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!
Wednesday, July 28, 2010
Philosophy - #5
Brain-twister # 5 is the "surprise day" paradox. This post is out of order, but I decided to write it quickly since Spiked Math made this comic. Instead of re-writing the deconstruction of logic, I'm just going to cheat and point you to the comments section - the mathematicians explained it better than I could =) They also touch on the "interesting numbers" paradox discussed earlier.
Wednesday, July 14, 2010
Philosophy - #10 and 9
Both brain-twisters #10 and #9 suffer from an inconsistent definition. Let's take them one at a time, starting with #9.
Take a set of numbers. The "interesting" number is the smallest (in this puzzle). But then, after removing that one, there is still a smallest number. Keep removing the interesting (smallest) number, the argument goes, and you will be left with no "uninteresting" numbers.
But, when you remove the smallest number, you've created a new set! Start again with the initial set; there is a smallest number, but the rest are still uninteresting IN THAT SET. If there are "n" numbers, "n-1" of them are uninteresting. Just stop there. If you remove that number from the set, you've changed the system.
#10 has the same problem. (And actually, I really liked that this problem was included. I used to think along these lines in elementary school when I was home sick for the day. My mom insisted that I go to school unless I had a fever. I used to wonder when the exact minute was that I was officially "sick." And then I'd wonder if I could break it down further - into the exact second, or millisecond, or microsecond that I transitioned from being "well enough to go to school" to "sick enough to stay home." Apparently I was a 6-yr-old philosopher. But I digress.)
Few things in life are in a simple binary state. If someone showed me a millions grains of sand, I'd probably say, "That's a heap of sand!" If someone showed me a single grain of sand, I'd certainly NOT say, "That's a heap of sand!" But this isn't a simple on / off switch. Just as, when I was a child, illness would come on gradually, with an intermediate "take a nap, then see if you're well enough to go to school late" phase, there is a gradient between "heap" and "single grain." Maybe a hundred thousand grains is a "pile." And ten thousand grains is a "lump." There might even be a single-grain difference between what I'd call a "pile" and what I'd call a "lump." But I wouldn't necessarily keep calling a collection of sand a "heap" until I got down to one grain. At some point along that gradient, I'd go into the next label.
Take a set of numbers. The "interesting" number is the smallest (in this puzzle). But then, after removing that one, there is still a smallest number. Keep removing the interesting (smallest) number, the argument goes, and you will be left with no "uninteresting" numbers.
But, when you remove the smallest number, you've created a new set! Start again with the initial set; there is a smallest number, but the rest are still uninteresting IN THAT SET. If there are "n" numbers, "n-1" of them are uninteresting. Just stop there. If you remove that number from the set, you've changed the system.
#10 has the same problem. (And actually, I really liked that this problem was included. I used to think along these lines in elementary school when I was home sick for the day. My mom insisted that I go to school unless I had a fever. I used to wonder when the exact minute was that I was officially "sick." And then I'd wonder if I could break it down further - into the exact second, or millisecond, or microsecond that I transitioned from being "well enough to go to school" to "sick enough to stay home." Apparently I was a 6-yr-old philosopher. But I digress.)
Few things in life are in a simple binary state. If someone showed me a millions grains of sand, I'd probably say, "That's a heap of sand!" If someone showed me a single grain of sand, I'd certainly NOT say, "That's a heap of sand!" But this isn't a simple on / off switch. Just as, when I was a child, illness would come on gradually, with an intermediate "take a nap, then see if you're well enough to go to school late" phase, there is a gradient between "heap" and "single grain." Maybe a hundred thousand grains is a "pile." And ten thousand grains is a "lump." There might even be a single-grain difference between what I'd call a "pile" and what I'd call a "lump." But I wouldn't necessarily keep calling a collection of sand a "heap" until I got down to one grain. At some point along that gradient, I'd go into the next label.
Philosophy - #11 and #1
I'm going to start by looking at the first and last brain-twisters. In my opinion, they're kind of the same argument. The contradiction comes from a lack of clarity in the definitions.
If an object is "unmovable," by definition that means that NOTHING can move it. Similarly, if a force is "unstoppable," that means that NOTHING can arrest its motion. Clearly these two things cannot exist in the same universe, so the hypothetical situation doesn't even make sense. It would (almost) be like saying, "What if gravity were both attractive and repulsive?" According to what we know of physics, this can't happen. The question is meaningless.
