Which is most helpful response?

1. Snotty lectures about how infinity isn't really a number, so you can't even ask that sort of question.

2. Reasonable explanations about how these terms aren't always well-defined, and the problems you run into trying to give them a definition.

3. Here are some ways in which mathematicians have tried to answer those questions in ways that still make sense!

Unfortunately, as the links above show, #1 is by far the most popular approach. #2 is not bad, but if you actually want somebody to stay interested in math, I think #3 is far superior.

Mathematics is a game. It's not a set of rules, it's about making up rules and seeing where they lead. And concepts like surreal numbers, projective geometry, and Hilbert spaces all use infinities in ways that are mathematically consistent but "bizarre". Surreal numbers in particular are a great example of taking seriously such ideas as infinitesimals and "infinity plus one" and giving them a concrete meaning instead of blowing such obvious ideas off as stupid. For that matter, the whole field of complex analysis results from taking seriously something that was previously just a "hack" to make the cubic function come out right: "what is the square root of -1?"

Some games don't work out, but that's probably just because nobody's been clever enough yet. Or in some cases, the game is provably not very interesting (which is sort of meta-interesting). But the next time somebody tries to get pseudo-sophisticated with you by explaining how your math question can't even be asked, treat them as you would any other bully.

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