be added to something else that exists, it would not make it any bigger. For if it were of no size and was added, it cannot increase in size. And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing. (Simplicius(a) On Aristotle's Physics,139.9)

But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited. (Simplicius(a) On Aristotle's Physics, 141.2)"

There is a third argument (that I won't go into) but these two can be summarized as follows: If you join or remove a size-less object...then it was nothing joined or removed to begin with. A "size-less object" seems like an oxymoron in terms to me and shows fallibility in the paradox because of this. Is he setting himself up for failure here?

3. The Argument from Complete Divisibility:

"… whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever, nothing impossible will have resulted if it has actually been divided … though perhaps nobody in fact could so divide it."

This is a tough one...I think what he was trying to ask the question: can a finite object be divided infinitely and still have magnitude?

4-7 are the paradoxes of motion as described in class.

8. The Paradox of Place:

"Zeno's difficulty demands an explanation; for if everything that exists has a place, place too will have a place, and so on ad infinitum. (Aristotle Physics, 209a23)"

Can we be in many places at once? I think so: I am at my desk, in my house, in Montgomery Village, on Earth, in the Solar System, etc. all at the same time. Beyond this concept I got a bit lost on how every "place has a place." Any clarification on this would be grand.

9. (My personal favorite) The Grain of Millet:

"… Zeno's reasoning is false when he argues that there is no part of the millet that does not make a sound; for there is no reason why any part should not in any length of time fail to move the air that the whole bushel moves in falling. (Aristotle Physics, 250a19)"

So you drop a huge sack of grain on the floor and it makes a loud "thud." Logically it is to follow that if you drop a 1/2 of a grain on the floor it should also make a sound. I wonder if the old saying "if a tree falls in the middle of the woods and no one is around does it still make a sound?" came from this? I like this one because it teaches us that our own senses can deceive us. Being a person with the highest regard to science but also a person of faith; I find this comforting somehow.

Apologies for the length.

To Set or Not to Set?

Professor Hamman and Class,

I forgot to get clarification on my question from class yesterday. I asked, "at what point does a set become too large to count, making it not a set?" From the reading, I can see that the answer is Russell's Paradox, but I'm not sure I understand when/at what point/what value a set turns in to NOT a set. I may have missed the answer to this in class. Help!

Mr. Cantor, An Extended Discussion

Sorry everyone for the lateness of my post. Seriously, I've written this post about five times now and then just erased everything and not posted anything. Even now as I write I don't think i've fully absorbed everything that I could've from our class discussions. I look at my notes and they're not complete thoughts, I do the readings and have a bunch of question marks, and I try and remember Prof. Hamman's words and it's like something doesn't click. I was originally going to open a discussion about the different sizes of infinity, but I found myself googling and re-reading aspects of Cantor's life on several occasions.

And so.... Georg Cantor.

Who was he? He was a mathematician who suffered from depression. The chronic depression he experienced really brought down his work. I think of it like when i'm having a really bad day. There is a rain cloud that looms over my head, and so when people I come across aren't that friendly or give me criticism, It produces more rain than if that cloud were nonexistent. I'm not discrediting Cantor's struggle during opposition, merely that I think, from what i've read, that he probably was more sensitive than most. On wikipedia (I'm going to use it as a valid source) it says he had a bipolar disorder. I think he could've overcome his opposition, lived longer, and brought the math world into understanding if he would've had a better attitude. His life, I felt, had a great deal of tragedy. He dealt with losing friends, questioning from his peers, not enough recognition, back to back death in the family, and confusion in spirituality. I don't know if you guys have seen A Beautiful Mind, but disregarding the imaginary friend deal, Cantor reminds me of Russell Crowe's character in that movie. My grandmother once told me that all geniuses usually live in their own world, that they're all a little crazy, and that their genius, as much a gift, is a curse because it sets them apart from the rest.

I really like this quote that I found online by him

"I have never proceeded from any 'Genus supremum' of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no "Genus supremum' of the actual infinite. What surpasses all that is finite and transfinite is no 'Genus'; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God."

I feel like that quote kinda encompasses my own views on the infinite and it's "boundeness" to be incomprehensible to the human mind.

One thing that I found really interesting about Cantor was his interest in Shakespeare. I personally love Shakespeare so this portion of my reading really caught my attention. Cantor not only came up with mathematical theories, but tried to prove the theory that Francis Bacon had written Shakespeare.

Some people believe that Shakespeare couldn't of been the real writer of the plays because the context within in them seemed too knowledgeable for someone of Shakespeare's education and social ranking.

Ultimately, Cantor's interest on the matter brought him no where. He was a much better mathematician than he was at breaking down Elizabethan literature.

My question for you: Do you have any "favorite" moments about Cantor's life or interests?

The End.