According to Wikipedia:
"Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape.I remember reading Richard Feynman's reaction to this in his memoir "Surely You're Joking Mr Feynman!",
However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.
A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".
"Then I [Feynman] got an idea. I challenged them [the mathematicians]: “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.”
“Impossible!
“Ha! 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."
There are no infinities in reality.
But Feynman's own diagram expansions are packed full of infinities. The challenge has been to get rid of them.
ReplyDeleteThe Real continuum plays (perhaps too much of) a role across physics, and it too is packed with infinities and unsolved puzzles like the Continuum Hypothesis.
Also in the Feynman story the "mathematicians" started talking about oranges, not mathematical definitions, theorems and axioms. Surely Feynman would have understood a geometric reference to balls in space? His mathematical intuition wasn't properly challenged in this example, as I guess that would be hard work, since this is Feynman!
Incidentally I have recently read a paper which is suggesting that the paradox works because the space really has "no" points. It is basically Locale Theory's take on the BT paradox, and I am not sure that there are any infinities involved.
(I am too busy with Backpropagation right now, to follow it up, but it is on the list.)
Yeah. Lots of *theories* have infinities .. I think that might have been what Feynman was driving at.
ReplyDeleteBest of luck with BP!