Near the beginning of The Port-Royal Logic there is a brief and somewhat odd discussion of the Pyrrhonists and the Academics, and I guess Michel de Montaigne:

We may indeed easily say outwardly with the lips that we doubt of all these things, because it is possible for us to lie ; but we cannot say this in our hearts. Thus Pyrrhonism is not a sect composed of men who are persuaded of what they say, but a sect of liars. Hence they often contradict themselves in uttering their opinion, since it is impossible for their hearts to agree with their language. We see this in Montaigne, who attempted to revive this sect in the last century ; for, after having said that the Academics were different from the Pyrrhonists, inasmuch as the Academics maintained that some things were more probable than others, which the Pyrrhonists would not allow, he declares himself on the side of the Pyrrhonists in the following terms : “The opinion,” says he, “of the Pyrrhonists is bolder, and much more probable.” There are, therefore, some things which are more probable than others. Nor was it for the sake of effect that he spoke thus : these are words which escaped him without thinking of them, springing from the depths of nature, which no illusion of opinions can destroy.

Antoine Arnauld, Pierre Nicole, The Port-Royal Logic, 1662, translated by Thomas Spencer Baines, 1861.

I love the last sentence here: “And don't you try to get out of it by claiming a sense of humor, either.” To me it hardly seems probable that Montaigne was not just saying that for effect. It's too perfect.

The principle of explosion

Pop quiz:

1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... = ?

It's obviously 0, right? Or maybe it's 1. In the 17th and 18th centuries, everyone apparently thought the correct answer was ½ (and it wasn't because they were stupid back then: this includes people like Leibniz and Euler).

Eh, so math is inconsistent. So what?

It is important to point out that it is not enough to consider at the same time two conflicting statements in order to develop in pupils' minds the awareness of an inconsistency and the necessity of second thoughts (Schoenfeld, 1985): the perception of some mutually conflicting elements does not always imply the perception of the situation as a problematic one (Tirosh, 1990).

Infinite series: from history to mathematics education (PDF), Giorgio T. Bagni.

Huh.

Now, maybe this is because math doesn't make a whole lot of sense to most kids to begin with. But I think the main cause is that kids, like the rest of us, are used to things being inconsistent sometimes. And they live with it. I mean, what are you going to do?

Well, let me tell you something. In math, you can't live with a contradiction.

The principle of explosion is built into the fundamental rules of logic, rules that both mathematicians and ordinary people use to reason with. Ex falso sequitur quodlibet: from a contradiction, anything follows. Or as an old friend of mine used to say, after you swallow the first pill, the rest go down real easy.

In math, if you accept a single contradiction, the entire system comes crashing down around you.

(Now there's such a thing as paraconsistent logic, in which inconsistencies are not so destructive. But it's quite different from ordinary logic, and not many people are familiar with it.)

In the above case, mathematicians eventually discovered a formal notion of “convergence” and found that the sum 1 - 1 + 1 - 1 + ... does not converge. That is, there's no answer, just as there's no answer for the sum 1 + 2 + 3 + 4 + ..., and for the same reason: you can go on for as long as you like, and your numbers are never going to converge on some specific value.

Is this a cop-out? It's a hard fact of life that some mathematical problems just don't have answers. The ancients considered 2 - 7 to be undefined, because the answer would be less than nothing, which was clearly nonsense. Today we have negative numbers, but other things, like division by zero, are still undefined. Given all that, maybe it's not so surprising if expressions that end in the innocent-looking “+ ...”, as if to say “oh don't mind me, I'm just a little infinite series, tra la”, sometimes fall into this category.

Concerning gifts (and other puzzles)

What can you conclude from the following three premisses?

1. If something is not gift-wrapped, it's not a gift.
2. Nothing that's gift-wrapped is entirely unlike a box of chocolates.
3. Life is a gift.

Lewis Carroll published a book of about a hundred puzzles like this one. Read it online: introduction; puzzles. My nephew IM and I stumbled upon them in Memphis last week. He pretty much knocked them out of the park one at a time.

They're fairly easy to make, if you know some logic and some algebra. Here are a few more (but Lewis Carroll's are the most sublime nonsense—you should probably try those instead.)

Concerning fashion

1. Anyone lacking impeccable fashion sense might wear a rhinestone sombrero.
2. Anyone who might wear a rhinestone sombrero can't dance.
3. All penguins can dance.

Concerning animals

1. Animals that are active during the day are either featherless or tasty—or both.
2. No creature is both nocturnal and naturally funny.
3. Chickens have feathers.
4. Chickens are naturally funny.

Concerning the inhabitants of this town

1. All the monsters in this town are carnivores.
2. A carnivore would eat anything made of meat.
3. No creature in this town would eat any other creature in this town.
4. Humans are made of meat.

Concerning monsters

1. Only monsters can make the ground tremble.
2. Two-year-olds and raccoons get into everything (two-year-old raccoons doubly so).
3. If something gets into everything, but it doesn't emit terrifying shrieks, it must be a raccoon.
4. All firebreathing creatures are monsters.
5. Opera singers can make the ground tremble.
6. Raccoons are not human.
7. A creature that isn't a monster doesn't have slavering fangs.
8. If a creature emits terrifying shrieks, then either it breathes fire, it has slavering fangs, or it's an opera singer.

Alice in Puzzle-Land

“How do I know for sure that I'm awake?” asked Alice. “Why can't it be that I'm now asleep and dreaming all this?”

“Ah, that's an interesting question and one quite difficult to answer!” replied the King. “I once had a long philosophical discussion with Humpty Dumpty about this. Do you know him?”

“Oh, yes!” replied Alice.

“Well, Humpty Dumpty is one of the keenest arguers I know—he can convince just about anyone of just about anything when he puts his mind to it! Anyway, he almost had me convinced that I had no valid reason to be sure that I was awake, but I outsmarted him! It took me about three hours, but I finally convinced him that I must be awake, and so he conceded that I had won the argument. And then—”

The King did not finish his sentence but stood lost in thought.

“And then what?” asked Alice.

“And then I woke up!” said the King, a bit sheepishly.

—Raymond Smullyan, Alice in Puzzle-Land. This is a fun book of logic puzzles ranging from cute to outrageously intricate. A fine gift for the mathematician on your list (though I hear The Annotated Alice is even better).