REFERENCES
http://faculty.washington.edu/
smcohen/320/ZenoArrow.html
http://faculty.washington.edu/
smcohen/320/ZenoArrow.html
http://plato.stanford.edu/entries/
zeno-elea/#StaDic
http://homes.soic.indiana.edu/
donbyrd/Teach/Math/
Zeno+Footraces+InfiniteSeries.pdf
http://ed.ted.com/lessons/what-is-zeno
-s-dichotomy-paradox-colm-kelleher
http://www.iep.utm.edu/zeno-par/
#SH1a
Zeno of Elea
Paradox of Motion
Sabrina C. Walker
Math 320
Biography
490—430 B.C.E.; Elea in Magna Graecia,
Member of the Eleatic School founded by Parmeni-des
Most of what we know about Zeno and his works comes from Plato’s Parmenides and Aristotle’s Physics.
Before Zeno, Greek thinkers favored presenting their philosophical views through poetry. Zeno began the shift away from poetry toward a prose that contained explicit premises and conclusions.
Apparently, Zeno planned to overthrow Nearchus the tyrant, but was arrested and tortured. His tor-turers wanted his to reveal the other conspirators; Zeno refused and said he had a secret for Near-chus. When Nearchus leaned in to hear it, Zeno bit his ear and held on till his death.
The Stadium
“Fourth is the one about the things in the stadium mov-
ing from opposite directions, being of equal bulk, along-
side things of equal size, with some moving from the
end of the stadium and some from the middle, at equal
speed, in which case he supposes it turns out that half
the time is equal to its double”
Zeno mistakenly supposes that a moving body pass-es both moving and stationary objects with equal speed.
Aristotle argues that how long it takes to pass a body depends on the speed of the body; for exam-ple, if the body is coming towards you, then you can pass it in less time than if it is stationary.
“If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.”
1. When the arrow is in a place just its own size, it’s at rest. 2. At every moment of its flight, the arrow is in a place just
its own size. 3. Therefore, at every moment of its flight, the arrow is at
rest. The argument falsely assumes that time is composed of
“nows” (indivisible instants). There is no such thing as motion (or rest) “in the now” (at an instant). Weakness in Aristotle’s solution: it seems to deny the possibility of motion or rest “at an instant.” But instantaneous velocity is a useful and important concept in physics.
The Arrow
Founded the Eleatic School of Philosophy
His only surviving work is a poem called “On Nature”.
“Parmenides rejected pluralism and the reality of any kind of change: for him all was one indivisible, unchanging reality, and any appearances to the contrary were illusions, to be dispelled by reason and revelation. “
Paradox—a statement or proposition that, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems senseless, logically unacceptable, or self-contradictory.
It is believed that he is the reason behind Zeno’s paradoxes. According to Plato, Zeno sought to defend his mentor by attacking his critics (ad hominem).
Parmenides and
the Beginning of Paradoxes
Dichotomy Paradox
“The Paradox of Cutting in Two”
“The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal.” Getting to the basics, an object cannot cover a
finite distance by moving during an infinite se-quence of time intervals.
Space is assumed to be divisible and time indi-visible.
“In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”
Zeno’s flaw in this argument was his unstated assumption that the sum of an infinite series (or at least an infinite series like this, where every term is greater than zero) cannot be finite.
The Achilles
