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Achilles and the Tortoise
"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."
The Dichotomy Paradox
"That which is in locomotion must arrive at the half-way stage before it arrives at the goal."
The Arrow Paradox
"If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless."
Originally posted by Athenion
I think you're missing the point. Mathematically, they are still paradoxes (although you are right in your statement that the first two are essentially the same).
A problem with using calculus to try and solve Zeno's paradoxes is that this only addresses the geometry of the situation, and not its dynamics. What is at the core of Zeno's paradoxes is the idea that one cannot finish the act of sequentially going through an infinite sequence, and while calculus shows that the sum of an infinite number of terms can be finite, calculus does not explain how one is able to finish going through an infinite number of points, if one has to go through these points one by one. Indeed, saying that there are an infinite number of points or intervals within some finite interval is of course the very assumption in the Achilles and Dichotomy Paradoxes, and it is this assumption regarding the geometry of the situation that leads to a paradox regarding its dynamics.
It should also be noted that calculus-based solutions that are offered often object to the claim that "it must take an infinite amount of time to traverse an infinite sequence of distances". However, Zeno's paradox doesn't contemplate the time it would take for Achilles to catch the Tortoise; it simply points out that in order for Achilles to catch up with the Tortoise, Achilles must first perform an infinite number of acts, which seems to be impossible in and of itself: time has nothing to do with it. Thus, calculus-based solutions to Zeno's paradoxes often make the paradox into a straw man.
In short, trying to use calculus to resolve the paradox simply reaffirms the idea that space and time are infinitely divisible, and thus still suffers from the basic question as to how one can possibly reach the end of an endless series.
Originally posted by Athenion
I think you're missing the point. Mathematically, they are still paradoxes (although you are right in your statement that the first two are essentially the same). Even taking the motion of your hand as a whole movement, it can be mathematically broken down to 1/2 spacial movements, which means you should never reach your destination, and yet you do. Hence the paradox. What happens in real life, does not match up with what happens mathematically, and somehow, both are correct. Unless of course we have a fundamental misunderstanding of the nature of time, as Lynds explains.
Originally posted by Yarium
Mind you, mathematically, though you also always have to travel to the next half-way point, that next half-way point is reached in HALF the amount of time as the last point!
So as the number of points approaches infinite, the amount of time it takes to get to that point approaches 0.
In the end, when points DOES equal infinite, the time to reach that point is Zero, and the destination is reached instantaneously.
The paradox is resolved!
Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Many philosophers, and certainly engineers, generally went along with the mathematical results.
Nevertheless, Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this he followed Leopold Kronecker, an earlier 19th century mathematician. Some claim that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has not resolved all problems involving infinities, including Zeno's.
As a practical matter, however, no engineer has been concerned about them since knowledge of the calculus became common at engineering schools. In ordinary life, very few people have ever been much concerned.