Never Ending Topic (Final summary)

Basically, we all started this class wondering what infinity was all about and everyone thought of it as something beyond our way of life. We compared it to nature by using the movements of the wind, sea waves, grass, human population, and water molecules. We also talked about some form of attributes of infinity which were boundless, irreducible, abstract, incomprehensible and endless, just to mention a few.
After further discussion, Professor Hamman started introducing us to simple way of relating infinity which he taught us One-to-One Correspondence by using two infinity set. Hilbert Hotel also showed us how an infinity hotel can accept each and everyone in the bus who needed a place to stay and the hotel never gets full. Cantor’s array of rational numbers and power set gave us more insight on infinity and we were able to create different subset from the original infinity set which were real numbers and natural numbers.
Zeno’s paradox which was the master of all infinity made everyone thought beyond their scope and Russell Paradox made it easier for us to get the details of the paradox and resolve it on our own. M. C. Escher and other great artist expressed their art in a way that we could express infinity through art work and we were able to identify them ourselves.
Finally, our final project work, the video and our poster summarized all we did in class but unfortunately one of our mirrors got broken with all our effort that evening but we were all happy to end the class with something we can continue to learn more about it. To believe in infinity or not to believe in infinity is the question?
On a personal note, I wish everyone all the best in the exams. We had a good team and everyone contributed in their own special ways. Professor Hamman, thank you for your time. Happy Hanukkah, Merry Christmas to you all and see you next semester if you are not transferring yet!!

Last class extension

We learned to look at infinity from different angles: mathematics, philosophy, artistic. We also discuss how sound and music relate to it. While doing research for this extension, I was so excited to learn that for the Pythagoreans, numbers, music and harmony were considered among the first principles guiding the universe.

Relating to mathematics, the Greeks first understood harmonics that vibrating strings and columns of air produced overtones. And, Pythagoras, specifically, “described the arithmetic ratios of the harmonic intervals between notes, for examples: Octaves, two-to-one ... fifths, three-to-two ... and fourths, four-to- three” (Rockmore) Pythagoras also noted that if two commensurate strings were strummed to vibrate, then the tones that they produced would be pleasing in harmony. Thus, the Greeks believed that all the harmonious things in the world must be based on whole number ratios. And all measurements must be rational.

Diagonal of a square

The Pythagoreans’ belief now changed because they discovered that the diagonal of a square is never commensurable with its side. No matter how many units they divided up each side of the square into, there was always a small amount leftover when they tried to measure the diagonal with this basic unit. 

 

In another words, “in any measuring system that gave a whole number of units to the side, the diagonal would have a length that has to be expressed as an infinite decimal expansion” (Rockmore). So, square root of 2 is irrational and therefore cannot be expressed as a ratio of two integers.

Response to Catherine's Email Post

I think your art teacher was right. She just didn't or couldn't go any further with it. She left out the 'reflection' part of it. She should have gone one step further. For instance, the leaves on the trees are green. They are reflecting the green light that doesn't get absorbed so they look green to us.

I like this idea of different sounds cancelling each other out and can create silence. I never really thought about it that way. The saying "if a tree falls in the woods and no one is around to hear it.." makes me laugh. Of course it makes a sound! We humans don't decide what makes a sound just because of our presence! I don't have a definitive answer for you, but I can say that you have piqued my interest and I am going to look in to it further.

Oh and thanks for the music reference! Gotta love The Police!

In response to Vorleak's post

Awesome post Vorleak. I'm really glad you posted about this because I got inspired and asked musician friends of mine on their opinion. My boyfriend who composes classical music told me about the "Shepard Tone," named after Robert Shepard.  In order to explain it technically, we would all need to understand composition terminology, but essentially, it is an auditory illusion that gives the impression as though a sound or upward/downward tone will reach its end. When you listen to a shepard tone, you either hear a falling or a rising in its tone and expect it to end but it doesn't. IT is explained as the following:

"The Shepard Tone fundamentally is based on sine waves. You start with a Sine wave say on note A4 which sits at 440 Hz and you have it glissando down to A3 at 220 Hz over a period of time. During the same time you have another glissando starting on A5 at 880 Hz and dropping down to 440 Hz.
If you were to repeat this cycle the glissando starting on 440 Hz would pick up where the glissando starting on 880 Hz left off. This creates the continued sensation of a falling pitch. However, if you repeat the cycle then you will quickly jump back to 880 Hz and will noticably hear it. So what do we do? In order to achieve a smooth and seemingly endless cycle we need to fade in the upper most glissando and fade out the lower most."
http://audio.tutsplus.com/tutorials/sound-design/sound-design-falling-forever-the-shepard-tone/
Here is a video of it if you want to listen. I think it's definitely a good representation of infinity through sound. Hear the illusion?

