1 of 5
C.J. Gorrell
Math 274
The History of Infiniti
Before entering any in depth view of the history of math, it is important to note
that there were two paths I could have taken in formulating this work.
The first method
would be to take advantage of the infinite monkey theorem.
This theorem states that if a
monkey were to randomly hit keys on a typewriter for a very long time (for an infinite
period of time), then he would most likely have typed a replica of any work such as a
Shakespearian play.
Therefore by this method, I would just have to randomly type for a
very long time, and this essay would have been eventually produced any way.
The other
method would be to do the research, and write the essay straight out.
I chose the later.
Infinity is an extremely vast subject, whose progression still continues today.
The
classical ideas of infinity, started with Aristotle, and continued up until the era of Cantor
and Weierstrass.
While infinity was a subject across many subjects other than
mathematics, it was such a powerful tool in the development of mathematics that it is
worth only considering the mathematical importance in the sum of this analysis.
The start of the study of infinity started in the 4
th
century BC, with ancient India
and ancient Greece.
The Grecians however, made the most significant progress on the
subject at this time.
Starting with Zeno an early contributor to Infinity, and perhaps the
first to introduce a paradox in the analysis of infinity.
Zeno observed that in an analysis
of movement, that for a man to span a distance, he must first span half the distance, and
then half the remaining distance.
By repeating this procedure, Zeno observed that by his
analysis it would become impossible for anyone to span and finite distance.
1
This
became Zeno’s most famous paradox, however he had numerous others.
The paradoxes
of Zeno became a popular subject in the arguments between Aristotle and Plato.
Plato first introduced the idea of infinity through the subject of religion.
He
defined infinity through the comparison of God to man.
Plato’s student, Aristotle, takes
on the subject of his teacher, and defines to separate identities to infinity.
These are
potential infinity and actual infinity.
2
Actual infinity or Aperion as Aristotle labels it, is
divine, and in a sense beyond the comprehension of man.
He therefore states that actual
infinity cannot exist.
Instead Aristotle says that man only deals with potential infinities.
Aristotle uses potential infinity to define the use of infinity in the mathematics of his
time.
For example, the method of exhaustion as used by Archimedes was considered
1
1
Victor J. Katz,
A History of Mathematics
, 3
rd
Ed.
(Boston: Pearson, 2009), 4547
2
2
Paolo Zellini,
A brief History of Infinity
, 3
rd
ed. (London, Penguin, 2005) Pg. 31
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2 of 5
C.J. Gorrell
Math 274
The History of Infiniti
potential infinity because the method of exhaustion was not considered to have any
reference to actual existence.
3
Aristotle’s theorems on infinity lasted until the 19
th
century, when they were
disproved be Weierstrass.
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