Mark's Blog: The intriguing paradox

The intriguing paradox

The word ‘paradox’ comes from the Greek paradoxon which means ‘contrary to expectations, existing belief or perceived opinion’.[i]

Simply put, a paradox usually contains two statements that are both true, but which, in general, cannot both be true at the same time; a contradiction exists. Put otherwise, what seems to be reasonable actually engenders conflicting inclinations about what to believe, and the result is cognitive tension.[ii]

A paradox might be created in error, or created intentionally, to serve as humour, for instance. It can, and usually is, not created in error, with the express intention of arresting attention and provoking fresh thought.[iii] With this intent the best paradoxes raise questions about what kinds of contradictions can occur, and what species of seeming impossibilities are, in fact, possible.[iv]

My first introduction to paradox was in junior school. It was the idea of the biggest military tank in the universe crashing at the speed of light into an impenetrable wall. What would happen? Yes, yes, it can’t happen but if it did, what would happen?

Adults describe this soberly as an unstoppable force colliding with an immoveable object, aka the irresistible force paradox.

My quest for the truth was unabated: Yes, but, one still has to be stronger than the other. The tank is unstoppable in all cases, as is the wall impenetrable in all cases. Which is it?

Thinking back, it makes sense that paradoxes may arise because theory refuses to be subordinate to reality.[v]

Some examples of paradox

This statement is false. (No wider context is given.)
As I said before, I never repeat myself. (The repetition has just occurred.)
I’m in total control, don’t let my wife find out. (This implies that when the greater authority, the wife, finds out, he will not be in total control.)
I’m not schizophrenic, and neither am I. (A play on dual personalities.)
I can resist anything but temptation.[vi] (The conflict lies with resist combined with temptation.)
All animals are equal, but some are more equal than others.[vii] (This is a famous quote, and Orwell ingeniously presents the ruling pigs’ proclamation as paradox.)
My nose will grow now, says Pinocchio. (If the statement is true, then his nose will not grow since it only grows when he is lying.)
Do you promise not to keep this promise? (A commitment not to uphold a commitment.)
On one side of a piece of paper is written The next statement is true. On the other side is written The previous statement is false.[viii] (The reader oscillates between the two statements, looking for which is most true.)
A crocodile has stolen your baby, which he’ll either eat or return. IF you correctly guess his plans THEN he’ll return it, otherwise he’ll eat it. (This is interesting as the timing is important: the baby might not be eaten yet, but knowing how a crocodile operates, the baby has probably been eaten.)
You travel back in time, with the intent to return to the current date. Once back in time you kill your father. (So how can you return to current date if you don’t exist? If you return and hear the postman is dead... )
Answer truthfully (yes or no) to this question: Will the next word you say be ‘no’? (Where the intent of the answer is mismatched with the verbal message.)

Paradox in literature

In literature, the paradox is an anomalous juxtaposition of incongruous ideas for the sake of striking exposition or unexpected insight. It functions as a method of literary composition and analysis, which involves examining apparently contradictory statements and drawing conclusions either to reconcile them or to explain their presence.[ix]

These are old fond paradoxes to make fools laugh i' the alehouse. (Shakespeare, Othello)
You undergo too strict a paradox, Striving to make an ugly deed look fair. (Shakespeare, Timon of Athens)
I must be cruel to be kind. (Shakespeare, Hamlet)
Catch-22 (Joseph Heller’s war novel)
The barbershop paradox (Lewis Carroll, 1894, A logical paradox)

Paradox in poetry

The poet may make use of paradox to create a remarkable thought or image out of words.[x] Paradox can serve as a device that holds both tensions of error and truth, subtly qualifying the ordinary meaning of words.[xi]

William Wordsworth The child is father of the man
Wallace Stevens Nothing that is not there and the nothing that is
Sir Philip Sidney Absent presence
Alfred, Lord Tennyson Falsely true
John Milton Darkness visible
Walt Whitman Transparent night
Lord Byron Melancholy merriment
Alexander Pope, describing man as a Great lord of all things, yet a prey to all
Ben Jonson Liquid marble[xii]

If this area interests you, research ‘paradox poetry’.

Paradox in art

Paradox in art is not about interpreting the written word (as in literature and poetry) but about interpreting visual information. This is a complicated field. Dykstra sees art as paradoxical by nature, reflecting the past and creating the future. It defies entropy, but maybe that’s what humans do, too: reflect and create.[xiii] There’s an interesting quote by Theodor Adorno, ‘It goes without saying that nothing about art goes without saying.’ [xiv]

With Escher one sees a collection of stairwells (Penrose Stairs)[xv] which have no beginning or end. The interpretation of the structure conflicts with the expectation of moving upwards or downwards. Similarly, his Waterfall refers.[xvi] There are other Escher drawings, e.g. of a hand which is drawing a hand, which, in turn, is drawing the original hand (Drawing Hands).[xvii]
Art paradox can become complex when the artist’s intent, the image, the message, the viewer’s reaction come together, e.g. Vincent Castiglia’s art is re-interpreted when the viewer discovers he paints exclusively in human blood.[xviii] An anti-war poster calling for peace amongst nations but (one later finds out) painted using the blood of one’s enemies, is another example. Blood is one fluid but other body fluids are also known to be used in art.[xix]

Perhaps the greatest paradox in art is the inability to separate money from art.[xx] [xxi]

Paradox in religion

I have found the paradox, that if you love until it hurts, there can be no more hurt, only more love. Mother Teresa.[xxii]
Free will exists yet co-exists with divine preordination. (This is a philosophic issue whereby having free will conflicts with a theistic form of determinism. One solution is the assumption of soft determinism, in which results still occur just without clockwork precision.)
The omnipotence paradox. If God is omnipotent, can he create a rock so heavy that he cannot lift it? If he cannot lift it then he cannot be omnipotent.

