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For example, how do you explain why the perimeter of this staircase does not converge to $\sqrt2$? Or, why isn't $\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}$?

I would say, the reason is simply because they cannot be proved. But non-mathematicians don't find such explanation satisfactory. They seem to be more satisfied by sophisticated explanation. For the staircase paradox, my friend who studies physics reasoned that it's because the perimeter is of dimension 1 not 2. It makes no sense to me, but they found such explanation more sensible.

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Those are not paradoxes. – Javier May 28 '12 at 14:49

It seems like you mean "unintuitive results", rather than paradoxes. In which case I just try and give them some intuition about why their ideas don't work, without going into the math of the correct way. For the staircase example:

To get the length of a curve, we have to be able to break that curve down into little lengths (most people would at least agree that this should be possible) and add those up. If I give you a V-shape and ask you to find its length, you wouldn't find a length across the point, you'd find the length on either side and add them together (if they're more technical and/or attentive I visualize the limiting process more explicitly with curves). But when you keep breaking the line like that, you break it in infinitely many places: there are no more little pieces to add up like there would have been if you stopped, because everywhere you look will have broken parts. So 'length' isn't something that carries over to the "infinitely broken line" you made.

There's no point in telling them the right way if they don't understand why their way isn't it, so that's generally what I start with. That's good enough for most people, and often vaguely suggests the form that the right way should take. Only if they ask further questions do I go into the details of the right way to do it.

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In the case of $\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}$ at least, there is no paradox if you understand the limits of your rules.

You and the non-mathematician learned at some point that $\sqrt{ab}=\sqrt{a}\sqrt{b}$ for nonnegative numbers a and b. If one has forgotten this last part, then they might be amazed by $\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}$, because they have blindly applied a rule where it is not applicable.

The same thing might happen if you pointed out that $ab \geq 0$ for all natural numbers. If someone forgets the last part and is amazed by the apparent paradox $(-1)5 \geq 0$ , does this really deserve to be called a paradox?

Paradoxes do not have anything to do with "not being able to be proven", they are roughly just instances of where something you would expect to be true is not, in the system you are working in.

Like the "twin paradox" in physics: if you are (erroneously) under the impression that time is absolute, you are stumped, but if you believe in relativity then everything is OK.

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