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
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.1300Ass
– after 1358)
Buridan’s
AssRoger
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
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) →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 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.
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)