This Lecture Will Surprise You: When Logic is Illogical Tony Mann, 19 January 2015 Three lectures on Paradox 19 January This Lecture Will Surprise You: When Logic is Illogical

16 February When Maths Doesn't Work: What we learn from the Prisoners' Dilemma 16 March Two Losses Make a Win: How a Physicist Surprised Mathematicians I guarantee that

you will be surprised Zhuang Zhou and the Butterfly Raymond Smullyan

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

Proof by Contradiction Proposition: If n2 is odd then n must be odd Proof: Suppose n is an even integer such that n2 is odd Then n = 2k for some integer k But n2 = (2k)2 = 4k2 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 A Pair o Docs Smullyans Interview Lie

Would you be prepared to lie? The Liar Paradox This sentence is false.

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

Golf and Tennis A volunteer please! 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 My Prediction The volunteer will write No on the card.

Buridans John Buridan (c.1300Ass after 1358) Buridans

AssRoger Buridan and Pierre Buridans Where are the snows Ass

of yesteryear? O est la trs sage Helos, Pour qui fut chastr et puis moyne Pierre Esbaillart Sainct-Denys? Pour son amour eut cest essoyne. Semblablement, o est la royne Qui commanda que Buridan

Fust jett en ung sac en Seine? Mais o sont les neiges d'antan! Franois Villon Ballade des dames du temps jadis Buridans science Theory of Impetus ( Newtons First Law)

Theory of money Buridan on self-reference I say that I am the greatest mathematician in the world

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

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

Plato is guarding a bridge. If Socrates makes a true statement Plato will let him cross. If Socratess statement is false, Plato will throw him in the river. Socrates says, You will throw me in the river.

Buridans Ass Don Quixote A Puzzle You meet two islanders, A and B.

A says At least one of us is a liar. What are A and B? 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? 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? Buridans Ass Witches in sixteenth-century France

Buridans Protagoras and Ass 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 dont have to pay. If Protagoras wins, I have lost and have nothing to pay

Buridans Ass 1946 State v. Jones, Ohio 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 Buridans Ass 1946 State v. Jones, Ohio

1) If Jones is guilty then Harris must also be guilty 2) Jones cannot be convicted solely on the evidence of a criminal accomplice A paradox of infinity

{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, Secure foundations for mathematics

Russells Paradox The set of all sets is a set. Therefore it is a member of itself. The set of all teapots is not a teapot,

so it is not a member of itself. Russells Paradox Let S be the set of all sets that are not members of themselves

Is S a member of itself? Russells Barber Paradox In a certain village, the barber shaves everyone who does not shave themselves Who shaves the barber?

Grelling-Nelson Paradox Some adjectives describe themselves eg short or polysyllabic Call them autologous Some adjectives dont describe themselves eg long or monosyllabic

Call them heterologous Is heterologous heterologous? Berrys Paradox (1906) Ways to tweet the number one 1 One

Zero factorial 4 3 Berrys Paradox (Twitter version) What is the smallest integer that cannot be identified in a tweet of no

more than 160 characters? Quines Paradox Yields falsehood when preceded by its quotation yields falsehood when

preceded by its quotation. Smullyans Charlatan Paradox Is a bogus charlatan a charlatan or not?

Another dubious proof A: Both these statements are false. B: I am the worlds greatest mathematician

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

Implication If A then B or A implies B, AB is true unless A is true and B is false

Another dubious proof If there were a Nobel Prize for mathematics then, as the greatest mathematician in the world,

I would deserve to win it. Currys Paradox If this statement is true, then I am the greatest mathematician in the world.

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

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 A Kill-Ease Taught-Us David Hilbert

In mathematics, there is no ignorabimus We must know we shall know! The Goldbach Conjecture Every even integer is the sum of

at most two primes Gdels Theorems A logical system can prove that it itself is consistent if and only if

it is not consistent Gdels Theorems In a consistent logical system there are true statements which cannot be

proved within that system Gdels Theorems Gdel's Incompleteness Theorem demonstrates that it is impossible for the Bible to be both true and complete.

Turing and the Halting Problem I guarantee that you will be surprised

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

Thank you for listening [email protected] @Tony_Mann Acknowledgments and picture credits Thanks to Noel-Ann Bradshaw

and everyone at Gresham College Picture credits Photograph of lecturer: Noel-Ann Bradshaw; T-shirt: www.thinkgeek.com 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 / esk sportovn, Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Czech Republic license Buridans Ass cartoons: Cham, Le Charivari, 1859, Wikimedia Commons; W.A. Rogers, New York Herald, c.1900, Wikimedia Commons Clement VI: Henri Sgur, Wikimedia Commons Franois 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 Dor, 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. Further Reading Douglas Hofstadter, Gdel, 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 Gdelian Puzzle Book: Puzzles,

Paradoxes and Proofs (Dover, 2013) Francesco Berto, There's Something About Gdel!: The Complete Guide to the Incompleteness Theorem (Wiley-Blackwell, 2009) Scott Aaronson, Quantum computing Since Democritus (Cambridge University Press, 2013)