Zeno - Achilles
Channel: Gridmath
Date: 15/03/2016

Short summary,  dealing with the limit concept, infinity, sum of an infinite series with some philosophy and more modern thinking. Copyright- Everything mine is marked cc-by, for anything else, if it exists, Fair Use is claimed.

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Zeno - Philosophy

Original direct link unavailable -  Jeff Speaks (2011)
One can think of Zeno’s strategy like this: he begins with the assumption space and time must be either continuous or discrete. He then proceeds to show that either assumption leads to the conclusion that motion is impossible.

Original direct link unavailable
Brian Palmer (2014) - No matter how quickly Achilles closes each gap, the tortoise will always open new, smaller ones and remain just ahead of the Greek hero…Joseph Mazur,…describes the paradox as “a trick in making you think about space, time, and motion the wrong way.”

Original direct link unavailable - Theo  (May 9 '12 at 5:45)  Achilles's and the turtle is no paradox at all, but a refutation of the hypotheses that the space is continuous. Zeno's arrow paradox is a refutation of the hypothesis that the space is discrete. Together they form a paradox and an explanation is probably not easy…The heart of the Zeno's argument is that there is no logical way to decide… Granted, this argument lies at the boundary of math, physics and perhaps philosophy.

http://www.decodedscience.org/zenos-p... Comments - Usedtobe Fishfry (2014) Now it is true that *in math* we can define the limit of a convergent series of real numbers, and then prove that .111… = 1/9 exactly. But that proof depends on a rigorous definition of the real numbers; which in turn depends on the axioms of Zermelo-Fraenkel set theory; one axiom of which is that there exists an infinite set…at the very least, far beyond our current ability to determine one way or the other…We do not know whether spacetime is “really” continuous or discrete; or if the question itself is even meaningful.

Zeno - Math

http://www.gutenberg.org/files/21016/... - Richard Dedekind - Continuity and Irrational Numbers, 1872 ( last line -Text p5 - PDF p10/78) - “And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought... effected in accordance with..." (Dedikind Cuts, Convergence, and/or Nested Intervals - per Courant-Robbins p72 fine print).


What is Mathematics? By Richard Courant, Herbert Robbins, (Originally Published - 1941)
(p 64) The infinite enters only in the unending 'procedure' and not as an actual quantity.
(p 305-6) Certainly the intuitive idea of a continuum has a psychological reality in the human mind. But it cannot be called upon to resolve a mathematical impossibility;

Original direct link unavailable -  Dr. Jeff Cruzan (2012) - Series are usually used to find an alternate route to the value of a function that's difficult to find directly — we want convergence to some solution.

https://www.khanacademy.org/math/inte... Khan Academy (2013)- Deriving the sum of an infinite geometric series

http://homes.soic.indiana.edu/donbyrd... 
Donald Byrd (2012) - Zeno+Footraces+InfiniteSeries.pdf

http://www.iep.utm.edu/zeno-par/ - Internet Encyclopedia of Philosophy - Zeno’s Paradoxes

https://youtu.be/ZxXZZXFzU58?t=162 - Fretboard @ 2:42

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