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This Lecture Will Surprise You: When Logic is Illogical Tony Mann, 19 January 2015 This Lecture Will Surprise You: When Logic is Illogical Tony Mann, 19.

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Presentation on theme: "This Lecture Will Surprise You: When Logic is Illogical Tony Mann, 19 January 2015 This Lecture Will Surprise You: When Logic is Illogical Tony Mann, 19."— Presentation transcript:

1 This Lecture Will Surprise You: When Logic is Illogical Tony Mann, 19 January 2015 This Lecture Will Surprise You: When Logic is Illogical Tony Mann, 19 January 2015

2 16 March – Two Losses Make a Win: How a Physicist Surprised Mathematicians 16 February – When Maths Doesn't Work: What we learn from the Prisoners' Dilemma 19 January – This Lecture Will Surprise You: When Logic is Illogical Three lectures on Paradox

3 I guarantee that you will be surprised

4 Zhuang Zhou and the Butterfly

5 Raymond Smullyan

6 Paradox “a statement that apparently contradicts itself and yet might be true” Wikipedia

7 Proof by Contradiction Proposition: If n 2 is odd then n must be odd Proof: Suppose n is an even integer such that n 2 is odd Then n = 2k for some integer k But n 2 = (2k) 2 = 4k 2 is divisible by 2, so it is both even and odd This contradiction means our assumption (that n could be even) must be false So we have proved n must be odd

8 A Pair o’ Docs

9 Smullyan’s Interview Lie “Would you be prepared to lie?”

10 The Liar Paradox This sentence is false.

11 The Cretan Paradox One of themselves, even a prophet of their own, said, The Cretians are always liars … Titus, I:12

12 Golf and Tennis

13 A volunteer please!

14 My Prediction I will make a prediction about an event which will take place shortly My volunteer will write “Yes” if they think my prediction will be correct and “No” if they think it will be wrong

15 My Prediction The volunteer will write “No” on the card.

16 Buridan’s Ass John Buridan (c.1300 – after 1358)

17 Buridan’s Ass Buridan and Pierre Roger

18 Buridan’s Ass “Where are the snows of yesteryear?” Où est la très sage Heloïs, Pour qui fut chastré et puis moyne Pierre Esbaillart à Sainct-Denys? Pour son amour eut cest essoyne. Semblablement, où est la royne Qui commanda que Buridan Fust jetté en ung sac en Seine? Mais où sont les neiges d'antan! François Villon Ballade des dames du temps jadis

19 Buridan’s science Theory of Impetus (≈ Newton’s First Law) Theory of money

20 Buridan on self-reference I say that I am the greatest mathematician in the world

21 Buridan on self-reference The fool hath said in his heart, There is no God. Psalm 14, I

22 Buridan on self-reference Proposition Someone at this moment is thinking about a proposition and is unsure whether it is true or false

23 Buridan on self-reference Plato is guarding a bridge. If Socrates makes a true statement Plato will let him cross. If Socrates’s statement is false, Plato will throw him in the river. Socrates says, “You will throw me in the river”.

24 Buridan’s Ass Don Quixote

25 A Puzzle You meet two islanders, A and B. A says “At least one of us is a liar.” What are A and B?

26 A Puzzle I found two of the islanders sitting together. I asked “Is either of you a truth-teller?” When one of them answered, I could deduce what each of them was. How?

27 A Puzzle E and F are two islanders. E said “We are both of the same type” F said “We are of opposite types.” What are E and F?

28 Buridan’s Ass Witches in sixteenth-century France

29 Buridan’s Ass Protagoras and Euathlus Euathlus owes Protagoras a fee when he wins his first case. Protagoras sues him. Protagoras: If I win, I get my fee If Euathlus wins, he must pay me because he has won the case Euathlus: If I win, I don’t have to pay. If Protagoras wins, I have lost and have nothing to pay

30 Buridan’s Ass State v. Jones, Ohio 1946 Jones is accused of carrying out an illegal abortion The only evidence against him is that of Harris on whom he allegedly performed the operation

31 Buridan’s Ass State v. Jones, Ohio )If Jones is guilty then Harris must also be guilty 2) Jones cannot be convicted solely on the evidence of a criminal accomplice

32 {1, 4, 9, 16, 25, 36, 49,64, 81, 100, …} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …} {1,4,9,16,25,36,49…} {1,2,3,4,5,6,7,…} A paradox of infinity

33 Secure foundations for mathematics

34 The set of all teapots is not a teapot, so it is not a member of itself. The set of all sets is a set. Therefore it is a member of itself. Russell’s Paradox

35 Is S a member of itself? Let S be the set of all sets that are not members of themselves Russell’s Paradox

36 Who shaves the barber? In a certain village, the barber shaves everyone who does not shave themselves Russell’s Barber Paradox

37 Some adjectives don’t describe themselves – eg “long” or “monosyllabic” Call them “heterologous” Some adjectives describe themselves – eg “short” or “polysyllabic” Call them “autologous” Grelling-Nelson Paradox Is “heterologous” heterologous?

