Paradox

Paradox

Paradox: when you can't have it both ways...

• etymology:
  • from Greek paradoxon for "contrary opinion
    • para = prior
    • dox = opinion

• definition:
  • a conflicting or self-contradictory opinion or situation
  • creates an absurdity, a puzzle or something unlikely
  • = a problem that
    • has no solution
    • the solution is never-ending
    • or the solution yields an outcome that negates the original problem

Paradox uses

• paradoxes are logically "invalid" or "invalid arguments"
  • since they can't be solved
  • like an irrational number that goes on forever
• however, paradoxes are useful thought experiments

Classic paradoxes

Buridan's bridge paradox

• Plato: "If your next statement is true, I will allow you to cross the bridge. If your next statement is false, I will throw you in the water"
• Socrates: "You will throw me in the water."

Free Will paradox

• if God knows what will happen to us, how can contradict it?
  • and if we cannot contradict it, there is no free will

Irresistible force paradox

• when an unstoppable force hits an immovable object

Government Temporary Powers paradox

• nothing lasts longer than a "temporary" government power or program

Omnipotence paradox

• if God is omnipotent (all powerful), can He make a rock so big He can't move it?

Plato's Beard paradox

• if something does not exist, is not that non-existence a form of existence?
  • see https://en.wikipedia.org/wiki/Plato%27s_beard

Problem of Evil paradox

• if God is good, then how can evil exist?

Russell's paradox

• "a list of all lists that do not contain themselves"

Ship of Theseus

• if a ship were, over time, repaired so much that every part was replaced, would it be the same ship it was originally?

Zeno's paradoxes

• Zeno of Elea was a Greek philosopher, c. 495-430 BC who lived in the Greek colonies on Italy
  • he preceded Socrates and Plato
• he was taught by Parmenides, who is thought to have visited Athens and influenced Socrates when he Socrates was a young man
  • Parmenides distinguished between "Aletheia," for truth and "Doxa" or opinion, or the world of our senses
  • considered the founder of "ontology", or the study of existence and reality
  • Plato and Aristotle credited Zeno as inventor of the "dialectic," or process of "reasoned discourse" by which a truth is established (aka, "Socratic method")
• Zeno developed his paradoxes to defend Parmenides' teachings
• click EXPAND for the story told by the ancient Greek biographer, Diogenes Laërtius, about Zeno biting the tyrant's ear

Diogenes Laertius wrote that Zeno opposed the rule of Nearchus the tyrant of Elea (Greek colony in Italy) and conspired to overthrow him. Nearchus discovered the plot and had Zeno tortured to reveal the names of his co-conspirators. Zeno refused to say, but told Nearchus that he had a secret to divulge and that he would whisper it in Nearchus' ear, whereupon, Zeno bit Nearchus' ear and didn't let go until he was killed, at which point the ear came off in Zeno's grip. See Zeno of Elea (wikipedia)

• Plato discusses Zeno's paradoxes in "Parmenides," a dialogue between Socrates and Zeno's teacher, Permenides
  • the point being that a paradox demonstrates an absurdity, such as

Or suppose one of us to have a portion of smallness; this is but a part of the small, and therefore the absolutely small is greater; if the absolutely small be greater, that to which the part of the small is added will be smaller and not greater than before.

from Parmenides by Plato (Project Gutenberg)

Achilles and the tortoise paradox

• "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.
  • as recounted by Aristotle, Physics VI:9, 239b15

click EXPAND for explanation from Wikipedia entry: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_of_motion

In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. As Aristotle noted, this argument is similar to the Dichotomy.[13] It lacks, however, the apparent conclusion of motionlessness.

Zeno's indivisibility of time

* many of his paradoxes rely on the idea that time and space are infinitely divisible and thus yield absurd conclusions * therefore, according to Zeno (and Parmenides) motion is an illusion ** ironically, video or motion pictures are made up of still images that create the illusion of motion when shown in rapid succession

Arrow paradox

* at any given instant, a flying arrow is in a certain place, so, since its entire flight is made up on such motionless instants, motion is impossible. ** * a described by Aristotle in Physics VI:9, 239b5 If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion. 

Dichotomy paradox

* if you keep walking half-way to somewhere, you will never get there * Aristotle described it in Physics VI:9, 239b10 as: That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

Moving rows paradox

* given three rows of four individuals (rows A, B, & C) ** if row A is not moving, and rows B and C are moving in oppositive directions past row A at equal speeds ** since B and C both move past A at one speed, they move past each other at double the speed they move past the stationary row A ** thereby half of a time is equal to its double *

Paradox of place

* since everything has a place, and a place is a thing, then every place has a place, ad infinitum (onward forever)

Paradox of the grain of millet

* if a single grain of millet (a seed) makes no sound upon falling, yet 1,000 grains that fall do make a sound, how can 1,000 nothings create a sound? See * Zeno’s Paradoxes | Internet Encyclopedia of Philosophy (utm.edu)

Science & technology paradoxes

Information or black hole paradox

* from physicist Steven Hawking * a black hole does not absorb every particle, so over time it will disappear into nothing * how can that be? ** see Information paradox simplified (physicsworld.com)

Visual paradoxes

>> Escher to do

Riddles

* while not paradoxes (because they can be solved), riddles present interesting intellectual scenarios for students

The truth-teller & the liar riddle

* two monsters guard a fork in the road ** one path leads to perdition, the other to salvation ** one monster always lies and the other always tells the truth ** you are permitted to ask each monster one question ** what do you ask in order to learn which path is the one to salvation? click EXPAND for the solution

* If you ask each one which path to salvation the other would say, they bill both tell you the path to perdition; so choose the other for salvation
** If Path A is salvation and Path B is perdition:
*** then Liar Monster will say the other will say that the path to Salvation is Path B (which leads to perdition)
*** while Truthful Monster will say that the Lying Monster will say the path to Salvation is Path B, as well
*** therefore, Path A is the path to salvation

Assorted or humorous paradoxes

Cat and buttered toast paradoxes

* cats always land on their feet ** may be supported by the "cat righting reflex" which is the ability of cats to right-themselves mid-air, thus landing on their feet
and
* buttered toast always lands with the butter-side down ** an experiment showed that buttered toast will land butter-side down 81% of the time (see Buttered cat paradox - Wikipedia

Intentionally blank page

* when a published or printed document states, "intentionally blank page" in order to indicate that the blank page in the document is there on purpose ** then the page is no longer blank

List of paradoxes in other articles here

* If life is unfair for everybody, wouldn't that make it fair? ** (w/ thanks to Henry) * * >> to do : list/ links * also from : https://en.wikipedia.org/wiki/List_of_paradoxes