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That's impossible! A simple mathematical paradox

11 points by acangiano 19y ago

slapshot 19y ago
Not at all a paradox. A paradox would require two inconsistent things to be simultaneously true. This is just an answer that's counter-intuitive. It's entirely consistent, but surprising.
- acangiano 19y ago
  From wikipedia: "A paradox can be an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition; or it can be, seemingly opposite, an apparent contradiction that actually expresses a non-dual truth (cf. Koan). Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. [...] A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a person's fifth birthday is the day he turns twenty, if born on a leap day. Likewise, Arrow's impossibility theorem involves behaviour of voting systems that is surprising but true." But yes, it's a puzzle with a counterintuitive result.
  - mixmax 19y ago
    Like the birthday paradox: http://en.wikipedia.org/wiki/Birthday_paradox http://en.wikipedia.org/wiki/Birthday_paradox
    - acangiano 19y ago
      Exactly. Arguably that's the etymological meaning of the term. (It comes from Greek, "para" which means "contrary to" and "doxa" which is "opinion").
- cperciva 19y ago
  I don't even find this counter-intuitive. Maybe my intuition has been shaped by spending most of my life doing mathematics. :-)
  - tdavis 19y ago
    I have to agree here... a meter is a meter is a meter -- regardless of how long a rope it's being added to. The relative difference between the ropes is still only one meter when taking into account the scaled proportions. I hope the other "paradoxes" in this book are more impressive ;)
KiwiNige 19y ago
The extra 16cm of Radius is lots compared to a golf ball, and nothing compared to earth. I have to scale these in my mind to be able to compare the two, and the relative gap is what I expected. But I'm still surprised that the gap is exactly the same in both.
tectonic 19y ago
See also the fascinating Benford's Law: http://en.wikipedia.org/wiki/Benford's_law http://en.wikipedia.org/wiki/Benford's_law
smanek 19y ago
Oh, one of the neater mathematical 'paradoxes' I've seen is finding the sum of a divergent series (obviously you can't treat infinity this way ... but it's a fun trick). (excuse the notation, I'm doing this with plain ASCII characters ...): Let the sum of the geometric series 1 + 2 + 4 + 8 + 16 + ... be called (=) S. Now multiply both sides by 2: 2S = 2 + 4 + 8 + 16 + 32 + .... Now subtract the second equation from the first: S - 2S = (1 + 2 + 4 + 16 + ...) - (2 + 4 + 8 + 16 + 32 + ...) Notice that all the terms except for the '1' cancel out on the right: -S = 1 And simplify: S = -1 Therefore: 1 + 2 + 4 + 8 + 16 + ... = -1 ;-)

11 points by acangiano 19y ago

Not at all a paradox. A paradox would require two inconsistent things to be simultaneously true. This is just an answer that's counter-intuitive. It's entirely consistent, but surprising.

acangiano 19y ago
From wikipedia: "A paradox can be an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition; or it can be, seemingly opposite, an apparent contradiction that actually expresses a non-dual truth (cf. Koan). Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. [...] A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a person's fifth birthday is the day he turns twenty, if born on a leap day. Likewise, Arrow's impossibility theorem involves behaviour of voting systems that is surprising but true." But yes, it's a puzzle with a counterintuitive result.
- mixmax 19y ago
  Like the birthday paradox: http://en.wikipedia.org/wiki/Birthday_paradox http://en.wikipedia.org/wiki/Birthday_paradox
  - acangiano 19y ago
    Exactly. Arguably that's the etymological meaning of the term. (It comes from Greek, "para" which means "contrary to" and "doxa" which is "opinion").
cperciva 19y ago
I don't even find this counter-intuitive. Maybe my intuition has been shaped by spending most of my life doing mathematics. :-)
- tdavis 19y ago
  I have to agree here... a meter is a meter is a meter -- regardless of how long a rope it's being added to. The relative difference between the ropes is still only one meter when taking into account the scaled proportions. I hope the other "paradoxes" in this book are more impressive ;)

KiwiNige 19y ago

The extra 16cm of Radius is lots compared to a golf ball, and nothing compared to earth. I have to scale these in my mind to be able to compare the two, and the relative gap is what I expected. But I'm still surprised that the gap is exactly the same in both.

tectonic 19y ago

See also the fascinating Benford's Law: http://en.wikipedia.org/wiki/Benford's_law http://en.wikipedia.org/wiki/Benford's_law

smanek 19y ago

Oh, one of the neater mathematical 'paradoxes' I've seen is finding the sum of a divergent series (obviously you can't treat infinity this way ... but it's a fun trick). (excuse the notation, I'm doing this with plain ASCII characters ...): Let the sum of the geometric series 1 + 2 + 4 + 8 + 16 + ... be called (=) S. Now multiply both sides by 2: 2S = 2 + 4 + 8 + 16 + 32 + .... Now subtract the second equation from the first: S - 2S = (1 + 2 + 4 + 16 + ...) - (2 + 4 + 8 + 16 + 32 + ...) Notice that all the terms except for the '1' cancel out on the right: -S = 1 And simplify: S = -1 Therefore: 1 + 2 + 4 + 8 + 16 + ... = -1 ;-)