The bonus braintwister is actually addressed in Stephen Hawking's "A Brief History of Time." The simple answer is that the universe is not static. There's also dust in the way from a lot of stars (like the ones in nebulae).
Speaking of which, if you have a chance, I highly recommend seeing "Hubble" in IMAX. A while back, they took an IMAX camera up on a shuttle flight where the astronauts were repairing the Hubble Space Telescope, so there's actual footage of space and Earth in the background. It's pretty spectacular. And they they took some of the pictures from Hubble and did something awesome with computers so it looks like you're flying into nebulae and you can see the stars inside. I was blown-away.
Friday, August 6, 2010
Philosophy - #2, 3 and 4 (and #6)
Braintwisters #2, 3 and 4 are essentially variations of the following paradox:
1) Statement 2 is true.
2) Statement 1 is false.
If 1 is true, then 2 is true. But if 2 is true, then 1 is false. But if 1 is false, then 2 is false. But if 2 is false, then 1 is true...
It's just wordplay; there's no solution.
#6, though different, is lumped in with this post because both explanations are short. It assumes that the donkey would be caught indefinitely due to never being able to make a rational decision between two equal things. But guys (and gals), in the real world, there's no "perfectly equal" scenario. Maybe the wind blows from the west. The donkey sniffs the food coming from that direction a little stronger than it does the food-smells from the east, so it heads west. Maybe a fly itches the donkey's neck, and it moves its head to the right. Since it's already looking to the right, it ambles in that direction. There will always be some little disturbance that tilts the balance one way or another. Sort of like how an inverted pendulum will never balance upside-down, even though there's an equilibrium position in that arrangement. Mathematically it's fine, but physically it won't work.
1) Statement 2 is true.
2) Statement 1 is false.
If 1 is true, then 2 is true. But if 2 is true, then 1 is false. But if 1 is false, then 2 is false. But if 2 is false, then 1 is true...
It's just wordplay; there's no solution.
#6, though different, is lumped in with this post because both explanations are short. It assumes that the donkey would be caught indefinitely due to never being able to make a rational decision between two equal things. But guys (and gals), in the real world, there's no "perfectly equal" scenario. Maybe the wind blows from the west. The donkey sniffs the food coming from that direction a little stronger than it does the food-smells from the east, so it heads west. Maybe a fly itches the donkey's neck, and it moves its head to the right. Since it's already looking to the right, it ambles in that direction. There will always be some little disturbance that tilts the balance one way or another. Sort of like how an inverted pendulum will never balance upside-down, even though there's an equilibrium position in that arrangement. Mathematically it's fine, but physically it won't work.
Philosophy - #7 and 8
Braintwisters number 7 and 8 are what originally inspired me to write these posts. Both demonstrate a lack of understanding of math and physics.
Number 8 was the one that really made me do a headdesk. The "paradox" only occurs if one ignores velocity. At any given instance the arrow is in one specific place (as stated); however, it has a non-zero velocity!
A similar thing happens when you throw a ball in the air (sort of...I'm gonna take a derivative here...) When the ball reaches the peak of its flight, there's an instant where it hovers - it has zero velocity. But we know from experience that the ball will fall back to Earth. That's because the ball has a non-zero acceleration (in this case, it's the acceleration due to gravity: 9.8 m/s^2). If you look at a "snapshot" of the ball when it's at the peak, you might think that it will hover forever, but that's only because you're not looking at the full picture (so to speak). You've left out acceleration.
It's kind of the same thing here. The arrow is instantaneously in one place, but it keeps moving because of its velocity. If you don't include velocity you're not really looking at the whole system.
Likewise, #7 falls into a mathematical trap. It's essentially a related-rate problem (Calc 101 - Mom always said that calculus was useful). Achilles' velocity is greater than the tortoise's velocity. So eventually he will overtake the tortoise, even though the tortoise keeps moving. You can calculate the distance each has traveled (distance = rate x time) at any particular instant. At the time where distance_Achilles is greater than distance_tortoise, Achilles has passed the tortoise, even though distance_tortoise keeps increasing with time. Distance_Achilles increases with time, too - but at a faster rate.
Sorry if this was garbled; it's been a few years since I took Calc I and I'm not sure if I'm explaining it clearly. If you're interested, take a math class! Yay math!
Number 8 was the one that really made me do a headdesk. The "paradox" only occurs if one ignores velocity. At any given instance the arrow is in one specific place (as stated); however, it has a non-zero velocity!
A similar thing happens when you throw a ball in the air (sort of...I'm gonna take a derivative here...) When the ball reaches the peak of its flight, there's an instant where it hovers - it has zero velocity. But we know from experience that the ball will fall back to Earth. That's because the ball has a non-zero acceleration (in this case, it's the acceleration due to gravity: 9.8 m/s^2). If you look at a "snapshot" of the ball when it's at the peak, you might think that it will hover forever, but that's only because you're not looking at the full picture (so to speak). You've left out acceleration.
It's kind of the same thing here. The arrow is instantaneously in one place, but it keeps moving because of its velocity. If you don't include velocity you're not really looking at the whole system.
Likewise, #7 falls into a mathematical trap. It's essentially a related-rate problem (Calc 101 - Mom always said that calculus was useful). Achilles' velocity is greater than the tortoise's velocity. So eventually he will overtake the tortoise, even though the tortoise keeps moving. You can calculate the distance each has traveled (distance = rate x time) at any particular instant. At the time where distance_Achilles is greater than distance_tortoise, Achilles has passed the tortoise, even though distance_tortoise keeps increasing with time. Distance_Achilles increases with time, too - but at a faster rate.
Sorry if this was garbled; it's been a few years since I took Calc I and I'm not sure if I'm explaining it clearly. If you're interested, take a math class! Yay math!
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