This Lecture Will Surprise You When Logic is

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 n 2 is odd then n must be odd Proof: Suppose n is an even integer such that n 2 is odd Then n = 2 k for some integer k But n 2 = (2 k)2 = 4 k 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

A Pair o’ Docs

Smullyan’s 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.

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

Buridan’s Ass. Roger Buridan and Pierre

Buridan’s “Where are the snows Ass 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

Buridan’s science Theory of Impetus (≈ Newton’s 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 Socrates’s statement is false, Plato will throw him in the river. Socrates says, “You will throw me in the river”.

Buridan’s 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?

Buridan’s Ass Witches in sixteenth-century France

Buridan’s Protagoras and Ass 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

Buridan’s 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

Buridan’s 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

Russell’s 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.

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

Russell’s 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 don’t describe themselves – eg “long” or “monosyllabic” Call them “heterologous” Is “heterologous” heterologous?

Berry’s Paradox (1906)

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

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

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

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

Another dubious proof A: Both these statements are false. B: I am the world’s 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”, A→B 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.

Curry’s 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 A→B, can we deduce that B is true?

What the Tortoise said to Achilles 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 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

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

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

Gödel’s Theorems “Gödel'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 wasn’t met. That was your surprise!

Thank you for listening a. mann@gre. ac. uk @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 Mc. Ilroy: Tour. Pro. Golf. Clubs, 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.

Further Reading Douglas Hofstadter, Gödel, Escher. Bach: an Eternal Golden Braid (Penguin, 20 th 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)