Last thoughts on a never-ending topic

"When the doors of perception are cleansed, man will see things as they truly are, infinite" _ William Blake
As i wrote this quote down, i had a feeling that it was for us. So when we first came in this class, we had the same ideas of what infinity was: endless, boundless, uncountable, incomprehensible, etc (the attribute list). As we learned more about the views and the sizes of infinity, about how to use one-to-one correspondences and the power sets to discover it, about philosophical problems with infinity, our "doors of perception" seem more opened.

After we got our first papers back and were assigned to be responsible for the final paper, professor Hamman explained some of the reading Escher on Escher. Next, he questioned if anybody ever related infinity with music. Some said yes and some said no. Then an argument came up between the answer: Is silence a sound? Again, some said yes some said no. Somebody on the Yes side basically mentioned that we see color because our mind can detect the wave radiation from light. If there's no light, the color is black. So black is a color. Using the same argument, we may say that the lack of sound (silence) is still perceived as a sound. But people one the No side argued that silence is independent. It is not created by anybody/anything so it's not a sound.

Then, we discussed how different it is when there's sound under water. The really high pitch of the whale was an example. We also talked about how fun and weird it is when we can make bubbles if we try to talk/breathe under water. After that, somehow, we ended up with what Escher said: "Infinity cannot be portrayed by sound."

Professor introduced us to the works of Ivars Peterson, a mathematician and computer writer.

 Triune Twists, at Philadelphia's City Hall

Fragment of Infinity, Designed by Ivars Peterson using the Sculpture Generator

 

Old Recycle Symbol, designed by Ivars Peterson

Petersons uses the power of math and computer to do all the work. Amazing. So from now on, we may want to notice all things that depict infinity wherever we go (sculptures, mobius strips, mirror, etc). 

 

 

 

 

Infinity in music

We had been spending time in class, looking at infinity from Philosophical, math, and art point of view. However, we haven’t spent much time, discussing what infinity is like in music. Is there such a thing as infinity in music or even in sound over all? In order to know if infinity exists in sound or not, one has to understand what sound is. This raised a question in class last week as to what considered as sound. Is silence a sound? According to the web, sound is a vibration that travel through the air or another medium and can be heard when they reach a person's or animal's ear. Thus, can silence be heard? Of course it can!! Have you ever heard of the expression “read inside your head?” It’s completely silence, yet there’s sound that read what was written on the page. Have you ever dreamt of people speaking in your dream? That’s too, completely silence from the outside, yet sound exists in the inside.  

One of the most amazing things that humans discover is music for deaf people. Deaf people sense vibration in the part of the brain that other people use for hearing – which helps explain how deaf musicians can sense music, and how deaf people can enjoy concerts and other musical events. "These findings suggest that the experience deaf people have when ‘feeling’ music is similar to the experience other people have when hearing music. The perception of the musical vibrations by the deaf is likely every bit as real as the equivalent sounds, since they are ultimately processed in the same part of the brain," says Dr. Dean Shibata, assistant professor of radiology at the University of Washington. To us, it might just be sound and sign languages, but to them, it is music. 

Just because it is silence, doesn’t mean you can’t hear the sound. Sound is everywhere around you; it’s outside of you, inside of you; it’s infinite. And when there’s sound, there is music. Thus, music is infinite.

Response to Catherine's Post Questions

Thanks Catherine! Ok, so I'm going to take a stab at the questions from what I know so far. The things that make up a tessellation are the shape, order/flow and colors. This is why Escher feels the fish tessellation best depicts the infinite. As far as I know, the non-regular tessellations don't necessarily need to be a paradox (see sidewalk illustration below). I think I like them more when they do though! I'm a glutton for punishment I guess. As for what distinguishes a tessellation from any other pattern has to do with shape, order/flow, color and ,I think, most importantly are the angles. If the "math" is incorrect and the angles aren't accurate I the tessellation sort of falls apart. Ok so I did some research on the beach art and it's an optical illusion. HOWEVER, some of those optical illusions ARE tessellations and OH do I have a treat for your eyes:

Can you see the black dots between the squares?! Cool huh? This could be considered a tessellation (from what I've read) but it also an optical illusion. So the beach paintings that we have all seen could qualify as tessellations (probably the non-regular kind). I'm not certain on that though.

Oh, the FedEx arrow on the truck.....I learned about that about 8 years ago....so sneaky! Thanks for the questions!

Escher The Magnificent!

