Hi Jim, Rather than giving the somewhat meta reply that you propose, I think that Mr. Apollinax can reply as follows: "Yes, one of you was correct, but this was just by chance. Indeed, you would have done equally well if each of you had simply guessed that your number was larger." "My claim was simply that neither of you can deduce, with certainty, which number is larger. This is clearly true, since whichever of you happened to be correct could also have been incorrect." Indeed, it seems that if you and I whisper, say, 7 and 8, then Apollinax's claim is true: neither of us can deduce which is larger. This is like being surprised when I hang you on Wednesday. Cris On Dec 9, 2014, at 2:05 PM, James Propp <jamespropp@gmail.com> wrote:
It's interesting to pursue the analysis a bit further.
Under the most obvious model of what happens, you and I separately conclude that neither of us picked 1, neither of us picked 2, neither of picked 3, etc. My "etc." stops 1 short of the number that I picked; having deduced that your number can't be smaller than mine, I deduce that your number is larger. Meanwhile, your "etc." stops 1 short of the number that you picked; having deduced that my number can't be smaller than yours, you deduce that my number is larger.
Let's suppose that we announce to Mr. Apollinax our conclusions and our reasoning processes.
What's curious is that one of us reached a correct conclusion and one of us reached an incorrect conclusion. And so it would seem that one of us DID deduce which of the two numbers was larger (contrary to Mr. Apollinax's assertion "neither of you can deduce which one is larger"). So Mr. Apollinax told an untruth. Since you and I both treated this untruth as if it were true, and based deductions on it, it isn't surprising that one of us reached a true conclusion and the other reached a false conclusion. And it might not seem there is more to be said. Or is there?
We could push the thought-experiment one step further, and imagine confronting Mr. Apollinax with the accusation that he told an untruth. I can imagine him replying "Ah, but your deduction was invalid, because it was based upon a false premise, namely, the untruth that I told!" Assume for purposes of this paragraph that we accept Mr. Apollinax's sophistry, and interpret "neither of you can deduce" to mean "neither of you can deduce, using valid principles of deduction applied to valid premises". Then we have the amusing paradox that Mr. Apollinax's pronouncement "neither of you can deduce which one was larger" appears to be vindicated after all, precisely because it is false.
I'd be curious to know what Tim Chow ( http://www-math.mit.edu/~tchow/unexpected.pdf) makes of this.
Jim Propp
On Thu, Dec 4, 2014 at 7:58 PM, Cris Moore <moore@santafe.edu> wrote:
You, I, and Mr. Apollinax are traveling on a train. He leans over to you and says, with a leer, "whisper a natural number into my ear." You do so. He then turns to me and asks me to do the same. I do.
He then claps his hands and says "Marvelous! Your numbers are distinct, and neither of you can deduce which one is larger."
Get it?
- Cris
p.s. I heard this from Thilo Gross...
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