Feynman's Orange Riddle: Impossible or Not?

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Discussion Overview

The discussion revolves around Feynman's anecdote regarding a riddle posed by topology students about cutting an orange into pieces and reassembling it to be as large as the sun. Participants explore the implications of this riddle, particularly in relation to the theorem of immeasurable measure and the Banach-Tarski paradox, while also reflecting on the nature of mathematical intuition and definitions.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants reference Feynman's argument that cutting an orange into pieces cannot yield a larger object, emphasizing the limitations of physical objects versus mathematical concepts.
  • Others mention the Banach-Tarski theorem as a related mathematical concept that challenges intuitive thinking about volume and measure.
  • A participant expresses curiosity about the existence of the theorem of immeasurable measure and its implications, suggesting it could be a theoretical construct.
  • One participant notes that mathematicians often create examples that defy intuition to illustrate the complexities of mathematical definitions.
  • Another participant shifts the topic slightly to discuss Feynman's ability to calculate cube roots faster than an abacus user, inviting further exploration of mathematical techniques.

Areas of Agreement / Disagreement

Participants generally agree on the intriguing nature of the riddle and its connection to mathematical paradoxes, but there is no consensus on the existence or implications of the theorem of immeasurable measure. The discussion remains open-ended regarding the interpretations of these mathematical concepts.

Contextual Notes

Some assumptions about the definitions of measure and the nature of physical objects versus mathematical abstractions are not fully explored. The discussion also lacks a resolution on the validity of the theorem of immeasurable measure.

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I'm reading "Surely you're joking, Mr. Feynman!" which is one of the funniest and interesting books I've ever read!

However, in one paragraph, Feynman is talking with some topology students about a riddle:

The topology students: "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."...

I understood Feynman argument, but I didn't get the idea of the theorem of immeasurable measure. How can you get an orange as big as the sun?
 
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It's related to this joke:

Q: What's an anagram of Banach-Tarski?

A: Banach-Tarski Banach-Tarski.

Google for it.
 
The point of the story is that mathematicians are always trying to show that intuitive thinking will lead you astray. So they love coming up with examples of something you'd think should be impossible and then they "prove" it's true. Feynman's point it that if they really are honest about their definitions, things really do make sense if you just look at them the right way.
 
I was just reading that book, and was writing my Theory of Knowledge essay.
I also was wondering if the theorem of immeasurable measure existed and if so, what is it?
Feynman is correct in saying that the topologists asked him about splitting an orange finite number of times, so for an orange it was impossible, but as a theory it intrigued me as it could be a 'real' theory where you assume nothing.
 
As long as we're on the topic of Feynman, in the same book was a story of how he was able to calculate cube roots using pad and pencil faster than someone using an abacus.

Does anyone here know how to calculate cube roots on an abacus?
 

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