(For what it's worth, I'm working really hard here not to throw in a discussion of coordinate systems and how motion is always measured relative to something else.)
Similarly, the "omnipotent" being needs a good definition of "omnipotent." What do you mean when you ask the question? Could this omnipotent being increase its own power? In which case the conflict dissolves - the omnipotent being could create a stone it could not (initially) lift, and then increase its power until it could lift the stone. There needs to be a baseline of power against which to measure the "unliftable stone;" but this contradicts the idea of a truly "omnipotent" being. Again, the argument is meaningless without further clarification.
If an object is "unmovable," by definition that means that NOTHING can move it. Similarly, if a force is "unstoppable," that means that NOTHING can arrest its motion. Clearly these two things cannot exist in the same universe, so the hypothetical situation doesn't even make sense. It would (almost) be like saying, "What if gravity were both attractive and repulsive?" According to what we know of physics, this can't happen. The question is meaningless.
(For what it's worth, I'm working really hard here not to throw in a discussion of coordinate systems and how motion is always measured relative to something else.)
Similarly, the "omnipotent" being needs a good definition of "omnipotent." What do you mean when you ask the question? Could this omnipotent being increase its own power? In which case the conflict dissolves - the omnipotent being could create a stone it could not (initially) lift, and then increase its power until it could lift the stone. There needs to be a baseline of power against which to measure the "unliftable stone;" but this contradicts the idea of a truly "omnipotent" being. Again, the argument is meaningless without further clarification.
Philosophy
I'm generally not shy about my hatred for philosophy. I think it's because I'm scientifically-minded; I like to test theories, not just debate them. And a lot of times it seems like philosophers argue about semantics and language so abstract that what they're saying has no real meaning. Other times, it seems like philosophers argue about things that could be tested or understood scientifically.
I'm not gonna say that all philosophy is bunk; I understand that philosophy helps exercise logical and critical thinking muscles. However, a lot of times I read a philosophical argument or brainteaser and have to roll my eyes. For example, I recently read this article on "philosophical brainteasers" and they all seem to fall into one of two categories:
1) things that people think are paradoxes because they don't properly understand math and science, or
2) arguments based on contradicting definitions.
Over the next several posts, I'll go through them and explain my thoughts on each of the paradoxes. If there's a lack of science / math understanding, I'll point that out and do my best to correct the misunderstanding. If it's a semantics argument, I'll point that out too.
I realize that a lot of this is going to sound pretty snarky. And like I said, I have respect for philosophy as a means to sharpen logic and critical thinking. But I also hate it when people equate something scientific with something mystical or abstract. (I'm thinking mostly of brain twister #8. I did a facepalm as soon as I read that one. Basic misunderstanding of physics going on there.)
And yeah, it'll probably take a while for me to get through all of them. I'm not exactly regular about posting stufff...
(although as I have zero followers I don't think it's really an issue. If you've stumbled upon my blog, HI! leave a comment so I know my words aren't going into the great black hole of cyberspace.)
I'm not gonna say that all philosophy is bunk; I understand that philosophy helps exercise logical and critical thinking muscles. However, a lot of times I read a philosophical argument or brainteaser and have to roll my eyes. For example, I recently read this article on "philosophical brainteasers" and they all seem to fall into one of two categories:
1) things that people think are paradoxes because they don't properly understand math and science, or
2) arguments based on contradicting definitions.
Over the next several posts, I'll go through them and explain my thoughts on each of the paradoxes. If there's a lack of science / math understanding, I'll point that out and do my best to correct the misunderstanding. If it's a semantics argument, I'll point that out too.
I realize that a lot of this is going to sound pretty snarky. And like I said, I have respect for philosophy as a means to sharpen logic and critical thinking. But I also hate it when people equate something scientific with something mystical or abstract. (I'm thinking mostly of brain twister #8. I did a facepalm as soon as I read that one. Basic misunderstanding of physics going on there.)
And yeah, it'll probably take a while for me to get through all of them. I'm not exactly regular about posting stufff...
(although as I have zero followers I don't think it's really an issue. If you've stumbled upon my blog, HI! leave a comment so I know my words aren't going into the great black hole of cyberspace.)
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