# How would you have answered Richard Feynman's challenge?

Gold Member
Summary:
As a student of the Princeton physics department, he used to challenge the students of the math department: "I bet there isn't a single theorem that you can tell me what the assumptions are and what the theorem is in terms I can understand where I can't tell you right away whether it's true or false."

fresh_42
Mentor
2021 Award
I would choose Banach-Tarski.

martinbn
I would choose Banach-Tarski.
That is the example Feynman gives
I challenged them: "I bet there isn't a single theorem that you can tell me what the assumptions are and what the theorem is in terms I can understand where I can't tell you right away whether it's true or false." It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes?" "No holes." "Impossible! There ain't no such a thing." "Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms." "But we have the condition of continuity: We can keep on cutting!" "No, you said an orange, so I assumed that you meant a real orange." So I always won. If I guessed it right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.

ps On a unrelated thought, and anagram of "Banach-Tarski" is "Banach-Tarski Banach-Tarski".

pbuk and PeroK
fresh_42
Mentor
2021 Award
I like to reason along those lines if people come here and ask: Is <any mathematical object> real? I like to say that there isn't even such a thing as a circle in real life. Latest under the electron microscope such a circle is all but round. Nevertheless, we perfectly deal with circles in all our real-life mechanics.

pbuk
pbuk
Gold Member
all but round
anything but round. The phrase 'all but round' implies 'round except for a very small but finite amount' rather than 'not really round at all'.

fresh_42
pbuk
Gold Member
• There is no set whose cardinality is strictly between that of the integers and the real numbers.
and if he got on well with that I'd go with
• This theorem cannot be proved using the axioms of ZFC.
Can't believe that MO didn't come up with either of these: just goes to show that PF is a much better forum!

Edit: note that each of these can be easily answered, but if I had time with Feynman then I would not waste it trying to catch him out, I would use it to hear what he had to say about the implications of these interesting questions.

Last edited:
martinbn
• There is no set whose cardinality is strictly between that of the integers and the real numbers.
and if he got on well with that I'd go with
• This theorem cannot be proved using the axioms of ZFC.
Can't believe that MO didn't come up with either of these: just goes to show that PF is a much better forum!

Edit: note that each of these can be easily answered, but if I had time with Feynman then I would not waste it trying to catch him out, I would use it to hear what he had to say about the implications of these interesting questions.
This is one of the answers on MO.

S.G. Janssens
TeethWhitener
$$\sum_{n=1}^{\infty}{\frac{1}{n}}$$