Zeno of Elea & the Evolution of Infinity - UMass Amherstpeople.math.umass.edu/~tevelev/391A_2015/zeno.pdfZeno’s Goals There is some contention around the goals of zeno and his paradoxes:

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 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

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.)