The Banach-Tarski one is not an explanation; it’s an analogy that makes a non-mathematician feel they’ve read an explanation. This phenomenon is dangerous in math for the same reason it’s dangerous in quantum mechanics.
(In fact, it’s the same thing that bugged me when I was sitting through Christmas Mass out of respect for my parents; the priest was making surface analogies about the love of God, and the congregation felt that they were hearing explanations of theology or even evidence for the existence of God’s love. Is there a name for this phenomenon?)
Smullyan’s intuitive example of quining and Goedel was good, though.
The Banach-Tarski one is not an explanation; it’s an analogy that makes a non-mathematician feel they’ve read an explanation. This phenomenon is dangerous in math for the same reason it’s dangerous in quantum mechanics.
(In fact, it’s the same thing that bugged me when I was sitting through Christmas Mass out of respect for my parents; the priest was making surface analogies about the love of God, and the congregation felt that they were hearing explanations of theology or even evidence for the existence of God’s love. Is there a name for this phenomenon?)
Smullyan’s intuitive example of quining and Goedel was good, though.