Caltech number theorist Tom Apostol devised this elegant proof of the irrationality of .

Suppose the number is rational. Then there must be an isosceles right triangle with minimum integer sides (here, triangle ABC with sides n and hypotenuse m).

By drawing two arcs as shown, we can immediately establish triangle FDC — a smaller isosceles right triangle with integer sides.

This leads to an infinite descent. Hence n and m can’t both be integers, and is irrational.

You’re presented with the four cards above. Each has a number on one side and a color on the other. Which card(s) must be turned over to test the idea that if a card shows an even number on one face, then its opposite face is red?

In a 1966 study by Peter Wason, fewer than 10 percent of respondents correctly indicated the 8 and brown cards.

Interestingly, respondents perform significantly better when they’re presented with the same task in the context of policing a social rule (e.g., the rule is “If you are drinking alcohol then you must be over 21” and the cards are marked “27,” “16,” “drinking Coke,” “drinking beer”). About 90 percent of people perform this task correctly — supporting the idea that our facility for such tasks evolved to catch cheaters in a social environment.

123456789 = ((86 + 2 × 7)⁵ – 91) / 3⁴
987654321 = (8 × (97 + 6/2)⁵ + 1) / 3⁴
14459929 = 1⁷ + 4⁷ + 4⁷ + 5⁷ + 9⁷ + 9⁷ + 2⁷ + 9⁷
595968 = 4⁵ + 4⁹ + 4⁵ + 4⁹ + 4⁶ + 4⁸
397612 = 3² + 9¹ + 7⁶ + 6⁷ + 1⁹ + 2³

36428594490313158783584452532870892261556 = 3⁴² + 6⁴² + 4⁴² + 2⁴² + 8⁴² + 5⁴² + 9⁴² + 4⁴² + 4⁴² + 9⁴² + 0⁴² + 3⁴² + 1⁴² + 3⁴² + 1⁴² + 5⁴² + 8⁴² + 7⁴² + 8⁴² + 3⁴² + 5⁴² + 8⁴² + 4⁴² + 4⁴² + 5⁴² + 2⁴² + 5⁴² + 3⁴² + 2⁴² + 8⁴² + 7⁴² + 0⁴² + 8⁴² + 9⁴² + 2⁴² + 2⁴² + 6⁴² + 1⁴² + 5⁴² + 5⁴² + 6⁴²

A boy at the edge of a pond pulls a toy boat ashore. If he pulls in one yard of string, will the boat advance by more or less than one yard?

Surprisingly (to me), it will cover more than one yard. Because the boy is above the level of the water, he won’t pull in the entire length of string — length c will remain when the boat reaches shore. The length he’ll pull in, then, is a – c.

In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side, so b + c > a and b > a – c — so the boat travels a greater distance than the length of string pulled in.

Epimenides, a Cretan, says that all Cretans are liars. Is this a paradox? Not really: If we suppose that the statement is true then we’re led to a contradiction, but we can consistently suppose it to be false.

But, A.N. Prior writes, “We thus reach the peculiar conclusion that if any Cretan does assert that nothing asserted by a Cretan is true, then this cannot possibly be the only assertion made by a Cretan — there must also be, beside this false Cretan assertion, some true one. Yet how can there be a logical impossibility in supposing that some Cretan asserts that no Cretan ever says anything true, and that this is the only assertion ever made by a Cretan?”

Alonzo Church first raised this point in 1946: “Without factual information about other statements by Cretans, it has been proved by pure logic (so it seems) that some other statement by a Cretan, not the famous statement of Epimenides, must once have been true.”

The paradox, Prior writes, is that “such examination makes it seem possible to settle an empirical question on logical grounds.”

Draw two circles with three tangents as shown.

AB will always equal CD.

A psychologist at a girl’s college asked the members of his class to compliment any girl wearing red. Within a week the cafeteria was a blaze of red. None of the girls were aware of being influenced, although they did notice that the atmosphere was more friendly. A class at the University of Minnesota is reported to have conditioned their psychology professor a week after he told them about learning without awareness. Every time he moved toward the right side of the room, they paid more attention and laughed more uproariously at his jokes, until apparently they were able to condition him right out the door.

— W. Lambert Gardiner, Psychology: A Story of a Search, 1970

Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the following conjunctive proposition:

(d) Jones is the man who will get the job, and Jones has ten coins in his pocket.

Smith’s evidence for (d) might be that the president of the company assured him that Jones would in the end be selected, and that he, Smith, had counted the coins in Jones’s pocket ten minutes ago. Proposition (d) entails:

(e) The man who will get the job has ten coins in his pocket.

Let us suppose that Smith sees the entailment from (d) to (e), and accepts (e) on the grounds of (d), for which he has strong evidence. In this case, Smith is clearly justified in believing that (e) is true.

But imagine, further, that unknown to Smith, he himself, not Jones, will get the job. And, also, unknown to Smith, he himself has ten coins in his pocket. Proposition (e) is then true, though proposition (d), from which Smith inferred (e), is false.

In our example, then, all of the following are true: (i) (e) is true, (ii) Smith believes that (e) is true, and (iii) Smith is justified in believing that (e) is true. But it is equally clear that Smith does not know that (e) is true; for (e) is true in virtue of the number of coins in Smith’s pocket, while Smith does not know how many coins are in Smith’s pocket, and bases his belief in (e) on a count of the coins in Jones’s pocket, whom he falsely believes to be the man who will get the job. [Does Smith know that the man who will get the job has ten coins in his pocket?]

— Edmund L. Gettier, “Is Justified True Belief Knowledge?”, Analysis, 1963

Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the Democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason:

— If a Republican wins the election, then if it’s not Reagan who wins it will be Anderson.
— A Republican will win the election.

Yet they did not have reason to believe

— If it’s not Reagan who wins, it will be Anderson.

— Vann McGee, “A Counterexample to Modus Ponens,” Journal of Philosophy, September 1985