How would you have answered Richard Feynman's challenge?

[Keith_McClary]

TL;DR 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."

Fun question on mathoverflow:
How would you have answered Richard Feynman's challenge?

 

[fresh_42]

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".

 

[fresh_42]

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]

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'.

 

[pbuk]

I'd start with

• 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.

 

[martinbn]

I'd start with

• 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.

 

[TeethWhitener]

One that struck me as really strange at first sight is that the harmonic series diverges:
$$\sum_{n=1}^{\infty}{\frac{1}{n}}$$
but if you remove all the terms with a 9 in the denominator from the sum, the new series converges (to a relatively small number).

The reason is that for larger and larger denominators, it becomes overwhelmingly likely that the denominator will contain a 9, and so the vast majority of the terms in the sum will be removed. This reasoning can be taken further to show that if you remove all the terms with any finite string of numerals in the denominator, the series will converge. It's surprising to a lot of people because we generally have terrible intuition about 1) really large numbers and 2) probability and statistics, and this fact plays on both of those intuitions.