Paradox in war

War is peace. Freedom is slavery. Ignorance is strength.[xxiii]
No other activity in history has been better at producing peaceful and prosperous societies than war.[xxiv]
The stability–instability paradox. When two countries each have nuclear weapons, the probability of a direct war between them greatly decreases, but the probability of minor or indirect conflicts between them increases.

Both rational sides understand that a nuclear war is all-destructive, so major conflicts are avoided but minor conflicts are not.[xxv] (One might wonder, given a 100 year period of détente, whether more people died in indirect conflicts than had there been a direct war. One cannot tell, given the theoretical possibility of massive nuclear destruction in the short term, but over 100 years of constant indirect conflicts the same number of people, or more, may have been killed.)

Paradox in mathematics

Accuracy paradox. Predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy.
Friendship paradox. For almost everyone, their friends have more friends than they do.
Birthday paradox. What is the chance that two people in a room have the same birthday? (It depends on the number of people in the room. One might say 366 people in a room is a good place to start. Actually, given 23 people in a room there’s a 50% chance; 75 people in the room and there’s a 99,9% chance.[xxvi])
Monty Hall problem. An unintuitive consequence of conditional probability. (This is a popular and complicated problem and the paradox lies in the applied logic. There are a few interpretations of this problem; caveat emptor.[xxvii])
Russell's paradox. Does the set of all those sets that do not contain themselves contain itself? (This is named after the British philosopher Bertrand Russell.)
Thomson's lamp. After flicking a lamp on and off infinitely often, is it on or off?
Coastline paradox. The perimeter of a landmass is, in general, ill-defined.[xxviii]
Infinity paradox. Is infinity an odd or even number? (If it were odd, then 2 times infinity would make it an even number, this is because 2 infinities are bigger than 1, as we all know.)

Paradox in philosophy

I know one thing, that I know nothing. Socrates.
The paradox of the court (aka The counterdilemma of Euathlus). Protagoras the Sophist took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his instruction after he wins his first court case. Euathlus received said instruction but decided not to enter the law profession, so Protagoras sued him for the amount owed.

Protagoras argued that if he won the case he would be paid his money. If Euathlus won the case, Protagoras would still be paid because, according to the original contract, Euathlus had won his first case.

Euathlus countered this, claiming that if he won, then by the court's decision he would not have to pay Protagoras. If, on the other hand, Protagoras won, then Euathlus would still not have won a case and thus not be obliged to pay.

Who is right?

The liar paradox. It’s not clear if this is attributed to Chrysippos[xxix] or Epimenides of Knossos,[xxx] but it goes like this: A Cretan (citizen of Crete) sails to Greece and makes a statement to the public: ‘All Cretans are liars’. One wonders if this statement is true, except that it was uttered by a Cretan. When he made this statement, was he speaking the truth, or, as a lying Cretan, lying?
Achilles and the tortoise paradox. This is one of Zeno of Elea’s paradoxes. Achilles is in a footrace with a tortoise. Achilles has generously granted the tortoise a few paces headstart. The race begins, and both the tortoise and Achilles advance. Achilles reaches half the distance to the tortoise and then reaches the point where the tortoise stood when the race began (Point 1). The tortoise has since advanced a bit, and thus Achilles continues running and reaches half the distance of the tortoise’s new advancement, and then reaches the point where the tortoise was when Achilles reached Point 1; let’s call this Point 2. But the tortoise has advanced again and Achilles first reaches the halfway mark and then comes to Point 3, at which time the tortoise is at Point 4. In short, the tortoise will win the race. (The paradox lies with the assumption that Achilles’ progress is contingent on, not independent of, the tortoise’s progress.)

Philosophical paradoxes didn’t end with the Greeks; here are two from Karl Popper.

The paradox of tolerance. Unlimited tolerance must lead to the disappearance of tolerance. If unlimited tolerance is extended even to those who are intolerant, if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and all tolerance with them.[xxxi]
The paradox of freedom. Freedom, if we mean the absence of any constraining control, must lead to very great restraint, since it makes the bully free to enslave the meek.[xxxii]

There are many paradoxes in many fields, e.g. Artificial Intelligence, Biology, Chemistry, Decision theory, Economics, Linguistics, Literature, Logic, Mathematics, Mysticism, Perception, Philosophy, Physics, Poetry, Politics, Psychology, Religion, Sociology, Time travel, Sociology, etc.[xxxiii] One wonders if the extent of paradoxes is proof of our ineptness, a want of accepting an argument regardless of its soundness, or perhaps credit to the ever-inquiring mind.

In the above examples one can see what comprises a paradox: The listener has an assumption of what the speaker means, takes in the simple statements, which appear devoid of ambiguity, and begins to conclude. But wait! Contradiction. Followed by logical conflict, followed by a search for the truth. And this problem is not simply forgotten because we love solving mysteries.

Paradoxes are entertaining. They’re also devastating when they ridicule the theory you’ve spent most of your life compiling. Paradoxes test definitions, assumptions, logic, theories, formulae; they redefine intent and meaning. The result is an enhanced intimacy, a more meaningful discourse, and a greater truth. Most importantly, paradoxes are necessary: Paradox is thus a much deeper and universal concept than the ancients would have dreamed. Rather than an oddity, it is a mainstay of the philosophy of science.[xxxiv]

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Thursday, March 2, 2017

The intriguing paradox

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