M.C. Escher was one of the most profound graphic artists. His fascination with: division of the plane, near vs. far and high vs. low is mindboggling and beautifully reflected in his artwork. I could ramble on and give a complete history on him and compare him to other graphic artist in this genre, but the artists fall short of what he created and I feel the artwork speaks for itself.
However, I would like to give you a little information on the types of tessellations and what they are composed of.

· Regular Tessellations – triangles, squares or hexagons
· Semi-Regular Tessellations – regular polygons
· Non-Regular Tessellations – have no particular shape (infinite # of this kind)

Escher (Regular Tessellation)

Computer Aided Triangulation (Regular Tessellation)

Side Walk Pattern (Non-regular Tessellation)

Natural Beehive/without bees (Semi-Regular Tessellation)

Escher, Drawing Hands 1948 (Non-Regular Tessellation)

It’s interesting to see how infinity plays a role in each of these renderings. Some were created by man (whether by hand or by technology) and others occur in nature. Where does the tessellation/paradox begin or end? Does it move inward or outward? The last photo in particular is my favorite. Which hand started first? A visual paradox!

Twisted Thinking

So after a brutal few weeks of Zeno and paradoxes, we took it down a notch, or so we thought, and examined M.C. Escher's artwork and his different depictions of the infinite. It is note worthy that mathematicians found his art appealing to mathematics and he then picked mathematicians brains to think of ways on how trick the eyes of his viewers and challenge their perceptions. The first piece we looked at was the "Concentric Rinds."

We discussed how it shows a sphere within a sphere with points, and related it to the idea that it is infinite inside the finite. This could relate to Zeno's paradox and argument that you can't do an infinite amount of tasks in a finite amount of time. The next piece we looked at was the tessellation of horseman:

Some of us agreed that it very much resembled the Stadium paradox, perhaps because it is about bodies moving in opposite directions with equal speed. We also related it to having a one to one correspondence in that you are able to match a black horseman with a white horseman.

Then Professor Hamman demonstrated the Mobius Strip using a piece of paper: Bring the ends of a strip together to make a loop and put a half twist in the loop, so that the top surface of

the strip meets the bottom surface of the strip. Tape the ends together.

He showed us how it now only has one side by starting at any point on the strip and drawing a line, following the length of the strip. Without picking up the point of the pencil, you'll go around the loop twice and end up back where you started. What happens if we divide it down the middle all along its length? Naturally, some of us might have assumed that you'd get two Mobius strips, but that's not what happened. We got one long, twisted strip.When we cut it in half, it becomes a more complex twist, and if we had continued cutting it, there would be lots of complex twists and rings and loops.

We asked, how do they stay together and never separate? What does it have to do with continuity? It is deceptively simple in its app

earance and exceptionally twisted in its properties.

We then compared the Fish and the Lizard tessellations which expresses the idea of infinity by starting with the largest tessellations on both the inside (fish) and the outside (lizard).

Lastly, we looked at his "Waterfall" depiction:

Upon initial glance, everything seems to be in place, but when you look closely, the water fall flows uphill. We all know that water naturally flows downward, though. What is wrong with it? Well, nothing is wrong necessarily, it is just that our minds naturally equate the lines he drew with what our senses perceive of the real world. There's a visual paradox.

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.

How far can we go?

As of last class, "infinite" is no longer enough to describe any set fully.  While we had ℝ, ℕ, and many other infinite sets, these only comprised two distinct sizes of infinity.  In order to find a new size of infinity, we had to prove that there existed an infinite set from which no 1:1 correspondence could be formed with either ℝ or ℕ.  Instead of just testing sets though, which wouldn't be very practical, we tried finding a function that could take any infinite set and, from it, produce an infinite set of higher cardinality.  To do this, we considered the power set of a given set, which is a set consisting of every possible subset of the original set. And lo and behold, it worked! We proved that with any set, finite or infinite, its power set is larger than it is.  We can then repeat this as many times as we want on the resulting power sets, yielding a new size of infinity each time, leaving us with many, many of sizes of infinity.  However, we still cannot classify the number of sizes of infinity properly yet.  As far as we know right now, it is countable, but we may yet prove otherwise.

