Zeno of Elea & the
Evolution of Infinity
Kornilowicz, Gabriel
Chu, Dan
Zeno and his Paradoxes
● Born in 490 BCE in Elea, Italy
● Student of the Eleatic
philosopher Parmenides
● Upon his arrival in Athens with
his teacher he introduced his
book of paradoxes, of which
very little remains
Source: https://probaway.wordpress.com/category/philosophers-squared/
Parmenides
● Born 515 BCE in Elea, Italy
● Greek philosopher who
focused on Monism
○ The belief that there exists
only one ‘thing’ in the
universe
Source: http://www.cplong.org/digitaldialogue/digital_dialogue_17_parmenides/
Zeno’s Goals
There is some contention around the goals of zeno and his paradoxes:
●
●
There is the belief that he was merely trying to prove his teachers Monistic
arguments
There is also the belief that instead of directly proving his teacher, his
paradoxes existed to show how ridiculous the oppositions beliefs were,
thus ‘indirectly’ defending his teacher.
The Arrow
●
●
●
Time is made up of an infinite number of points
An object must occupy a space equal to itself at all times
So, an arrow in flight at any and all moments is not moving
y
1
1
speed=0
y
1
2
2
speed=0
y
2
t1
t2
t
The Arrow
We know that in calculus the derivative of
the position of an object with respect to
time is the velocity:
y
1
1
y
|velocity|>0
1
2
2
y
|velocity|>0
2
t1
t2
t
Dichotomy (The Racetrack)
●
●
●
Before a runner can reach the end,
they must first travel half
Before a runner can reach the
halfway, they must first travel onefourth
Before a runner can reach the onefourth, they must first travel one-eighth
● Therefore the runner must travel
an infinite distance
Source: https://1badnavajo.files.wordpress.com/2013/04/track.jpg
Dichotomy (The Racetrack)
∞
●
●
●
n=1
Zeno believed this to converge to infinity
Actually converges to one
Lack of calculus
1
1/2
1/4
1/8
1/16
1/32
...
Achilles and the Turtle
●
●
●
●
Imagine that Achilles and a turtle are having a race
○ Achilles is much faster than the turtle
The turtle receives a head start
When Achilles begins running, he must first catch up to where the turtle
once was
○ By the time he does this, the turtle has moved on
○ He must then catch up with the turtle once more
■ But by this time, the turtle has moved again
■ Etc
Zeno assumes that space and time are infinitely divisible
Achilles and the Turtle
●
●
Source: http://www.iep.utm.edu/wp-content/media/Achilles_Tortoise.jpg
Straw man argument
Distance Achilles's has to travel is
not actually infinite
○ Imagine Achilles travels:
○ d1 = catch up to p1
○ d2 = catch up to p2
○ d3 = catch up to p3
○ d1 + d2 + d3 + … != ∞
■ Will eventually converge
to a constant
Infinity Post - Zeno
●
●
Zeno’s paradoxes caused Aristotle to redefine the concept of infinity
Producing the concepts of
○ Actual Infinity
○ Potential Infinity
Actual Infinity: Aristotle described it as a set with a beginning and end that at
this moment contains an infinite number of terms, he did not believe this could
be achieved in nature
Example: N
Potential Infinity: Aristotle believed it to be a process that will continuously
operate for an infinite amount of time
Example: Splitting a group in two continuously
Aristotle on Zeno’s Paradoxes
●
●
Aristotle believed that Zeno was incorrect in saying, for example with the
dichotomy, a person must walk through an actually infinite number of
points
He said that you instead traverse a potentially infinite number of points
We now know that Zeno’s fault did not lie with his interpretation of actual
infinity, in fact there are an actually infinite number of points that need to be
traversed. He was incorrect in thinking that one cannot traverse this.
Infinity Today
Potential Infinity:
Actual Infinity:
Cat paradox
Infinity Today
●
Computer Science
○ Growth functions
○ Both n^3 and n^2 will reach
infinity, but which one does it
faster?
○ Sometimes used as “some
large number”
Source: http://i50.tinypic.com/f23nuh.jpg
Works Cited
●
Dowden, Bradley. "Zeno’s Paradoxes." Internet Encyclopedia of Philosophy. University of Tennessee Martin, n.
d. Web. 01 Mar. 2015.
●
Huggett, Nick. "Zeno's Paradoxes." Stanford University. Stanford University, 30 Apr. 2002. Web. 01 Mar. 2015.
●
Palmer, John, "Zeno of Elea", The Stanford Encyclopedia of Philosophy (Spring 2012 Edition), Edward N. Zalta
(ed.)
●
Palmer, John, "Parmenides", The Stanford Encyclopedia of Philosophy (Summer 2012 Edition), Edward N. Zalta
(ed.)