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Math Department, Mission
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Angel Serran

Professor Walton

10 May 2010

Final

Infinity

The idea that, mathematically, there are infinitely many
infinities is exactly what I have found to be extremely interesting. Although
many believe that the concept of infinity has one meaning: the largest number
or amount that you can ever come up with, the world of mathematics can prove
that definition otherwise. The idea of the
largest number you can think of suggests that an infinite amount of
something continuously goes on for an eternity until you reach that number. Through
a mathematical point of view, the idea of reaching that extremely large number
will no longer become a part of infinity for it will prove that an end does not
exist at all.

Can you recall the times during your childhood when
counting to the highest number we knew became most intriguing to us? “I can
count to infinity!” some of us would shout when we were too young and naïve to
understand the meaning of the term “infinity.” As children, our “intention[s]
were to get to the end, to name the ‘biggest number.’ But this [was] a task
where a sense of completeness [was] never going to be achieved” (Clegg 7).  Today, many of us use this term so loosely
that over time, it is as if we have lost the understanding of infinity. Some
may

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refer to material
things or other concepts such as love or change when thinking of something that
is infinite. What exactly does infinite mean anyway?

Because the media advertises that “diamonds are forever,”
couples who plan on getting married always choose diamond rings. Now, why do
people choose diamonds? In today’s society, diamonds symbolize eternity which
links to the thought that the couple’s marriage will last forever; an infinite
marriage. In reality, marriages do not last forever. Sometimes people get
divorced or decide to separate even though the couple had diamond rings to
signify their marriage. Once this is realized, their interpretation of a
diamond then changes as they find that diamonds do not really last forever. The
meaning of infinite gets lost as time goes by and that is when people realize
that in real life, infinity does not exist when it comes to material things.

There are people in this world who fall in love too
quickly. And there are those who suddenly begin to believe that they are
literally in love with someone. Believing this idea makes one think that they
will always be in love with that specific person.

There is another type of love that may sound similar and
it is called unconditional love. Unconditional love is: loving someone no
matter what the circumstances are. This type of love usually goes on between
family members. For example, a mother may unconditionally love her son or
daughter no matter what emotional pains they may cause. This is a type of love
that seems to survive any situation. Because of this, many are blinded by the
thought that infinitely loving another actually exists.

Returning to the idea of those who are in love, falling
in love with someone and believing that they are never going to stop loving
that person allows them to come to a conclusion that they

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will infinitely love
their significant other. But this idea of infinitely loving another is a
misunderstanding.

The idea of loving someone forever
becomes contradicted by the concept of change and how everything is always
changing. Nothing ever stays the same and that includes the love for another
person whether it being that: you will no longer love them, your love for them
will transform into another type of love, or the love you have will be
expressed in a different way.

Most things in our world are constantly changing every
single day. There are an infinite number of changes that are made throughout
the world. Every spring season when plants and flowers begin to bloom and grow,
a flower’s petals are never the same exact shape they were last spring. A
tree’s leaves are never the same exact hue they were before they fell off last
autumn. Every little detail in the environment, humans, animals, and even
plants continues to change even though you may not realize it. So, when trying
to acknowledge which interpretation of the term “infinite” is the right one,
how do we know which definition to trust when everything seems to infinitely be
changing?

There are plenty of interpretations of the term
“infinite,” but when it comes to the interpretation of the term through a
mathematical view point, that is where everyone’s definitions become a bit hazy
and their perception of the concept goes dim. People have often used the term
loosely when trying to describe anything that they believe can last forever. In
the previous paragraphs I thoroughly explained the ideas of materialistic
things, such as diamonds, love, and the concept of change that some of us may
believe to be infinite. Numerous ideas that have nothing to do with mathematics
come into our minds when we think of anything that infinitely

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goes on forever. But,
are we even aware of what the term actually means? Before I get into any
mathematical details, let me introduce the literal definition of the word
through the use of a simple resource; a dictionary.

