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Paradoxes. The human mind has a capacity to accept paradoxes Humans like to classify things Paradoxes arise when an idea appears to fit in a class from one angle, but appears outside the class when viewed from a different perspective. Paradoxes often come up in mathematics.
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Paradoxes • The human mind has a capacity to accept paradoxes • Humans like to classify things • Paradoxes arise when an idea appears to fit in a class from one angle, but appears outside the class when viewed from a different perspective. • Paradoxes often come up in mathematics. • Mathematicians even try to classify paradoxes.
Soren Kierkegaard • One must not think ill of the paradox, for the paradox is the passion of thought, and the thinker without the paradox is like the lover without passion: a mediocre fellow. But the ultimate potentiation of every passion is always to will its own downfall, and so it is also the ultimate passion of the understanding to will the collision, although in one way or another the collision must become its downfall. This, then, is the ultimate paradox of thought: to want to discover something that thought itself cannot think.
Moderation in all things, including moderation • Paradoxes of self reference • Is the previous slide blank? • This is a paradox of self reference • This sentence is false (liar paradox) • liar paradox is attributed to Eubulides of Miletus in the 4th century BC. • Rule: there is an exception to every rule.
More sophisticated paradoxes of self reference • Buridan’s paradox (ca. 1300 – after 1358) • Imagine the following scenario:Socrates wants to cross a river and comes to a bridge guarded by Plato • Plato: Socrates, if in the first proposition which you utter, you speak the truth, I will permit you to cross. But surely, if you speak falsely, I shall throw you into the water. • Socrates: You will throw me into the water.
Resolution of the dilemma • Is the proposition uttered by Socrates: "You are going to throw me into the water" true, or is it false? • Is Plato's promise true or is it false? • How can Plato fulfill his promise?” • … “ought implies can” (Kant)
Fitch's paradox of knowability • The existence of an unknown truth is unknowable… • If all truths were knowable, it would follow that all truths are in fact known.
Suppose p is a sentence which is an unknown truth; that is, the sentence p is true, but it is not known that p is true. • Example p: “Every even integer greater than two is the sum of two prime numbers.” (Goldbach’s conjecture) • In such a case, the sentence "the sentence p is an unknown truth" is true • if all truths are knowable, it should be possible to know that "p is an unknown truth". • But this isn't possible, because as soon as we know "p is an unknown truth", we know that p is true, rendering p no longer an unknown truth, • Therefore: the statement "p is an unknown truth" becomes a falsity. • The statement "p is an unknown truth" cannot be both known and true at the same time. • Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "something is an unknown truth” • Conclusion: there must be no unknown truths, and thus all truths must be known.
The barber’s paradox • In a village, the barber shaves everyone who does not shave himself, but no one else. • The question that prompts the paradox is this: • Who shaves the barber?
Consequences to set theory • The problem is self-referential statements. • Is there a set of all sets? Is there a set of all sets that are not members of themselves? • Example: the set of all green dogs is not a member of itself. • The set of all things that are not green dogs is a member of itself, by its defining conditions. • To resolve the paradox: the barber cannot be in the category of men, in order that the condition make sense.
Logical paradoxes I: Zeno’s paradox of Achilles and the tortoise
That which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle) • Achilles is in a footrace with the tortoise. • Achilles allows the tortoise a head start of 100 meters. • each racer starts to run at some constant speed • after some finite time, Achilles will have run 100 meters, • During this time, the tortoise has run 10 meters. • after some finite time, Achilles will have run 10 more meters, • During this time, the tortoise has run 1 more meter • ETC • Whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. • There are an infinite number of points Achilles must reach where the tortoise has already been. • He can never overtake the tortoise.
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” —Aristotle, Physics VI:9, 239b
Other paradoxes of the infinite • Adding an element to an infinite set results in a set with the same number of elements. • There is an infinite number of infinite numbers. • If a hotel with infinitely many rooms is full, it can still take more guests. • Banach–Tarski paradox: Cut a ball into a finite number of pieces, re-assemble the pieces to get two balls, both of equal size to the first.
Does the set of all those sets that do not contain themselves contain itself?
Homework problem • Uncle Joe and Uncle Jim are walking to the barber shop. There are three barbers who live and work in the shop—Mo, Larry, and Curly—but not all of them are always in the shop. Curly is a good barber, and Uncle Jim is keen to be shaved by him. He knows that the shop is open, so at least one of them must be in. He also knows that Mo is a very nervous man, so that he never leaves the shop without Larry going with him. • Uncle Joe insists that Curly is certain to be in, and then claims that he can prove it logically. Uncle Jim demands the proof. Uncle Joe reasons as follows. • Suppose that Curly is out. If Curly is out, then if Mo is also out Larry would have to be in—since someone must be in the shop for it to be open. However, we know that whenever Mo goes out he takes Larry with him, and thus we know as a general rule that if Mo is out, Larry is out. So if Curly is out then the statements "if Mo is out then Larry is in" and "if Mo is out then Larry is out" would both be true at the same time. • Uncle Joe notes that this seems paradoxical; the hypotheticals seem "incompatible" with each other. So, by contradiction, Curly must logically be in. • Must Curly be in? Explain.
Some paradoxes to resolve • A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week, but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on a Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on a Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all. The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.
The Paradox of the Court is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his instruction after he had won his first case (in some versions: if and only if Euathlus wins his first court case). Some accounts claim that Protagoras demanded his money as soon as Euathlus completed his education; others say that Protagoras waited until it was obvious that Euathlus was making no effort to take on clients and still others assert that Euathlus made a genuine attempt but that no clients ever came. In any case, Protagoras decided to sue Euathlus 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 according to the original contract, because Euathlus would have won his first case. • Euathlus, however, claimed 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 therefore not be obliged to pay. The question is: which of the two men is in the right?
