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Zeno's Paradox's
- Thread starter Rex
- Start date
Mr Spinkles
Mr
Zeno's paradox doesn't really work though....it requires that either Achilles slows down by half every second, or we slow down time. If you slow down time, or if Achilles goes slower, of course Achilles will 'never' reach the tortoise! But if Achilles maintains his speed, and the tortoise maintains his speed, yes, Achilles does catch the tortoise.
Rex
Founder
Not according to Zeno. If everyone keeps moving then it's basically cut but a 1/2 and you will always be able to debate by 2.
Example. Lets say you want to get from point 1 to point 2 . No matter how fast you go you will never get their b/c you can always divide by 2. Using the infinite number line.
Example. Lets say you want to get from point 1 to point 2 . No matter how fast you go you will never get their b/c you can always divide by 2. Using the infinite number line.
Pah
Uber all member
Zeno's theory fails in reality if the speed of each is held constant. Then distance to the finish line, not the opponent, is the factor that determines the winner regardless of any "head start". That is the reality. Zeno's theory presumes a condition not applicable to reality and Mr. Sprinkles is correct.
The theory can be proven mathmatically but so can the speed/distance relationship.
Zeno's theory leads to a denial of all motion, an unreal condition of matter. An atom would collaspe (or fly apart?) if motion (orbit) was not charateristic of the particles of an atom.
-pah-
The theory can be proven mathmatically but so can the speed/distance relationship.
Zeno's theory leads to a denial of all motion, an unreal condition of matter. An atom would collaspe (or fly apart?) if motion (orbit) was not charateristic of the particles of an atom.
-pah-
standing_on_one_foot
Well-Known Member
Right. Which is why it's a paradox, because the logic seems to work and yet it obviously doesn't. Which is why I like calculus and limits and stuff, coz it explains that kinda stuff.
Mr Spinkles
Mr
Also, Zeno's Paradox assumes that between two points there are an infinite number of points in between (classical physics)....but in reality there is a distinct number of positions a particle can occupy between two points (quantum physics). I think that is what pah was saying, too.
trishtrish10
Active Member
my guardian angel has slowed time for me so i could make it to an appointment on time.
Buttons*
Glass half Panda'd
Zeno's paradox was trying to explain that you cannot trust your senses, and that what can sometimes be seen is not what is. He based this off Pythagoreans and their abstract geometry...
Sometimes when you use the pythagorean theorem, if you actually go and measure the hypotenuse, you wont find the number that comes up with the forumula (A squared + B squared = C squared). As in the instance of a right triangle with the measurements: height = 1, length = 1. The theorem would turn out to be a non-whole number. If you physically took a ruler and measured the hypotenuse, then, you would not get the exact same number as you would with the abstract method of the pythagoreans.
So Pythagoreans would say that you can only trust your abstract thought, and not what appears to be. Zeno's point is similar: Not everything is as it appears. So even if it appears that Achilles beat the tortoise, this would actually be false because of the mathematics and abstract calculations used for this specific paradox.
When it comes to calculus in solving this paradox, there is still an issue. Within a finite measurement, there are infinate possibilities. I've only heard the one argument against it:
If you place polls at every step you take, say you put down one hundred polls, and you take a flashlight. The objective is that at every even poll, you turn the light on, and at every odd poll, you turn the light off. Then by the time you get to your very last poll, the light will be on. This indicates that there is still another poll ahead because there are only two settings on the light. So the journey is never complete.
This is Zeno's point. The journey is never complete, no one can finish something... and because of this, he rejects the idea of motion and change. Because if there are infinate journeys within one journey, then you will never finish the one set ahead of you. There is always something else.
So it's more of a matter if you follow abstract thought only then time is not a matter. Time does not enter into what is eternal and universal, like numbers. So how could one possibly apply that thought to observation, apparent reality? The two conflict. Both things are very real, but you have to decide if there is one you side with more, to determine how you see the universe and what you believe to be real and true.
Sometimes when you use the pythagorean theorem, if you actually go and measure the hypotenuse, you wont find the number that comes up with the forumula (A squared + B squared = C squared). As in the instance of a right triangle with the measurements: height = 1, length = 1. The theorem would turn out to be a non-whole number. If you physically took a ruler and measured the hypotenuse, then, you would not get the exact same number as you would with the abstract method of the pythagoreans.
So Pythagoreans would say that you can only trust your abstract thought, and not what appears to be. Zeno's point is similar: Not everything is as it appears. So even if it appears that Achilles beat the tortoise, this would actually be false because of the mathematics and abstract calculations used for this specific paradox.
When it comes to calculus in solving this paradox, there is still an issue. Within a finite measurement, there are infinate possibilities. I've only heard the one argument against it:
If you place polls at every step you take, say you put down one hundred polls, and you take a flashlight. The objective is that at every even poll, you turn the light on, and at every odd poll, you turn the light off. Then by the time you get to your very last poll, the light will be on. This indicates that there is still another poll ahead because there are only two settings on the light. So the journey is never complete.
This is Zeno's point. The journey is never complete, no one can finish something... and because of this, he rejects the idea of motion and change. Because if there are infinate journeys within one journey, then you will never finish the one set ahead of you. There is always something else.