According to the third edition of Webster’s II
Dictionary, the definition of infinite
is “without boundaries or limits; immeasurably great; going beyond or greater
than any arbitrarily large value.” Now that I have defined the literal
definition through the use of a dictionary, I will present an explanation for
the word. When it comes to looking at infinity through a mathematical angle,
infinity is no longer that extremely large number that you can think of, but it
becomes every number larger than the one before it. Infinity is not a specific
number, since we cannot literally measure how large it can grow, but it is a
sort of characteristic that allows us to question whether or not infinity is a
fixed value.

Many of us tend to get lost when attempting to make sense
of what we are taught when it comes to infinity. In calculus classes, infinity
is used when making specific graphs that express a line is continuously going
in the direction that its arrow is pointing. Teachers may use certain symbols
when talking about infinity, but may never thoroughly explain what the term or
symbol itself means. Because teachers have not given us a complete explanation
of infinity, we are left thinking that infinity must be a specific number we
have not reached yet. Since we have been told that specific symbols represent
infinity, then that leaves us with the conclusion that infinity must be a fixed
amount! But, these thoughts are surely wrong.

When taking higher level math classes you will notice
when something is infinite. Infinity is given a specific symbol to represent
its continuity. For example, there is the side-ways

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eights that looks like
this:  ∞    and there are those three dots that suggest
the numbers continue that look like this:
…   . There is the Aleph Null symbol,  Ào   , that deals with infinity as well. Many may
think that because infinity has symbols that represent it, it is a fixed
number, has an ending or does not continue after that value, but infinity is
not any of this. If you were to think of the largest number, then adding that number
to its self, infinity would still be larger than that amount. So, although
fixed symbols are used to represent infinity, infinity itself is not a fixed
number at all!

During the first few weeks of this extraordinary math
class, I came upon an educational article called, “Infinite Reflections,”
written by Peter Suber while scrolling through the multimedia, educational
Merlot website. Peter Suber did a great job thoroughly explaining the different
angles of infinity, but what struck me the most was his discussion on whether
or not, we as humans, literally experience something infinite.

Although a majority of the article spent most of its time
on the mathematical history of infinity, I found it intriguing to read about
real ways that real people may have experienced something infinite. For
example, Peter Suber suggested that if there had been a way to divide time,
space or matter into separate pieces, then we would easily be able to
experience infinity, but only the infinity of the specific part we are exposed
to. Even though, in reality, we may never have the ability to be exposed to the
actual state of infinity because it is immeasurably large, I find it pretty
amazing that there might never be a way for us to see infinity as a whole.

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Through Peter Suber’s article, I have come to realize
that through the language of mathematics, we have the ability to describe and
symbolize this concept of infinity. What I find so mind boggling is the fact
that we still cannot physically reach it.

There was a math problem that came up in class that
proves the point I am attempting to make right now. The math problem was:  Ào   + 1 = ?. With this math problem at hand, the
answer was written on the board;  Ào   +1=  Ào   . The given answer tells us that, no matter
how much we try to enlarge this amount that is given to us through the Aleph
Null symbol, infinity will never have a limit. Even if we changed the problem
to: Ào    + Ào    = ?, the answer would still have turned out
to be equivalent to  Ào    .

I began reading the first few pages of a book called
“Infinity The Quest to Think the Unthinkable” written by a man named Brian
Clegg. There were a few lines that made me think way beyond mathematics into
some sort of other realm that allowed me to realize infinity really was not a
number, but an extremely complex idea.

When human beings contemplate the infinite, it is almost
impossible to avoid things                       theological,
whether in an attempt to disprove or prove the existence of something more,          something greater than the physical
universe. Infinity has this strange ability to be many           things at once. It is both practical and mysterious (Clegg
2).

Due to many contradictions of teachers telling me one
thing, but seeing something else about infinity written on the board, a strong
foundation of understanding the term had been lost. Now that I am currently
getting a higher level of education, what I have concluded is: infinity, in

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fact, is not a fixed
value or a fixed amount. Although the term has fixed symbols, infinity is more
of an idea that can be thought of over and over again without ever reaching an
existing ending.

