Category: Mathematics
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Ten weary, footsore travelers
Anonymous Ten weary, footsore travelers,All in a woeful plight,Sought shelter at a wayside innOne dark and stormy night. “Nine beds — no more,” the landlord said,“Have I to offer you;To each of eight a single room,But the ninth must serve for two.” A din arose. The troubled hostCould only scratch his head,For of those tired…
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A Positive Reminder
James Albert Lindon (b. 1914, d. December 16, 1979) A carpenter named Charlie Bratticks,Who had a taste for mathematics,One summer Tuesday, just for fun,Made a wooden cube side minus one. Though this to you may well seem wrong,He made it minus one foot long,Which meant (I hope your brains aren’t frothing)Its length was one foot…
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Belling the Cat (The Mice in Council)
A community of mice is terrorized by a marauding house cat that pounces on them stealthily. Desperate to find a solution, the mice convene a council to discuss how to protect themselves. Many ideas are debated. Finally, a young mouse proposes a bold plan: hang a bell around the cat’s neck. That way, the bell…
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Significant Paradoxes: Decision Theory and Game Theory
52 Prisoner’s Dilemma Two accomplices are interrogated separately. Each may Co-operate (stay silent) or Defect (betray). Mutual co-operation yields 2 years each; mutual defection 5 years each; unilateral betrayal frees the defector and jails the co-operator for 10 years. Because defection strictly dominates, rational agents defect, landing at 5 + 5 instead of 2 +…
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Significant Paradoxes: Epistemology and Probability
38 Hempel’s Raven Scientific hypotheses are usually universal claims—for example, “All ravens are black.” By standard logic that statement is equivalent to its contrapositive “All non-black things are non-ravens.” Hempel pointed out a tension between that equivalence and ordinary confirmation. A sighting of a black raven clearly supports the hypothesis, yet the contrapositive implies that…