So it's more of a matter if you follow abstract thought only then time is not a matter. Time does not enter into what is eternal and universal, like numbers. So how could one possibly apply that thought to observation, apparent reality? The two conflict. Both things are very real, but you have to decide if there is one you side with more, to determine how you see the universe and what you believe to be real and true.
Xeno's parodoxes assume no alteration of time or any other aspect of reality. Logically they stand.
The quantam/relativistic universe is anti-intuitive. Reality is entirely different from our perception of it. We perceive three dimensions of a tiny sliver of an 11-dimentional universe, and presume to believe we are actually experiencing Reality.
The quantam/relativistic universe is anti-intuitive. Reality is entirely different from our perception of it. We perceive three dimensions of a tiny sliver of an 11-dimentional universe, and presume to believe we are actually experiencing Reality.
ChrisP
Veteran Member
Well yeah, but then noone gets Tortoise soup...Seyorni said:
All this being well and good (and thank the Teenage Mutant Ninja Turtles Calculus is back in High School), as long as we can walk to the coffee shop, why does this matter?
Most people realise they create their own reality in one sense or another.
Seyorni said:
As illusory as everything else that is temporal, yes, but from certain fleeting mystical states one can detect the vibrant waves of force in ones own body, especially in the subtle ranges of perfect stillness, if ones mind is sufficiently attuned by meditation. The appearance of motion is universal: but sound is the precursor of matter and the solidarity of reality. In short, motion is the only constant. Reality rocks in silence!
Zeno could not prove it by contradiction, due to reductio ad absurdum. For me, as above, the paradox is apparent but an illusion itself.
Mike182
Flaming Queer
i understand Xeno's paradox to be a challenge to logical thinking.... if we look at a journey from the point of view of Xeno's argument, for you to travel 1/2 of the journey, no matter how long, is only 1/2 of the journey, and to travel that 1/2, you must travel 1/2 of that etc etc etc. the length of time it takes does not factor into the argument, isn't that why it's a paradox 
Fluffy
A fool
Heya Seyorni,
Only if we assume that an infinite sequence of terms can only be added together to give an infinite amount. Under limit theorum, this is clearly not the case, however.
If I have a convergent series (such as a series where each term is half the previous term), then as I add each successive term, I will tend towards a given number. Therefore, if I extend my series into infinity, I will reach this finite number meaning that an infinite series has a finite solution. A finite distance can be traversed in a finite amount of time hence this paradox is no paradox.
For those not familiar with mathematics, this can be illustrated to make it more clear. If I have a series of terms such as the following: 1+1/2+1/4+1/8 then I can come up with a rule that governs how the next term in the series is discovered. If you look at the example, you can deduce that the next term in the series will be 1/16 since as I go through each term, the value is halved. If I total these values then I get the following:
1+1/2+1/4+1/8=1.875
1+1/2+1/4+1/8+1/16=1.9375
1+1/2+1/4+1/8+1/16+1/32=1.96875
As can be seen, as each term is added, the total gets closer and closer to the number 2. However, since each increment becomes progressively smaller, a finite series governed by this rule will never reach 2. Zeno correctly identified that instead of a finite number of points between 2 other points, there are in fact an infinite number of points. If we add an infinite series of this nature, it will equal exactly 2.
Since 2 is a finite number, we can see that it is possible to add a infinite series of terms and produce a finite total. If we did the same for the amount of time taken for each distance, we would also reach a finite total since this series is also convergent.
What Zeno observed was entirely correct. The only problem was that he got the wrong conclusion from that oberservation. Paradox solved.
Seyorni said:
Only if we assume that an infinite sequence of terms can only be added together to give an infinite amount. Under limit theorum, this is clearly not the case, however.
If I have a convergent series (such as a series where each term is half the previous term), then as I add each successive term, I will tend towards a given number. Therefore, if I extend my series into infinity, I will reach this finite number meaning that an infinite series has a finite solution. A finite distance can be traversed in a finite amount of time hence this paradox is no paradox.
For those not familiar with mathematics, this can be illustrated to make it more clear. If I have a series of terms such as the following: 1+1/2+1/4+1/8 then I can come up with a rule that governs how the next term in the series is discovered. If you look at the example, you can deduce that the next term in the series will be 1/16 since as I go through each term, the value is halved. If I total these values then I get the following:
1+1/2+1/4+1/8=1.875
1+1/2+1/4+1/8+1/16=1.9375
1+1/2+1/4+1/8+1/16+1/32=1.96875
As can be seen, as each term is added, the total gets closer and closer to the number 2. However, since each increment becomes progressively smaller, a finite series governed by this rule will never reach 2. Zeno correctly identified that instead of a finite number of points between 2 other points, there are in fact an infinite number of points. If we add an infinite series of this nature, it will equal exactly 2.
Since 2 is a finite number, we can see that it is possible to add a infinite series of terms and produce a finite total. If we did the same for the amount of time taken for each distance, we would also reach a finite total since this series is also convergent.
What Zeno observed was entirely correct. The only problem was that he got the wrong conclusion from that oberservation. Paradox solved
.