This is an engaging, if verbose, article in the NYT about logical paradoxes.

There’s not much new ground covered here, but the author touches on some of the classics–the Liar’s Paradox (Is the statement “This sentence is false” true or false?), Zeno’s Paradox (you’ll never get from point A to point B because first you have to get halfway to B, say C, then halfway between C and B, say D, and so on).

The author notes that writers and philosophers love paradoxes.  Students love paradoxes, too.  It’s always enjoyable making a student act out Zeno’s paradox by making them get infinitely close to the board, or arguing about whether there are more even numbers or integers.  And of course, everyone loves arguing about whether .999999999….  really equals 1 .

Categories: Teaching

#### patrick honner

Math teacher in Brooklyn, New York

#### Sean · December 8, 2010 at 6:18 pm

Yes, very good post. I remembered this being a well-known paradox when you asked for the longest cevian in a triangle. I wanted to say that there’s no such thing as “closest,” but couldn’t remember the name of the paradox to support it with. Also, the topic of .99999999… being equal to 1 is something that is always interesting to discuss. Whenever someone says 1e-18 “is zero,” I find that person to be wrong. After all, 1e-18 times 1e18 = 1, but 1e18 times 0 = 0, and it’s safe to say that 1 is not 0.

#### Ahmed Gouda · December 10, 2010 at 12:21 am

Stephen Hawking did something on time travel. Though he technically could be wrong, he said that time travel, at least backwards, is impossible. If you make a portal, the radiation travels through, and will eventually come back through time and overly radiate the universe. And then their is the question of stable time loops. An unstable time loop can’t happen. I imagine if time travel existed, it would exist in the form of self-fulfilling loops. For example, although I wouldn’t be able to kill myself, I’d say be able to be my own father as long as I provided younger me with the incentive to become me in the future. It really all comes down to how time really works though. Does altering something in the past, if that is even possible, make an extra time line? I’m a bit doubtful though. If something can go back in time, it would increase the matter in the past. If you have say a table with balls, sort of like in pool, all moving constantly, no friction etc. Then lets say I rewind the clock and add/remove a ball. The entire movement of several balls would change in the future. (Butterfly effect I think?)

On an unrelated note the paradox of Pinocchio saying his nose will grow is amusing.

#### MrHonner · December 12, 2010 at 8:59 pm

The paradox of Pinocchio is outstanding! I’ve never heard that before. I’m glad I stuck with your comment through all the time travel stuff–usually I just zone out when one of you starts talking about parallel universes.

#### notezactme · December 16, 2010 at 11:30 pm

If I vow to help only those who don’t help themselves, am I allowed to help myself?

If I help myself, I broke my vow because I am helping myself.
If I don’t help myself, I broke my vow because I am not helping myself.

Thus there is a contradiction in saying that I will “help only those who don’t help themselves.”

To me, that means pure “selflessness” doesn’t exist.

Follow