Not only have I found the idea of infinity so
interesting, but I have found the process of trying to fully understand
infinity most intriguing. There are so many variables that have been given to
us throughout our years of mathematical education. There have been plenty of
equations thrown at us, whether we understood them or not, to help us find
answers. There have been a number of symbols that kept everything simple when
going through a process of finding answers. But when it came to trying to
understand the idea of infinity, although there were variables, equations and
symbols given to us to sort of keep this idea at its simplest form, it still
remained at a complexity.

The whole purpose of this paper was to explain exactly
what it was that I found so interesting about mathematics. Although my paper
may not be at its best advantage because some may feel differently about the
concept of infinity, I strongly believe that infinity has the ability to lead
interests in others things. It has this portal into complexity that allows
people to think beyond what you would normally think about in a math class or
the first thing you would have thought of when you said the word infinity; huge
number!

I wanted to share my interest of this black hole we call
infinity because it seemed to me that the ideas, the questions and the thoughts
about it never come to an end. I remember my senior year in high school when I
attempted to take an AP Calculus class and infinity became such a common answer
that I began to wonder why did I ever decide to take a math class where a
majority of its answers never came to an end? A couple weeks later I dropped
out of the AP

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Calculus class. I then
remained living my senior year without the hassle of a math class I did not
need, but the thought of something never coming to an end still lingered within
my thoughts.

Although I would not have been able to realize just how
much infinity did remain deep inside my thoughts, now that I look back at it, I
never truly understood the real meaning of this complex idea we call infinity
which is probably why I had decided to give up on that particular class my
senior year. Just as I had read in Brian Clegg’s book, Infinity the Quest to
Thinking the Unthinkable, “infinity has this strange ability to be many things
at once” (Clegg 2). After reading those lines, it gradually began to make sense
as to why this extremely complex idea remained so unclear to me.

In high school I strongly believed that infinity was just
one specific amount and had one specific definition that everyone understood.
But finally coming to a point where I literally read these lines that were
telling me that it was completely okay to think that infinity was more than
just one thing, allowed me to put all the puzzle pieces together and figure out
that infinity was considered to be everything!

Through the process of putting this paper together, I
have found that, although infinity may be complex when it comes to explaining
its different parts, it really is much simpler in comparison to other numbers
or amounts that come to an end. In mathematics, when something comes to end,
such as a line segment or where two lines meet, you would be asked to find that
end. But when it comes to infinity, there are no questions asked; infinity
remains as is.

What a paradox this idea of infinity is. We can prove its
existence, but we cannot reach it. We give it fixed symbols and use it as some
sort of cliché that we are able to count to (I can

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count to infinity!),
when in reality we humans cannot even grab hold of infinity itself. Even though
we claim infinity is not fixed, why do we continue to label it with fixed
symbols? Why are we so fixed on believing that it is nothing but an idea? Why
do these ideas continuously have us trying to figure out if infinity really
does have an ending? We practically give the idea symbols just so that we
humans can be relieved and say that it apparently does have some sort of ending
when it does not.

The questions and all the thoughts that linger in the
back of our heads never come to an end as we begin to learn something new about
the idea of infinity. Although I have finally established that infinity is an
idea that includes everything in our universe, the questions never stop and
neither do the thoughts of trying to acknowledge the endless possibilities of
what else infinity could be. The farther and farther I go into analyzing this
amazingly intriguing idea, I learn so much more about life. I have finally
found something genuinely interesting in mathematics and that is infinity.

Sources

1.
Webster’s II Dictionary (New Third
Edition) published by the Penguin Group in 2005

2.
Infinity The Quest to Thinking the
Unthinkable written by Brian Clegg

3.
“A Brief History of Infinity”
(Firstscience.com) written by Brian Clegg http://www.firstscience.com/home/articles/big-theories/a-brief-history-of-infinity_1223.html

4.
“Infinite Reflections” (Merlot.org)
written by Peter Suber http://www.earlham.edu/~peters/writing/infinity.htm#experience

5.
“What is Infinity?” (Mathisfun.com) http://www.mathsisfun.com/numbers/infinity.html

Infinity (Principia
Cybernetica Web) http://pespmc1.vub.ac.be/infinity.html