From these lecture notes by Harvey Friedman comes one of the best metamathematical anecdotes I’ve ever heard (and yes, I’ve heard my share). Apparently Friedman was attending a talk by the “ultra-finitist” Alexander Yessenin-Volpin, who challenged the “Platonic existence” not only of infinity, but even of large integers like 2100. So Friedman raised the obvious “draw the line” objection: in the sequence 21,22,…,2100, which is the first integer that Yessenin-Volpin would say doesn’t exist?
Yessenin-Volpin asked Friedman to be more specific.
“Okay, then. Does 21 exist?”
Yessenin-Volpin quickly answered “yes.”
“What about 22?”
After a noticeable delay: “yes.”
After a longer delay: “yes.”
It soon became clear that Yessenin-Volpin would answer “yes” to every question, but would take twice as long for each one as for the one before it.