Infinite Hotel- Extended Discussion

Hi everyone,
When the class started, we elaborated more on some perspective on infinity and we jumped on infinity hotel which gave us an idea of infinite and how it works. After reading my notes I realize some of the attribute come into place when you compare the hotel room situation. Some of them were limitless, irreducible, endless and boundless. I chose this 4 because no matter how the hotel room was filled, it could still take infinite number of people in the bus.
It is limitless because it has beginning but you can never know the end and there is no last person at the hotel room. No matter how you break it down there is more room available to be occupied, small or big doesn't define it so it's irreducible. It is endless or boundless because we added as many infinite buses as we like but they all had a room at the hotel.
We used different method like n-2, 2n-2, 6n-2 etc to make it easy to move people from room one and then it went on and on but at the end of the class I felt like they all come down to some of the attribute or perspective that the class listed. In my point of view infinity is undefined no matter how we think about it; there is always more to it. I was asking myself that if everything in this World is infinite how will the World be like to live in? If no one dies and everyone keeps on living will the World be a better place to live after all? Thanks a lot
Yaa

Infinity+Infinity+Infinity

It seems Infinity is a bit slippery and just out of reach at ALL TIMES. It is bounded, unbounded, AND order matters? "The Rule of Infinity" is a concept that became a little less foggy(However, not at first!) for me when Professor Hamman proposed the infinite bus/hotel room scenario. The idea that infinity plus infinity (or even infinity + infinity + inifinity) really can be partially compartmentalized in a predictable way, can be accounted for and has value is astounding to me. I feel like I have started to grasp what the bounded aspects of infinity represent and how order plays a role and little about the unbounded possiblities it holds.

Extended discussion on Infinity- Class 1

As we were ending our debate on infinity, I had mentioned that perhaps we assign things as infinite when we've come to our limit of being able to comprehend it. In other words, when we realize we're limited with what we can see or, where logic can no longer be attainable. I think classifying something as infinite almost saves you from going insane! Jk. Even if you believe that everything has to have finite attributes or will eventually come to it's end, the limitation in comprehending how far infinite goes or when something will stop still makes whatever is finite, infinite in itself. And so perhaps finite and infinite are not mutually exclusive, but one in the same.

Thinking about what it means for anything to be infinite has proven to be difficult, obviously. If you illustrate your thinking with human analogies, God, Time, Space, Cells, grass, sand, etc, you will most likely come to see it as not possible to prove either, or, as become more confused. The quote Professor Hamman posted, "The Infinite! No other question has ever moved so profoundly the spirit of man" very much asserts this. I think that the topic of infinity inevitably raises spiritual and philosophical discussion. Where this may not be good enough for mathematicians because math solves problems and is governed by laws, infinity is above and beyond laws, and to "solve" or figure it out is almost foolish and will become an infinite project in itself!

I asked my father casually the other day, "What do you think of infinity?" He answered simply and calmly, "Eternal now." I was shocked and moved, and was satisfied with his answer.

Basically, what hearing that did for me was remind me to stop thinking about infinity, or, to stop mental chatter all together and just be present to my experience in the moment. This is very much the awakening and enlightened realization that many spiritual traditions such as Buddhism and Daoism speak of. It doesn't surprise me that my father would say that, coming from a traditional Buddhist upbringing. It also reminded me of Jessica's tattoo which had a clock with wings and in the middle of the clock it says, "NOW." Very cool, Jessica! I think the whole concept, perception, or reality of infinity is asking us to be present and accept what is happening NOW, and it does that by not allowing us to ever really know. I think that scientists who go mad ultimately surrender and accept that infinity is called infinity for a reason! But, maybe I'm wrong.

Infinity- What is it?

After a brief introduction of our classmates the big question was asked, "What is infinity?" An hour into our discussion we could still not figure this out. To explore this thought we took apart some thoughts that were used to try and describe infinity. Can infinity be counted? We analyzed this thought by relating infinity to something that's countable yet not knowable. Nobody knows how much grass there is on earth, but we do know that there is a finite number of it. Nobody knows how many humans there on earth, but for a half second there is a definite number around the globe. From this the class agreed that finite number of things could be "infinite" because although the finite number exists, it is not knowable. One classmate described infinity as the category that things are tossed into when they can never be knowable. Through this thought we explored the big bang theory. There is a finite amount of matter but how much of it is unknown. Therefore, infinity could maybe be measured. How do we measure it? This is still unknown because there's the counter argument that maybe it is not measurable. Like a circle, we can accept that it keeps going but we cant understand its infinite rotation. Because the human mind is meant to accept a beginning and an end through our own reality, it's hard to accept something as being infinite. And so we try to find it's end, or infer that a never ending process of change keeps something from going out of existence. Is change an attribute of infinity? Cause and effect is something all humans can relate to and accept. But to think that something had no cause and has always been caused further confusion in our class. How can you measure something if you don't know where it begins? If this is not explainable does it just get tossed into the category of infinity? Or maybe, the beginning is knowable and before we keep moving forward we need to go back and answer the start. As I write this I notice how many circles our class discussion went in. We started with not knowing, moved to analyzing why it is or isn't knowable, and ended with not knowing the answer to that as well. Have no fear classmates, we are not alone. No one has figured this out yet.

In conclusion, after our first discussion our class knows nothing about infinity and its attributes and knows little about the finite.