Mathematics in Borges

∞

Infinity is the state of being

limitless or endless in extent

or size.

One model of infinity that Borges

included in his works is Cantor's infinity,

which was created by Mathematician Georg

Cantor. Cantor's infinity is the idea that something

is infinite when a part is the same size as the whole.

The concept of infinity also allows for

the existence of recursive objects. These

are objects that supposedly encompass

everything. Yeah.

Borges also toys with the idea that a number

can be so large that it can be percieved as infinite.

That is, the number is so large that its size is

inconcievable and acts the same as it would if it was

infinitely large.

Set Theory

Set theory is a branch of mathematics that deals with sets, or groups of objects. A set can have no duplicates.

Sets are defined by the objects that they contain, which would make sets that contain the exact same objects one and the same.

For example, the set {a,b,c,d} is equal to the set {c,b,a,d}. On the other hand, the set {a,b,c,d} is not equal to the set {a,b, {c,d}} because the latter only has three elements and the former has four elements.

Sets can be either or finite. For example, and ordered pair would be a finite set because it has a countable amount of numbers. A function on the other hand is an infinite set because it has a limitless amount of numbers within it's set.

The Hebrew letter Aleph (א) is used to show the cardinality (or size) of an infinite set. Aleph-zero is the smallest set, representing all natural numbers, and aleph-one represents all countable numbers.

Tessellations

A tessellation is a group of figures that fills a plane without any gaps (like the one we have made here with the hexagons).

Tessellations can also be three-dimensional. These are often reffered to as honeycombs.

Probability

Probability is the likelihood

that something will happen.

The classical definition of the probability that

an event would occur was the number of favorible outcomes

over the number of posssible outcomes.

The infinite monkey theorem is a probability theorem

states that if a monkey were to randomly mash on

a keyboard for an infinite amount of time,

it would eventually type out the complete works

of Shakespeare (or any other literary work).

The Library

of Babel

The Library itself is a set, and all of the

books represent different numbers. The library

only contains one copy of every book, just as

a set contains no duplicates.

The actual physical structure

of the library is a tesselation made up

of hexagons (like this one).

The infinite monkey theorem is displayed

in the library. The library suppossedly contains

every book that could possibly exist. This means

that, just by randomly grouping letters into the

restraints set up in the library, every book ever made

would happen to exist, just by chance.

In the story, the library is supposed to go on infinitely. However, given the provided specifications for the books (410 pages, 40 lines on each page, 80 characters in each line, and 25 characters to choose from), there actually is a finite number of books that the library could hold. The number would end up being 25^1,312,000 books. (25^(410 x 40 x 80)). On the other hand, it is speculated that our universe would only be able to hold 10^84 books. That means that the universe of the library would be ridiculously large, and would act as though it is infinite, even though it actually isn't.

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The Lottery of

Babylon

The story Lottery of Babylon flirts with the idea that everything in life is based completely on chance.

In the story, all the the citizens of Babylon have their lives dictated by "The Company". The company runs a lottery that has absolutely no justification for the outcomes. Therefor, the probability that one specific outcome would occur is one over infinity.

On Exactitude

in Science

On Exactitude in Science, a

map is made that is the exact same size

as the empire that it depicts. While neither

the empire nor the map are actually infinite,

this idea that a part is the same size as the

whole, is a Cantor's infinity.

Also, the idea of a perfect map that takes up

the exact same amount of space as the empire

it depicts is a metaphor for a tessellation.

The Garden of The Forking Paths

This story discusses the three types of books that would go on infinitely.

The first is a story who's ending is also it's beginning, creating an infinite loop.

The second is one which is still being written and has the possibility of any outcome.

The last type of infinite story is one where every possible outcome of an event in the story happens, creating an infinite amount of new branches for outcomes to happen in. This would be a recursive object.

Book of

Sand

In the book of sand, Borges speaks about a book who's number of pages is limitless

Each page can be divided in half to create a new page, thus never ending or beginning. The pages are a metaphor for the infinite amount of numbers between each whole number we know. If the front cover equals zero and the back cover equals one, then there will be an infinite number of pages in between the two, no matter how small the number.

The book is also an infinite set, having an infinite amount of pages inside of it.

The Aleph

In the story, the Aleph is an infinite set.

It contains everything in the entire universe,

an infinite amount of data, making it a set.

The Aleph is also a recursive object, because

in the story it is literally an object that conatains

everything (a recursive object).

Sources:

>http://logicophilosophicus.org/guillermo-martinez/borges-and-mathematics.html -- --->http://www.vivaria.net/experiments/notes/documentation/press/clippings/atlantajournal(usa-05-10).jpg

>http://plato.stanford.edu/entries/set-theory/#5

>http://scidiv.bellevuecollege.edu/math/infinity.html

Mathematics in Borges