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 GoalsThere 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
tt1 t2
y1
y2
speed=0
speed=0
1
2
2
1
The ArrowWe know that in calculus the derivative of the position of an object with respect to time is the velocity:
y
t1 t2
y1
y2
t
|velocity|>0
|velocity|>0
1
2 2
1
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 one-fourth
● Before a runner can reach the one-fourth, they must first travel one-eighth● Therefore the runner must travel
an infinite distanceSource: https://1badnavajo.files.wordpress.com/2013/04/track.jpg
Dichotomy (The Racetrack)● Zeno believed this to converge to infinity● Actually converges to one● Lack of calculusn = 1
∞
1/2 1/4 1/8
1/16 1/32 ...
1
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● 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
Source: http://www.iep.utm.edu/wp-content/media/Achilles_Tortoise.jpg
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● 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.)