You say you know your brother.
Yet when your brother is hooded you are unable to identify him.
Therefore you both do and do not know your brother.
— Eubulides
Science & Math
The Corkscrew
Thin Thinking
Some of the figures (particularly the holy ones) in El Greco paintings seem unnaturally tall and thin. An ophthalmologist surmised that the painter had a defect of vision that caused him to see people this way.
The zoologist Sir Peter Medawar pointed out that we can reject this conjecture on purely logical grounds. What was his insight?
“The Unprovable Liar”
‘What I am saying cannot be proved.’
Suppose this statement can be proved. Then what it says must be true. But it says it cannot be proved. If we assume it can be proved, we prove it cannot be proved. So our supposition that it was provable is wrong. With that road closed to us, let’s try the only other one available — let’s suppose it cannot be proved. Since that is precisely what it says, then it is true after all. And this ends our proof of the above statement!
— Gary Hayden and Michael Picard, This Book Does Not Exist, 2009
Consultation
A letter from Lewis Carroll to 14-year-old Wilton Rix:
Honoured Sir,
Understanding you to be a distinguished algebraist (i.e. distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.
If x and y are each equal to ‘1,’ it is plain that 2 × (x2 – y2) = 0, and also that 5 × (x – y) = 0.
Hence 2 × (x2 – y2) = 5 × (x – y).
Now divide each side of this equation by (x – y).
Then 2 × (x + y) = 5.
But (x + y) = (1 + 1), i.e. = 2.
So that 2 × 2 = 5.
Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.
I trust you will pity me and will kindly explain the difficulty to
Your obliged, Lewis Carroll
Tall Order
From Lewis Carroll:
Men over 5 feet high are numerous.
Men over 10 feet high are not numerous.
Therefore men over 10 feet high are not over 5 feet high.
Perspective
“How are you going to teach logic in a world where everybody talks about the sun setting, when it’s really the horizon rising?” — Cal Craig, quoted in Howard Eves, Mathematical Circles Revisited, 1971
Run Toppers
Let’s play a game. We’ll each name three consecutive outcomes of a coin toss (for example, tails-heads-heads, or THH). Then we’ll flip a coin repeatedly until one of our chosen runs appears. That player wins.
Is there any strategy you can take to improve your chance of beating me? Strangely, there is. When I’ve named my triplet (say, HTH), take the complement of the center symbol and add it to the beginning, and then discard the last symbol (here yielding HHT). This new triplet will be more likely to appear than mine.
The remarkable thing is that this always works. No matter what triplet I pick, this method will always produce a triplet that is more likely to appear than mine. It was discovered by Barry Wolk of the University of Manitoba, building on a discovery by Walter Penney.
The Airy Box
It’s said that British Astronomer Royal G.B. Airy once discovered an empty box at the Greenwich Observatory in London.
He wrote EMPTY BOX on a piece of paper and put it inside.
“Attached to the outside, such a label is true,” write Gary Hayden and Michael Picard in This Book Does Not Exist. “Placed inside the box, it makes itself false. Alternatively, suppose the label says: ‘The box this label is inside is empty.’ Outside of any box, the subject of this sentence fails to refer — there is no box inside which the label is located. However, once inside an otherwise empty box, the sentence becomes false.”
Faith and Sugar
A placebo has no pharmaceutical properties; if it works, it works only because of my own belief in its efficacy.
If I know that I’m taking a placebo, it will be ineffective.
So while the placebo cures me only because I believe it will, I can’t believe that it will cure me only because I believe it will.
(From City University philosopher Peter Cave.)