5 ms·

Exactly. There's nothing paradoxical about this at all, just a bit counter-intuitive.

archgoon 16y ago
Well, he does say 'seemingly paradoxical', and look what happens when you try and express this (naively) in English: "The shorter stick is typically (about) 2/5 the length of the longer stick, but the longer stick is typically much more than 3 times the length of the shorter stick." To be honest, I can't figure out how to summarize the two results in English correctly (short of saying 'the mean of x/y~=2/5, but the mean of y/x does not exist, where x is the shorter and y the longer length')
- _delirium 16y ago
  There've been some psychological studies on averageness, and I believe one common case (though I'd have to look it up) is that people think of "typical" as a representative example in the middle of a distribution, something like a peak of a distribution and/or a median, and then imagine the distribution as a cloud of "stuff on either side of that". So the intuitive answer to the question would require there to exist a "typical" stick. Once you pick one specific stick, of course things work intuitively: the stick's dimensions are x=A, y=B, its shorter/longer ratio is min(A,B)/max(A,B), and its longer/shorter ratio is max(A,B)/min(A,B). An interesting question might be whether there's a definition of "typical stick" that in any useful way provides the answer that's intuitively wanted here.
  - wlievens 16y ago
    Our brains aren't exactly wired to handle uniform distributions too, I guess. Nature is full of normal distributions anyway.
- joshu 16y ago
  I think the notion of the average ratio makes little sense. Maybe a geometric mean? I dunno.
- jacobolus 16y ago
  Your problem here is that “typically” and “the mean” are two very different concepts. Taking the mean of a set of ratios is frankly not very meaningful, and so saying that the shorter stick is typically about 2/5 the length of the longer is highly misleading.
celoyd 16y ago
Strong counter-intuitiveness is a widely accepted use of the word “paradox”, even by logicians. (See, e.g., W. V. Quine’s The Ways of Paradox.)
- michael_dorfman 16y ago
  Hmm. Well, Quine and I will just have to agree to disagree, then. But in this case, I would hardly call the counter-intuitiveness "strong."
  - Confusion 16y ago
    Now I wonder whether you are brushing off Quine out of the conviction that it is a matter of opinion anyway and that his actual argument therefore doesn't matter? In which case I would like to point out the advantages of challenging your own opinions. Sometimes opinions turn out to have been, well, let's say: much harder to defend than originally thought and even though you can defend them as 'mere opinion', you just don't want to anymore. Decent philosophical treatises have at least changed my 'opinions' about some things.
    - michael_dorfman 16y ago
      On the contrary: I think Quine is being sloppy with language, as many philosophers reserve "paradox" to stand for something far stronger than mere counterintuitiveness. At the moment, I'm in the middle of writing a dissertation on Nagarjuna-- now, there was a master of paradox-- and I agree completely with you that decent philosophical can change a reader's opinions profoundly. I just don't put Quine into that categorization. Naturally, YMMV.

Well, he does say 'seemingly paradoxical', and look what happens when you try and express this (naively) in English: "The shorter stick is typically (about) 2/5 the length of the longer stick, but the longer stick is typically much more than 3 times the length of the shorter stick." To be honest, I can't figure out how to summarize the two results in English correctly (short of saying 'the mean of x/y~=2/5, but the mean of y/x does not exist, where x is the shorter and y the longer length')

_delirium 16y ago
There've been some psychological studies on averageness, and I believe one common case (though I'd have to look it up) is that people think of "typical" as a representative example in the middle of a distribution, something like a peak of a distribution and/or a median, and then imagine the distribution as a cloud of "stuff on either side of that". So the intuitive answer to the question would require there to exist a "typical" stick. Once you pick one specific stick, of course things work intuitively: the stick's dimensions are x=A, y=B, its shorter/longer ratio is min(A,B)/max(A,B), and its longer/shorter ratio is max(A,B)/min(A,B). An interesting question might be whether there's a definition of "typical stick" that in any useful way provides the answer that's intuitively wanted here.
- wlievens 16y ago
  Our brains aren't exactly wired to handle uniform distributions too, I guess. Nature is full of normal distributions anyway.
joshu 16y ago
I think the notion of the average ratio makes little sense. Maybe a geometric mean? I dunno.
jacobolus 16y ago
Your problem here is that “typically” and “the mean” are two very different concepts. Taking the mean of a set of ratios is frankly not very meaningful, and so saying that the shorter stick is typically about 2/5 the length of the longer is highly misleading.

celoyd 16y ago

Strong counter-intuitiveness is a widely accepted use of the word “paradox”, even by logicians. (See, e.g., W. V. Quine’s The Ways of Paradox.)

michael_dorfman 16y ago
Hmm. Well, Quine and I will just have to agree to disagree, then. But in this case, I would hardly call the counter-intuitiveness "strong."
- Confusion 16y ago
  Now I wonder whether you are brushing off Quine out of the conviction that it is a matter of opinion anyway and that his actual argument therefore doesn't matter? In which case I would like to point out the advantages of challenging your own opinions. Sometimes opinions turn out to have been, well, let's say: much harder to defend than originally thought and even though you can defend them as 'mere opinion', you just don't want to anymore. Decent philosophical treatises have at least changed my 'opinions' about some things.
  - michael_dorfman 16y ago
    On the contrary: I think Quine is being sloppy with language, as many philosophers reserve "paradox" to stand for something far stronger than mere counterintuitiveness. At the moment, I'm in the middle of writing a dissertation on Nagarjuna-- now, there was a master of paradox-- and I agree completely with you that decent philosophical can change a reader's opinions profoundly. I just don't put Quine into that categorization. Naturally, YMMV.