Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

But you can also apply this configuration to the halfway of the distance. In other words, this paradox assumes that we can move and, based on that, it derives a conclusion that we cannot move, since a distance has a half distance(i.e can be divided into parts).

So this statement should be wrong obviously but it has been called a "paradox". What is the problem in my explanation?

• Your text is the opposite of your diagram. Which is your intended meaning? – user4894 May 3 '16 at 17:49
• @user4894 My intention :) : So this statement should be wrong obviously but it has called a "paradox".What is the problem in my explanation ? – onurcanbektas May 3 '16 at 17:51
• @user4894 the diagram is added for just example for Zeno's paradox – onurcanbektas May 3 '16 at 17:53
• Diagram shows runner first going 1/2, then to the 3/4 point, then to the 7/8 point, etc. Your text notes that before getting to 1/2 the runner must first get to 1/4, but must first get to 1/8, ... Do you see that your diagram is the exact opposite of what you wrote? – user4894 May 3 '16 at 18:09
• To clarify what @user4894 is saying, the diagram shows a situation where (according to the paradox) the person will never arrive, but your text describes a situation where the person can't begin moving. – Era May 3 '16 at 18:58

Zeno's paradox occurs in a continuous space. It isn't about taking individual step, it's about a series of events which must occur before you reach your destination if you are moving continuously.

It's impossible to get to the destination without first getting half way there. It's impossible to get half way there without first getting a quarter of the way there. And so on.

The theory was that it would take an infinite number of such events occurring, in order, before you could get to your destination. Thus, it "should be impossible."

If you indeed have a discrete world, such as "I am two steps away from my destination," and you only consider discrete transitions from the start of your walk, to half way, to completed, Zeno's paradox does not apply. However, if someone were to stand in your way 1.3 steps into your "discrete" walk and try to impede your motion, you would find this discrete model is not a good model of reality. Reality is typically considered more continuous than that.

Zeno's paradox still has its place in philosophy, but the physical application of it regarding movement in the real world has been displaced by calculus. In calculus, we can use limits to manage these infinite strings of events in a way which modern physicists and engineers have found sufficiently valid that we no longer mind these sorts of paradoxes which create an infinite string of infinitely small events (or, to be more precise, we have very strict criteria as to which strings of events are physically realizable and which ones are not, typically formalized as epsilon-delta proofs. Zeno's original paradox is physically realizable)

• Since physics has no evidence that the real world is perfectly modeled by the real numbers; and at least some evidence that it's not; how does calculus resolve Zeno's paradoxes? I've never understood this point. The fact that calculus is good at approximating the world doesn't seem sufficient. – user4894 May 5 '16 at 6:49
• @user4894 Zeno's paradox depends on the existence of at least rational numbers, so any time his paradox is invoked, you can at least assume that rational numbers are being used. If you're comfortable extending that to real numbers (which the Greeks had to be, once they learned irrational numbers existed), then you're in a position to accept calculus's answer to the paradox using limits. For purposes of the paradox, the real world is not actually involved, just a mathematical one. However, the reason the axioms of calculus are accepted today is because they do a good job of explaining – Cort Ammon May 5 '16 at 14:55
• why we dont experience Zeno's paradox like behaviors in the real world, without having to require the world to be modeled discretely. Calculus can "declare" the meaning of a limit, and can state the limit of the math of Zeno's paradox without any assumptions. The assumption comes in when we state that either Zeno's view or calculus' limits have meaning in the real world. And in this case, calculus's results are consistent with our experiences, and Zeno's is not, so it's typically an easy choice to make. – Cort Ammon May 5 '16 at 14:57
• Your words: "the physical application of it regarding movement in the real world [my emphasis] has been displaced by calculus ..." Followed by: "For purposes of the paradox, the real world is not actually involved ..." You see the problem? Calculus (and the modern theory of the real numbers, based on set theory) are not at issue here. Zeno was talking about the real world. And you yourself are inconsistent on this point. Also, there are arbitrarily small rationals, so the rationals are just as inconsistent with known physics as the reals are. – user4894 May 5 '16 at 17:04
• @user4894 I actually was rather careful with wording on that sentence. Zeno's paradox has not been "resolved" or "falsified" or any other strong word. However, the ideas behind calculus have been so extraordinarily useful for predicting real world behaviors that the issue of Zeno's Paradox has been displaced. And yes, there are arbitrarily small rationals. The cause of my cautious wording was that I've never looked at the assumptions behind epsilon-delta proofs of limits to see if they depend in any way on irrational numbers or not. – Cort Ammon May 5 '16 at 21:07

I think that what you're having trouble with is the hypothetical nature of the argument. There is something in the argument like, "Suppose we were to take a step ..." This is not tantamount to being able to take a step being a premise or underlying assumption of the argument.

There is a common form of argument known in philosophy as reductio ad absurdum and in mathematics as proof by contradiction. This argument is not the same, but I think that reductio illustrates what (it seems to me) you are having trouble with. In a reductio, you begin by asserting the claim you are trying to disprove as a logical hypothesis (not the same as a scientific hypothesis). Then, you derive from this premise an absurdity (contradiction) showing that your initial hypothesis is impossible.

The Paradox says, "If I were to take a step, I would first have to travel half that distance." Ultimately, it concludes by saying we cannot take a step. That conclusion is not based on an actual assumption that we can take a step, but rather some reasoning we did when thinking hypothetically about taking a step.