38 Berry’s Paradox (1906)

39 Ways to tweet the number one “1” “One” “Zero factorial” “4 – 3”

40 What is the smallest integer that cannot be identified in a tweet of no more than 160 characters? Berry’s Paradox (Twitter version)

41 “Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation. Quine’s Paradox

42 Is a bogus charlatan a charlatan or not? Smullyan’s Charlatan Paradox

43 A: Both these statements are false. B: I am the world’s greatest mathematician Another dubious proof

44 If there were a Nobel Prize for mathematics then, as the greatest mathematician in the world, I would deserve to win it. Another dubious proof

45 “If A then B” or “A implies B”, A→B is true unless A is true and B is false Implication

46 If there were a Nobel Prize for mathematics then, as the greatest mathematician in the world, I would deserve to win it. Another dubious proof

47 If this statement is true, then I am the greatest mathematician in the world. Curry’s Paradox

48 If A is true, and A→B, can we deduce that B is true? What the Tortoise said to Achilles

49 If A is true, and A→B, can Achilles deduce that B is true? He needs to know also that (A & A →B) →B and (A & A →B&((A & A →B) →B) →B and so on What the Tortoise said to Achilles

50 “Taught-Us” “A Kill-Ease”

51 “In mathematics, there is no ignorabimus” “We must know – we shall know!” David Hilbert

52 Every even integer is the sum of at most two primes The Goldbach Conjecture

53 A logical system can prove that it itself is consistent if and only if it is not consistent G ödel’s Theorems

54 In a consistent logical system there are true statements which cannot be proved within that system G ödel’s Theorems

55 “Gödel's Incompleteness Theorem demonstrates that it is impossible for the Bible to be both true and complete.” G ödel’s Theorems

56 Turing and the Halting Problem

57 I guarantee that you will be surprised

58 Perhaps something in this lecture surprised you. If not, you expected a surprise guaranteed by your lecturer, and your expectation wasn’t met. That was your surprise! Were you surprised?

59 Thank you for

60 Thanks to Noel-Ann Bradshaw and everyone at Gresham College Picture credits Photograph of lecturer: Noel-Ann Bradshaw; T-shirt: Monarch butterfly: Kenneth Dwain Harrelson licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license Zhuang Zhou: Wikimedia Commons, public domain Raymond Smullyan: Wikipedia, with permission “Pair o’ Docs”: Microsoft Clip Art Vacuum cleaner advert: National Geographic, via Wikipedia (out of copyright) Rory McIlroy: TourProGolfClubs, Wikimedia Commons, licensed under the Creative Commons Attribution 2.0 Generic license Petra Kvitova: Pavel Lebeda / Česká sportovní, Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Czech Republic license Buridan’s Ass cartoons: Cham, Le Charivari, 1859, Wikimedia Commons; W.A. Rogers, New York Herald, c.1900, Wikimedia Commons Clement VI: Henri Ségur, Wikimedia Commons François Villon: stock image used to represent Villon in 1489, Wikimedia Commons Isaac Newton: Sir Godfrey Kneller, Wikimedia Commons Don Quixote title page: Wikimedia Commons Don Quixote illustration: Gustave Doré, Wikimedia Commons Witches: Hans Baldung, 1508, Wikimedia Commons Protagoras: Salvator Rosa (1663/64), Wikimedia Commons Bertrand Russell: Wikimedia Commons Gottlob Frege: Wikimedia Commons Teapot: Andy Titcomb, Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Street barber: Amir Hussain Zolfaghary, licensed under the Creative Commons Attribution 3.0 License. Willard Van Orman Quine: copyright owner Dr. Douglas Quine, Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Lewis Carroll: Wikimedia Commons Achilles statue in Corfu: Dr K, Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Giant tortoise: Childzy, Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. David Hilbert, Wikimedia Commons Goldbach signature – Wikimedia Commons Alan Turing statue, Bletch;ey Park: Sjoerd Ferwerda, Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Acknowledgments and picture credits

61 Further Reading Douglas Hofstadter, Gödel, Escher. Bach: an Eternal Golden Braid (Penguin, 20th anniversary edition, 2000) Raymond Smullyan, What is the Name of this Book? (Prentice-Hall, 1978: Dover, 2011) and The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs (Dover, 2013) Francesco Berto, There's Something About Gödel!: The Complete Guide to the Incompleteness Theorem (Wiley-Blackwell, 2009) Scott Aaronson, Quantum computing Since Democritus (Cambridge University Press, 2013)


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